interference alignment and dof analysis of interference
Post on 16-Feb-2022
10 Views
Preview:
TRANSCRIPT
Interference Alignment and DOF Analysis ofInterference and Interference Broadcast
Channels
by
Jhanak Parajuli
A thesis submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
in Electrical Engineering
Approved Dissertation Committee:
Prof. Dr. Eng. Giuseppe Abreu(Jacobs University Bremen)
Prof. Dr. Ing. Werner Henkel(Jacobs University Bremen)
Dr. Antti Tolli(CWC, University of Oulu)
Date of Defense: September 22, 2016.
Computer Science & Electrical Engineering, Jacobs University Bremen
ii
Dedication
“Behind every child who believes in himself/herself is a parent who believed first.”
Matthew Jacobson
I dedicate this thesis to my parents:
Jayaram Parajuli
and
Chuna Devi Parajuli.
iii
iv
Acknowledgements
I would like to express my deep thanks and sincere gratitude to all who have helped me in
one way or other during my complete Ph.D program. I am very very grateful to my Ph.D.
supervisor Prof. Dr. Giuseppe Abreu, whose constant motivation, strong support, quality
guidance, valuable discussions and strict requirements could make my work a success. I am
very thankful for the convenient and friendly academic environment that he provided during
my Ph.D and masters studies at Jacobs University Bremen. I could really grow academically
during this time. Thanks to the university for such a wonderful environment . I am also very
grateful to Prof. Dr. Ing. Werner Henkel and Dr. Antti Tolli, who have agreed to be on my
dissertation committee.
I am also thankful to the European Union FP7 project BUTLER (uBiquitous, secUre
inTernet-of-things with Location and contEx-awaReness) and H2020 project HIGHTS (High-
precision positioning for cooperative Intelligent Transport systems) for providing me economic
assistance, without which I could not imagine my Ph.D degree. My special thanks to Dr.
Stefano Severi for his constant efforts to work on new projects which made it really easy
for me and I could focus on my Ph.D work. I am very thankful to my previous and present
work-group and colleagues Samip, Satya, Tayo, Simona, Yohannes, Remun, Ali, Andrei,
Cristian, Jonathan and all other who are involved in our work-group in one way or the other.
Fruitful discussions during our group meeting always helped me a lot.
My sincere thanks to my Nepali friends in Jacobs University, Samip, Asman, Anup, Riwaj,
Shailesh, Dileep, Chhabi, Ujjwal, Dev and my friends outside the university Shankar, Ashish,
Sudha, Bishal, Gyanendra, Aabhushan, Jwalanta , Gunjan and Binod who supported me in
every possible ways they could and helped me feel homely every time. A very very sincere
thanks to my very nice and lovely host family in Bremen (Gerlind, Axel and Marius) who
helped me a lot during my initial days in Bremen. I always appreciate their help.
My special thanks to the MIT open courseware, edX and coursera that allows a framework
to take a number of online courses and make myself clear on the topics which I had not
understood before. It was really very helpful many times. Also, a very special thanks to the
badminton club members in the university and members of my house in Cigarrenmanufaktur,
especially to Shailesh, Viri, Satya and Harsha.
Last but not the least, I am very thankful to my parents, who always devoted their life for
v
giving me better education and helping me in every possible ways they could; and to all my
family members, my brothers, sisters, brother-in-laws and sister-in-laws for providing me a
strong mental and economic support when needed. Thanks to all my well-wishers and to all
those who helped me and I forgot to mention here.
vi
Statutory Declaration
I, Jhanak Parajuli, hereby declare, under penalty of perjury, that I am aware of the
consequences of a deliberately or negligently wrongly submitted affidavit, in particular the
punitive provisions of § 156 and § 161 of the Criminal Code (up to 1 year imprisonment or a
fine at delivering a negligent or 3 years or a fine at a knowingly false affidavit).
Furthermore I declare that I have written this PhD thesis independently, unless where
clearly stated otherwise. I have used only the sources, the data and the support that I have
clearly mentioned.
This PhD thesis has not been submitted for the conferral of a degree elsewhere.
Bremen, October 10, 2016.
Signature
vii
viii
Abstract
Multi-user wireless communication system is interference limited and determining the capacity
region for such network is a challenging problem. Information theoretically, even the capacity
region of a two-user Gaussian interference channel (IC) remains an open problem since
1970s. However, recently a new cooperative interference management technique, called the
interference alignment (IA), has been proposed and shown to achieve the optimum degrees of
freedom (DOF) in interference networks such as multiple access channel (MAC), broadcast
channel (BC), IC and the X-channel (XC). Since DOF is the first order approximation of the
capacity at high signal to noise ratios (SNRs), IA has been a topic of tremendous interests to
the wireless communication engineers and information theorists.
Interference alignment is a technique in which all the interfering signals observed from
different transmitters are aligned onto the same direction or onto a common subspace while
maintaining independence with the desired signal in a particular receiver. In a K-user single
input single output (SISO) interference network with K > 2, IA allows all K − 1 interference
signals to be aligned over only 1 dimension as opposed to K−1 dimensions in other spectrum
sharing techniques like frequency division multiple access (FDMA) and time division multiple
access (TDMA). Thus, each user can achieve maximum of 12
DOF, which is much higher than1K
in traditional FDMA and TDMA systems.
Aligning K − 1 independent interference signals observed from different sources onto a
single direction becomes mathematically challenging as K increases. Initial ideas in [1]
considered infinite uses of the channel and proved that 12
DOF is mathematically achievable
for any SISO-IC. Later, it is observed that for the multiple input multiple output (MIMO)
settings, the spatial dimensions can be exploited and infinite channel uses are not required
for the alignment, but the interference can be aligned by designing proper precoding matrices
at all the transmitters [2].
However, one of the main drawbacks of these approaches are the requirement of instanta-
neous channel state information at the transmitter (CSIT) and the explosion of parameters
as K increases. In fact, a robust generalized IA algorithm for any interference network still
remains an open problem.
In this thesis, we aim to address some of these existing challenges in the interference
management of multi-user networks by providing following three fold contributions:
ix
• Improvement of the existing IA algorithms: Using cooperation among the receivers, it
is shown that a proper precoding and zero-forcing matrix can be designed iteratively by
decreasing the interference power and increasing the desired signal power simultaneously
for a K-user MIMO-IC. Such precoding matrices improves the system capacity measured
in bits per second per channel use than previously existing algorithms. Simulation
results are provided.
• Relaxation of the requirement of CSIT: A space-time transmission scheme is designed
which uses only the delayed CSIT instead of instantaneous CSIT and such scheme is
shown to achieve greater than one DOF in any multi-cell multiple input single output
(MISO) BC. Such interference broadcast channel (IBC) requires both the inter cell
interference (ICI) and the inter user interference (IUI) be canceled properly. It is shown
that by proper transmission strategy, the achievable DOF converges to 85
when the
number of users per cell becomes very large.
• Proposition of a new alignment technique: A new grouping based IA technique is
proposed where the receivers within a group cooperatively align the interference from a
common source onto an overlapping space. By doing so, the receivers can take benefit
from the unused spatial dimensions of each other. Such concept of alignment is termed
as vertical alignment. Proper precoding matrices are designed to align the ICI onto a
common overlap space in two cell MIMO BC and optimal overlapping dimensions are
determined for ring and star topologically arranged three user MIMO IC.
DOF analysis of three user MIMO IC with any arbitrary number of transmit and receive
antennas is a challenging problem and very few works are done in this regard. By
determining the optimal overlapping between any two adjacent receivers, a relatively
new and simple chain based DOF analysis technique is proposed, where the DOF can
be expressed in terms of the length of such chain. The scheme is termed as receiver
chain alignment (RCA).
x
Contents
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Interference Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Basic Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Precoding Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 Asymptotic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 Ergodic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.3 IA Based on Real Numbers . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.4 Lattice-Based Approach . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.5 Topology Based Approach . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 MIMO Interference Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 From Global to Local CSIT . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.1 Minimization of Leakage Interference . . . . . . . . . . . . . . . . . . 23
2.4.2 Maximization of Signal to Interference plus Noise Ratio . . . . . . . . 26
2.4.3 Alternating Minimization of Interference . . . . . . . . . . . . . . . . 27
2.5 Feasibility of Interference Alignment . . . . . . . . . . . . . . . . . . . . . . 29
3 Interference Alignment with Receiver Cooperation and Greedy Transmission . . . 33
3.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2 Proposed Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Coalition Game Theory and Full Cooperation . . . . . . . . . . . . . . . . . 39
3.4 Simulation Results and Feasibility . . . . . . . . . . . . . . . . . . . . . . . . 40
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4 Space-Time Transmission with Delayed CSIT . . . . . . . . . . . . . . . . . . . . 45
4.1 Single Cell Two-user MISO BC with Delayed CSIT . . . . . . . . . . . . . . 47
4.2 K-user MISO BC with Delayed CSIT . . . . . . . . . . . . . . . . . . . . . . 51
4.3 Two cell MISO IBC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
xi
4.4 Space-Time Transmission Scheme . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4.1 Case I ( M=1, K=1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.4.2 Case II (M=1, K=2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.4.3 Case III (M=2, K=1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4.4 Case IV (M=2, K=2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4.5 Case V (M=3, K=1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4.6 Case VI (M=3, K=3) . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4.7 Case VII (Generalized Scheme for M = K) . . . . . . . . . . . . . . . 76
4.5 Improved Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.6 More Than Two-Cell MISO-IBC . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5 Interference Alignment and Optimal Overlapping in MIMO IBC and IC . . . . . 85
5.1 MIMO Interference Broadcast Channels . . . . . . . . . . . . . . . . . . . . 86
5.1.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.1.2 Designing Post-processing Matrices . . . . . . . . . . . . . . . . . . . 88
5.1.3 Designing the Precoding Matrices . . . . . . . . . . . . . . . . . . . . 89
5.2 MIMO Interference Channels . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2.2 Determining the Rank of N (Φ) . . . . . . . . . . . . . . . . . . . . . 98
5.2.3 Ring Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2.4 Star Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6 DOF Analysis of Three User MIMO IC via Receiver Chain Alignment . . . . . . . 111
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.2 Receiver Chain Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.3 DoF Analysis and Achievability . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.3.1 Case 1 : 0 ≤ N−MM≤ 1
2. . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.3.2 Case 2 : 12< N−M
M≤ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.3.3 Case 3 : 1 < N−MM≤ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.4 Information theoretic outer bound for DOF . . . . . . . . . . . . . . . . . . 119
6.5 Achievability of the DOF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7 Conclusions and Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
xii
7.2 Discussion and Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
A Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
B Proof of Theorem 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
C Proof of Theorem 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
xiii
xiv
List of Figures
2.1 A K-user single input single output interference channel, where the solid lines
represent a direct or the desired links and the dashed line represent the cross
or interference links. All the receivers receive one desired link and K − 1
interference links. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 A geometric explanation of IA with four interference aligned in two dimensions
and one desired signal in the orthogonal direction. Green is the desired signal
while blue, red, black and magenta are the interference signal. . . . . . . . . 10
2.3 A partially connected 5 user SISO IC. A solid line represents the desired
channel while the dashed lined represents the interference channel. . . . . . . 19
2.4 Alignment conflict graph for the network in Figure 2.3. Solid line represents
the alignment graph and the dashed line represents the conflict graph. . . . . 19
2.5 Interference alignment in three user MIMO IC. The solid line represents a
desired channel matrix and the dashed and dotted lines represent the interfer-
ence channel matrices. For ease of representation, the alignment of multiple
spatial dimensions is represented by an arrow. For example, dashed green
arrow represents a d dimensional subspace observed at RX1 from TX3. . . . 21
2.6 A K-user MIMO IC with M transmit and N receive antennas. Vjs are the
precoding matrices and Uis are the zero-forcing matrices, ∀ i, j = {1, 2, · · · , K}.Solid lines represent desired channels and dotted and dashed lines represent
interference channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.7 For K = 3 user MIMO IC with M = 7 transmit antennas and N = 9 receive
antennas, only d = 4 DOF is achievable for a proper system. . . . . . . . . . 32
3.1 Sum rate achieved measured in bits/sec/Hz as a function of SNR measured
in dB for different transmit antennas, receive antennas and feasible DOF
that achieves IA for such case. The results are compared for the proposed
cooperative algorithm, minimization of leakage interference (MLI) algorithm
and maximization of signal to interference plus noise ratio (max-SINR) algorithm. 42
xv
3.2 Feasibility of interference alignment for the proposed cooperative and MLI
algorithms. The feasibility is measured as the percentage of the leakage
interference after each iteration measured for the given total data streams.
The proposed cooperative algorithm has less leakage interference than the MLI
for M = N = 3 and M = N = 4 with different number of total transmit streams. 43
4.1 A two user MISO BC with two antennas at the transmitter. The two indepen-
dent data streams are transmitted as a vector x(t) at any time instance t and
the channel vector is represented by hi1(t) and hi2(t), ∀i = {1, 2}. . . . . . . 48
4.2 A two-cell MISO interference broadcast channel with M transmit antennas in
each base station and K single antenna users per cell. The solid line represents
the signal received from the same cell and the dashed line represents the inter
cell interference received from the adjacent cell. The dotted line represents
that the delayed channel state information (CSI) feedback is provided from
any receiver to the transmitter. The two cells are connected via back-haul
connection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3 DOF of two-cell MISO IBC with delayed and instantaneous CSIT for odd and
even number of users. For odd number of users, the DOF converges to 43
and
for the even number of users achievable DOF is always 43. . . . . . . . . . . . 79
4.4 Achievable DOF for two cell MISO IBC with delayed and instantaneous CSIT.
The improved transmission scheme achieves better DOF which converges
to 85
unlike the earlier approach that converges to 43. The DOF with the
instantaneous CSIT converges to 2. . . . . . . . . . . . . . . . . . . . . . . 82
5.1 Two cell MIMO Broadcast Channel with two users per cell and arbitrary
M antennas at each base station and N antennas per user. The solid line
represents the desired signal plus IUI, while the dashed line represents the ICI
observed by each user. The channels from base station j to user k in cell `
is represented as H(`)kj , while Vj are the precoding matrices and Uk` are the
interference suppressing matrices or the post processors at user k in cell `. . 87
xvi
5.2 Plot of the total interference dimensions and the overlap dimensions for given
transmit streams d varying with the number of overlapping rows. The upper
part plots the total number of free dimensions after each user receives d dimen-
sional interference signal varying with the overlapping rows of the channels
between two users. The lower part plots the dimensions of the common region
where all interferences are aligned by two users. . . . . . . . . . . . . . . . . 90
5.3 Plot of the rank(N (Φ)
)with the number of overlapping rows per user when
different number of data streams are transmitted. The optimal number of
overlapping rows are the number of overlapping rows when the rank(N (Φ)
)=
d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.4 MAC (left side) and BC (right side) duality for the three-user MIMO IC where
only the interference signals are shown. U1 is determined with the knowledge
of V2 and V3 and V1 is determined with the knowledge of U2 and U3. . . . 93
5.5 Relationship between Φ and Z. . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.6 Interference dimension for different M . . . . . . . . . . . . . . . . . . . . . 100
5.7 An example of ring topology for antennas overlapping. Each antenna is indexed
with the alphabets. ‘a,b,c,d’ are the antennas in RX1, ‘e,f,g,h’ are the antennas
in RX2, ‘i,j,k,l’ are the antennas in RX3. All receivers share different antennas
with the adjacent receivers. The antenna elements in the dashed box are the
antennas shared between two receivers. . . . . . . . . . . . . . . . . . . . . . 101
5.8 Rank of N (Φj) for different M , when different antennas overlap between
the adjacent receivers. The null space is decreasing as the overlapping rows
increase. The graph shows two regions and the point where the graph changes
as shown by the dashed line gives the optimal number of overlapping rows. 104
5.9 An example of star topology for antennas overlapping in three user MIMO IC.
Each antenna is indexed with the alphabets. Same antennas overlap for all
the receivers. For example, the antennas in the dashed box ‘a’,’e’,’i’ overlap in
all three receivers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.10 Rank of N (Φ) for different transmit antennas varying with the number of
overlapping antennas in the ring topological overlap structure. The graph
shows two regions and the point where the region changes shows the optimal
overlapping rows as shown by the dashed line. . . . . . . . . . . . . . . . . . 108
xvii
6.1 RCA for M = N = 4 and M = N = p (general case) in 3-user IC with optimal
overlapping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 RCA for M = 4, N = 5 with optimal overlapping. . . . . . . . . . . . . . . . 114
6.3 RCA for M = 4, N = 7 with optimal overlapping. . . . . . . . . . . . . . . . 116
6.4 RCA for M=6 and N=10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.5 RCA for M = 2, N = 5 in 3-user IC for optimal overlapping. . . . . . . . . 118
6.6 DOF plot for different transmit and receive antennas under different DOF
achievable schemes. The ‘*’ line represents the DOF achieved using the
proposed RCA scheme, the ‘o’ line represents the DOF achieved with the
scheme proposed by Jafar et. al [3] and the ‘square’ line represents the DOF
achieved with subspace alignment chain (SAC) scheme proposed by Wang et.
al [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.7 Overlap interference and transmission in K = 3 user IC with M = 2 and N = 3.122
6.8 An example analysis of alignment due to antenna sharing. . . . . . . . . . . 125
7.1 Alignment error for different total overlap dimensions due to two receivers.
The error is decreasing as the overlap is increasing. . . . . . . . . . . . . . . 134
xviii
List of Tables
4.1 Two desired equations are observed by each user in both the cells after 9
channel uses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2 The signal received by all the users in both the cells during time instances
t = 1, 2, 3. The red color signal represents the signal to be swapped between
the first and second user in each cell and the blue colored signal represents the
signal to be swapped between the first and the third users in each cell. . . . 69
4.3 Three desired equations are observed by each user in both the cells after 15
time instances. All users can solve three independent desired data streams
using these three equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
xix
xx
Acronyms
AMI alternating minimization of interference.
AWGN additive white Gaussian noise.
BC broadcast channel.
CDI channel direction information.
CFF characteristic function form.
CQI channel quality information.
CSI channel state information.
CSIR channel state information at the receiver.
CSIT channel state information at the transmitter.
DOF degrees of freedom.
DPC dirty paper coding.
FDMA frequency division multiple access.
GDOF generalized degrees of freedom.
i.i.d independent and identically distributed.
IA interference alignment.
IA-AM interference alignment using alignment matrix.
IBC interference broadcast channel.
IC interference channel.
ICI inter cell interference.
xxi
IMAC interference multiple access channel.
IUI inter user interference.
MAC multiple access channel.
max-SINR maximization of signal to interference plus noise ratio.
MIMO multiple input multiple output.
MISO multiple input single output.
MLI minimization of leakage interference.
PFF partition function form.
RCA receiver chain alignment.
SAC subspace alignment chain.
SINR signal to interference plus noise ratio.
SISO single input single output.
SNR signal to noise ratio.
SVD singular value decomposition.
TDD time division duplexing.
TDMA time division multiple access.
XC X-channel.
xxii
Chapter 1
Introduction
Modern wireless communication system users demand higher data rate and better quality
of service, unconcerned with the fact that the transmit power, bandwidth and the design
complexity are expensive and limited resources and these demands continue to increase in
the following years. Wireless communication engineers and researchers are developing new
ideas and tools to meet these demands. Inherently, any wireless communication comes up
with three major challenges-noise, multi-path fading and interference. While nothing much
can be done about the noise, a tremendous amount of research from the last few decades are
concerned about mitigating multi-path fading and interference.
One of the ground-breaking ideas that utilized the inherent multi-path property of wireless
channels to mitigate fading is the introduction of multiple antennas at the transmitter and/or
at the receiver. By transmitting or receiving the signal that carry same information multiple
times through different independent channel paths, a multiple version of the same signal with
different fades are observed at the receiver that provides more freedom to choose the best
signal for detection. Such spatial diversity of the signal increases the reliability or quality of
service of the wireless system [5].
Further, by using multiple antennas both at the transmitter and at the receiver, not only
diversity gain is achieved but multiple data streams can be transmitted and decoded at the
same time, thus, increasing the degrees of freedom (DOF) for communication [5–7]. This
multi-antenna scheme called multiple input multiple output (MIMO) exploits the spatial
domain to improve the capacity and spectral efficiency significantly and several data streams
are spatially multiplexed onto the MIMO channel [5–7]. The multiplexing gain of the MIMO
channel is proportional to the minimum of the number of transmit and receive antennas for a
given total transmit power, under the assumptions of independent fading channels and the
noise, i.e, the capacity of a point to point MIMO channel with N transmit and N receive
antennas increases by N folds than that of the point to point single input single output
(SISO) channel with the same transmit power [8]. Here the independent fading channel means
that an independent transmission path exists between each transmit and receive antenna due
1
Introduction
to rich scattering environment such that the channel matrix between all transmit and receive
antennas is a full rank matrix.
In the real world communication, multiple users transmit and receive simultaneously. One of
the ways to achieve multi-user communication is the sharing of the available resources between
the users, for example, time division multiple access (TDMA) uses only a fraction of time per
user and frequency division multiple access (FDMA) divides the available frequency among
the users. This, however, reduces the DOF by a fraction of 1K
if there are K users transmitting
and receiving simultaneously. On the other hand, when all the transmitters transmit at the
same time in the same frequency, the interference from the adjacent transmitters become
unavoidable due to the superposition nature of the wireless channel and the desired signal
can not be decoded properly at the desired receiver. Such channels are called interfering
channels. Interfering wireless channels such as two way communication channel [9], broadcast
channel (BC) [10], multiple access channel (MAC) [11] and interference channel (IC) [12,13]
are well-known among the researchers since 1970s.
However, till date a very little is known about the information theoretic capacity of such
channels and a general solution for most of the problems are still open. Only the capacity
region of a degraded BC and general MAC is well-known and given as the closure of convex
hull of the rate vectors that satisfy user-by-user power constraint for the given product input
distribution [14]. Quite a few results are known about the capacity of MIMO MAC and
MIMO BC for constant and fading channels with different degree of channel state information
at the transmitter (CSIT) and channel state information at the receiver (CSIR) as in in [15].
The results are obtained using Costa’s dirty paper coding (DPC) [16] and the concept of
MAC-BC duality [17].
The capacity region of even a two-user Gaussian IC is not solved in general, except that
it is known for a very strong interference regime [12, 18]. Also, lately using simple Han
Kobayashi type scheme [18], Etkin and Tse has shown that the capacity of two user Gaussian
IC is approximated to within one bit per second per Hertz of the bound for all channel
parameters [19]. This finding highlighted the fact that approximation of the capacity region
is one of the ways to solve the existing open problems regarding Gaussian IC and the most
preliminary form of capacity characterization is its linear approximation, which is the DOF
characterization as can be observed directly from the Shannon’s capacity expression [20] for
a very high signal to noise ratio (SNR) as
C = DOF log2(1 + SNR) = DOF log2(SNR) + DOF log2(1 +1
SNR). (1.1)
2
Introduction
When SNR→∞, DOF log2(1 + 1/SNR)→ 0 and log2(SNR)→ k, some constant, then
C ≈ kDOF⇒ C ∝ DOF. (1.2)
One of the important aspects of this observation is that time, frequency and space all
offer DOF in terms of the orthogonal dimensions and it is easier to analyze multi-user single
antenna and multi-antenna networks in this regard. Hence, a majority of the research works
regarding the interference network is focused on the DOF characterization instead of capacity
characterization though the DOF characterization is not a trivial problem in itself. The
initial work in DoF characterization conjectured that the K-user single antenna IC has only
1 DOF while the achievable best outer bound was found to be K2
[21]. Interference alignment
(IA) was introduced to reduce the gap between this inner and outer bound and to answer the
question ‘what is the optimal achievable DOF of a K-user IC?’ [1].
By aligning the interference from all the unwanted transmitters in a single direction, all
K − 1 unwanted signal occupy only 1 dimension instead of K − 1 dimensions as thought
previously. Hence, it behaves as if there is only one interference signal and one desired signal
per transmitter-receiver pair, thus achieving 12
DOF per pair and K2
total DOF. The alignment
is proved to be achievable for K-user time-varying IC and shown that every user almost surely
achieves reliable communication at rates approaching one-half of the achievable capacity at
very high SNRs by using the channels infinite times in [1]. This result is interesting because
whatever is the number of users each user can use half of the resources all the time. In other
words, everyone gets half of the cake irrespective of the number of people and the total DOF
increase from previously thought is
DOFinc = (K
2− 1)× 100%. (1.3)
This shows that the DOF increase is tremendous as the number of transmit-receive pairs
(sometimes referred to as ‘users’ in this article) increase, for example, if K = 3, DoFinc = 50%,
if K = 10, DoFinc = 400% and if K = 50, DoFinc = 2400%. The result is a breakthrough
achievement in the field of information theory and communication systems and it clearly
contributes an important message that the “K-user IC is not interference limited”.
Although it is mathematically possible that every user gets half of the cake, this is a huge
practical challenge in itself because it is achievable only under idealistic conditions such as
deterministic channels or infinite channel uses or global channel information as in [1, 22,23].
These challenges provide plethora of opportunities for the researchers to work in this topic
and a huge number of research articles have been published in designing IA algorithms and
obtaining the feasibility conditions of IA in a very few years time under different network
3
Thesis Outline
scenarios and channel conditions such as [2, 3, 24–45].
More antennas at the transmitter and the receiver provide higher spatial DOF. Hence, on
one hand, it is easier to achieve IA in a MIMO channel because it does not require infinite
channel uses but on the other hand the achievable DOF is limited by M and N , where M
represents the number of transmit antennas and N represent the number of receive antennas.
Also, the complexity increases as the number of users grow. This makes K-user MIMO-IA
an interesting problem to the researchers. The main focus of this Ph.D. thesis is to obtain
the in-depth understanding of IA techniques and algorithms with regard to MIMO ICs, to
design a new and better alignment algorithm for the K-user MIMO-IC under suitable channel
assumptions and to analyze the achievable DOF for such networks. As such a complete DOF
expression for a general K-user MIMO-IC is still an open problem. Hence, most of our works
are based on three user MIMO-IC. Perfect CSIT is a very optimistic assumption for wireless
channel. In this regard, we also design a space time transmission scheme to obtain the DOF
of multiple input single output (MISO) and MIMO interference broadcast channel (IBC)
using delayed CSIT. Following section outlines the structure of the thesis.
1.1 Thesis Outline
After the introduction in chapter 1 where we outlined the motivation behind the invention
of interference alignment algorithms for multi-user interfering networks, the rest of the
dissertation is structured as follows:
Chapter 2 is a brief overview of the well-known interference alignment algorithms and
detailed description of the idea of interference alignment. In this chapter, we describe the basic
concept of IA with examples, we consider K-user SISO IC and explain in detail various well
known precoding techniques to achieve the alignment of interference such as the asymptotic
approach as proposed by Cadambe and jafar [1], the ergodic approach as proposed by Bobak
and Nazar [26], IA using the concept of Diophantine approximation which deals with the
approximation of real numbers by rational numbers as proposed by Motahari et al. [30,46],
the lattice based IA approach which deals with the structured coding and further introduces
the concept of generalized degrees of freedom (GDOF), a more generic way to describe the
DOF as a function of some parameter α and the topology based alignment approach, where
the knowledge of network topology is helpful in a partially connected network to achieve
better DOF using the concept of IA. Furthermore, chapter 2 also deals with the concept of
IA in multi-user MIMO IC, we discuss the algorithms that require global CSIT and some
4
Thesis Outline
of the well-known iterative algorithms that require only the distributed CSIT to achieve
the alignment. Last but not the least, this chapter describes the feasibility conditions for
achieving IA. Only those networks, where IA is feasible are the networks of our interest.
Chapter 3 describes a proposed cooperative IA algorithm, which outperforms some of the
well-known algorithms such as minimization of leakage interference (MLI) and maximization
of signal to interference plus noise ratio (max-SINR) [2]. The main idea of this algorithm is
to use the hybrid (cooperative + greedy ) optimization approach. Cooperation at the receiver
side helps to design the zero-forcing matrix and the greedy approach at the transmitter side
helps to design the precoding matrix. The other benefit of this approach is that the receivers
can estimate the precoding matrices themselves. Using coalition game theoretic approach,
we show that the full cooperation is always optimal.
Chapter 4 deals with the DOF analysis of single cell MISO BC and multi-cell MISO BC
with the help of delayed CSIT. Delayed CSIT is the CSIT of all previous time instances till
t − 1 observed at any time instance t. With the help of the CSIT from the previous time
instances, the interference signal could be aligned by using proper transmission technique
and hence the achievable DOF is improved. The DOF expression for the single cell K user
MISO BC is derived and the concept is extended to two-cell MISO BC. With two cells,
there are not only inter user interference (IUI) but also inter cell interference (ICI), which
is a great challenge to mitigate. However, we show that by proper space-time transmission
scheme ICI can be mitigated and gain in DOF can be achieved. We consider a number of
example transmission schemes and generalize the result with a conjecture. We also provide
an improved transmission strategy to obtain better results.
In chapter 5, we define and introduce the concept IA, where a group of receivers create
a common overlap subspace to align the interference observed from any transmitter. We
termed such an alignment approach as a vertical alignment because the interfered receivers
align the interference observed from a single transmitter unlike a single receiver aligning the
interference observed from multiple transmitters in traditional IA schemes. We show that it
is a very useful scheme in MIMO IBC and we also show that it is a useful idea to study the
optimal overlapping dimensions when the receivers have the common channel information
and independent channel information. Optimal overlapping dimensions are determined with
the help of a so called alignment matrix as defined in non linear dimensionality reduction [47].
In chapter 6, we characterize the achievable DOF of any three-user MIMO IC for any
arbitrary number of transmit and receive antennas when they are not equal by creating
an optimal overlap space between the receivers and forming a simple chain as long as the
5
Notation
optimal overlap is possible, called the receiver chain alignment. We provide an information
theoretic proof and also show that it is possible to design the precoder as long as the optimal
overlapping space is created.
Chapter 7 presents the conclusions and state some of the future works that can be done
with the help of this thesis.
1.2 Notation
(1) C1 Field of complex scalars.
(2) R1 Field of real scalars.
(3) Cd×1 Field of complex vectors of dimension d.
(4) CN×M Field of complex matrices with dimensions N ×M .
(5) RN×M Field of real matrices with dimensions N ×M .
(6) X A matrix X.
(7) x A vector x.
(8) x A scalar x.
(9) X† Moore Penrose pseudo inverse of a matrix X.
(10) Ir Identity matrix of dimensions r × r.(11) XT Transpose of a matrix X.
(12) XH Conjugate transpose of a matrix X.
(13) Tr(X) Trace of a matrix X.
(14) ||X||2 Euclidean norm of a matrix X.
(15) ||X||F Frobenius norm of a matrix X.
(16) span(X) Range or linear combination of column vectors of a matrix X.
(17) N (X) Null space of a matrix X.
(18) X = {·, ·, ·} A set X with a number of elements {·, ·, ·}.(19) X ∪Y Union of two sets X and Y.
(20) X ∩Y Intersection of two sets X and Y.
(21) E(X) Expected value of a random variable X.
(22) min(x, y) Minimum of the two scalars x and y.
(23) max(x, y) Maximum of the two scalars x and y.
(24)(nk
)Binomial coefficient.
6
Chapter 2
Interference Alignment
2.1 Basic Concept
Consider a K-user SISO IC as shown in Figure 2.1, where the signal from transmitter
j to receiver i is represented by hij ∈ C1, a complex scalar, ∀i = {1, 2, · · · , K} and
∀j = {1, 2, · · · , K}. The desired channels are represented by the solid line and the interfering
channels are represented by the dashed lines. Let xj ∈ C1 be the transmitted complex symbol
from any transmitter j, then the received signal at each receiver is the superposition of the
signals transmitted from each transmitter, affected by the fading channel and the noise, as
given by :
yi = hiixi +K∑
j=1,j 6=i
hijxj + ni, (2.1)
where, hiixi is the desired part and the rest K−1 parts inside summation are the interfering
parts while ni is the noise part, which means that there are K−1 unwanted symbols and only
1 desired symbol per receiver. In elementary linear algebraic sense, there are K unknowns to
be solved per receiver with only one desired unknown. Hence, in order to solve one desired
unknown, we need at least K linear equations per receiver with the channel coefficients
drawn from the continuous distribution. In other words, we need K independent signaling
dimensions, e.g., we need to use independent channels K times, to decode 1 desired symbol,
thus limiting the DOF to 1K
. But, we can obviously do better than this. Let us consider that
each receiver uses the channels T times (T < K) such that the system is under-determined
with T equations and K unknowns. For any receiver i, this system of linear equations can be
written as
yi(1) = hi1(1)x1 + hi2(1)x2 + · · ·+ hiK(1)xK + ni(1),
yi(2) = hi1(2)x1 + hi2(2)x2 + · · ·+ hiK(2)xK + ni(2),
......
...
yi(T ) = hi1(T )x1 + hi2(T )x2 + · · ·+ hiK(T )xK + ni(T ), (2.2)
7
Basic Concept
TX1
TX2
TXK
RX1
RX2
RXK
h11
h22
hKK
h1K
h12hK1
h21
h2KhK2
Figure 2.1: A K-user single input single output interference channel, where the solid linesrepresent a direct or the desired links and the dashed line represent the cross or interferencelinks. All the receivers receive one desired link and K − 1 interference links.
8
Basic Concept
which can be further expressed in the vector form as
yi = hi1x1 + hi2x2 + · · ·+ hiKxK + ni, (2.3)
where the vectors yi =[yi(1) yi(2) · · · yi(T )
]T, hij =
[hij(1) hij(2) · · · hij(T )
]Tand ni =
[ni(1) ni(2) · · · ni(T )
]T.
From equation (2.3), we observe that the received signal space is the linear combination of
T dimensional channel vectors. Since only one data stream, lets say x1, is to be decoded at
RX1, we want the vector hi1 to be independent of all the other vectors hij,∀j 6= 1, i.e., we
should not be able to express hi1 as the linear combination of hij,∀j 6= 1 or mathematically,
hi1 /∈ span({hij,∀j 6= 1}), (2.4)
where span(A) represents the range or linear combination of column vectors of A.
If span({hij,∀j 6= 1}) spans all the available dimensions, i.e., min(T,K − 1), then x1 is not
still decoded because there are no free dimensions available to resolve x1. Thus, we want to
consolidate the span({hij,∀j 6= 1}) in as much smaller dimension as possible so that there is
enough free dimension to resolve x1. This approach of consolidating the interference subspace
onto the lower dimensional space is achieved by aligning or overlapping the interference
vectors as much as possible, which is called interference alignment.
Consider a system with K = 5 and T = 3, also assume that the channel states are real and
the signal observed at any receiver i in all the time instances, which desires to decode x1 are
given by
yi(1) = −3x1 + 2x2 + 3x3 + x4 + 5x5, (2.5)
yi(2) = −2x1 + 4x2 + x3 − 3x4 + 5x5, (2.6)
yi(3) = −4x1 + 3x2 + 5x3 + 2x4 + 8x5. (2.7)
Although, this is an under-determined system, x1 is solvable because the coefficients of
the interfering signals x5 is the linear combination of coefficients of interfering signal x2 and
x3 and also the coefficients of x2 are the linear combination of the coefficients of x3 and x4.
This dependence allows all the four interference signal to be aligned over two dimensions as
is clearly observed in the Figure 2.2, where all the interference (shown by blue, red, magenta
and black colored vectors) span only two dimensions while allowing an independent dimension
for the desired signal x1 ( shown by green colored vector). Thus, the whole signal space can
be projected on a plane that zero-forces the interference to obtain the desired data.
At the same time, the other receivers j 6= i,∀j = {1, 2 · · ·K} also receive a linear combina-
tion of all the transmit signals from all transmitters. But interestingly, the linear combination
9
Basic Concept
−2y
2
0
4
0
−2
−4
−2 x
24
0
−4
z 2
4
6
8
Figure 2.2: A geometric explanation of IA with four interference aligned in two dimensionsand one desired signal in the orthogonal direction. Green is the desired signal while blue, red,black and magenta are the interference signal.
of the interference signal observed by all these receivers are different because of the fact that
each one of them desire a data stream different from the other. Hence, the requirement of
one receiver for alignment do not conflict with the requirement of the other receivers. This
interesting feature, called relativity of alignment, is an essential premise to ensure IA [1,48].
For the given example with deterministic and well-defined channel states, the IA principle
looks very simple and the alignment is easily achieved. But in real world scenario, the channel
states are completely random and can not be controlled by the designer. Hence, it is a great
challenge to make the coefficients of all the interfering signals as much dependent as possible
to consolidate them in the lower dimensional space. One of the ideas is to precode all the
transmitted signal before transmitting such that the interfering signal align with each other
as much as possible in each receiver. This, however, requires that all the transmitters need
to know all the channel states (global CSIT) perfectly beforehand and the channel need to
remain constant within the transmission period. This is a very optimistic assumption in any
wireless communication scenario and is a major drawback of the IA scheme.
Also, in the example scheme, only 53
DOF is achieved while the outer bound on the DOF of
any 5-user SISO-IC is 52
[1,21] and the obvious question is “ How close can we go to the outer
bound ?” This depends on how effectively all the receivers align the available interference. If
all the receivers can align the available interference in 2T time instances in order to decode
10
Precoding Techniques
T desired streams by each of them, then the outer bound is achieved. In other words, if each
receiver aligns all the interference over one-half of the available space leaving other half for
the desired streams, then the outer bound is achieved. In [1], Cadambe and Jafar showed
that this condition is not strictly achievable but it can be achieved with some relaxation in
the number of channel uses. For 3-user SISO-IC, [1] shows that 3n+ 1 streams of data can
be decoded in 2n+ 1 channel uses for any positive integer n, thus allowing 3n+12n+1
DOF, which
achieves the outer bound of 32
as n→∞.
2.2 Precoding Techniques
In this section, we briefly discuss about some of the well-known precoding techniques that
have been proposed in the literature to achieve IA. This provides us a better understanding
in this topic and acts as a framework for further discussions in the future chapters.
2.2.1 Asymptotic Approach
In K-user SISO-IC, one of the approaches to achieve alignment is by beamforming over
multiple symbol extensions of the time-varying wireless channel as proposed in [1]. By
extending the channel over (n+ 1)N + nN symbols, (n+ 1)N + (K − 1)nN independent data
streams can be decoded, where N = (K − 1)(K − 2)− 1 and n is any natural number, thus
providing K2
DOF asymptotically as
limn→∞
(n+ 1)N + (K − 1)nN
(n+ 1)N + nN= lim
n→∞
(n+ 1)N − nN(n+ 1)N + nN
+KnN
(n+ 1)N + nN(2.8)
= limn→∞
(1 + 1n)N − 1
(1 + 1n)N + 1
+K
(1 + 1n)N + 1
(2.9)
=K
2. (2.10)
For K = 3, N = 1 and (n+1)+2n = 3n+1 data streams are decoded for (n+1)+n = 2n+1
symbol extensions of the channel, which means that one of the transmitters transmits (n+ 1)
data streams and the rest transmit n data streams free of error in 2n+ 1 channel uses. This
is achieved by the proper alignment of interference signal at the receivers. Hence, the aim of
this approach is to design the precoders at each transmitter that achieves certain alignment
conditions.
Assume that each transmitted message is precoded by the same precoding vector v and
Hij is the diagonal channel matrix obtained from the symbol extensions of time-varying
11
Precoding Techniques
channel observed at receiver i corresponding to transmitter j. The signal observed at any
receiver i without considering the noise is then given by
yi = Hiivsi +K∑
j=1,j 6=i
Hijvsj, (2.11)
where vsj represents the precoded symbol at transmitter j. The first part is the desired
signal and the second part in the summation is the interference part.
For simplicity, the channels are normalized over Hii and the received signal is
yi = vsi +K∑
j=1,j 6=i
Tijvsj, (2.12)
where Tij is the diagonal channel normalized with respect to Hii.
Let V be the space spanned by the desired part (corresponding to the precoding vector v)
and assume that same space is set aside by any receiver for the interference part then the
total space observed by the receiver i and the space set aside for interference is given by
N =⋃
i=1,i 6=j
(V ∪TijV). (2.13)
This is true for all receivers k 6= i. The total interference space is the union of the
interference space observed at all the receivers and the space set aside by each receiver.
There are total of N = K(K − 1) observed interference space and a space V set aside by
each receiver. If we represent all the normalized channels as T1,T2, · · · ,TN , then the total
interference space is
NT = (V ∪T1V ∪T2V ∪ · · · ∪TNV). (2.14)
The aim of IA is to consolidate all the observed interference space to the space set aside by
each receiver. This is possible only if
V ≈ T1V ≈ T2V ≈ · · · ≈ TNV. (2.15)
In order to achieve (2.15), any random V is initially chosen and updated iteratively such
that the next V contains all the interference space previously observed. Since V corresponds
to the precoding vector v, we obtain v by updating V. As in [36], choose V1 = 1, the vector
of all 1s then
N1 = 1 ∪T11 ∪T21 ∪ · · · ∪TN1. (2.16)
Clearly, the interference space is N + 1 dimensional, while the desired space V1 is 1
dimensional and the ratio dim(V1)dim(N1)
= 1N+1
, which is much less than 1. Now set the new desired
12
Precoding Techniques
space V2 = N1, which is N + 1 dimensional, then the new interference space is
N2 = 1 ∪T11 ∪T21 ∪ · · · ∪TN1 ∪T1(1 ∪T11 ∪T21 ∪ · · · ∪TN1) ∪T2(1∪T11 ∪T21 ∪ · · · ∪TN1) ∪ · · · ∪TN(1 ∪T11 ∪T21 ∪ · · · ∪TN1), (2.17)
= 1 ∪T11 ∪T21 ∪ · · · ∪TN1 ∪T11 ∪T211 ∪T1T21 ∪ · · · ∪T1TN1 ∪T21
T2T11 ∪T221 ∪ · · · ∪T2TN1 ∪ · · · ∪TN1 ∪TNT11 ∪TNT21 ∪ · · · ∪T2
N1. (2.18)
Further , assuming that the diagonal interference channels are commutative, i.e., TiTj =
TjTi,∀i 6= j and i, j = {1, 2, · · · , N}, we have
N2 = 1 ∪T11 ∪T21 ∪ · · · ∪TN1 ∪T211 ∪T1T21 ∪ · · · ∪T1TN1∪
T221 ∪ · · · ∪T2TN1 ∪ · · · ∪T2
N1, (2.19)
= 1 ∪ · · · ∪Ti1 ∪ · · ·TiTj1 ∪ · · · ∪T2i1,∀i 6= j. (2.20)
The dimensions of N2 is (N + 1) +N(N − 1)/2 +N = (N + 1)(N + 2)/2 and the ratio
dim(V2)
dim(N2)=
N + 1
(N + 1)(N + 2)/2=
2
N + 2, (2.21)
which is still much less than 1 and we continue the next iteration for which V3 = N2 and
obtain N3 similarly as before whose dimensions is (N + 1)(N + 2)(N + 3)/6 and the ratiodim(V3)dim(N3)
= 3N+3
, which is still much less than one but better than previous ratios. Hence,
continuing similarly, it is observed that after n iterations the ratiodim(Vn)
dim(Nn)=
n
N + n, (2.22)
which approaches 1 as n→∞.
2.2.2 Ergodic Approach
The ergodic IA, initially presented by Bobak Nazar et al. in [26], considers that the channel
coefficients are independent, time-varying and are drawn from the distribution with uniform
phase. In a K-user SISO IC, all the transmitters transmit at time t and wait for the next
transmission until the complementary channel matrix occurs at time tc. The complementary
channel matrix for the given channel matrix H(t) =
h11 h12 · · · h1K
h21 h22 · · · h2K
......
. . ....
hK1 hK2 · · · hKK
, observed at all
the K receivers corresponding to all the K transmitters at any time t, is
13
Precoding Techniques
Hc(tc) =
h11 −h12 · · · −h1K
−h21 h22 · · · −h2K
......
. . ....
−hK1 −hK2 · · · hKK
, (2.23)
such that H(t) + Hc(tc) is a diagonal matrix, which means all the interferences cancel
automatically if same data streams are transmitted at time t over channel H and at time tc
over channel Hc. Thus, at any receiver i,
yi(t) + yi(tc) = 2hii + ni(t) + ni(tc), (2.24)
where yi(t) and yi(tc) are the signal received at time t and tc respectively and ni(t) and
ni(tc) are the noise at time t and tc respectively.
However, the assumption that the exact complementary channel occurs after certain time
is very unrealistic and will happen with zero probability for any channel states distributed
continuously. Thus, the channel coefficients are quantized and matched up as closely as
possible with the available complementary channel. The error probability decreases for finer
quantization and the targeted rate is achieved in the limit as described in detail in [26].
2.2.3 IA Based on Real Numbers
The idea of real IA is to transform the single antenna system into pseudo multiple antenna
systems with infinitely many pseudo antennas, and simultaneously align interference at all
receivers. Hence, unlike ergodic approach channel need not be time-varying. In [30], it is
shown that K2
DOF is achievable for K-user Gaussian IC by selecting an appropriate transmit
directions at all the transmitters.
Consider a 3-user SISO Gaussian IC, the received signal at each receiver is given by
y1 = h11x1 + h12x2 + h13x3 + n1, (2.25)
y2 = h21x1 + h22x2 + h23x3 + n2, (2.26)
y3 = h31x1 + h32x2 + h33x3 + n3, (2.27)
where xi are the input symbols, hij are the channels from transmitter j to receiver i and
ni are the additive white Gaussian noise (AWGN) at receiver i.
Since the linear operations at the transmitter and the receiver do not affect the capacity
region of the channel, the following transmission strategy is adopted in [30], to generate an
equivalent standard channel with respect to DOF:
14
Precoding Techniques
(i) Transmitter 1 transmits x1 = h23h12h21
x1,
(ii) Transmitter 2 transmits x2 = h13x2 and
(iii) Transmitter 3 transmits x3 = h12x3,
then the modified received signal is represented as
y1 =h11h23h12
h21
x1 + h12h13x2 + h13h12x3 + n1,
⇒ y1 =h11h23h12
h21h12h13
x1 + x2 + x3 + n1, (2.28)
y2 =h21h23h12
h21
x1 + h22h13x2 + h23h12x3 + n2,
⇒ y2 = x1 +h22h13
h23h12
x2 + x3 + n2, (2.29)
y3 =h31h23h12
h21
x1 + h32h13x2 + h33h12x3 + n3,
⇒ y3 = x1 +h21h32h13
h31h23h12
x2 +h33h12h21
h31h23h12
x3 + n3. (2.30)
Further, consider
G0 =h13h21h32
h12h23h31
=(h13
h12
)(h21
h23
)(h32
h31
), (2.31)
G1 =h11h12h23
h12h21h13
=(h11
h13
)(h23
h21
), (2.32)
G2 =h22h13
h12h23
=(h13
h12
)(h22
h23
), (2.33)
G3 =h33h12h21
h12h23h31
=(h21
h23
)(h33
h31
). (2.34)
Here, G0 is important because it is the product of the ratios of all the cross channels in
all receivers and in [30, 46], it is proved that for 3-user IC, if G0 is rational, then 32
DOF
is achievable. Hence, only G0 is considered as a generator function to create the transmit
directions. Since each transmit direction is independent of the other, the transmit directions
for d pseudo antennas which carry d streams are the basis set {1, G0, G20, · · · , Gd−1
0 } over the
rational numbers and the transmitted data is represented as the linear combination given by
xj = A(sj1 +d−1∑i=1
Gi0sj(i+1)), (2.35)
where A is any real number and sji are independent data streams transmitted from
15
Precoding Techniques
transmitter j. Thus, for 3 user IC, we have
x1 = As11 + A(G0s12 +G20s13 + · · ·+Gd−1
0 s1d), (2.36)
x2 = As21 + A(G0s22 +G20s23 + · · ·+Gd−1
0 s2d), (2.37)
x3 = As31 + A(G0s32 +G20s33 + · · ·+Gd−1
0 s3d). (2.38)
Substituting the values of x1, x2 and x3 from (2.36), (2.37) and (2.38) in (2.28), we obtain
y1 =AG1(s11 +G0s12 +G20s13 + · · ·+Gd−1
0 s1d) + A(s21 + s31) + AG0(s22 + s32)+
AG20(s23 + s33) + · · ·+ AGd−1
0 (s2d + s3d) + n1, (2.39)
where all the interference from transmitter 2 and transmitter 3 are aligned over d inde-
pendent directions. Similarly substituting the values of x1, x2 and x3 from (2.36), (2.37)
and (2.38) in (2.29), it is easily possible to align all the interferences from transmitter 1
and transmitter 3 over d independent dimensions as before. But substituting the values of
x1, x2 and x3 from (2.36), (2.37) and (2.38) in (2.30), the alignment is observed over d+ 1
dimensions as given by
y3 =As11 + A(G0s12 +G20s13 + · · ·+Gd−1
0 s1d) +G0(As21 + A(G0s22 +G20s23 + · · ·
+Gd−10 s2d)) +G3(As31 + A(G0s32 +G2
0s33 + · · ·+Gd−10 s3d)) + n3, (2.40)
=As11 + AG0(s12 + s21) + AG20(s13 + s22) + · · ·+ AGd−1
0 (s1d + s2(d−1))+
AGd0s2d + AG3(s31 +G0s32 +G2
0s33 + · · ·+Gd−10 s3d) + n3. (2.41)
But since there are only d basis vectors, one of the dimensions is dependent which is evident
from the fact that Gd0 can be represented as a linear combination of {1, G0, G
20, · · · , Gd−1
0 }with rational coefficients. Hence, in all three receivers the interference signal is aligned over d
dimensions and the desired signal occupy other d dimensions independent of the interference
signal. In [46], it is proved that if such conditions exist, the achievable DOF for almost all
realization of the system is given by
dtot =total interference dimensions
maximum received dimensions=d+ d+ d
2d=
3
2. (2.42)
On the other hand, if G0 is not rational but a transcendental number, it is again proved
in [46] that the total achievable DOF is given by
dtot =3n+ 1
2n+ 1, (2.43)
as also given by [1] for any integer n.
16
Precoding Techniques
2.2.4 Lattice-Based Approach
The structured codes are used to achieve IA in some K-user IC, called the lattice based IA
approach. Since the sum of the lattice points, which are the codewords is also a lattice point,
a new codeword, it is possible to decode the sum even if the individual codewords are not
decodable [49]. When the interference is strong and decoding the interference helps to decode
the desired signal, lattice codes are useful. In [49], it is shown that the capacity of many
to one IC is achieved within a constant bits per second per Hertz regardless of the channel
parameters using the lattice alignment.
When some signals are significantly stronger or weaker than the other, the DOF is not a
proper metric as it forces all the channels to be equally strong. In such case, a new metric
called GDOF is used which is a function of some parameter α that captures the signal
strength of particular channel [19,50–52]. Like DOF, GDOF is mathematically defined as :
GDOF(α) = limSNR→∞
C(SNR, α)
log(SNR), (2.44)
where C(SNR, α) is the channel capacity as a function of SNR and α, the ratio of interference
channel (in dB) to the desired channel (in dB).
Since layered code structure can be used when signal levels are different, lattice alignment
is used to optimize the GDOF of such IC. However, the wireless channel is random and
uncertain in practice. Hence the random codes are preferred than the structured codes most
of the time for wireless channel. Robust lattice alignment for wireless channels is still an
open problem and only few works are available in this regard such as [53].
2.2.5 Topology Based Approach
Requirement of complete channel knowledge is one of the major drawbacks of IA, because
wireless channels are uncertain and difficult to predict. It is relatively easier to predict the
channel states at the receiver side but more difficult to predict the channel states at the
transmitter side. Hence, a number of research works on IA are focused on relaxing the
requirement of complete perfect CSIT, such as [27,28,39–41,54–56].
The concept of blind IA is introduced in [27], where the alignment is observed only based
on the knowledge of autocorrelation of the channels seen by different receivers, whereas IA
is achieved with imperfect CSIT in [54]. Other approaches such as in [28,39–41] uses the
knowledge of CSIT from the previous time instances to achieve the alignment of interference.
One of the relatively new approaches that relaxes the explicit requirement of CSIT is
17
MIMO Interference Alignment
introduced by Jafar [45], which uses the knowledge of the network topology to align the
interference at the receiver using the concept of index coding [57,58].
For a partially connected network, alignment graph and the conflict graph is created from
which alignment set and number of internal conflicts are determined. Alignment graph is the
graph of all the interference messages which need to be aligned and conflict graph is the graph
of all the messages that need to be kept separate. For better understanding, consider a 5-user
partially connected SISO IC as shown in Figure 2.3, where wj represents the message from
any transmitter j and wi is the message desired by any receiver i, the solid line represents the
desired channel and the dashed line represents the interference channel and the corresponding
alignment conflict graph for the network is shown in Figure 2.4, where the solid line represents
the alignment graph and the dashed line represents the conflict graph.
As seen from the figure w1 and w5 need to be aligned at receiver 3 and w3 and w4 need to
be aligned at receivers 1 and 2. Hence, w1, w5 and w3, w4 are connected by solid lines. Thus
there are total of three alignment sets {w1, w5}, {w3, w4} and {w2}. If two messages in the
same alignment graph have conflict, it is called internal conflict, e.g., if w3 and w4 are also
connected by dashed line in Figure 2.4 or if receiver 4 receives the signal from transmitter 3
in Figure 2.3 then the internal conflict exists between w3 and w4. However, there exists no
internal conflict for the given network.
When there exists no internal conflict in the network, a symmetric DOF of 12
per user is
always achievable without the knowledge of CSIT [45]. For all the other networks, where 12
DOF is not achievable, internal conflict exists and the symmetric DOF per user is expressed
in terms of minimum internal conflict distance (∆) as given by:
DOF ≤ ∆
2∆ + 1, (2.45)
where the minimum conflict distance ∆ is defined as the minimum number of alignment
graph edges traversed by the messages nodes with internal conflict to go from one message
node to the other.
2.3 MIMO Interference Alignment
Multiple antennas at the transmitter and the receiver offer DOF in terms spatial dimensions.
Spatial dimensions are interesting because of their distributed nature and can be as large as
the number of transmit and receive antennas. Also, spatial dimensions allow operations such
as beam-forming to direct (beam) the interference signal onto the null space while keeping
18
MIMO Interference Alignment
1
2
3
4
5
1
2
3
4
5
w1
w2
w3
w4
w5
w1
w2
w3
w4
w5
~
~
~
~
~
Figure 2.3: A partially connected 5 user SISO IC. A solid line represents the desired channelwhile the dashed lined represents the interference channel.
w1
w2
w3 w4
w5
Figure 2.4: Alignment conflict graph for the network in Figure 2.3. Solid line represents thealignment graph and the dashed line represents the conflict graph.
19
MIMO Interference Alignment
the desired signal independent of interference signal. This nulling of interference is called
spatial zero-forcing [59]. Spatial dimensions provide spatial DOF as high as min(M,N) in a
single user MIMO channel, where M and N are the number of transmit and receive antennas
respectively [5].
Mathematically, beam-forming is achieved by sending the coded information to the receiver
with prior knowledge of all the channel states. This operation of coding before transmission is
called precoding. Multiple streams transmitted via spatial dimensions are coded by multiple
precoding vectors individually. All these precoding vectors form columns of the precoding
matrix. Since precoding is performed for beaming the signal streams to particular direction,
the signal amplitude is preserved while changing the phase of the signal. Hence, a precoding
matrix is assumed to be a unitary matrix. A number of literature such as [5,8,14] provide an
extensive detail on precoding in point to point MIMO channels.
Like a multi-user SISO channel, the major limitation of a multi-user MIMO channel is the
interference. More spatial dimensions allow more independent interference streams to be
observed at any receiver. If the total number of spatial dimensions available at any receiver
is less than the total number of interference observed by that receiver, it is unable to decode
the desired signal and no reliable communication is possible. Thus, MIMO IA is a technique
by which all the observed interference are allocated as minimum dimensions as possible
thus allowing the possible maximum dimensions for the desired streams, which increases
the spatial DOF of the system. This is achieved by designing proper precoding matrix at
each transmitter which aligns all the interference signal received by a receiver onto the same
subspace, meaning that the spatial dimensions allocated for the interference signal from all
interferer is the same as the spatial dimensions allocated for a single interferer [2, 3, 32].
Consider a simple example as depicted in Figure 2.5, where alignment of interference
is observed in a three user MIMO IC. Here, Hij ∈ CN×M is a channel matrix from any
transmitter j to receiver i and Vj is a precoding matrix whose dimensions depend on the
number of transmit streams from transmitter j. The precoding matrices Vj,∀j = {1, 2, 3}are unknown and they need to be designed such that alignment of interference is achieved in
all three receivers. When M = N , each channel can be separated into M parallel channels
and the whole network acts as KM user SISO IC, the achievable DOF for which is given by
DOF =KM
2. (2.46)
.
But for the cases when M 6= N , it is not trivial to determine the achievable DOF. In fact,
20
MIMO Interference Alignment
TX 1
TX 2
TX 3
RX 1
RX 2
RX 3
V1
V2
V3
H11
H22
H33
H21
H31
H32
H12H
13 H23
H13V3
H12V2
H11V1
H22V2
H32V2
H21V1
H23V3
H33V3
H31V1
Figure 2.5: Interference alignment in three user MIMO IC. The solid line represents a desiredchannel matrix and the dashed and dotted lines represent the interference channel matrices.For ease of representation, the alignment of multiple spatial dimensions is represented by anarrow. For example, dashed green arrow represents a d dimensional subspace observed atRX1 from TX3.
21
MIMO Interference Alignment
the alignment conditions increase exponentially with the increasing number of users and
designing a proper precoding matrix becomes a great challenge. Even for a simple three user
MIMO IC, the alignment is achieved when the following conditions are satisfied:
(I) At RX1,
span(H12V2) = span(H13V3), (2.47)
(II) At RX2,
span(H21V1) = span(H23V3), (2.48)
(III) At RX3,
span(H31V1) = span(H32V2), (2.49)
where span(A) represents the linear combination of the column vectors of A. From a very
simple observation, the linear combination of column vectors of two matrices are same when
two matrices are equal. Hence
(I) At RX1,
H12V2 = H13V3, (2.50)
⇒V2 = (H12)†H13V3, (2.51)
(II) At RX2,
H21V1 = H23V3, (2.52)
⇒V1 = (H21)†H23V3, (2.53)
(III) At RX3,
H31V1 = H32V2, (2.54)
Substituting the values of V2 and V1 from (2.51) and (2.53) in (2.54), we obtain
H31(H21)†H23V3 = H32(H12)†H13V3, (2.55)
⇒V3 = (H23)†H21(H31)†H32(H12)†H13V3, (2.56)
⇒[IM − (H23)†H21(H31)†H32(H12)†H13
]V3 = 0, (2.57)
where IM represents the M × M identity matrix and V3 is given by the eigenvectors
corresponding to the zero eigenvalues of A =[IM − (H23)†H21(H31)†H32(H12)†H13
].
Clearly, the solution to V3 exists only if the null space of A exists and the dimensions of V3
is the dimensions of the null space. This means the complete knowledge of A is required in
22
From Global to Local CSIT
order to determine V3. Since A is a function of all the interfering channel matrices to all the
receivers, the transmitters need to know all channels globally to precode the transmit streams
in order to achieve IA. Since wireless channels are time-varying, global channel knowledge at
the transmitter is unlikely almost surely.
In the next section, we present some of the IA algorithms to design the precoding matrices
that do not require the CSIT globally but locally. Local CSIT, here refers to the CSIT from one
transmitter to all the receivers, i.e., TX1 knows only Hi1 and not Hi2 and Hi3; ∀i = {1, 2, 3}.
2.4 From Global to Local CSIT
Local CSIT is easier to obtain because of the reciprocity of wireless channels. Reciprocity of
wireless channels means that the channels observed at any receiver i from the transmitter j
are the same as the channels observed in the reciprocal network where the receiver i acts as
a transmitter and the transmitter j acts as a receiver. Transmitter j can obtain the channel
states from all the receivers i = {1, 2, · · · , K}, by allowing the receivers to transmit in the
reciprocal direction in different synchronized time slot. Thus, local CSIT is available naturally
in time division duplexing (TDD) systems. A number of iterative algorithms such as MLI,
max-SINR [2], alternating minimization of interference (AMI) [32] and interference alignment
using alignment matrix (IA-AM) [60] are proposed to design the optimum precoder matrices
for MIMO IC in the literature. In this section, we discuss MLI, max-SINR and AMI, while
IA-AM is discussed in detail in chapter 5.
2.4.1 Minimization of Leakage Interference
The MLI algorithm iteratively minimizes the leakage interference at each receiver by designing
zero-forcing matrix at the receiver and precoding matrix at the transmitter. Leakage
interference is the amount of interference power still observed after zero-forcing the available
interference. The precoding matrices in all transmitters are randomly chosen initially, based
on which zero-forcing matrices are determined in all the receivers. Assuming that the zero-
forcing matrices acts as precoding matrices in the reciprocal network, the actual precoding
matrices are iteratively updated by minimizing the amount of leakage interference [2].
Consider a K-user MIMO IC with M transmit and N receive antennas as shown in
Figure 2.6. The channels from transmitter j to receiver i is represented by a matrix
Hij ∈ CN×M . Each element hk`,∀k = {1, 2, · · · , N}, ` = {1, 2, · · · ,M} of Hij are independent
23
From Global to Local CSIT
and identically distributed (i.i.d) with zero mean and |hk`|2 variance. The precoding matrices
are represented by Vj ∈ CM×d, ∀j = {1, 2, · · · , K} and the zero-forcing matrices are
represented by Ui ∈ CN×d, ∀i = {1, 2, · · · , K}, where d is the total independent data
streams transmitted from any transmitter j. Such system where the number of all transmit
antennas are equal, the number of all receive antennas are equal and also the number of
independent data streams transmitted from all transmitters are equal is called a symmetric
system.
The total leakage interference power at any receiver i is the sum of the leakage interference
power due to all interfering transmitters j 6= i, which can be mathematically expressed as:
Fi = Tr[UHi Q
(f)i Ui
], (2.58)
where
Q(f)i =
K∑i 6=j
P
dHijVjV
Hj HH
ij , (2.59)
is the interference covariance matrix in the forward channel, P is the transmit power and
Tr[A] represents the trace of matrix A.
In order to satisfy the power constraint, the precoding matrix Vj is assumed to be an
orthonormal matrix, i.e., VHj Vj = Id and for any randomly initialized Vj, ∀j = {1, 2, · · · , K},
the optimum Ui is obtained by solving the following optimization problem:
minUi
Fi(Ui,Q(f)i ), (2.60)
s.t. VHj Vj = Id, (2.61)
UHi Ui = Id. (2.62)
The interference covariance Q(f)i ∈ CN×N is a positive semi-definite matrix and can be
eigen-decomposed. Hence, the optimum Ui is given by the eigenvectors corresponding to the
d smallest eigenvalues of Qi [2],
Ui = eigvd(Q(f)i ). (2.63)
The precoding matrix is obtained in all the transmitters by observing the dual network
where each receiver acts as a transmitter and each transmitter acts as a receiver. In that
case, previously determined zero-forcing matrix Ui plays the role of a new precoding matrix
and new zero-forcing matrix is obtained by minimizing the leakage interference in the reverse
direction. The reverse leakage interference is given by:
Rj = Tr[VHj Q
(r)j Vj
], (2.64)
24
From Global to Local CSIT
TX1
TX2
TXK
RX1
RX2
RXK
V1
V2
VK
H11
H22
HKK
H21HK1
HK2
H 12
H1K
H 2K
U1
U2
UK
Figure 2.6: A K-user MIMO IC with M transmit and N receive antennas. Vjs are theprecoding matrices and Uis are the zero-forcing matrices, ∀ i, j = {1, 2, · · · , K}. Solid linesrepresent desired channels and dotted and dashed lines represent interference channels.
25
From Global to Local CSIT
where
Q(r)j =
K∑j 6=i
P
dHjiUiU
Hi HH
ji , (2.65)
is the interference covariance matrix observed in the reverse direction.
The optimum Vj is thus obtained by solving the following optimization problem:
minVj
Rj(Vj,Q(r)j ), (2.66)
s.t. VHj Vj = Id, (2.67)
UHi Ui = Id, (2.68)
the solution to which is given by the eigenvectors corresponding to the d minimum
eigenvalues of Q(r)j . This Vj acts as a new precoding matrix to determine new Ui and the
operation is repeated iteratively till the leakage interference converges to a very small value
and the convergence is guaranteed for such optimization as proved in [2].
Here, we note that the alignment is possible by choosing d streams of data both in the
forward and the backward directions. This requires that feasible number of independent data
streams to be transmitted (the feasible DOF) be known beforehand and the alignment is not
observed if chosen d is greater than the feasible d.
2.4.2 Maximization of Signal to Interference plus Noise Ratio
In the MLI algorithm, only the interference signal observed in each receiver is minimized and
the desired signal is not considered at all. However in the max-SINR algorithm signal to
interference plus noise power ratio is maximized to design the precoding and zero-forcing
matrices at the transmitters and at the receivers respectively [2].
One of the main benefits of this scheme is that it is even valid for small SNR and the
precoding vectors need not be orthogonal to each other as before. In fact, orthogonal precoding
vectors are sub-optimal [2]. Hence, the precoding vector per data stream is determined. This
also requires the knowledge of feasible data streams beforehand. The signal to interference
plus noise ratio (SINR) of the tth stream of any receiver i is given by :
SINRit =(u
(i)t )
HHiiv
(t)i (v
(t)i )
HHHii u
(t)i
(u(t)i )
HBitu
(t)i
, (2.69)
where u(t)i is the tth column vector of any zero-forcing matrix Ui obtained at receiver i, v
(t)i
is the tth column vector of any precoding matrix Vi obtained at any transmitter i and the
matrix Bit is the interference covariance observed by receiver i due to tth column of precoding
26
From Global to Local CSIT
matrices Vj, ∀j = {1, 2, · · · , K} as given by :
Bit =K∑j=1
d∑`=1
Hijv(`)j (v
(`)j )
HHHij −Hiiv
(t)i (v
(t)i )
HHHii + IN , (2.70)
where the dimensions of all the channels are same as defined in the previous algorithm.
We can then solve for the tth column of the zero-forcing matrix Ui as the unit vector u(t)i
which maximizes (2.69) as given by :
u(t)i =
(Bit)−1Hiiv
(t)i
||(Bit)−1Hiiv(t)i ||
. (2.71)
The dual network with reciprocal channel states is considered as in the MLI algorithm to
determine the tth column of any precoding matrix Vi, which is obtained by considering u(t)i
as the new precoding vector and the operation is repeated iteratively till convergence. The
convergence for this algorithm is guaranteed though global optimum is not achieved due to
non-convex nature of the objective function as proved in [2].
2.4.3 Alternating Minimization of Interference
Unlike the previous algorithms, this alternating minimization of interference approach pro-
posed by Peters and Heath in [32] uses the concept of projection to determine the optimal
subspace where all the interference is aligned. The zero-forcing matrix or sometimes called
interference suppression matrix Ui at any receiver i is the basis of the optimal subspace
where all the interference observed by receiver i is aligned.
The precoding matrices designed at each transmitter helps to align all the interference.
Since the precoding matrices and the zero-forcing matrices are not known initially, they
are alternately optimized assuming one is known at an instance. Consider that all the
precoding matrices are orthonormal and are randomly initialized, then the receiver i observes
d dimensional interference HijVj, ∀j 6= i from all transmitters j 6= i. Here, we assume that
all the symbols and dimensions are same as defined in previous algorithms and the system
is symmetric, i.e, all transmitters have M antennas, all receivers have N antennas and all
transmitters transmit d independent streams of data.
At any receiver i, Ui ∈ CN×d is the basis of the subspace where all the interference is
aligned. Since there are total of (K − 1)d interference streams, all interference projected onto
d dimensional subspace is aligned. Thus, the aligned interference due to any receiver j is
UiUHi HijVj, ∀j 6= i and the optimum Ui is obtained by minimizing the total error between
the initial and the projected interference streams as given by,
27
From Global to Local CSIT
minUi
K∑j=1,j 6=i
||HijVj −UiUHi HijVj||2F , (2.72)
s.t. VHj Vj = Id, (2.73)
UHi Ui = Id, (2.74)
where ||A||F represents the Frobenius norm of matrix A.
Using the relation between Frobenius norm and the trace of a matrix, we can further
express (2.72) as
minUi
K∑j=1,j 6=i
Tr[HijVj −UiU
Hi HijVj)(HijVj −UiU
Hi HijVj)
H], (2.75)
which is further simplified to
minUi
K∑j=1,j 6=i
Tr[HijVj(HijVj)
H −UiUHi HijVj(HijVj)
H], (2.76)
= maxUi
K∑j=1,j 6=i
Tr[UiU
Hi HijVj(HijVj)
H], (2.77)
= maxUi
Tr[UHi
( K∑j=1,j 6=i
HijVjVHj HH
ij
)Ui
], (2.78)
whereK∑
j=1,j 6=i
HijVjVHj HH
ij is the positive semi-definite interference covariance matrix and
the d dimensional optimum basis Ui that maximizes the trace is given by the eigenvectors
corresponding to the d dominant eigenvalues ofK∑
j=1,j 6=i
HijVjVHj HH
ij .
Further, the optimum precoding matrix is obtained by minimizing ||HijVj−UiUHi HijVj||2F
over Vj with the distributed channel knowledge and with the knowledge of Uis obtained
previously as given by
minVj
K∑i=1,i 6=j
||HijVj −UiUHi HijVj||2F , (2.79)
s.t. VHj Vj = Id, (2.80)
UHi Ui = Id, (2.81)
28
Feasibility of Interference Alignment
which is further simplified to
minVj
K∑i=1,i 6=j
||(IN −UiUHi )HijVj||2F , (2.82)
= minVj
K∑i=1,i 6=j
Tr[(IN −UiU
Hi )HijVj((IN −UiU
Hi )HijVj)
H], (2.83)
= minVj
Tr[VHj
K∑i=1,i 6=j
HHij (IN −UiU
Hi )HijVj
], (2.84)
under the same constraints.
SinceK∑
i=1,i 6=j
HHij (IN −UiU
Hi )HijVj is a positive semi-definite matrix, the d dimensional
optimum Vj is given by the eigenvectors corresponding to the d minimum eigenvalues ofK∑
i=1,i 6=j
HHij (IN −UiU
Hi )HijVj . Using this Vj , (2.72) is optimized to get new Ui which is used
to solve (2.82) to obtain new Vj, and the process is repeated iteratively till the objective
converges to a small value. However, the global optimum is not guaranteed [32].
In all these distributed algorithms, the achievable DOF or the number of independent data
streams to be transmitted, are defined beforehand and when this is not known precisely, then
the algorithms do not converge to the desired value. So one of the challenges in designing IA
algorithms is to determine the achievable DOF for the given system or to determine if the
given system is feasible for achieving IA. In the next section, we discuss some of the criteria
and conditions for the feasibility of IA.
2.5 Feasibility of Interference Alignment
Interference alignment achieved by designing precoding matrices at the transmitters and
zero-forcing matrices at the receivers if needed is a linear operation and it is practically a
feasible assumption. Such linear IA is not feasible in all the systems, with any number of
users ( transmitters and receivers ) and transmit and receive antennas. In all the infeasible
systems, we can not achieve the benefits of IA. Hence the feasibility of IA is investigated in a
number of literature such as [2, 32, 34,61,62] to state a few.
In the distributed algorithms like MLI,max-SINR and AMI [2,32] as discussed before, IA
is obtained only if the following conditions are feasible at any receiver i and these are the
29
Feasibility of Interference Alignment
necessary and sufficient conditions to achieve IA:
UHi HijVj = 0, ∀j = {1, 2, · · · , K} (2.85)
rank(UHi HiiVi) = d, (2.86)
where we again consider the symmetric system with M transmit and N receive antennas
and d independent data streams transmitted from each transmitter, Ui ∈ CN×d,VM×dj
and Hij ∈ CN×M represent the zero-forcing matrix at receiver i, the precoding matrix at
transmitter j and the channel matrix corresponding to receiver i and transmitter j respectively.
Hence, the feasibility of IA is determined by the non-trivial solutions of system of polynomial
equations obtained from conditions (2.85) and (2.86). In fact, condition (2.86) is easily
achieved when the desired channels are sufficiently independent of the interference channels
by choosing d dimensional Ui and Vi. Thus, the non-trivial solutions of system of polynomial
equations obtained from condition (2.85) is enough to determine the feasibility of linear IA
and the multi-variable polynomial equations are solvable in general only if the number of
equations do not exceed the number of variables ( Bezout’s theorem ). IA problem where the
number of equations exceed the number of variables are termed as improper and the rest as
proper in [34].
The total number of equations obtained from condition (2.85) observed over any receiver i
due to a signal from transmitter j is d× d = d2, due to d rows of Ui and d columns of Vj.
Since there are K − 1 such transmissions observed by receiver i, total number of equations
per user for this symmetric system is given by
Ne = (K − 1)d2. (2.87)
On the other hand, the total number of variables observed at any receiver also depends on
the number of transmit and receive antennas in order to incorporate the available aligned
variables. Then the total number of variables per user observed in the symmetric system is
given by the following equation [34],
Nv = d(M +N − d). (2.88)
Thus for a proper system, Ne ≤ Nv which implies that
(K − 1)d2 ≤ d(M +N − d), (2.89)
⇒ d ≤ M +N
K + 1, (2.90)
is the feasible condition of linear interference alignment for symmetric network . However,
the feasibility condition for any type of generic interference network is still an open problem
and in fact is proved to be an NP-hard problem by Razaviyayn et al. in [63].
30
Feasibility of Interference Alignment
The distributed algorithms such as MLI and AMI achieve IA only for the proper system.
Based on the proposed idea of precoding matrix and zero-forcing matrix design, we observe
that interference is iteratively aligned and zero-forced. In that case, the information about the
interference signal is contained in the eigenvalues of sum of interference covariance matrices.
In each iteration, the amount of interference observed in the projection space (AMI algorithm)
or leakage interference (MLI algorithm) is measured in terms of sum of d minimum eigenvalues
of interference covariance (MLI algorithm) or the projection matrices (AMI algorithm), which
decreases and approaches zero for a proper system. When the system is improper, the
zero-forcing of interference is not possible.
This fact is depicted in depicted in Figure 2.7, which plots the sum of d minimum
eigenvalues of iteratively zero-forced and aligned interference covariance matrix with the
number of iterations. For M = 7 and N = 9 and K = 3, the achievable d is 7+93+1
= 4.
Hence, interference is completely zero-forced only when d = 4 and for d = 5 and d = 6, there
still exists a large amount of interference power. As long as interference is not completely
zero-forced it is not possible to decode the d dimensional transmitted data. Hence, we achieve
d = 4 DOF at maximum with M = 7, N = 9 and K = 3.
31
Feasibility of Interference Alignment
0 20 40 60 80 100 120 140 160 180 2000
5
10
15
20
25
30
35
40
Feasibility of Interference Alignment.
(M = 7, N = 9, K = 3)
No. of iterations
Sum
ofdminim
um
eigenvalues
d = 4d = 5d = 6
Figure 2.7: For K = 3 user MIMO IC with M = 7 transmit antennas and N = 9 receiveantennas, only d = 4 DOF is achievable for a proper system.
32
Chapter 3
Interference Alignment with Receiver
Cooperation and Greedy Transmission
In this chapter, we propose an improved interference alignment (IA) algorithm with hybrid
optimization, in which both leakage interference is minimized and desired signal power is
maximized simultaneously, unlike minimization of leakage interference (MLI) which minimizes
only leakage interference and maximization of signal to interference plus noise ratio (max-
SINR) which maximizes only signal to interference plus noise ratio (SINR) per transmit
stream of data. Through receiver cooperation the precoding matrix computed at transmitters
need not be known at each receiver, but rather are implicitly estimated cooperatively. The
cooperative structure of the algorithm benefits from the fact that only knowledge of the
sum of interference covariance matrices is enough to characterize the alignment problem
statistically. The choice of full cooperation amongst all the receivers is shown to be optimum
via a cooperative game theoretic argument. Simulation results show that the proposed
algorithm with receiver cooperation and hybrid optimization outperforms the well known
distributed IA schemes (MLI and max-SINR) proposed by Gomadam et al. [2].
All the distributed IA algorithms we discussed in section 2.4, i.e., MLI, max-SINR and
alternating minimization of interference (AMI) algorithms assume that the receivers have
perfect knowledge of the latest form of the precoding matrices at each iteration. Nevertheless,
these solutions do have the property that each receiver can act independently by aligning and
suppressing their own interference iteratively. This interesting feature of these algorithms
motivate us to think further about cooperation amongst receivers and the benefits of such
cooperation in K-user multiple input multiple output (MIMO) interference channel (IC).
In fact, it has been shown in [64] that cooperation amongst receivers can help mitigate
interference in the two-user IC.
We consider that the receivers cooperate by sharing amongst themselves their backward
channel state information (CSI), which enables each receiver to estimate the precoding matrix
at each iteration of IA optimization problem independently. This cooperation at at receivers is
33
System Model
aimed to minimize the leakage interference after zero-forcing the received signal by designing
proper zero-forcing matrices. However we consider a greedy approach at the transmitter
side. All transmitters aim to aim to maximize their own desired signal power and do not
care about the interference they cause to the other receivers. Such cooperative optimization
at receiver and greedy (non-cooperative) optimization at the transmitter side is termed as
“hybrid optimization”.
Such hybrid optimization arises a number of questions for further analysis. One of the
important questions is concerned with the amount of cooperation and the other question is
concerned with the convergence of such problems. Using coalition game theory [65,66], we
show that the spectrum efficiency of such scheme is maximized only if all the receivers are
the part of the cooperation and considering this iterative scheme as a class of evolutionary
process and relating to similar mathematical problem such as population migration problem as
described in [67], we show that such optimization converges after certain number of iterations.
The interesting observation is that such hybrid optimization gives better results in terms of
the data rate of the system (bits/sec/Hertz) and the feasibility, than the MLI and max-SINR
algorithms [2].
3.1 System Model
Consider a linear IA scheme over the K user MIMO IC with Mj transmit and Ni receive
antennas, ∀ i, j = {1, 2, · · · , K} in each transmitter-receiver pair (j, i). Let dj denotes the
independent data streams transmitted from transmitter j, i.e,degrees of freedom (DOF)
corresponding to the j-th transmitter, which is outer bounded by dj ≤ min(Mj, Nj). Thus for
any transmitter-receiver pair (j, i), there are dj desired links ∀j = i and (K − 1)dk interfering
links, such that each dk ≤ min(Mk, Nk), ∀k 6= i.
The received signal at any receiver i is the sum of a desired signal and K − 1 interfering
signals from all interfering transmitters j 6= i plus noise, as given by
yi = HiiVisi +∑j 6=i
HijVjsj + zi, (3.1)
where yi ∈ CNi×1 is the received signal vector at receiver i, Hij ∈ CNi×Mj is the MIMO
channel matrix corresponding to transmitter j and receiver i, Vj ∈ CMj×dj is the precoding
matrix at any transmitter j, sj ∈ Cdj×1 is the transmit symbol vector at any transmitter j,
and zi ∈ CNi×1 is the additive white Gaussian noise (AWGN) vector at receiver i, with zero
mean and unit variance. Each element of Hij is independent and identically distributed (i.i.d)
34
System Model
with zero mean and |hkl|2 variance, where ∀k = {1, 2, · · · , Ni} and ∀` = {1, 2, · · · ,Mj}.The transmit power at transmitter j is constrained by the available power Pj as :
E(sjsHj ) ≤ Pj. (3.2)
Also, the power constraint does not change after precoding the transmit signal, thus
E[||Vjsj||2
]= Tr
[E(Vjsj)(Vjsj)
H]
= Tr[VjE(sjs
Hj )VH
j
]≤ Pj (3.3)
The condition (3.3) is satisfied only if the precoding matrix is a unitary matrix, hence
VHj Vj = Idj . (3.4)
As discussed in section 2.4, linear IA schemes are mechanisms to design the precodeing
matrices Vj at any transmitter j that helps to beamform the transmitted signal in a particular
direction such that it aligns with the other interference signal in the undesired receivers
and the zero-forcing matrix or the interference suppressing matrix Ui at any receiver i that
zero-forces the aligned interference signal observed at any receiver i.
Since Vj and Ui behave exactly the same way one in the forward direction and another in
the backward direction assuming the reciprocity of the channels as observed in [2], Ui is also
orthonormal with the same number of columns as Vj. Hence,
UHi Ui = Idi . (3.5)
Our aim is to design Ui and Vj, ∀i, j = {1, 2, · · · , K} that satisfy the feasible IA conditions
as discussed in section 2.5. We employ the similar approach of Gomadam et al. [2] to use the
forward and backward channel and propose a hybrid iterative algorithm with cooperation
amongst the receiver and greedy transmission from the transmitters to achieve the following
linear feasible IA conditions:
UHi HijVj = 0, ∀ i 6= j, (3.6)
rank(UHi HiiVi
)= di, (3.7)
under the constraints
UHi Ui = Idi , (3.8)
VHi Vi = Idi , ∀i = 1, 2, · · · , K. (3.9)
Since MIMO channels are assumed non-degenerate, i.e., they are sampled from i.i.d
complex Gaussian random variables with zero mean and unit variance, they are full rank
with probability one and such assumption is sufficient to state that if equation (3.6) is
achievable, then equation (3.7) is also achievable with probability one. In other words, when
the interference subspace is nulled, the total desired subspace is contained in the signal space.
35
Proposed Algorithm
3.2 Proposed Algorithm
The main drawbacks of distributed algorithms such as MLI and max-SINR as discussed in
section 2.4 is that they require both full knowledge of forward channel and precoding matrices
Vj ’s in order to design the receive supressors Ui’s. Worse, both full knowledge and reciprocal
CSI of the backward channel, as well as the latest form of Ui’s are needed in order to design
the next precoding matrix Vj’s. In other words, the price payed for the distributiveness of
the schemes is considerable overhead and a rather strong limitation to channels with long
coherence time.
In addition to the aforementioned limitations, these distributed algorithms as proposed
in [2] are also in a sense incomplete, as each are designed to address only part of the
parameters of interest, namely, the interference and the received signal power. To clarify, the
MLI algorithm minimizes the interference that still remains after zero-forcing in both the
forward and backward channel and does not consider the desired signal power. Similarly, the
max-SINR algorithm maximizes SINR per data streams in both forward and reverse channel.
Thus, we propose a hybrid IA algorithm that contributes to mitigate all the aforementioned
limitations by simultaneously minimizing the leakage interference interference in the forward
step and maximizing the desired signal power in the backward step. The cooperation employed
amongst the receivers ensure that the exact knowledge of precoding matrices is not required
to be known at all the receivers and the zero-forcing matrices at all the transmitters [68].
Forward Step: Cooperative Interference Minimization
Let xj , Vjsj be the transmitted signal at the j-th transmitter, then the signal received at
the i-th receiver as given by (3.1) can be re-written as
yi = Hiixi +∑j 6=i
Hijxj + zi. (3.10)
Assume that each receiver has knowledge of the CSI between itself and corresponding
transmitters, and consider that the receivers cooperate with each other by sharing such CSI
amongst them, which means each receiver has the perfect knowledge of the channel matrix
H ∈ C(∑Ki=1Ni×
∑Kj=1Mj) as given by
H =
H11 H12 · · · H1K
H21 H22 · · · H2K
......
. . ....
HK1 HK2 · · · HKK
. (3.11)
36
Proposed Algorithm
Also, assume that the receivers cooperatively share the received signal amongst themselves
such that every receiver has the information of the received signal given by the vector
y ∈ C(∑Ki=1Ni)×1 expressed as
y =[yT1 yT2 · · · yTK
]T, (3.12)
where each yi is given by (3.10), ∀i = {1, 2, · · · , K}.With knowledge of channel matrix H and received signal vector y, the receivers cooperatively
estimate the transmitted signal x ∈ C∑Kj=1Mj×1 from all the transmitters using the least
square estimate [67] as given by
x = (HHH)−1HHy, (3.13)
which contains the estimates of the transmitted signal from each transmitter as given by
x =[xT1 xT2 · · · xTK
]T. (3.14)
Now, the receiver i designs a Ni × di dimensional zero-forcing matrix Ui based on the esti-
mated knowledge of xj,∀j = {1, 2, · · · , K} that minimizes the estimated leakage interference
observed at the receiver i as given by
Ii(UHi , {Qij}) =
∑j 6=i
Pjdj
Tr(UHi HijXjX
Hj HH
ijUi),
=∑j 6=i
Tr(UHi
Pjdj
HijXjXHj HH
ij︸ ︷︷ ︸Qij
Ui
),
=∑j 6=i
Tr(UHi QijUi), (3.15)
where Pj is the average transmit power of the j-th transmitter and the implicitly defined
quantities Qij are the interference covariance matrices of the i-th receiver, corresponding to
the interference estimate vectors xj expressed in terms of the corresponding diagonal matrices
Xj = diag(xj).
Notice that unlike [2], Ii(UHi , {Qij}) is an estimated leakage interference, which is computed
at each receiver without exact knowledge of all Vj’s but only the cooperative knowledge of
all the channel states and the received signal vector.
The i-th receiver then proceeds to solve the following optimization problem:
minUi
Ii(UHi , {Qij})
s.t. UHi Ui = Idi . (3.16)
Since Qij ∈ CNi×Ni is the positive semi-definite matrix, the solution to this trace min-
imization problem is well known as in [2, 32] and is obtained as follows. Consider the
37
Proposed Algorithm
eigendecomposition of the sum∑
j 6=i Qij, namely∑l 6=k
Qij = W ·ΛUi·WH , (3.17)
where the Ni eigenvectors of the matrix W are in descending order, i.e., W = [w1, · · · ,wNi ].
Then the optimum interference suppressing matrix at receiver i, U∗i is given by
U∗i =[wNi−di+1
,wNi−di+2
, · · · ,wNi
]. (3.18)
At this point we may remark that since the solution of the leakage interference minimization
problem actually depends only on the sum∑
l 6=k Qkl, the cooperative scheme here discussed
can be efficiently implemented with the help of a coordinator which collects all Hkl’s and
yk’s, calculates the sums∑
l 6=k Qkl for each k, and retransmits that information to the
corresponding receiver.
Backward Step: Power Maximization
While in each of the schemes in [2], the same optimization problem is solved on both forward
and backward stages of the iterative procedure, we remark here that this does not need
to be necessarily so. Relying on the fact that the receivers are capable of zero-forcing the
interference, a greedy power maximization approach can be employed at the backward stage
of the algorithm, such that the j-th transmitter performs
maxVj
Tr(VHj HH
ijUjUHj HijVj),
s.t. VHj Vj = Idj . (3.19)
Again, it is known that dj dimensional precoding matrix that solves this problem is
built from the dj largest eigenvectors of the desired covariance matrix. That is, given the
eigendecomposition
HHijUjU
Hj Hij = P ·ΛVj
·PH , (3.20)
with eigenvectors P = [p1, · · · ,pMj] in increasing order, the optimum precoding matrix is
obtained as
V∗j =[p1,p2, · · · ,pdj
]. (3.21)
This V∗j is now used to precode d independent data streams in the next iteration. The
receivers estimate the received signal and design optimum U∗i , with that knowledge the
transmitter again maximizes its desired power and design new V∗j and two steps iterate until
they converge. The convergence is guaranteed as long as the number of transmitted data
38
Coalition Game Theory and Full Cooperation
streams are feasible for IA as given by the feasibility condition in [34]
d ≤∑K
j=1Mj +∑K
i=1 Ni
K(K + 1), (3.22)
when all the transmitters transmit the same number of independent data streams d.
The exact feasibility condition for different number of transmitted independent data streams
is still an open problem. Hence, we consider a symmetric system for the simulation. The
algorithm can be stopped after the norm of the difference of consecutive precoding/zero-forcing
matrix is inferior to a pre-determined tolerance value.
3.3 Coalition Game Theory and Full Cooperation
Before we continue to show results on the performance of the cooperative interference
alignment described above, let us take the time to address the optimality of the full cooperation
assumed in section 3.2. This question can be approached from a game theoretical perspective,
specifically under the prism of Coalition Game Theory [65,66]. Let us be clear on some of
the terminologies that we have used in the later discussion.
Terminlogy: In a coalition game, several players form a coalition and act jointly to
improve the individual payoff. If there are no restrictions in the distribution of the payoff
among the members forming the coalition, the payoff is called transferable. When all the
available players form a single coalition, the coalition is called a grand coalition. Finally,
a coalition game is said to have a characteristic function form (CFF) if the value of the
coalition does not depend on outside coalitions. Otherwise, the coalition game is said to have
a partition function form (PFF).
Let R = {r1, r2, · · · , rK} be the set of all receivers and S ⊆ R consist of a coalition
(subset) of |S| players, where |.| represents the cardinality of the set. Suppose the function v
associates to each non-empty subset S the real value v(S), which is the total payoff available
for partition among the members of S. Then the coalition game with transferrable payoff
is denoted by 〈R, v〉 in CFF [66]. A coalition game 〈R, v〉 with transferable payoff v is
superadditive if for any two disjoint coalitions S1, S2 ⊆ R,
v(S1 ∪ S2
)≥ v(S1
)+ v(S2
). (3.23)
The core of a coalition game is a solution concept in CFF, where the set of players do
not break the coalition but act jointly to make the coalition better. When the core exists
and is non-empty, it indicates that the coalition formed is stable and can be considered as a
solution.
39
Simulation Results and Feasibility
Assuming no cost is incurred in forming a coalition amongst receivers, the payoff assigned
can be considered as the sum rate achieved by any receiver ri, which is given by
v({ri}) = I(xj,yi
), (3.24)
where I(xj,yi
)= H
(xj)− H
(xj|yi
)is the maximum mutual information between the
transmitter j and receiver i with the maximum taken over all possible input distributions,
H(xj)
is the input uncertainty, and H(x|y)
is the conditional uncertainty.
Assume that the receivers form a coalition S whose value is not influenced by the action of
players outside the coalition. Such a game is called coalition game with no externalities. The
payoff of a coalition S achieved by all the members is
v(S) =∑i∈S
v({ri}) = I(xS ,yS
), (3.25)
where I(xS ,yS
)is the maximum mutual information between transmitters and receivers
that are members of S.
Under the given system model and for the game with no externalities, the payoff of a
coalition v(S)
is superadditive, i.e., for any two disjoint coalitions S1 and S2, the following
condition is satisfied [65]
I(xS1∪S2 ,yS1∪S2
)≥ I
(xS1 ,yS1
)+ I
(xS2 ,yS2
). (3.26)
This is achieved by expanding I(xS1∪S2 ,yS1∪S2
)as follows using the chain rule of mutual
information [14,65] :
I(xS1∪S2 ,yS1∪S2
)=I(xS1 ,yS1
)+I(xS1 ,yS2|yS1
)+ I
(xS2 ,yS2|xS1
)+ I
(xS2 ,yS1 |yS2 ,xS1
),
(3.27)
and further expressing the mutual information in terms of entropy [14] as
I(xS2 ,yS1|yS2 ,xS1
)= H
(xS2)−H
(xS2|yS2 ,xS1
), (3.28)
≥ I(xS2 ,yS2
), (3.29)
and comparing the obtained results.
In [65], it is also shown that in such games the core always exists, which means that grand
coalition always gives improved and stable results, which finally implies the optimality of the
choice of full receiver cooperation in the proposed algorithm.
3.4 Simulation Results and Feasibility
We consider a three-user symmetric MIMO ICwith : M = N = 2, d = 1 and M = N = 4, d =
2. The independent streams of data transmitted from each transmitter d is determined before
40
Simulation Results and Feasibility
transmission so that IA is feasible for the given system. In each case, we plot the achieved
sum rate measured in bits/sec/Hertz against the signal to noise ratio (SNR) measured in
deciBel (dB) as shown in Figure 3.1 and compare the results for our proposed algorithm with
MLI and max-SINR algorithms proposed in [2].
Interestingly, we observe that the proposed algorithm performs better than MLI for both
the cases and this algorithm achieves the same sum rate as max-SINR algorithm for the
case M = N = 2, d = 1; and achieves better sum rate for the case M = N = 4 and d = 2.
This shows that the estimation error is not high when all the receivers share the channel
information with each other and maximizing the direct desired power can improve the sum
rate. Other interesting observation here is that when there are more than one data streams,
the max-SINR algorithm provides better rate at low SNR but the MLI algorithm provides
better rate at high SNR, while the proposed cooperative performs better for all SNRs. This
result is justified because, max-SINR maximizes SINR per data stream [2], unlike MLI and
proposed cooperative approach. Thus, when there is only one data stream max-SINR provides
the same result as the proposed cooperative algorithm.
Since the leakage interference power is minimized in each iteration of the IA algorithms,
the feasibility of IA can be measured by determining the percentage of interference in the
desired signal space, which is mathematically measured as
ρi =
∑Λ∗Ui∑
j 6=iTr(Qij)
, (3.30)
where Λ∗Uiis the diagonal matrix of di minimum eigenvalues as given by equation (3.17).
When the percentage of interference calculated as in equation (3.30) is zero, perfect
alignment is achieved. The maximum total data streams for which the percentage of leakage
interference is zero is the total achievable DOF for the given system.
In Figure 3.2, the percentage of leakage interference is plotted against the achievable total
DOF in K = 3 user MIMO IC for the proposed cooperative and well-known MLI algorithm
in [2] at 0 dB transmit power for the symmetric cases with M = N = 3 and M = N = 4. We
observe here that though both algorithms perform better for feasible total number of data
streams, the proposed cooperative algorithm is slightly better than the MLI algorithm.
It can also be observed that the achieved sum DOF nearly meet the theoretical limit of
achievable sum DOF for the particular case of linear IA with all transmitters and all receivers
having the same number of antennas, as given by [34]
d = |s| ≤ M +N
K + 1. (3.31)
41
Simulation Results and Feasibility
0 5 10 15 20 25 30 35 40 45 500
10
20
30
40
50
60
70
80
90
Sum Rate against Received SNR
(Number of Users: K=3)
Sum
Rate
Received SNR in dB
Proposed CooperativeMinimum LeakageMaximum SINR
DoF=2;M=4;N=4
DoF=1;M=2;N=2
Figure 3.1: Sum rate achieved measured in bits/sec/Hz as a function of SNR measured in dBfor different transmit antennas, receive antennas and feasible DOF that achieves IA for suchcase. The results are compared for the proposed cooperative algorithm, MLI algorithm andmax-SINR algorithm.
42
Simulation Results and Feasibility
3 4 5 6 7 8
0
1
2
3
4
5
6
7
Sum DoF against percentage of leakage interference
(Number of Users: K=3)
Percentage
ofinterference
indesired
sign
alspace
Total number of transmit streams in the network
Proposed CooperativeMinimum Leakage
M=3;N=3
M=4;N=4
Figure 3.2: Feasibility of interference alignment for the proposed cooperative and MLIalgorithms. The feasibility is measured as the percentage of the leakage interference aftereach iteration measured for the given total data streams. The proposed cooperative algorithmhas less leakage interference than the MLI for M = N = 3 and M = N = 4 with differentnumber of total transmit streams.
43
Conclusions
Specifically, for M = 4 and K = 4, the outer bound on the achievable sum DoF, as given
by equation (3.31) yield DoF = 4+43+1× 3 = 6, which is achieved accurately by the proposed
algorithm. In turn, for M = 3 and K = 3, the achievable outer bound on the sum DoF
is given by DoF = 3+33+1× 3 = 4.5, which is nearly achieved, in the sense that only a small
amount of residual interference remains at that point.
3.5 Conclusions
We proposed a cooperative and hybrid distributed interference alignment algorithm based on
the MLI and max-SINR schemes presented in [2]. The hybrid aspect of our algorithm arises
in the fact that both leakage interference is minimized (in the forward step) and desired signal
power at there receiver is maximized (at the backward step) simultaneously. Employing
receiver cooperation, the requirement that precoding vectors computed at transmitters be
communicated to receivers at each step is mitigated.
The choice of full cooperation amongst all the receivers is shown to be optimum via a
cooperative game theoretic argument. Simulation results show that the proposed algorithm
with receiver cooperation and hybrid optimization outperforms the well known distributed
interference alignment scheme proposed by Gomadam et al. [2].
An interesting outcome of the approach is that, since only knowledge of the sum of
interference covariance matrices is enough to characterize the alignment problem statistically,
a robust (stochastic) version of the distributed alignment concept can be envisioned, motivating
future work.
44
Chapter 4
Space-Time Transmission with
Delayed CSIT
One of the major drawbacks of most of the interference alignment (IA) algorithms described
in the previous chapters are the requirement of perfect channel state information at the
transmitter (CSIT). Proper precoding and zero-forcing can achieve the optimal degrees of
freedom (DOF) only if the instantaneous CSIT is known perfectly. However, perfect CSIT is
a very optimistic assumption when the channels are varying rapidly with time and it is no
more possible to achieve the benefits of IA.
Previously, it was believed that CSIT from the previous time instances could not improve
the achievable DOF of the given multi-user multi-antenna channels. But recently, Maddah Ali
and Tse published an article [41, 69] with a curious result which states that the knowledge of
delayed or the outdated CSIT improves the DOF of the multiple input single output (MISO)
broadcast channel (BC). They showed that the per-user DOF of 23
is easily achievable in
two-user MISO BC only with the knowledge of delayed CSIT using the idea of retrospective
IA, which is the improvement from previously thought 12
per-user DOF [41,69].
Knowledge of delayed CSIT means that at time t, the transmitters know all the channel
states till time t − 1 but does not know the channel state at time instance t. This is a
valid model when the channel changes from symbol to symbol with each channel state being
independent and identically distributed (i.i.d).
In this chapter, we discuss the transmission scheme that improves the previously thought
achievable DOF in any K-user MISO BC as proposed in [41, 69] with the help of delayed
CSIT. We will further consider another scheme that improves the achievable DOF by not only
considering the delayed CSIT but also assuming that some portion of imperfect instantaneous
CSIT is available at any time instance t as proposed in [43, 70]. Following the same basis,
we propose our own space-time based transmission scheme to show that better DOF is
achievable in K user multi-cell MISO BC or simply a K-user interference broadcast channel
(IBC) with the knowledge of perfect delayed CSIT. The proposed space-time transmission
45
Space-Time Transmission with Delayed CSIT
scheme is based on retrospective interference alignment for two-cell interfering MISO BC
with M transmit antennas and K single antenna users in each cell. We suggest that using
the improved transmission scheme, the total DOF achieved converges to 85
in the case when
M = K and delayed CSIT is known perfectly. This is an interesting observation because it
shows that better than 1 DOF is achievable even if the CSIT is not known instantaneously.
Even the DOF of the MISO IBC with perfect instantaneous CSIT is not well explored
in literature. Only few works such as [71, 72] study the achievable DOF in any B cell M
antennas transmitter and K single antenna users MISO BC using the concept of IA. The
authors in [71]design the precoders using IA and zero-forcing to align the inter-cell and
intra-cell interference and achieve the optimum DOF which is outer bounded by BK2
, when
M = K. As such they show that for any β = max(M,K)min(M,K)
, the DOF of any B MISO IBC with
M transmit antennas and K single antenna users per cell is given by :
d =
BK if B ≤ β & M ≥ K,
BK1+ 1
β
if B ≥ β + 1 & M ≥ K,
BM if B ≤ β & M < K,
BM1+ 1
β
ifB ≥ β + 1 & M < K.
(4.1)
This satisfies the DOF achieved per user in a single input single output (SISO)interference
channel (IC), i.e., 12
as in [1].
Also, the authors in [72] have developed the transmission scheme based on subspace IA
to align interference over multiple dimensions and analyze the achievable DOF per cell for
B-cells interfering system with K users in each cell using multiple sub-carriers. The authors
have proved that B cells BC is the dual of B cells interference multiple access channel (IMAC)
and show that the following DOF is achievable for both channels with K users per cell:
d =K(
B−1√K + 1
)B−1, (4.2)
which approaches to unity as K approaches to infinity.
Some other works like [73,74] have been proposed to analyze the achievable DOF of multiple
input multiple output (MIMO) IBC based on IA and linear beam-forming schemes. The
authors in [74] have proved that the achievable DOF per user for a B cell MIMO IBC with
M transmit antennas per base-station and N receive antennas per user with K users per cell
is outer bounded by :
d ≤ max( M
Kη + 1,
Nη
Kη + 1
), (4.3)
46
Single Cell Two-user MISO BC with Delayed CSIT
where η ∈{pq
: p ∈ {1, 2, · · · , B − 1}, q = {1, 2, · · · , (B − p)K}}
.
For B = 2, p = {1} and q = {1, 2, · · · , K} and η ∈ {1, 12, · · · , 1
K} and d is outer bounded
by MK+1
for η = 1 and M2
for η = 1K
.
The main challenges in the analysis of any multi-cell MISO or MIMO BC or multiple access
channel (MAC) are the following:
• All the base stations transmit without co-operation. So, for the cell edge users, the
system acts as an interference channel.
• Both inter cell interference (ICI) and inter user interference (IUI) are to be managed
when there are multiple users in each cell.
Due to these challenges even in the presence of the instantaneous CSIT, the DOF analysis
of the multi-cell system in any generic channel is an open problem and is one of the interesting
topics for information theorists and communication engineers.
Next, we consider a transmission scheme to analyze the DOF for a simple network scenario
with single cell MISO BC with two transmit antennas and only two single antenna users per
cell using delayed CSIT.
4.1 Single Cell Two-user MISO BC with Delayed CSIT
Consider a MISO BC with two transmit antennas and two single antenna users as shown
in Figure 4.1, where the channel is independently varying at each time instance. The
channel from the transmitter to user i, ∀i = {1, 2} at time instance t is represented as
hi(t) =[hi1(t) hi2(t)
]and the precoded transmit streams from the transmitter at any time
instance t is x(t), which contains two independently encoded streams of data, u(i)1 (t) and
u(i)2 (t), intended to user i, ∀i = {1, 2}.Using the following transmission scheme, the delayed CSIT is useful to transmit and decode
four independent streams of data in three channel uses, thus gaining 43
total DOF.
• Time instance t = 1 : During this time instance which requires one time slot, the
transmitter transmits two independent data streams both intended to user 1 from two
antennas, i.e.,
x(1) =[u
(1)1 u
(1)2
]T. (4.4)
47
Single Cell Two-user MISO BC with Delayed CSIT
1
2
x(t)
[h 11(t)
h 12(t)]
[h21 (t) h22 (t)]
Figure 4.1: A two user MISO BC with two antennas at the transmitter. The two independentdata streams are transmitted as a vector x(t) at any time instance t and the channel vectoris represented by hi1(t) and hi2(t), ∀i = {1, 2}.
48
Single Cell Two-user MISO BC with Delayed CSIT
Thus, the signal observed at both the users at t = 1 is expressed as:
y1(1) =[h11(1) h12(1)
]x(1) + z1(1), (4.5)
= h11(1)u(1)1 + h12(1)u
(1)2 + z1(1), (4.6)
y2(1) =[h21(1) h22(1)
]x(1) + z2(1), (4.7)
= h21(1)u(1)1 + h22(1)u
(1)2 + z2(1), (4.8)
where zi(1) is the additive white Gaussian noise (AWGN) at time instance t = 1.
• Time instance t = 2: During this time instance which requires one time slot, the
transmitter transmits two independent data streams both intended to user 2 from two
antennas, i.e.,
x(2) =[u
(2)1 u
(2)2
]T. (4.9)
The signal received by both the receivers at time instance t = 2 is thus given by
y1(2) =[h11(2) h12(2)
]x(2) + z1(2), (4.10)
= h11(2)u(2)1 + h12(2)u
(2)2 + z1(2), (4.11)
y2(2) =[h21(2) h22(2)
]x(2) + z2(2), (4.12)
= h21(1)u(2)1 + h22(1)u
(2)2 + z2(2), (4.13)
• Time instance t = 3: From the previous instances, we observe that y2(1) and y1(2)
are the interference signal for user 2 and user 1 respectively. However, y2(1) and y1(2)
contain the desired data streams for user 1 and user 2. During this time instance which
uses one time slot, the transmitter transmits the linear combination of y2(1) and y1(2)
minus noise, because the transmitter has the knowledge of all the channel states till
time t = 2, as given by:
x(3) =[h11(2)u
(2)1 + h12(2)u
(2)2 + h21(1)u
(1)1 + h22(1)u
(1)2 0
]T. (4.14)
The signal received by both the receivers at time instance t = 3 is then given by
y1(3) =[h11(3) h12(3)
]x(3) + z1(3), (4.15)
= h11(3)(h11(2)u
(2)1 + h12(2)u
(2)2 + h21(1)u
(1)1 + h22(1)u
(1)2
)+ z1(3), (4.16)
49
Single Cell Two-user MISO BC with Delayed CSIT
and
y2(3) =[h21(3) h22(3)
]x(3) + z2(3), (4.17)
= h21(3)(h11(2)u
(2)1 + h12(2)u
(2)2 + h21(1)u
(1)1 + h22(1)u
(1)2
)+ z2(3), (4.18)
where zi(3) is the AWGN at time instance t = 3.
Further, we can express the signal received by any user i, ∀i = {1, 2} over all the time
instances in the form of matrix equation as,y1(1)
y1(2)
y1(3)
=
h11(1) h12(1)
0 0
h11(3)h21(1) h11(3)h22(1)
︸ ︷︷ ︸
rank= 2
x(1) +
0 0
h11(2) h12(2)
h11(3)h11(2) h11(3)h12(2)
︸ ︷︷ ︸
rank= 1
x(2) +
z1(1)
z1(2)
z1(3)
,(4.19)y2(1)
y2(2)
y2(3)
=
h21(1) h22(1)
0 0
h21(3)h21(1) h21(3)h22(1)
︸ ︷︷ ︸
rank= 1
x(1) +
0 0
h21(2) h22(2)
h21(3)h11(2) h21(3)h12(2)
︸ ︷︷ ︸
rank= 2
x(2) +
z2(1)
z2(2)
z2(3)
,(4.20)
Clearly, we observe here that the two data streams desired by user 1, i.e., x(1) =[u
(1)1 u
(1)2
]Tis spanned by rank 2 matrix over two independent dimensions, while the
undesired data streams are spanned by a rank 1 matrix over a single dimension and similarly,
the two data streams desired by user 2, i.e., x(2) =[u
(2)1 u
(2)2
]Tis spanned by rank 2 matrix
over two independent dimensions while the undesired data streams are spanned by rank 1
matrix over a single dimension. Hence both receivers can decode total of 4 data streams in
three channel uses.
The achievable DOF is measured as:
d =total decodable data streams
total number of time slots required, (4.21)
=4
3, (4.22)
which is greater than 1 as previously believed. This result from [69] is further extended for
K user MISO BC as discussed in the section below:
50
K-user MISO BC with Delayed CSIT
4.2 K-user MISO BC with Delayed CSIT
Consider a MISO BC with K transmit antennas and K single antenna users. As in the case
for two-user MISO BC, linear combination of K streams are transmitted intended to each
user for the first K channel uses. Thus, there are total of K2 data streams to be decoded
and the DOF is obtained by determining the total number of channels required in K phases.
The nth phase determines the number of channels required to decode degree n messages.
As shown in [69], the phase j has(Kj
)sub-phases each with one time slot which takes
(K − j + 1)(Kj
)symbols with degree j and generate j
(Kj+1
)symbols with degree j + 1. Thus,
the DOF corresponding to any degree j for K user MISO BC with K transmit antennas is
given by:
dj(K,K) =total number of degree j symbols
total channel uses. (4.23)
The total number of degree j symbols is given by (K − j + 1)(Kj
)while the total number of
channel uses is the total number of time slots required, i.e,(Kj
)time slots for
(Kj
)sub-phases
and the extra time slots to decode j(Kj+1
)symbols with degree j + 1, hence
dj(K,K) =(K − j + 1)
(Kj
)(Kj
)+
j( Kj+1)
dj+1(K,K)
, (4.24)
=K − j + 1
1 + j (K−j)j+1
1dj+1(K,K)
, (4.25)
where dj+1(K,K) is the DOF of symbols with degree j + 1. Further simplifying, we can
express (4.25) as:
K − (j − 1)
j
1
dj(K,K)=
1
j+K − jj + 1
1
dj+1(K,K), (4.26)
=1
j+
1
j + 1+K − (j + 1)
j + 2
1
dj+2(K,K), (4.27)
The maximum possible degree of the symbol is K and there are at maximum one data
symbols of degree K, thus it is easy to verify that dK(K,K) = 1 and continuing the similar
pattern as (4.27) up until j + 1 = K, we can further simplify (4.27) as:
K − (j − 1)
j
1
dj(K,K)=
1
j+
1
j + 1+
1
j + 2+ · · ·+ 1
K. (4.28)
51
Two cell MISO IBC
We require the DOF for degree one symbols, hence j = 1 and
K1
d(K,K)= 1 +
1
2+
1
3+ · · ·+ 1
K, (4.29)
⇒d(K,K) =K
1 + 12
+ 13
+ · · ·+ 1K
, (4.30)
is the achievable DOF for K-user MISO BC with delayed CSIT.
This idea of DOF analysis of single cell MISO BC is extended to the case of two-cell MISO
IBC with delayed CSIT in [75], where we show that delayed CSIT is useful to mitigate ICI
and IUI with proper space-time transmission technique and more than unity DOF is easily
achievable in that case. We consider all the cases where the number of transmit antennas is
not necessarily equal to the number of users per cell. The space-time transmission scheme
and DOF analysis of two-cell MISO IBC is discussed in the next section.
4.3 Two cell MISO IBC
Consider a multi-cell MISO IBC with B = 2 cells, where a base station per cell with M
transmitting antennas transmit data to K single antenna users in each cell. Since there are
two base stations transmitting at the same time, the cell-edge users in both the cells receive
inter cell interference from the neighboring cell. Also, all the base-stations are broadcasting
the message to all the users at the same time, every user in the cell receive the signal
transmitted to K− 1 other users as an interference, called inter user interference. The system
model for this two-cell MISO IBC system with M transmit antennas per base station and K
users per cell is depicted in Figure 4.2.
We assume that each user has the perfect CSI for both the serving and the interfering
link. Since the transmitters have no instantaneous CSIT, the system is designed such that
each user feedbacks the available CSI to the serving base-station. In doing so, the user sends
the channel direction information (CDI) and the channel quality information (CQI) of both
the serving and the interfering links to the serving base-station. The base stations, thus,
exchange only the CDI and CQI for the interfering links by using the error and delay free
back-haul channel. Such information exchange technique is also discussed in detail in [76].
Let us denote the channel between the base-station b and user k in the `th cell at any
time-instant t by the vector h(`)kb (t) ∈ CM×1, the transmitted signal from the base-station b
intended to the user k in the same cell at time t by a vector s(b)k (t) ∈ CM×1, the received
signal at the user k in the `th cell at any time instant t by a scalar y(`)k (t) and the AWGN at
52
Two cell MISO IBC
Cell 1
M antennas
K users CSI feedback 1
CSI feedback 2
Cell 2 M antennas
K users
Back-haul connection
Inter-cell interference
Figure 4.2: A two-cell MISO interference broadcast channel with M transmit antennas ineach base station and K single antenna users per cell. The solid line represents the signalreceived from the same cell and the dashed line represents the inter cell interference receivedfrom the adjacent cell. The dotted line represents that the delayed channel state information(CSI) feedback is provided from any receiver to the transmitter. The two cells are connectedvia back-haul connection.
53
Space-Time Transmission Scheme
user k in the cell ` at time instant t as z(`)k (t). Each channel element is assumed to be i.i.d
with zero mean. The received signal at any user k at time instant t can be expressed as:
y(`)k (t) = [h
(`)k` (t)]
Hs
(`)k (t) +
K∑m 6=k,m=1
[h(`)k` (t)]
Hs(l)m (t) +
B∑b 6=`
K∑m=1
[h(`)kb (t)]
Hs(b)m (t) + z
(`)k (t). (4.31)
Following the similar information theoretic definition of DOF as in [1,14], the DOF per
cell is defined as:
d =1
Blim
SNR→∞
CT (SNR)
log SNR, (4.32)
where CT (SNR) is the total capacity of the system as a function of signal to noise ratio
(SNR).
This implies that this capacity pre-log factor can be obtained as the total number of
independent data streams or signals at the receivers per channel use. Since the instantaneous
CSIT is not available, proper precoding vectors at the transmitter side and zero-forcing vectors
at the receiver side are difficult to design. In that case, the spatio-temporal transmission
scheme to transmit the data streams from transmitter to the receiver is utilized to analyze
the DOF of the system. The delayed CSIT is useful in this regard and the DOF is expressed
in terms of the ratio of total number of independent data streams transmitted to the total
number of channel uses as also described in previous section. Hence, for MISO IBC with
delayed CSIT, the DOF per cell is measured as:
d =total independent streams decoded per cell
total number of channel uses per cell. (4.33)
In the following section, we present the space time transmission technique to analyze the
DOF of two-cell MISO IBC with the help of delayed CSIT for different values of transmit
antennas M and the number of users per cell K.
4.4 Space-Time Transmission Scheme
We consider various cases for different values of M and K before we generalize the result for
achievable DOF. Some of them are discussed here:
54
Space-Time Transmission Scheme
4.4.1 Case I ( M=1, K=1)
The simplest possible case is the one where each base-station with single antenna transmits
to a single antenna user in each cell. We observe that such system acts as a two-user SISO IC
and most of the fact about the DOF of such system is well known. It is easily observed that
even in the absence of any CSIT, the DOF of 12
is achievable and the outdated CSIT does
not help much to improve this DOF because it is the outer bound on the achievable DOF
for two-user SISO IC as we can not improve much with the help of IA. In other words, we
require at least two time instances to decode two transmitted data streams, thus, providing12
DOF per cell.
• At time instant t = 1, the signal received by each user is given by
y(1)1 (1) = h
(1)11 (1)s
(1)11 + h
(1)12 (1)s
(2)12 + z
(1)1 (1), (4.34)
y(2)1 (1) = h
(2)11 (1)s
(1)11 + h
(2)12 (1)s
(2)12 + z
(2)1 (1). (4.35)
• At time instant t = 2, each user feedbacks the CSI to the corresponding transmitter,
which share the interfering channel information with each other. Since each of them
have all the CSIT from previous time instances, the base-station 1 transmits the
side information observed by the user in cell 2, which is h(2)11 (1)s
(1)11 and similarly the
base-station 2 transmits the side information observed by the user in cell 1, which is
h(1)12 (1)s
(2)12 . Thus the received signal is
y(1)1 (2) = h
(1)11 (2)h
(2)11 (1)s
(1)11 + h
(1)12 (2)h
(1)12 (1)s
(2)12 + z
(1)1 (2), (4.36)
y(2)1 (2) = h
(2)11 (2)h
(2)11 (1)s
(1)11 + h
(2)12 (2)h
(1)12 (1)s
(2)12 + z
(2)1 (2). (4.37)
• Since we perform the DOF analysis at high SNR, we neglect the noise terms, which is
reasonable assumption. Thus, decoding is easily possible with the available information
in this case by canceling the ICI as given by:
s(1)11 =
y(1)1 (2)− h(1)
12 (2)y(1)1 (1)
h(1)11 (2)h
(2)11 (1)− h(1)
12 (2)h(1)11 (1)
, (4.38)
s(2)12 =
y(2)1 (2)− h(2)
11 (2)y(2)1 (1)
h(2)12 (2)h
(1)12 (1)− h(2)
11 (2)h(2)12 (1)
. (4.39)
4.4.2 Case II (M=1, K=2)
This is an example of a scheme where K > M and we observe here that the scheme can’t
achieve better than one DOF, which is not an improvement. The key reason is the lack of
55
Space-Time Transmission Scheme
independent channel coefficients to cancel out the ICI. The main idea of the proposed space-
time transmission scheme is the repetitive alignment of interferences by repetitive transmission
of the similar signal in different time instances. This is valid as long as independent channel
coefficients can be created to decode all the streams.
From now onwards, we remove the noise terms in each analysis for simplicity as the noise
plays no role in high SNR analysis and we consider following transmission scheme:
• At time instant t = 1, base-station 1 transmits the sum of the data streams intended to
both the users in cell 1, which is s(1)11 + s
(1)21 and the base-station 2 transmits the sum of
data streams intended to both users in cell 2, which is s(2)12 + s
(2)22 . The signals received
by all the users at t = 1 is given by:
y(1)1 (1) = h
(1)11 (1)
(s
(1)11 + s
(1)21
)+ h
(1)12 (1)
(s
(2)12 + s
(2)22
), (4.40)
y(1)2 (1) = h
(1)21 (1)
(s
(1)11 + s
(1)21
)+ h
(1)22 (1)
(s
(2)12 + s
(2)22
), (4.41)
y(2)1 (1) = h
(2)11 (1)
(s
(1)11 + s
(1)21
)+ h
(2)12 (1)
(s
(2)12 + s
(2)22
), (4.42)
y(2)2 (1) = h
(2)21 (1)
(s
(1)11 + s
(1)21
)+ h
(2)22 (1)
(s
(2)12 + s
(2)22
). (4.43)
• At time instance t = 2, both the base-stations use the delayed CSIT and transmit the
linear combination of the ICI observed by each cell to align those interference. We
consider only the signal received by the first user in each cell because all the other users
behave symmetrically and use extra channel. The signal received by the first user in
each cell after t = 2 is
y(1)1 (2) = h
(1)11 (2)h
(2)11 (1)
(s
(1)11 + s
(1)21
)+ h
(1)12 (2)h
(1)12 (1)
(s
(2)12 + s
(2)22
), (4.44)
y(2)1 (2) = h
(2)11 (2)h
(2)11 (1)
(s
(1)11 + s
(1)21
)+ h
(2)12 (2)h
(1)12 (1)
(s
(2)12 + s
(2)22
). (4.45)
This implies the following decoding condition
s(1)11 + s
(1)21 =
y(1)1 (2)− h(1)
12 (2)y(1)1 (1)
h(1)11 (2)h
(2)11 (1)− h(1)
12 (2)h(1)11 (1)
, (4.46)
s(2)12 + s
(2)22 =
y(2)1 (2)− h(2)
11 (2)y(2)1 (1)
h(2)12 (2)h
(1)12 (1)− h(2)
11 (2)h(2)12 (1)
. (4.47)
• Since it is not decoded perfectly, we repeat the transmission in the next time instant
56
Space-Time Transmission Scheme
t = 3, which gives the similar decoding conditions given by:
s(1)11 + s
(1)21 =
y(1)1 (3)− h(1)
12 (3)y(1)1 (1)
h(1)11 (3)h
(2)11 (1)− h(1)
12 (3)h(1)11 (1)
, (4.48)
s(2)12 + s
(2)22 =
y(2)1 (3)− h(2)
11 (3)y(2)1 (1)
h(2)12 (3)h
(1)12 (1)− h(2)
11 (3)h(2)12 (1)
. (4.49)
It is clearly visible that from (4.46), (4.47), (4.48) and (4.49), the decoding of all the four
streams is not possible with these four channel conditions. This shows that when there are
more streams to be transmitted than the number of transmit antennas, the decoding is not
possible with this proposed scheme. So, we consider only the cases when M ≥ K.
4.4.3 Case III (M=2, K=1)
When the number of transmit antennas is greater than the number of users in each cell, it is
possible to have enough independent channel coefficients and the decoding is easily possible.
For the case with M = 2 and K = 1, we show that using the space time transmission scheme
4 data streams can be transmitted and decoded using only 3 channel instances, thus providing
total DOF of 43
or per cell DOF of 23.
• At time instance t = 1, each base-station transmits two data streams from the available
two antennas and the signal received by the users in each cell is given by
y(1)1 (1) = h
(1)11 (1)
Hs
(1)11 + h
(1)12 (1)
Hs
(2)12 , (4.50)
y(2)1 (1) = h
(2)11 (1)
Hs
(1)11 + h
(2)12 (1)
Hs
(2)12 , (4.51)
where h(`)kb (t) ∈ CM×1 is the channel vector from base station b to user k at cell ` during
time instance t and s(`)kb ∈ CM×1 is the transmit signal vector intended to user k in cell
` transmitted from base station b.
• At time instance t = 2, each base-station has the knowledge of previous channel states.
Thus, both the base stations transmit in such a way that the ICI observed by the users
in each cell is aligned. Each base-station uses only one antenna for this transmission as
given by:
y(1)1 (2) = h
(1)11 (2)
(h
(2)11 (1)
Hs
(1)11
)+ h
(1)12 (2)
(h
(1)12 (1)
Hs
(2)12
), (4.52)
y(2)1 (2) = h
(2)11 (2)
(h
(2)11 (1)
Hs
(1)11
)+ h
(2)12 (2)
(h
(1)12 (1)
Hs
(2)12
). (4.53)
57
Space-Time Transmission Scheme
• At time instance t = 3, we use the concept of repetition of the transmission assuming
all the channel elements are i.i.d and the signal received at each user is, thus, given by:
y(1)1 (3) = h
(1)11 (3)
(h
(2)11 (1)
Hs
(1)11
)+ h
(1)12 (3)
(h
(1)12 (1)
Hs
(2)12
), (4.54)
y(2)1 (3) = h
(2)11 (3)
(h
(2)11 (1)
Hs
(1)11
)+ h
(2)12 (3)
(h
(1)12 (1)
Hs
(2)12
). (4.55)
• The desired signals can be easily decoded by each user with the available information
by simply observing the following conditions:[y
(1)1 (2)− h(1)
12 (2)y(1)1 (1)
y(1)1 (3)− h(1)
12 (3)y(1)1 (1)
]=
[h
(1)11 (2)h
(2)11 (1)
H − h(1)12 (2)h
(1)11 (1)
H
h(1)11 (3)h
(2)11 (1)
H − h(1)12 (3)h
(1)11 (1)
H
]s
(1)11 , (4.56)
and
[y
(2)1 (2)− h(2)
11 (2)y(2)1 (1)
y(2)1 (3)− h(2)
11 (3)y(2)1 (1)
]=
[h
(2)12 (2)h
(1)12 (1)
H − h(2)11 (2)h
(2)12 (1)
H
h(2)12 (3)h
(1)12 (1)
H − h(2)11 (3)h
(2)12 (1)
H
]s
(2)12 . (4.57)
4.4.4 Case IV (M=2, K=2)
The simplest example with more than one users per cell and more than one transmit antennas
per base station is the symmetric MISO IBC with M = 2 in all base-stations and K = 2 in
all cells. Using the similar space-time transmission scheme, we can easily verify in this case
that the total of 8 data independent streams can be transmitted over 6 channel uses, thus
achieving the total DOF of 86
= 43
and per cell DOF of 23.
• At time instant t = 1, both the base stations transmit all the data streams intended to
the first user in each cell. Hence base station 1 transmits two data streams intended to
user 1 at cell 1 using two independent antennas and base station 2 transmits two data
streams intended to user 1 at cell 2 using two independent antennas. Thus, the signal
received by all the users in both the cells at t = 1 is given by
y(1)1 (1) = [h
(1)11 (1)]
Hs
(1)11 + [h
(1)12 (1)]
Hs
(2)12 , (4.58)
y(1)2 (1) = [h
(1)21 (1)]
Hs
(1)11 + [h
(1)22 (1)]
Hs
(2)12 , (4.59)
y(2)1 (1) = [h
(2)11 (1)]
Hs
(1)11 + [h
(2)12 (1)]
Hs
(2)12 , (4.60)
y(2)2 (1) = [h
(2)21 (1)]
Hs
(1)11 + [h
(2)22 (1)]
Hs
(2)12 , (4.61)
where y(`)k (t) is a scalar signal received by a single antenna user k at cell ` during
time instance t, h`kb(t) ∈ C2×1 is the channel vector from base station b to user k in
58
Space-Time Transmission Scheme
cell ` during time instance t and s(`)kb ∈ C2×1 is the independent transmit data vector
transmitted from base station b intended to user k at cell `.
• At time instance t = 2, both the base stations transmit all the data streams intended
to the second user in each cell, which means base station 1 transmits two independent
data streams intended to user 2 in cell 1 and base station 2 transmits two independent
data streams intended to user 2 in cell 2. Hence, the signal received by all the users at
time instance t = 2 is given by:
y(1)1 (2) = [h
(1)11 (2)]
Hs
(1)21 + [h
(1)12 (2)]
Hs
(2)22 , (4.62)
y(1)2 (2) = [h
(1)21 (2)]
Hs
(1)21 + [h
(1)22 (2)]
Hs
(2)22 , (4.63)
y(2)1 (2) = [h
(2)11 (2)]
Hs
(1)21 + [h
(2)12 (2)]
Hs
(2)22 , (4.64)
y(2)2 (2) = [h
(2)21 (2)]
Hs
(1)21 + [h
(2)22 (2)]
Hs
(2)22 . (4.65)
• During t = 1, user 2 in both cells observe IUI plus ICI while user 1 in both cells observes
only ICI. Also during t = 2, user 1 in both cells observe IUI plus ICI while user 2 in
both cells observe only ICI.
At time instance t = 3, base-station 1 and base-station 2 have the knowledge of all the
previous channel states. Each user in the corresponding cell feedback the CSI to the
relevant base-station and the base stations then share all the information through a
back-haul connection. Hence base station 1 transmits the linear combination of all the
ICI observed by user 1 in cell 2 during time instance t = 1 using only one antenna and
base station 2 transmits the linear combination of all the ICI observed by user 1 in cell
1 during time instance t = 1 using only one antenna. The received signal at all the
users at time instance t = 3 is given by:
y(1)1 (3) = h
(1)11 (3)[h
(2)11 (1)]
Hs
(1)11 + h
(1)12 (3)[h
(1)12 (1)]
Hs
(2)12 , (4.66)
y(1)2 (3) = h
(1)21 (3)[h
(2)11 (1)]
Hs
(1)11 + h
(1)22 (3)[h
(1)12 (1)]
Hs
(2)12 , (4.67)
y(2)1 (3) = h
(2)11 (3)[h
(2)11 (1)]
Hs
(1)11 + h
(2)12 (3)[h
(1)12 (1)]
Hs
(2)12 , (4.68)
y(2)2 (3) = h
(2)21 (3)[h
(2)11 (1)]
Hs
(1)11 + h
(2)22 (3)[h
(1)12 (1)]
Hs
(2)12 . (4.69)
Please note here the scalar channels used during time instance t = 3 to indicate that
the linear combination of the data streams is transmitted from only one antenna.
• Similarly at time instance t = 4, the base-station 1 transmits the linear combination of
all the ICI observed by user 2 in cell 2 during time instance t = 2 and base station 2
59
Space-Time Transmission Scheme
transmits the linear combination of all the ICI observed by user 2 in cell 1 during time
instance t = 2. In this case also, both the base stations use only one antenna. The
received signal by all the users during time instance t = 4 is given by:
y(1)1 (4) = h
(1)11 (4)[h
(2)21 (2)]
Hs
(1)21 + h
(1)12 (4)[h
(1)22 (2)]
Hs
(2)22 , (4.70)
y(1)2 (4) = h
(1)21 (4)[h
(2)21 (2)]
Hs
(1)21 + h
(1)22 (4)[h
(1)22 (2)]
Hs
(2)22 , (4.71)
y(2)1 (4) = h
(2)11 (4)[h
(2)21 (2)]
Hs
(1)21 + h
(2)12 (4)[h
(1)22 (2)]
Hs
(2)22 , (4.72)
y(2)2 (4) = h
(2)21 (4)[h
(2)21 (2)]
Hs
(1)21 + h
(2)22 (4)[h
(1)22 (2)]
Hs
(2)22 . (4.73)
Also note the use of scalar channels during time instance t = 4, this is because of only
one antenna being used and the other being muted by the base-station.
• We employ the repetition of transmission technique to obtain other set of equations
to decode the desired streams. Hence at time instance t = 5, base station 1 and base
station 2 transmits the same linear combination as in time instance t = 3 through single
antenna and the signal observed by all the users in this case is given by :
y(1)1 (5) = h
(1)11 (5)[h
(2)11 (1)]
Hs
(1)11 + h
(1)12 (5)[h
(1)12 (1)]
Hs
(2)12 , (4.74)
y(1)2 (5) = h
(1)21 (5)[h
(2)11 (1)]
Hs
(1)11 + h
(1)22 (5)[h
(1)12 (1)]
Hs
(2)12 , (4.75)
y(2)1 (5) = h
(2)11 (5)[h
(2)11 (1)]
Hs
(1)11 + h
(2)12 (5)[h
(1)12 (1)]
Hs
(2)12 , (4.76)
y(2)2 (5) = h
(2)21 (5)[h
(2)11 (1)]
Hs
(1)11 + h
(2)22 (5)[h
(1)12 (1)]
Hs
(2)12 . (4.77)
• Similarly, at time instance t = 6, base station 1 and base station 2 transmit the same
linear combination of data streams as in time instance t = 4, which again helps to
mitigate the ICI by obtaining other independent set of equations in order to decode the
desired streams by all the users. The signal received by all the users at time instance
t = 6 is given by:
y(1)1 (6) = h
(1)11 (6)[h
(2)21 (2)]
Hs
(1)21 + h
(1)12 (6)[h
(1)22 (2)]
Hs
(2)22 , (4.78)
y(1)2 (6) = h
(1)21 (6)[h
(2)21 (2)]
Hs
(1)21 + h
(1)22 (6)[h
(1)22 (2)]
Hs
(2)22 , (4.79)
y(2)1 (6) = h
(2)11 (6)[h
(2)21 (2)]
Hs
(1)21 + h
(2)12 (6)[h
(1)22 (2)]
Hs
(2)22 , (4.80)
y(2)2 (6) = h
(2)21 (6)[h
(2)21 (2)]
Hs
(1)21 + h
(2)22 (6)[h
(1)22 (2)]
Hs
(2)22 . (4.81)
• The decoding of independent data streams is possible from the signals observed during
all the time instances by performing simple manipulations. Let us consider the decoding
for the user 1 in cell 1. By subtracting the signal received at time instance t = 1
60
Space-Time Transmission Scheme
multiplied by CSI at t = 3 from the signal received at time instance t = 3, we obtain
an equation which contains only the data streams intended to user 1 in cell 1. Further
by subtracting the signal received at time instance t = 1 multiplied by CSI at t = 5
from the signal received at time instance t = 5, we get another equation which also
contains only the data streams intended to user 1 in cell 1. Since there are two data
streams transmitted from base-station 1 intended to user 1 in cell 1, two independent
equations are enough to decode these two streams as explained mathematically by
following equation:
[y
(1)1 (3)− h(1)
12 (3)y(1)1 (1)
y(1)1 (5)− h(1)
12 (5)y(1)1 (1)
]=
[h
(1)11 (3)[h
(2)11 (1)]
H − h(1)12 (3)[h
(1)11 (1)]
H
h(1)11 (5)[h
(2)11 (1)]
H − h(1)12 (5)[h
(1)11 (1)]
H
]︸ ︷︷ ︸
A
s(1)11 . (4.82)
Since the channel coefficients are i.i.d and non-degenerate, A is a full rank matrix
almost surely with rank=2, and hence the two dimensional data vector s(1)11 is easily
decoded.
• Similarly, we can decode the data streams intended to user 2 in cell 1 by observing the
signals received at time instance t = 2, time instance t = 4 and time instance t = 6 as
given by:[y
(1)2 (4)− h(1)
22 (4)y(1)2 (2)
y(1)2 (6)− h(1)
22 (6)y(1)2 (2)
]=
[h
(1)21 (4)[h
(2)21 (2)]
H − h(1)22 (4)[h
(1)21 (2)]
H
h(1)21 (6)[h
(2)21 (2)]
H − h(1)22 (6)[h
(1)21 (2)]
H
]︸ ︷︷ ︸
B
s(1)21 . (4.83)
Also, B is full rank matrix for all i.i.d channel coefficients and hence we can easily
decode s(1)21 .
With the similar calculations, we can easily decode all the data streams intended to
user 1 and user 2 in cell 2. Hence, overall 8 data streams are decoded in just 6 channel
uses with the help of past channel states and DOF of 43
is achieved in two-cell MISO
IBC with M = K = 2.
The repetition scheme works fine for M = 2 and K = 2 and we achieved all the required
equations in all the users just by canceling the ICI and we not much analysis regarding the
cancellation and alignment of IUI is needed. But the repetition scheme is not suitable when
M > 2 because this scheme provides only two sets of independent equations. For M > 2,
the matrices constructed, i.e., A and B as observed in (4.82) and (4.83) are not full rank
61
Space-Time Transmission Scheme
matrices anymore. We verify this by considering the case M = 3 and K = 1 in the next
subsection.
4.4.5 Case V (M=3, K=1)
The transmission scheme for this case is observed as:
• Since there are three transmit antennas and only one user per cell, both the transmitters
transmit 3 independent streams of data at time instance t = 1 and the observed signal
is given by:
y(1)1 (1) =
[h
(1)11 (1)
]Hs
(1)11 +
[h
(1)12 (1)
]Hs
(2)12 , (4.84)
y(2)1 (1) =
[h
(2)11 (1)
]Hs
(1)11 +
[h
(2)22 (1)
]Hs
(2)12 , (4.85)
where y(`)k (t) is the scalar signal observed by user k at cell ` during time instance t,
h(`)kb (t) ∈ C3×1 is the channel vector from base station b to user k in cell ` during time
instance t and s(`)kb ∈ C3×1 is the independent transmit data vector from base station b
intended to user k in cell `.
• In order to cancel ICI, the base station in cell 1 transmits the linear combination of the
signal which is the ICI as observed by the user in cell 2 and the base station in cell 2
transmits the linear combination of the signal which is the ICI as observed by the user
in cell 1 using only one antenna in the time instance t = 2 and the signal received by
each user is given by:
y(1)1 (2) = h
(1)11 (2)
[h
(2)11 (1)
]Hs
(1)11 + h
(1)12 (2)
[h
(1)12 (1)
]Hs
(2)12 , (4.86)
y(2)1 (2) = h
(2)11 (2)
[h
(2)11 (1)
]Hs
(1)11 + h
(2)22 (2)
[h
(1)12 (1)
]Hs
(2)12 . (4.87)
• Similarly, the same operation is repeated in the next time instance t = 3 and the
independent time varying channels ensure that the signal received by the users is
different than that of the previous time instance and is given by:
y(1)1 (3) = h
(1)11 (3)
[h
(2)11 (1)
]Hs
(1)11 + h
(1)12 (3)
[h
(1)12 (1)
]Hs
(2)12 , (4.88)
y(2)1 (3) = h
(2)11 (3)
[h
(2)11 (1)
]Hs
(1)11 + h
(2)22 (3)
[h
(1)12 (1)
]Hs
(2)12 . (4.89)
• We still need another equation since 3 independent streams are transmitted. Repetition
based space time transmission scheme suggests that, we cancel ICI by transmitting the
62
Space-Time Transmission Scheme
linear combination of the interference as before over independent channels. The signal
received after t = 4 is thus given by:
y(1)1 (4) = h
(1)11 (4)
[h
(2)11 (1)
]Hs
(1)11 + h
(1)12 (4)
[h
(1)12 (1)
]Hs
(2)12 , (4.90)
y(2)1 (4) = h
(2)11 (4)
[h
(2)11 (1)
]Hs
(1)11 + h
(2)22 (4)
[h
(1)12 (1)
]Hs
(2)12 . (4.91)
• Decoding: Decoding is done by canceling the ICI observed by each user. The following
equations show the decoding in the case of user 1 in cell 1 as:y(1)1 (2)− h(1)
12 (2)y(1)1 (1)
y(1)1 (3)− h(1)
12 (3)y(1)1 (1)
y(1)1 (4)− h(1)
12 (4)y(1)1 (1)
=
h
(1)11 (2)
[(h
(2)11 (1)
]H− h(1)
12 (2)[h
(1)11 (1))
]Hh
(1)11 (3)
[(h
(2)11 (1)
]H− h(1)
12 (3)[h
(1)11 (1))
]Hh
(1)11 (4)
[(h
(2)11 (1)
]H− h(1)
12 (4)[h
(1)11 (1))
]H
︸ ︷︷ ︸C
s(1)11 , (4.92)
=
h(1)11 (2) −h(1)
12 (2)
h(1)11 (3) −h(1)
12 (3)
h(1)11 (4) −h(1)
12 (4)
︸ ︷︷ ︸
3× 2
[(h(2)11 (1)
]H[(h
(1)11 (1)
]H
︸ ︷︷ ︸2× 3
s(1)11 . (4.93)
Since s(1)11 is a three dimensional vector, we are unable to decode the data streams because
the rank of the matrix C is only 2 since it is the product of a 3 × 2 and a 2 × 3 matrices
as expressed by (4.93). Hence, for such case, it requires 3 time instances to determine two
equations and another 3 time instances to determine another equation, thus providing no
DOF gain.
However, we observe that for the cases with M = K, we can still achieve greater than
one DOF using the repetition transmission scheme and swapping of the equations required
by each user from other users where they act as an interference signal as explained with an
example case with M = K = 3 in the subsection below.
4.4.6 Case VI (M=3, K=3)
The transmission scheme for M = 3 and K = 3 is divided into the following three phases,
where each phase requires certain number of time instances.
63
Space-Time Transmission Scheme
Phase I: Transmission of M = K independent data streams intended to each user.
In this phase, each base station transmits independent data streams intended to each user
using one time instance from M antennas. Thus K = 3 users require 3 time instances. The
signal received by each user in the given time instance are as follows:
• At time instance t = 1, both the transmitters transmit all the three independent data
streams intended to the first user in each cell similar to the case of M = 2, K = 2 and
the signal received by all three users in both cell is given by:
y(1)1 (1) = [h
(1)11 (1)]
Hs
(1)11 + [h
(1)12 (1)]
Hs
(2)12 , (4.94)
y(1)2 (1) = [h
(1)21 (1)]
Hs
(1)11 + [h
(1)22 (1)]
Hs
(2)12 , (4.95)
y(1)3 (1) = [h
(1)31 (1)]
Hs
(1)11 + [h
(1)32 (1)]
Hs
(2)12 , (4.96)
y(2)1 (1) = [h
(2)11 (1)]
Hs
(1)11 + [h
(2)12 (1)]
Hs
(2)12 , (4.97)
y(2)2 (1) = [h
(2)21 (1)]
Hs
(1)11 + [h
(2)22 (1)]
Hs
(2)12 , (4.98)
y(2)3 (1) = [h
(2)31 (1)]
Hs
(1)11 + [h
(2)32 (1)]
Hs
(2)12 , (4.99)
where y(`)k (t) is a scalar signal received by a single antenna user k at cell ` during
time instance t, h`kb(t) ∈ C3×1 is the channel vector from base station b to user k in
cell ` during time instance t and s`kb ∈ C3×1 is the independent transmit data vector
transmitted from base station b intended to user k at cell `.
• At time instance t = 2, both the transmitter transmit three independent data streams
intended to the second user in each cell and using the similar notational pattern as
before, the signal received by all the users in both the cells is expressed as:
y(1)1 (2) = [h
(1)11 (2)]
Hs
(1)21 + [h
(1)12 (2)]
Hs
(2)22 , (4.100)
y(1)2 (2) = [h
(1)21 (2)]
Hs
(1)21 + [h
(1)22 (2)]
Hs
(2)22 , (4.101)
y(1)3 (2) = [h
(1)31 (2)]
Hs
(1)21 + [h
(1)32 (2)]
Hs
(2)22 , (4.102)
y(2)1 (2) = [h
(2)11 (2)]
Hs
(1)21 + [h
(2)12 (2)]
Hs
(2)22 , (4.103)
y(2)2 (2) = [h
(2)21 (2)]
Hs
(1)21 + [h
(2)22 (2)]
Hs
(2)22 , (4.104)
y(2)3 (2) = [h
(2)31 (2)]
Hs
(1)21 + [h
(2)32 (2)]
Hs
(2)22 . (4.105)
64
Space-Time Transmission Scheme
• Similarly, at time instance t = 3, both the transmitters transmit three independent
data streams intended to the third user and the signal observed by each user is given
by:
y(1)1 (3) = [h
(1)11 (3)]
Hs
(1)31 + [h
(1)12 (3)]
Hs
(2)32 , (4.106)
y(1)2 (3) = [h
(1)21 (3)]
Hs
(1)31 + [h
(1)22 (3)]
Hs
(2)32 , (4.107)
y(1)3 (3) = [h
(1)31 (3)]
Hs
(1)31 + [h
(1)32 (3)]
Hs
(2)32 , (4.108)
y(2)1 (3) = [h
(2)11 (3)]
Hs
(1)31 + [h
(2)12 (3)]
Hs
(2)32 , (4.109)
y(2)2 (3) = [h
(2)21 (3)]
Hs
(1)31 + [h
(2)22 (3)]
Hs
(2)32 , (4.110)
y(2)3 (3) = [h
(2)31 (3)]
Hs
(1)31 + [h
(2)32 (3)]
Hs
(2)32 . (4.111)
Phase II: Use of repetition transmission scheme
Repetition transmission scheme as discussed for the case M = K = 2 is used to cancel
ICI observed by the k−th user in cell 1 from the k−th user in cell 2 and obtain B desired
equations per user, where B is the number of cells. This phase requires BK = 6 time
instances. The transmission scheme and the signal received by each user in the given time
instance during this phase are listed below:
• Let L1 = [h(2)11 (1)]Hs
(1)11 be the linear combination of the ICI observed by the first user
in cell 2 due to the transmission from the base station in cell 1 in time instance t = 1
and L2 = [h(1)12 (1)]Hs
(2)12 be the linear combination of the ICI observed by the first user
in cell 1 due to the transmission from the base station in cell 2 at time instance t = 1.
During the time instance t = 4, the base station from cell 1 transmits L1 using a single
antenna and the base station from the cell 2 transmits L2 also using a single antenna.
65
Space-Time Transmission Scheme
The signal received by all the users is then given by:
y(1)1 (4) = h
(1)11 (4)L1 + h
(1)12 (4)L2, (4.112)
y(1)2 (4) = h
(1)21 (4)L1 + h
(1)22 (4)L2, (4.113)
y(1)3 (4) = h
(1)31 (4)L1 + h
(1)32 (4)L2, (4.114)
y(2)1 (4) = h
(2)11 (4)L1 + h
(2)12 (4)L2, (4.115)
y(2)2 (4) = h
(2)21 (4)L1 + h
(2)22 (4)L2, (4.116)
y(2)3 (4) = h
(2)31 (4)L1 + h
(2)32 (4)L2. (4.117)
• Similarly, let L3 = [h(2)21 (2)]Hs
(1)21 be the linear combination of ICI observed by the second
user in cell 2 due to transmission from the base station in cell 1 during time instance
t = 2 and L4 = [h(1)22 (2)]Hs
(2)22 be the linear combination of ICI observed by the second
user in cell 1 due to transmission fro the base station in cell 2 during time instance
t = 2. During t = 5 base station from cell 1 transmits L3 and the base station from cell
2 transmits L4 using a single antenna and the signal received by all the users is given
by:
y(1)1 (5) = h
(1)11 (5)L3 + h
(1)12 (5)L4, (4.118)
y(1)2 (5) = h
(1)21 (5)L3 + h
(1)22 (4)L4, (4.119)
y(1)3 (5) = h
(1)31 (5)L3 + h
(1)32 (4)L4, (4.120)
y(2)1 (5) = h
(2)11 (5)L3 + h
(2)12 (4)L4, (4.121)
y(2)2 (5) = h
(2)21 (5)L3 + h
(2)22 (4)L4, (4.122)
y(2)3 (5) = h
(2)31 (5)L3 + h
(2)32 (4)L4. (4.123)
• Also let L5 = [h(2)31 (3)]Hs
(1)31 and L6 = h
(1)32 (3)]Hs
(2)32 be the linear combination of ICI
observed by the third user in cell 2 due to transmission from base station in cell 1 and
the linear combination of ICI observed by the third user in cell 1 due to transmission
from base station in cell 2 during t = 3 respectively. During t = 6 the base station from
the cell 1 transmits L5 and the base station from the cell 2 transmits L6 using a single
66
Space-Time Transmission Scheme
antenna and the signal received by all the users is given by :
y(1)1 (6) = h
(1)11 (6)L5 + h
(1)12 (6)L6, (4.124)
y(1)2 (6) = h
(1)21 (6)L5 + h
(1)22 (4)L6, (4.125)
y(1)3 (6) = h
(1)31 (6)L5 + h
(1)32 (4)L6, (4.126)
y(2)1 (6) = h
(2)11 (6)L5 + h
(2)12 (4)L6, (4.127)
y(2)2 (6) = h
(2)21 (6)L5 + h
(2)22 (4)L6, (4.128)
y(2)3 (6) = h
(2)31 (6)L5 + h
(2)32 (4)L6. (4.129)
• During the next three time instances t = 7, 8, 9, the repetition transmission takes
place and L1, L2 , L3, L4, and L5, L6 are transmitted respectively over the independent
channels and each transmitter use only one antenna. The signal received by all the
users is obtained similarly as in previous three instances, only changing the channel
parameters. For example, the signal received by the first user in cell 1 and cell 2 during
t = 7 is given by:
y(1)1 (7) = h
(1)11 (7)L1 + h
(1)12 (7)L2, (4.130)
y(2)1 (7) = h
(2)11 (7)L1 + h
(2)12 (7)L2. (4.131)
This allows us to solve L1 and L2 in the first user of cell 1 and cell 2 independently,
using the signals received during time instances t = 4 and t = 7 as given by the following
for the first user of cell 1:
L1 =h
(1)12 (7)y1(4)− h(1)
12 (4)y1(7)
h(1)11 (4) + h
(1)11 (7)
, (4.132)
L2 =h
(1)11 (7)y1(4)− h(1)
11 (4)y1(7)
h(1)12 (4) + h
(1)12 (7)
. (4.133)
Similarly, all the users can solve for L1 and L2 using the two available equations. Also,
using the signal observed during the time instance t = 5 and t = 8, all the users can
solve for L3 and L4. Consider the signal received by the second user in each cell during
t = 8 as given by:
y(1)1 (8) = h
(1)11 (8)L3 + h
(1)12 (8)L4, (4.134)
y(2)1 (8) = h
(2)11 (8)L3 + h
(2)12 (8)L4, (4.135)
and L3 and L4 is obtained by the second user in cell 1 by solving the signals received
67
Space-Time Transmission Scheme
Cell 1 Cell 2
User 1 L1 = [h(2)11 (1)]Hs
(1)11 , L2 = [h
(1)12 (1)]Hs
(2)12 ,
y(1)1 (1)− L2 = [h
(1)11 (1)]Hs
(1)11 y
(2)1 (1)− L1 = [h
(2)12 (1)]
Hs
(2)12
User 2 L3 = [h(2)21 (2)]Hs
(1)21 , L4 = [h
(1)22 (2)]Hs
(2)22 ,
y(1)2 (2)− L4 = [h
(1)21 (2)]Hs
(1)21 y
(2)2 (2)− L3 = [h
(2)22 (2)]
Hs
(2)22
User 3 L5 = [h(2)31 (3)]Hs
(1)31 , L6 = h
(1)32 (3)]Hs
(2)32 ,
y(1)3 (3)− L6 = [h
(1)31 (3)]Hs
(1)31 y
(2)3 (3)− L5 = [h
(2)32 (3)]
Hs
(2)32
Table 4.1: Two desired equations are observed by each user in both the cells after 9 channeluses.
during time instance t = 5 and t = 8 as given by:
L3 =h
(1)12 (8)y1(5)− h(1)
12 (5)y1(8)
h(1)11 (5) + h
(1)11 (8)
, (4.136)
L4 =h
(1)11 (8)y1(5)− h(1)
11 (5)y1(8)
h(1)12 (5) + h
(1)12 (8)
. (4.137)
Using the similar approach all the users in both the cells can solve for L5 and L6. Hence
at the end of 9 channel uses, all users have the knowledge of two desired channels as
shown in table 4.1
Phase III: Swapping of desired signal from the adjacent users and further can-
celing the ICI.
Since two equations per user are not enough to solve three independent streams of data,
we need one more equation per user. The aim of the next phase is to obtain the required
equation in all the users. This is achieved by swapping the required information from the
adjacent users and again canceling the extra ICI observed due to the transmission in both
the cells. This phase requires 6 time instances. The signal received during each time instance
in this phase and the transmission strategy is listed below:
• By swapping the signal observed by the first user during the time instance t = 2 with
the signal observed by the second user during the time instance t = 1 in both the cells,
both the first and the second user obtains the required equations. However, the ICI is
68
Space-Time Transmission Scheme
Cell 1 Cell 2
User 1 y(1)1 (1) = [h
(1)11 (1)]
Hs
(1)11 + [h
(1)12 (1)]
Hs
(2)12 , y
(2)1 (1) = [h
(2)11 (1)]
Hs
(1)11 + [h
(2)12 (1)]Hs
(2)12 ,
y(1)1 (2) = [h
(1)11 (2)]Hs
(1)21 + [h
(1)12 (2)]Hs
(2)22 , y
(2)1 (2) = [h
(2)11 (2)]
Hs
(1)21 + [h
(2)12 (2)]Hs
(2)22 ,
y(1)1 (3) = [h
(1)11 (3)]Hs
(1)31 + [h
(1)12 (3)]
Hs
(2)32 y
(2)1 (3) = [h
(2)11 (3)]
Hs
(1)31 + [h
(2)12 (3)]Hs
(2)32
User 2 y(1)2 (1) = [h
(1)21 (1)]Hs
(1)11 + [h
(1)22 (1)]
Hs
(2)12 , y
(2)2 (1) = [h
(2)21 (1)]
Hs
(1)11 + [h
(2)22 (1)]Hs
(2)12 ,
y(1)2 (2) = [h
(1)21 (2)]
Hs
(1)21 + [h
(1)22 (2)]
Hs
(2)22 , y
(2)2 (2) = [h
(2)21 (2)]
Hs
(1)21 + [h
(2)22 (2)]
Hs
(2)22 ,
y(1)2 (3) = [h
(1)21 (3)]
Hs
(1)31 + [h
(1)22 (3)]
Hs
(2)32 , y
(2)2 (3) = [h
(2)21 (3)]
Hs
(1)31 + [h
(2)22 (3)]
Hs
(2)32 ,
User 3 y(1)3 (1) = [h
(1)31 (1)]Hs
(1)11 + [h
(1)32 (1)]
Hs
(2)12 , y
(2)3 (1) = [h
(2)31 (1)]
Hs
(1)11 + [h
(2)32 (1)]Hs
(2)12 ,
y(1)3 (2) = [h
(1)31 (2)]
Hs
(1)21 + [h
(1)32 (2)]
Hs
(2)22 , y
(2)3 (2) = [h
(2)31 (2)]
Hs
(1)21 + [h
(2)32 (2)]
Hs
(2)22
y(1)3 (3) = [h
(1)31 (3)]
Hs
(1)31 + [h
(1)32 (3)]
Hs
(2)32 y
(2)3 (3) = [h
(2)31 (3)]
Hs
(1)31 + [h
(2)32 (3)]
Hs
(2)32 .
Table 4.2: The signal received by all the users in both the cells during time instances t = 1, 2, 3.The red color signal represents the signal to be swapped between the first and second user ineach cell and the blue colored signal represents the signal to be swapped between the firstand the third users in each cell.
not completely canceled.
In the next time instance t = 10, the base station from cell 1 transmits [h(1)11 (2)]Hs
(1)21
and[h(1)21 (1)]Hs
(1)11 using two antennas; and also the base station from cell 2 transmits
[h(2)12 (2)]Hs
(2)22 and [h
(2)22 (1)]Hs
(2)12 using two antennas. These signals are shown by red
color in table 4.2. Thus the signal received during time instance t = 10 is given by:
y(1)1 (10) = h
(1)11 (10)[h
(1)11 (2)]Hs
(1)21 + g
(1)11 (10)[h
(1)21 (1)]Hs
(1)11
+ h(1)12 (10)[h
(2)12 (2)]Hs
(2)22 + g
(1)12 (10)[h
(2)22 (1)]Hs
(2)12 , (4.138)
y(1)2 (10) = h
(1)21 (10)[h
(1)11 (2)]Hs
(1)21 + g
(1)21 (10)[h
(1)21 (1)]Hs
(1)11
+ h(1)22 (10)[h
(2)12 (2)]Hs
(2)22 + g
(1)22 (10)[h
(2)22 (1)]Hs
(2)12 , (4.139)
y(1)3 (10) = h
(1)31 (10)[h
(1)11 (2)]Hs
(1)21 + g
(1)31 (10)[h
(1)21 (1)]Hs
(1)11
+ h(1)32 (10)[h
(2)12 (2)]Hs
(2)22 + g
(1)32 (10)[h
(2)22 (1)]Hs
(2)12 , (4.140)
69
Space-Time Transmission Scheme
y(2)1 (10) = h
(2)11 (10)[h
(1)11 (2)]Hs
(1)21 + g
(2)11 (10)[h
(1)21 (1)]Hs
(1)11
+ h(2)12 (10)[h
(2)12 (2)]Hs
(2)22 + g
(2)12 (10)[h
(2)22 (1)]Hs
(2)12 , (4.141)
y(2)2 (10) = h
(2)21 (10)[h
(1)11 (2)]Hs
(1)21 + g
(2)21 (10)[h
(1)21 (1)]Hs
(1)11
+ h(2)22 (10)[h
(2)12 (2)]Hs
(2)22 + g
(2)22 (10)[h
(2)22 (1)]Hs
(2)12 , (4.142)
y(2)3 (10) = h
(2)31 (10)[h
(1)11 (2)]Hs
(1)21 + g
(2)31 (10)[h
(1)21 (1)]Hs
(1)11
+ h(2)32 (10)[h
(2)12 (2)]Hs
(2)22 + g
(2)32 (10)[h
(2)22 (1)]Hs
(2)12 , (4.143)
where h(1)ij and h
(2)ij are the scalar channels from cell j to user i corresponding to antenna
1 (single antenna) observed in cell 1 and cell 2 respectively and g(1)ij and g
(2)ij are the scalar
channels from cell j to user i corresponding to antenna 2 (single antenna) observed in
cell 1 and cell 2 respectively.
Now, the first and the second user swaps the relevant signal as follows:
(I) First user in cell 1 performs the following operation,
y(1)1 (10)− h(1)
11 (10)y(1)1 (2) = g
(1)11 (10)[h
(1)21 (1)]Hs
(1)11︸ ︷︷ ︸
desired
+ g(1)12 (10)[h
(2)22 (1)]Hs
(2)12︸ ︷︷ ︸
ICI from the first user
+ h(1)12 (10)[h
(2)12 (2)]Hs
(2)22 − h(1)
11 (10)[h(1)12 (2)]Hs
(2)22︸ ︷︷ ︸
ICI from the second user
, (4.144)
which provides a desired signal but ICI from the first and second user in cell 2
exists. We can break [h(2)22 (1)]Hs
(2)12 as:
[h(2)22 (1)]Hs
(2)12 = y
(2)2 (1)− [h
(2)21 (1)]Hs
(1)11 , (4.145)
and substitute in (4.144) to obtain
y(1)1 (10)− h(1)
11 (10)y(1)1 (2) = g
(1)11 (10)[h
(1)21 (1)]Hs
(1)11 − g(1)
12 (10)[h(2)21 (1)]Hs
(1)11︸ ︷︷ ︸
desired
+
h(1)12 (10)[h
(2)12 (2)]Hs
(2)22 − h(1)
11 (10)[h(1)12 (2)]Hs
(2)22 + g
(1)12 (10)y
(2)2 (1)︸ ︷︷ ︸
ICI from the second user
.
(4.146)
(II) Second user in cell 1 performs the following operation
y(1)2 (10)− g(1)
21 (10)y(1)2 (1) = h
(1)21 (10)[h
(1)11 (2)]Hs
(1)21︸ ︷︷ ︸
desired
+h(1)22 (10)[h
(2)12 (2)]Hs
(2)22︸ ︷︷ ︸
ICI from the second user
+ g(1)22 (10)[h
(2)22 (1)]Hs
(2)12 − g(1)
21 (10)[h(1)22 (1)]Hs
(2)12︸ ︷︷ ︸
ICI from the first user
, (4.147)
70
Space-Time Transmission Scheme
which can be further simplified by substituting [h(2)12 (2)]Hs
(2)22 as:
[h(2)12 (2)]Hs
(2)22 = y
(2)1 (2)− [h
(2)11 (2)]Hs
(1)21 , (4.148)
to obtain
y(1)2 (10)− g(1)
21 (10)y(1)2 (1) = h
(1)21 (10)[h
(1)11 (2)]Hs
(1)21 − h(1)
22 (10)[h(2)11 (2)]Hs
(1)21︸ ︷︷ ︸
desired
+
g(1)22 (10)[h
(2)22 (1)]Hs
(2)12 − g(1)
21 (10)[h(1)22 (1)]Hs
(2)12 + h
(1)22 (10)y
(2)1 (2)︸ ︷︷ ︸
ICI from the first user
.
(4.149)
(III) First user in cell 2 performs the following operation
y(2)1 (10)− h(2)
12 (10)y(2)1 (2) = g
(2)12 (10)[h
(2)22 (1)]Hs
(2)12︸ ︷︷ ︸
desired
+ g(2)11 (10)[h
(1)21 (1)]Hs
(1)11︸ ︷︷ ︸
ICI from the first user
+ h(2)11 (10)[h
(1)11 (2)]Hs
(1)21 − h(2)
12 (10)[h(2)11 (2)]Hs
(1)21︸ ︷︷ ︸
ICI from the second user
, (4.150)
which can be further simplified by substituting [h(1)21 (1)]Hs
(1)11 as:
[h(1)21 (1)]Hs
(1)11 = y
(1)2 (1)− [h
(1)22 (1)]Hs
(2)12 , (4.151)
to obtain
y(2)1 (10)− h(2)
12 (10)y(2)1 (2) = g
(2)12 (10)[h
(2)22 (1)]Hs
(2)12 − g(2)
11 (10)[h(1)22 (1)]Hs
(2)12︸ ︷︷ ︸
desired
+
h(2)11 (10)[h
(1)11 (2)]Hs
(1)21 − h(2)
21 (10)[h(2)11 (2)]Hs
(1)21 + g
(2)11 (10)y
(1)2 (1)︸ ︷︷ ︸
ICI from the second user
.
(4.152)
(IV) Similarly, the second user in cell 2 performs the following operation:
y(2)2 (10)− g(2)
22 (10)y(2)2 (1) = h
(2)22 (10)[h
(2)12 (2)]Hs
(2)22︸ ︷︷ ︸
desired
+h(2)21 (10)[h
(1)11 (2)]Hs
(1)21︸ ︷︷ ︸
ICI from the second user
+ g(2)21 (10)[h
(1)21 (1)]Hs
(1)11 − g(2)
22 (10)[h(2)21 (1)]Hs
(1)11︸ ︷︷ ︸
ICI from the first user
, (4.153)
which can be further simplified by substituting [h(1)11 (2)]Hs
(1)21 as:
[h(1)11 (2)]Hs
(1)21 = y
(1)1 (2)− [h
(1)12 (2)]Hs
(2)22 , (4.154)
71
Space-Time Transmission Scheme
to obtain
y(2)2 (10)− g(2)
22 (10)y(2)2 (1) = h
(2)22 (10)[h
(2)12 (2)]Hs
(2)22 − h(2)
21 (10)[h(1)12 (2)]Hs
(2)22︸ ︷︷ ︸
desired
+
g(2)21 (10)[h
(1)21 (1)]Hs
(1)11 − g(2)
22 (10)[h(2)21 (1)]Hs
(1)11 + h
(2)21 (10)y
(1)1 (2)︸ ︷︷ ︸
ICI from the first user
.
(4.155)
• In the next time instance t = 11, transmit the signal
U11 = g(2)21 (10)[h
(1)21 (1)]Hs
(1)11 − g(2)
22 (10)[h(2)21 (1)]Hs
(1)11 + h
(2)21 (10)y
(1)1 (2)
from the base station in cell 1 using a single antenna and transmit the signal
U22 = h(1)12 (10)[h
(2)12 (2)]Hs
(2)22 − h(1)
11 (10)[h(1)12 (2)]Hs
(2)22 + g
(1)12 (10)y
(2)2 (1)
from the base station in cell 2 using single antenna. This cancels all the ICI observed
by the first user in cell 1 and the the second user in the cell 2. This requires some extra
previous signal received to be known at the receiver which can be easily obtained like
the CSIT. The signal received by the first user in cell 1 and the second user in cell 2
during t = 11 is given by:
y(1)1 (11) = h
(1)11 (11)U11 + h
(1)12 (11)U22, (4.156)
y(2)2 (11) = h
(2)21 (11)U11 + h
(2)22 (11)U22. (4.157)
Next, the first user in cell 1 and the second user in cell 2 can cancel the observed ICI
as follows:
y(1)1 (11)− h(1)
12 (11)(y
(1)1 (10)− h(1)
11 (10)y(1)1 (2)
)=h
(1)11 (11)U11 − h(1)
12 (11)(g
(1)11 (10)[h
(1)21 (1)]Hs
(1)11 − g(1)
12 (10)[h(2)21 (1)]Hs
(1)11
)=h
(1)11 (11)
[g
(2)21 (10)[h
(1)21 (1)]H − g(2)
22 (10)[h(2)21 (1)]H
]s
(1)11 − h(1)
12 (11)[g
(1)11 (10)[h
(1)21 (1)]H
− g(1)12 (10)[h
(2)21 (1)]H
]s
(1)11 + h
(1)11 (11)h
(2)21 (10)y
(1)1 (2). (4.158)
72
Space-Time Transmission Scheme
Thus,
y(1)1 (11)− h(1)
12 (11)(y
(1)1 (10)− h(1)
11 (10)y(1)1 (2)
)− h(1)
11 (11)h(2)21 (10)y
(1)1 (2)
=h(1)11 (11)
[g
(2)21 (10)[h
(1)21 (1)]H − g(2)
22 (10)[h(2)21 (1)]H
]s
(1)11 − h(1)
12 (11)[g
(1)11 (10)[h
(1)21 (1)]H
− g(1)12 (10)[h
(2)11 (1)]H
]s
(1)11
=[(h
(1)11 (11)g
(2)21 (10)− h(1)
12 (11)g(1)11 (10)
)[h
(1)21 (1)]H +
(h
(1)12 (11)g
(1)12 (10) −
h(1)11 (11)g
(2)22 (10)
)[h
(2)21 (1)]H
]s
(1)11 (4.159)
is the third equation required by the first user in cell 1.
Similarly,
y(2)2 (11)− h(2)
21 (11)(y
(2)2 (10)− g(2)
22 (10)y(2)2 (1)
)=h
(2)22 (11)U22 − h(2)
21 (11)(h
(2)22 (10)[h
(2)12 (2)]Hs
(2)22 − h(2)
21 (10)[h(1)12 (2)]Hs
(2)22
)=h
(2)22 (11)
[h
(1)12 (10)[h
(2)12 (2)]H − h(1)
11 (10)[h(1)12 (2)]H
]s
(2)22 − h(2)
21 (11)[h
(2)22 (10)[h
(2)12 (2)]H
− h(2)21 (10)[h
(1)12 (2)]H
]s
(2)22 + h
(2)22 (11)g
(1)12 (10)y
(2)2 (1)), (4.160)
and
y(2)2 (11)− h(2)
21 (11)(y
(2)2 (10)− g(2)
22 (10)y(2)2 (1)
)− h(2)
22 (11)g(1)12 (10)y
(2)2 (1))
=h(2)22 (11)
[h
(1)12 (10)[h
(2)12 (2)]H − h(1)
11 (10)[h(1)12 (2)]H
]s
(2)22 − h(2)
21 (11)[h
(2)22 (10)[h
(2)12 (2)]H
− h(2)21 (10)[h
(1)12 (2)]H
]s
(2)22 ,
=[(h
(2)22 (11)h
(1)12 (10)− h(2)
21 (11)h(2)22 (10)
)[h
(2)12 (2)]H +
(h
(2)21 (11)h
(2)21 (10)−
h(2)22 (11)h
(1)11 (10)
)[h
(1)12 (2)]H
]s
(2)22 , (4.161)
is the third equation required by the second user in cell 2.
• In the next time instance t = 12, the second user in cell 1 and the first user in cell 2
cancel the ICI observed by each of them by transmitting U21 from the base station in
cell 1 and U12 from the base station in cell 2 using a single antenna where
U21 = h(2)11 (10)[h
(1)11 (2)]Hs
(1)21 − h(2)
21 (10)[h(2)11 (2)]Hs
(1)21 + g
(2)11 (10)y
(1)2 (1),
U12 = g(1)22 (10)[h
(2)22 (1)]Hs
(2)12 − g(1)
21 (10)[h(1)22 (1)]Hs
(2)12 + h
(1)22 (10)y
(2)1 (2),
and the signal received by the second user in cell 1 and the first user in cell 2 during
73
Space-Time Transmission Scheme
t = 12 is given by:
y(1)2 (12) = h
(1)21 (12)U21 + h
(1)22 (12)U12, (4.162)
y(2)1 (12) = h
(2)11 (12)U21 + h
(2)12 (12)U12. (4.163)
Hence, the second user in cell 1 and the first user in cell 2 determines the required third
equation by canceling the ICI as follows:
y(1)2 (12)− h(1)
22 (12)(y
(1)2 (10)− g(1)
21 (10)y(1)2 (1)
)=h
(1)21 (12)U21 − h(1)
22 (12)(h
(1)21 (10)[h
(1)11 (2)]Hs
(1)21 − h(1)
22 (10)[h(2)11 (2)]Hs
(1)21
)=h
(1)21 (12)
[h
(2)11 (10)[h
(1)11 (2)]H − h(2)
21 (10)[h(2)11 (2)]H
]s
(1)21 − h(1)
22 (12)[h
(1)21 (10)[h
(1)11 (2)]H
− h(1)22 (10)[h
(2)11 (2)]H
]s
(1)21 + h
(1)21 (12)g
(2)11 (10)y
(1)2 (1), (4.164)
which is further simplified as:
y(1)2 (12)− h(1)
22 (12)(y
(1)2 (10)− g(1)
21 (10)y(1)2 (1)
)− h(1)
21 (12)g(2)11 (10)y
(1)2 (1)
=h(1)21 (12)
[h
(2)11 (10)[h
(1)11 (2)]H − h(2)
21 (10)[h(2)11 (2)]H
]s
(1)21 − h(1)
22 (12)[h
(1)21 (10)[h
(1)11 (2)]H
− h(1)22 (10)[h
(2)11 (2)]H
]s
(1)21
=[(h
(1)21 (12)h
(2)11 (10)− h(1)
22 (12)h(1)21 (10)
)[h
(1)11 (2)]H +
(h
(1)22 (12)h
(1)22 (10)−
h(1)21 (12)h
(2)21 (12)
)[h
(2)11 (2)]H
]s
(1)21 . (4.165)
Also,
y(2)1 (12)− h(2)
11 (12)(y
(2)1 (10)− h(2)
12 (10)y(2)1 (2)
)=h
(2)12 (12)U12 − h(2)
11 (12)(g
(2)12 (10)[h
(2)22 (1)]Hs
(2)12 − g(2)
11 (10)[h(1)22 (1)]Hs
(2)12
)=h
(2)12 (12)
[g
(1)22 (10)[h
(2)22 (1)]H − g(1)
21 (10)[h(1)22 (1)]H
]s
(2)12 − h(2)
11 (12)[g
(2)12 (10)[h
(2)22 (1)]H
− g(2)11 (10)[h
(1)22 (1)]H
]s
(2)12 + h
(2)12 (12)h
(1)22 (10)y
(2)1 (2), (4.166)
74
Space-Time Transmission Scheme
which is further simplified as:
y(2)1 (12)− h(2)
11 (12)(y
(2)1 (10)− h(2)
12 (10)y(2)1 (2)
)− h(2)
12 (12)h(1)22 (10)y
(2)1 (2)
=h(2)12 (12)
[g
(1)22 (10)[h
(2)22 (1)]H − g(1)
21 (10)[h(1)22 (1)]H
]s
(2)12 − h(2)
11 (12)[g
(2)12 (10)[h
(2)22 (1)]H
− g(2)11 (10)[h
(1)22 (1)]H
]s
(2)12
=[(h
(2)12 (12)g
(1)22 (10)− h(2)
11 (12)g(2)12 (10)
)[h
(2)22 (1)]H +
(h
(2)11 (12)g
(2)11 (10)−
h(2)12 (12)g
(1)22 (10)
)[h
(1)22 (1)]H
]s
(2)12 . (4.167)
Thus, after 12 channel uses all the four users have the required three equations to solve three
desired data streams. In order to obtain the third equation for the third user in both the cells,
same approach of swapping of desired signal and canceling of ICI is used. Let us assume that
the swapping takes place between the first user and the third user in both the cells. Note,
here that the swapping can also take place between the second and the third users in both
the cells.
Following all the approaches as we considered before for the swapping between the first
and second users in both the cells, we require next three time instances to obtain the third
equation for the third user in both cells. The first time instance t = 13 transmits the linear
combination of IUI from the first user and the third user in both cells. Next time instance
t = 14 is used to cancel ICI and obtain the third equation for the third user in cell 2 and
similarly other time instance t = 15 is used to obtain the third equation for the third user in
cell 1. The following are the third equations observed by the third user in both cells.
y(2)3 (14)− h(2)
31 (14)(y
(2)3 (13)− g(2)
33 (13)y(2)3 (1)
)− h(2)
33 (14)g(1)13 (10)y
(2)3 (1))
=[(h
(2)33 (14)h
(1)13 (13)− h(2)
31 (14)h(2)33 (13)
)[h
(2)13 (2)]H +
(h
(2)31 (14)h
(2)31 (13)−
h(2)32 (14)h
(1)11 (13)
)[h
(1)13 (2)]H
]s
(2)32 , (4.168)
and
y(1)3 (15)− h(1)
33 (15)(y
(1)3 (13)− g(1)
31 (13)y(1)3 (1)
)− h(1)
31 (15)g(2)11 (13)y
(1)3 (1)
=[(h
(1)31 (15)h
(2)11 (13)− h(1)
33 (15)h(1)31 (13)
)[h
(1)11 (2)]H +
(h
(1)33 (15)h
(1)33 (13)−
h(1)31 (15)h
(2)31 (15)
)[h
(2)11 (2)]H
]s
(1)31 . (4.169)
Thus, we show that in a two-cell MISO BC with three users per cell, we can decode total
of 18 streams of data in just 15 time instances only using the delayed CSIT and very few
75
Space-Time Transmission Scheme
delayed signal from previous time instances. The achievable DOF is 1815
= 65, which is greater
than 1 even though there are enough ICI and IUI. All three desired equations for each user
in both the cells are tabulated in the table 4.3.
4.4.7 Case VII (Generalized Scheme for M = K)
The three phase proposed scheme is valid for all the general number of users K such that
M = K as in the case of K = 3. In the general scheme, the first phase requires K time
instances to transmit 2K2 independent data streams from two cells such that K independent
data streams are intended to each user. Similarly, the second phase or the repetition phase
requires another 2K time instances to obtain 2 independent desired equations per user.
Next, each user needs to obtain K − 2 desired equations by swapping the desired streams
from the adjacent users. As observed for K = 3 case, we can only swap between two adjacent
users in the same cell at a time and each swapping provides one independent desired equation.
But there occurs the ICI from the adjacent cell and require two more time instances to cancel
all the ICI observed by both the users in both the cells. Thus, 4 users require three time
instances to obtain one desired equation. In other words, four desired equations are obtained
in three time instances.
Since there are 2K users and K − 2 desired equation per user, the total of 2K(K − 2)
desired equations are obtained in 34× 2K(K − 2) = 3× K(K−2)
2. However, we observed that
if K(K − 2) is not divisible by 2, it requires another extra three time instances as in the case
of K = 3. Hence, we can express the total time required in the third phase as :
t3 = 3×⌈K(K − 2)
2
⌉.
Based on this observation, we make a following conjecture on the achievable DOF of a
two-cell MISO IBC with delayed CSIT:
Conjecture 4.1. The total achievable DOF of a two-cell MISO-IBC with M transmit anten-
nas and K users in each cell, such that M = K, with the proposed space-time transmission
scheme is given by
dtot =2K2
K + 2K + 3⌈K(K−2)
2
⌉ . (4.170)
Thus, we observe here that if K is even (divisible by 2), the achievable DOF is
dtot =2K2
3K(1 + K−22
)=
4
3, (4.171)
independent of the number of users.
76
Space-Time Transmission Scheme
Cell 1 Cell 2
User 1 1.) L1 = [h(2)11 (1)]
Hs(1)11 , 1.) L2 = [h
(1)12 (1)]
Hs(2)12 ,
2.) y(1)1 (1)− L2 = [h
(1)11 (1)]
Hs(1)11 , 2.) y
(2)1 (1)− L1 = [h
(2)12 (1)]
Hs(2)12 ,
3.) y(1)1 (11)− h
(1)12 (11)
(y(1)1 (10)− h
(1)11 (10)y
(1)1 (2)
)3.) y
(2)1 (12)− h
(2)11 (12)
(y(2)1 (10)− h
(2)12 (10)y
(2)1 (2)
)−h(1)
11 (11)h(2)21 (10)y
(1)1 (2) =
[(h(1)11 (11)g
(2)21 (10) −h(2)
12 (12)h(1)22 (10)y
(2)1 (2)
)=[(
h(2)12 (12)g
(1)22 (10)
−h(1)12 (11)g
(1)11 (10)
)[h
(1)21 (1)]
H +(h(1)12 (11)g
(1)12 (10) −h(2)
11 (12)g(2)12 (10)
)[h
(2)22 (1)]
H +(h(2)11 (12)g
(2)11 (10)
−h(1)11 (11)g
(2)22 (10)
)[h
(2)21 (1)]
H]s(1)11 −h(2)
12 (12)g(1)22 (10)
)[h
(1)22 (1)]
H]s(2)12
User 2 1.) L3 = [h(2)21 (2)]
Hs(1)21 , 1.) L4 = [h
(1)22 (2)]
Hs(2)22 ,
2.) y(1)2 (2)− L4 = [h
(1)21 (2)]
Hs(1)21 , 2.) y
(2)2 (2)− L3 = [h
(2)22 (2)]
Hs(2)22 ,
3.) y(1)2 (12)− h
(1)22 (12)
(y(1)2 (10)− g
(1)21 (10)y
(1)2 (1)
)3.) y
(2)2 (11)− h
(2)21 (11)
(y(2)2 (10)− g
(2)22 (10)y
(2)2 (1)
)−h(1)
21 (12)g(2)11 (10)y
(1)2 (1) =
[(h(1)21 (12)h
(2)11 (10) −h(2)
22 (11)g(1)12 (10)y
(2)2 (1)) =
[(h(2)22 (11)h
(1)12 (10)
−h(1)22 (12)h
(1)21 (10)
)[h
(1)11 (2)]
H +(h(1)22 (12)h
(1)22 (10) −h(2)
21 (11)h(2)22 (10)
)[h
(2)12 (2)]
H +(h(2)21 (11)h
(2)21 (10)
−h(1)21 (12)h
(2)21 (12)
)[h
(2)11 (2)]
H]s(1)21 −h(2)
22 (11)h(1)11 (10)
)[h
(1)12 (2)]
H]s(2)22
User 3 1.) L5 = [h(2)31 (3)]
Hs(1)31 , 1.) L6 = h
(1)32 (3)]
Hs(2)32 ,
2.) y(1)3 (3)− L6 = [h
(1)31 (3)]
Hs(1)31 , 2.) y
(2)3 (3)− L5 = [h
(2)32 (3)]
Hs(2)32 ,
3.) y(1)3 (15)− h
(1)33 (15)
(y(1)3 (13)− g
(1)31 (13)y
(1)3 (1)
)3.) y
(2)3 (14)− h
(2)31 (14)
(y(2)3 (13)− g
(2)33 (13)y
(2)3 (1)
)−h(1)
31 (15)g(2)11 (13)y
(1)3 (1) =
[(h(1)31 (15)h
(2)11 (13) −h(2)
33 (14)g(1)13 (10)y
(2)3 (1)) =
[(h(2)33 (14)h
(1)13 (13)
−h(1)33 (15)h
(1)31 (13)
)[h
(1)11 (2)]
H +(h(1)33 (15)h
(1)33 (13) −h(2)
31 (14)h(2)33 (13)
)[h
(2)13 (2)]
H +(h(2)31 (14)h
(2)31 (13)
−h(1)31 (15)h
(2)31 (15)
)[h
(2)11 (2)]
H]s(1)31 h
(2)32 (14)h
(1)11 (13)
)[h
(1)13 (2)]
H]s(2)32
Table 4.3: Three desired equations are observed by each user in both the cells after 15 timeinstances. All users can solve three independent desired data streams using these threeequations.
77
Improved Scheme
If K is odd (not divisible by 2), K − 2 is also odd and K(K−2)2
is not divisible by 2. Thus⌈K(K−2)2
⌉can be simplified as⌈K(K − 2)
2
⌉=K(K − 2)− 1
2+ 1 =
K2 − 2K + 1
2,
and
dtot =2K2
3K + 3K2−2K+1
2
=4K2
3K2 + 3,
=4
3(1 + 1K2 )→ 4
3as K →∞. (4.172)
However, in all the cases, the achievable DOF is always greater than 1 and this fact is
depicted in Figure 4.3, where the achievable DOF is plotted with varying number of users
when the number of users are odd and even. We plot the achievable DOF with instantaneous
CSIT as observed in [72] and the achievable DOF using the proposed scheme with delayed
CSIT.
4.5 Improved Scheme
Further, we observe that the lower DOF achieved for the odd values of K is due to extra
time instances required during the third phase (swapping and ICI cancellation phase) and by
improving the transmission scheme, we can also achieve 43
and better DOF when K ≥ 3.
Let us again consider the case of K = 3 where we observe that during the swapping and
ICI cancellation phase the first user in each cell requires only one desired equation but obtains
two desired equation because it does the swapping with the the second user and also with the
third user in each cell. By using these extra equations, we can improve the achievable DOF.
By the improved transmission strategy, we perform following improvements in the respective
phases:
Phase I:
We transmit another three independent data streams intended to each user using extra three
time instances. Another independent data streams intended to user 1, user 2 and user 3 are
transmitted during the time instances t = 4, t = 5, and t = 6. Hence, the first phase now
requires 6 time instances.
78
Improved Scheme
0 5 10 15 20 25 30 35 40 45 50
0.8
1
1.2
1.4
1.6
1.8
2
Achie
vable
DO
F
0 5 10 15 20 25 30 35 40 45 501.3
1.4
1.5
1.6
1.7
1.8
1.9
2
Achie
vable
DO
F
DoF of two cell MISO IBC.(M = K, K is odd)
DoF of two cell MISO IBC.(M = K, K is even)
Number of Users (K)
Number of Users (K)
Delayed CSIT
Delayed CSIT
Instantaneous CSIT
Instantaneous CSIT
Figure 4.3: DOF of two-cell MISO IBC with delayed and instantaneous CSIT for odd andeven number of users. For odd number of users, the DOF converges to 4
3and for the even
number of users achievable DOF is always 43.
79
Improved Scheme
Phase II:
The second phase is the repetition scheme and also requires 6 time instances. Like in the
previous example, the linear combination of ICI observed by the first user in cell 2 due to
base station in cell 1 during t = 1 and the linear combination of all the ICI observed by
the first user in cell 1 due to base station in cell 2 during t = 1 is transmitted during t = 7.
Similarly, the linear combination of all the ICI observed by the second user in cell 2 due to
base station in cell 1 and the linear combination of all the ICI observed by the second user
in cell 1 due to the base station in cell 2 during t = 2 is transmitted in the time instance
t = 8 and the linear combination of all the ICI observed by the third user in both the cells
during t = 3 is transmitted in the time instance t = 9. Same transmission is repeated for
another three time instance. This allows us to obtain two equations observed by each user in
both the cells as described before. The only difference is that each equation has six variables
instead of three.
Phase III:
In this phase, we need to obtain four more equations instead of only one as in the previous
case. The swapping and cancellation of ICI between any two users within the cell in both
the cells require three time instances as observed previously. For example, the swapping and
ICI cancellation between the first and the second user in both the cells require three time
instances and provide one more equation to the first and second user in both cells. Similarly
the swapping and ICI cancellation between the first and the third user require another three
time instance and so does the swapping between the second and the third user in both the
cells. Thus, after 9 more time instances, all the users in both the cells have 2 more desired
equations.
Phase IV:
After the third phase, each user still requires two more equations. However, all the users
already have the linear combination of all the required equations as observed in the third
phase. Hence, we easily observe that by transmitting a linear combination of the desired
streams per user, we do not need more than another 6 time instances to obtain another 2
equations per user. Thus, we require total of 27 time instances to decode all 36 data streams,
which allows us to achieve 3627
= 43
total DOF.
80
Improved Scheme
For the general case with K users, following are the time instances required during each
phase:
(I) Phase I: It requires total of 2K time instances to transmit 4K2 data streams from two
cells.
(II) Phase II: This repetition phase also requires 2K time instances as discussed in the
previous example and two desired equations are obtained per user.
(III) Phase III: In this swapping and ICI cancellation phase, all the desired equations are
swapped between the adjacent users. Since there are K users and each time swapping
occurs between any two users, the total number of possible swapping is
ns =
(K
2
)=
K!
(K − 2)!2!=K(K − 1)
2,
As observed previously, each swapping and ICI cancellation requires 3 time instances,
ns swapping requires 3 × ns time instances and each user obtains all K − 1 desired
equations from the adjacent users.
(IV) Phase IV: Each user has already obtained 2+(K−1) = K+1 equations from the second
and third phase and at the same time these users also have the linear combination of
all the desired equations during the third phase. Since 2K streams were transmitted
per user, 2K − (K + 1) = K − 1 desired equations per user are to be obtained in the
fourth phase. We observe that all the K users require K(K − 1) time instances to do
so.
Hence, total time instances required during all the phases are
ttot = 2K + 2K +3K(K − 1)
2+K(K − 1)
= K(4 +
5
2(K − 1)
)=K
2(3 + 5K). (4.173)
Now, we can express the total achievable DOF as:
dtot =4K2
K2
(3 + 5K)
=8K
3 + 5K→ 8
5as K →∞. (4.174)
This improved achievable DOF for K ≥ 3 is depicted in Figure 4.4, which shows that the
DOF converges to 85
as K → ∞. The achievable DOF with instantaneous CSIT as given
by [72] is also plotted which converges to 2 as K →∞.
81
Improved Scheme
0 5 10 15 20 25 30 35 40 45 501.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
Achie
va
ble
DO
F
DoF of two cell MISO IBC.(M = K,K ≥ 3)
Number of Users (K)
Delayed CSITInstantaneous CSITDelayed CSIT (Improved)
Figure 4.4: Achievable DOF for two cell MISO IBC with delayed and instantaneous CSIT.The improved transmission scheme achieves better DOF which converges to 8
5unlike the
earlier approach that converges to 43. The DOF with the instantaneous CSIT converges to 2.
82
Conclusions
4.6 More Than Two-Cell MISO-IBC
As the number of cells increase, the ICI increase considerably and it becomes difficult to align
the interference in the absence of instantaneous CSIT. Even in the presence of instantaneous
CSIT, the multi-cell MISO is difficult to analyze because of the ICI and IUI. The DOF
analysis with delayed CSIT for B ≥ 3, where B is the number of cells, is an open problem
and an effective transmission scheme that provides a DOF gain greater than 1 is not known.
Hence, we leave the DOF analysis for multi-cell MISO IBC for B ≥ 3 with delayed analysis
as a future work.
Other important topic which has been less discovered is the multi-cell MIMO BC or MIMO-
IBC. Only a few works such as [73, 76–78] are well-known on MIMO IBC with instantaneous
CSIT and lately a few works such as [79,80] has been done with delayed CSIT. In the next
section, we present an interference alignment scheme in MIMO-IBC, where the users within
a cell or within a cluster collectively align the observed ICI onto a common subspace.
4.7 Conclusions
The space time transmission scheme to obtain the DOF of two-cell MISO IBC with delayed
CSIT is presented, mainly for the cases when the number of transmit antennas M is equal
to the number of user K in each cell. A detailed scheme is presented for M = K = 2 and
M = K = 3 and the achievable DOF measured in terms of ratio of number of independent
data streams to the number of time instances required is observed to be greater than one.
We presented an improved transmission scheme which shows that the DOF of two cell MISO
IBC converges to 85
as the number of users K increases. We also observed that it becomes
much harder to define a transmission scheme due to the introduction of ICI and occurs large
amount of delay to decode all the desired streams.
83
Conclusions
84
Chapter 5
Interference Alignment and Optimal
Overlapping in MIMO IBC and IC
In this chapter, we present an interference alignment (IA) scheme where a group of users
within a cell or within a cluster collectively align all the unwanted signal or interference onto
a common subspace. We assume that such group exists within a cell, where all the users
collectively assign a common overlap subspace to the observed inter cell interference (ICI)
such as in multiple input multiple output (MIMO) interference broadcast channel (IBC) or
the number of users in MIMO interference channel (IC) form different topological structures
to overlap the unwanted signal onto a common subspace.
We term such IA technique of aligning the interference from a same source onto a common
subspace by different users, as vertical alignment, which is unlike the conventional horizontal
alignment where the same user align all the interference coming from different sources in a
common subspace. This approach of vertical IA is not well studied in the literature except
few slightly different version of such scheme presented in [77] and [78].
One of the major challenges in the vertical IA is the determination of the common
overlapping subspace and the optimal dimensions of this overlapping subspace. In this
work, we present a scheme to determine the overlap subspace and also the dimensions of
optimal overlapping required between any two or more users to create such an overlap. We
observe that the common subspace (overlap region) exists between the users only if they
share common spatial dimensions.
The next generation communication in a high speed train and local buses or in a crowded
stadium can be a combination of wired plus wireless communication. All users can plug in
certain device, which allows all the users get connected and also provide all of them with
the common spatial dimensions. However, all the users observe independent channels from
the wireless transmitters at the same time. The concept of vertical alignment and the idea
of optimal overlapping plays a pivotal role in such scenarios where each user observes the
common channel and the private channels.
85
MIMO Interference Broadcast Channels
Some of the antennas in each users are dedicated to common channels and other to the
private channels. We assume that the knowledge of which antennas are common and which
are private are given by the topological matrices Si, ∀i = {1, 2, · · · , K}, where K is the
number of users. Si also determines how the antennas are topologically shared among the
users. In this regard, we also propose a ‘ring topological sharing scheme’ and ‘star topological
sharing scheme’ for the case of three user MIMO IC.
Next, we describe the idea of vertical alignment and optimal overlapping in two-cell MIMO
channels with two users per cell, where each user receive the ICI from the adjacent cell.
5.1 MIMO Interference Broadcast Channels
5.1.1 System Model
Consider the case of two-cell MIMO-IBC as depicted in the Figure 5.1, where two cells each
having a base station with M antennas serve two users with N antennas in each cell. Each
user observes both the inter user interference (IUI) and the ICI, which requires the precoders
Vj =[V1j V2j
]and post-processors Ukj,∀k = {1, 2} be properly designed at any cell j so
as to align both IUI and ICI as much as possible.
Here, all the users within a cell (two) are assumed to form a group and they can collectively
align the ICI observed from the unwanted transmitter onto a common subspace, denoted by
Gj for any cell j, ∀j = {1, 2}. In order to get a clear picture of the idea, let us consider there
are M = 6 transmit antennas in each base station and N = 4 receive antennas per user and
each transmitter is transmitting d = 3 independent data streams intended to both the users.
We assume the data streams are intended to both users just for simplicity. However different
users require different data streams from the transmitter.
This causes ICI of d = 3 dimensions to be observed by each user. Since there are N = 4
antennas per user, each user can resolve only N − d = 1 data streams. However, if two users
align all the ICI in a common d = 3 dimensional space, even with N = 4 antennas per user
total of 5 data streams can be resolved and this is a considerable gain from total of 2 data
streams in the absence of overlapping.
86
MIMO Interference Broadcast Channels
BS1
BS2
1
2
1
2
cell 1
cell 2
H11(1)
H21 (1)
H11 (2)H
21 (2)H 12
(1)
H 22(1)
H12(2)
H22 (2)
U11
U12
U21
U22
V1 = [V11 V21]
V2= [ V12 V22]
Figure 5.1: Two cell MIMO Broadcast Channel with two users per cell and arbitrary Mantennas at each base station and N antennas per user. The solid line represents the desiredsignal plus IUI, while the dashed line represents the ICI observed by each user. The channelsfrom base station j to user k in cell ` is represented as H
(`)kj , while Vj are the precoding
matrices and Uk` are the interference suppressing matrices or the post processors at user kin cell `.
87
MIMO Interference Broadcast Channels
5.1.2 Designing Post-processing Matrices
As discussed earlier, the challenge is again to determine the overlapped subspace and the
overlapped dimensions. Let Ukj,∀k = {1, 2} at any cell j be the orthonormal basis for the
projected signal space by user k, then both the users align the ICI onto the common subspace
which are spanned over the same dimensions. Hence, for the given two-cell system, the users
in cell 1 align the ICI observed from cell 2 onto a common subspace G1 such that
span(UH
11H(1)12 V2
)= span
(UH
21H(1)22 V2
)= G1. (5.1)
The common subspace region only exists when there is enough overlap between the signals
observed by the two users. Otherwise, the users obtain interference individually and can
not align them collectively. One of the ideas to create an overlap space is to assume that
the interference channel observed by both the users is correlated. We assume that there is a
strong correlation and consider that the two channels H(1)12 ∈ CN×M and H
(1)22 ∈ CN×M are
the sub-matrices of another bigger channel matrix H(1)2 ∈ C2N×M with enough overlapping
such that the common overlap space G1 exists between the first and second user in the cell 1.
Thus, the interference observed by both the users is a part of H(1)2 V2 as
H(1)12 V2 = S1H
(1)2 V2, (5.2)
H(1)22 V2 = S2H
(1)2 V2, (5.3)
where S1 ∈ R(N+No/2)×2N and S2 ∈ R(N+No/2)×M are considered as the 0 − 1 selection
matrices and No is the total number of overlapping rows.
If G1 exists then we can design U11 and U21 from (5.1), (5.2) and (5.3) and using the
property span(A) = span(AAH) [81] as
span(UH
11S1PSH1 U11
)= span
(UH
21S2PSH2 U21
), (5.4)
where
P = H(1)2 V2V
H2 H
(1)2
H. (5.5)
We can further express (5.4) as
UH11S1PSH1 U11 = UH
21S2PSH2 U21, (5.6)
⇒UH11S1PSH1 U⊥11 −UH
21S2PSH2 U⊥21 = 0, (5.7)
⇒[UH
11 UH21
] [S1PSH1 0
0 −S2PSH2
][U11
U21
]= 0, (5.8)
⇒UH1 ∆U1 = 0, (5.9)
88
MIMO Interference Broadcast Channels
where
U1 =
[U11
U21
],
∆ =
[S1PSH1 0
0 −S2PSH2
],
and we can determine U⊥1 by determining the null space of ∆, i.e., U1 ∈ N (∆).
For the given P and Sk,∀k = {1, 2}, U1 exists only if N (∆) exists. Since the dimensions of
P depends on the dimensions of V2, which is the number of interference streams transmitted
to both the users in cell 1 from cell 2, this dimension can vary up to min(M, 2N). The
N (∆) does not exist when ∆ is a full column rank matrix, but the overlapping ensures that
N (∆) always exists. Thus, we can determine U1 and also the basis of overlap subspace.
For different dimensions of V2, the rank of N (∆) and the available overlap dimensions are
plotted in the upper part and the lower part of the Figure 5.2 respectively.
The plot on the upper part of Figure 5.2 clearly shows that when the dimensions (d) of
V2 is less than N , N (∆) still exists even with no overlapping rows. This is because there
are N − d free dimensions available at each user independent of the d dimensions. The plot
on the lower part of Figure 5.2 shows that the common region or the overlap space exists
only when there are overlapping rows between the channels. We also observe that the N (∆)
contains the overlapped region for all d ≥ N .
5.1.3 Designing the Precoding Matrices
Similarly, the users in cell 2 design the respective post-processing or interference suppressing
matrices U12 and U22. Based on the knowledge of all the post-processing matrices Ukj, the
precoding matrix Vj is updated such that Vj aligns and zero forces all ICI. If we consider
V2 that zero forces all ICI, such V2 is contained in the null space of Φ, where
Φ = SH1 U11UH11S1 + SH2 U21U
H21S2, (5.10)
is a modified version of the so called alignment matrix as defined in [47,82–84]. The null
space of Φ, hereafter denoted N (Φ), is interesting because it allows all the interference
streams to be aligned on the common subspace. If the dimension of N (Φ) is greater than the
dimension of the allocated streams d, the overlap space affects the available free dimensions.
Hence the optimum dimension of the overlap space is that number of overlapping rows when
rank(N (Φ)
)= d. (5.11)
89
MIMO Interference Broadcast Channels
0 0.5 1 1.5 2 2.5 3 3.5 40
2
4
6
8
10
0 0.5 1 1.5 2 2.5 3 3.5 40
2
4
6
8
Dim. of N (∆) increasing with Overlapping Rows
(M = 6, N = 4, K = 2)
Dim. of Overlap space increasing with Overlapping Rows
(M = 6, N = 4, K = 2)
Number of overlapping rows
Number of overlapping rows
Ran
kof
N(∆
)Ran
kof
Overlap
Space
d = 3
d = 3
d = 4
d = 4
d = 5
d = 5
d = 6
d = 6
Figure 5.2: Plot of the total interference dimensions and the overlap dimensions for giventransmit streams d varying with the number of overlapping rows. The upper part plots thetotal number of free dimensions after each user receives d dimensional interference signalvarying with the overlapping rows of the channels between two users. The lower part plotsthe dimensions of the common region where all interferences are aligned by two users.
90
MIMO Interference Channels
In the Figure 5.3, we plot the rank(N (Φ)
)for different number of transmit streams d and
we observe that the optimum number of overlapping rows from each user is always equal to
dd2e. Intuitively, this analysis makes sense since the overlapping occurs between two users and
we assume that there are equal number of antennas overlapping from each user. After total
overlap dimensions of d is achieved, increasing the overlapping rows does not affect since all
interfering dimensions are aligned on the overlap space.
For example, the optimal number of overlapping rows per user is 2, when d = 3 and
d = 4. This is true because the overlap dimension of odd numbers is not created due to equal
antennas overlapping between two users.
5.2 MIMO Interference Channels
In this section, we present the idea of vertical alignment and overlapping subspace in the
case of three user MIMO IC. We achieve MIMO-IA by estimating the projection matrices at
each receiver, called the local projectors, embedding the orthogonal complements of the local
projectors onto some common dimensions and adding them to obtain the so called alignment
matrix, the null space of which contains the optimal precoders. Such concept of the alignment
matrix, hereafter denoted Φ is also introduced in dimensionality reduction for non-linear
manifold learning [47,82–84].
The embedding of the orthogonal complements of the local projectors is achieved by the
antenna selection matrices, which also provides flexibility in selecting different structural
topology for the receivers as well as introduces the notion of partial antennas sharing among
the receivers such as the ‘ring topological sharing’ and the ‘star topological sharing’ and
discussed before. Before that let us redefine and reformulate the problem of MIMO IA in
K-user MIMO IC in the next section .
5.2.1 Preliminaries
For K-user MIMO-IC, IA is achieved by designing the optimal post-processor matrices at
each receiver i, here denoted by U∗i and the optimal precoding matrices at each transmitter j,
here denoted by V∗j . The aim of IA is to align all K−1 interfering signals, each of dimensions
d in a common subspace and at the same time allocate d dimensions for the desired signal
independent of interference signal. Consider that U∗i is the optimal basis for the common
subspace, where all the interference is aligned.
91
MIMO Interference Channels
0 0.5 1 1.5 2 2.5 3 3.5 43
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
Optimal Overlapping Dimensions
(M = 6, N = 4, K = 2)
Number of overlapping rows
Overlap
Dim
ension
d = 3d = 4d = 5d = 6
Figure 5.3: Plot of the rank(N (Φ)
)with the number of overlapping rows per user when
different number of data streams are transmitted. The optimal number of overlapping rowsare the number of overlapping rows when the rank
(N (Φ)
)= d.
92
MIMO Interference Channels
Tx 2
Tx 3
Rx 1 Tx 1
Rx 2
Rx 3
V2
V3
U1 V1
U2
U3
Figure 5.4: Multiple access channel (MAC) (left side) and broadcast channel (BC) (rightside) duality for the three-user MIMO IC where only the interference signals are shown. U1
is determined with the knowledge of V2 and V3 and V1 is determined with the knowledge ofU2 and U3.
In order to design U∗i , the IC is treated as a MAC (left side of Figure 5.4), where the
receiver i observes all the interference signal from the transmitters j 6= i and the desired
signal from the transmitter i. Since U∗i spans d dimensional subspace, d needs to be properly
defined such that the alignment of interference is achievable. For a K user MIMO-IC with M
transmit and N receive antennas, the d that achieves IA is outer bounded by the following
expression [34]
d ≤ M +N
K + 1. (5.12)
Similarly, in order to design V∗j , the IC is treated as a BC (right side of Figure 5.4) and the
signal from any transmitter j is precoded in such a way that these signals align themselves
as much as possible in the subspace spanned by U∗i , ∀i 6= j. Since the alignment is possible
for only d independent streams, the precoding matrix consists of d column vectors.
Thus, the optimal precoders and the optimal post-processor matrices are obtained iteratively
by using the property of MAC/BC duality or the uplink/downlink duality. Let Vj ∈ CM×dand Ui ∈ CN×d be the randomly initialized precoder and post-processor for any transmitter
j and any receiver i respectively, then in order to maintain the power constraints Vj and Ui
93
MIMO Interference Channels
are orthonormal such that VHj Vj = Id and UH
i Ui = Id. Since d ≤ min(M,N), UiUHi 6= Id.
Let Pi = UiUHi be the local projection matrix observed at each receiver i and P⊥i = IN −Pi
be a projection matrix onto the orthogonal complement of the subspace spanned by Ui [81].
Also, the random channels from any transmitter j to any receiver i is represented by a
matrix Hij ∈ CN×M , where each element hnm is the independent and identically distributed
(i.i.d) Gaussian channel state corresponding to the n-th antenna element of receiver i and
m-th antenna element of transmitter j with zero mean and |hnm|2 variance. For any random
Vj and Hij, ∀i 6= j, HijVj is the span of d orthogonal transmitted interference signal.
Optimization of Ui
Ui is the basis of subspace where all these interference signal are projected. Hence these
interference signal are minimized as much as possible on the orthogonal complement to the
subspace spanned by Ui. Thus, Ui is obtained by minimizing the following:
minUi
K∑j=1,j 6=i
||P⊥i HijVj||2F , (5.13)
minUi
K∑j=1,j 6=i
trace[P⊥i HijVjV
Hj HH
ij
], (5.14)
minUi
K∑j=1,j 6=i
trace[HijVjV
Hj HH
ij −UiUHi HijVjV
Hj HH
ij
], (5.15)
under the constraints
UHi Ui = Id, VH
j Vj = Id, (5.16)
where (5.14) follows from (5.13) by norm-trace relationship and observing the fact that trace
is invariant under cyclic permutations, that is , trace(ABC) = trace(BCA) = trace(CAB)
for some matrices A,B and C; the property of projection matrix P2 = P for some projection
matrix P. Equation (5.15) follows immediately from (5.14). We can now equivalently
express (5.15) as
maxUi
K∑j=1,j 6=i
trace[UHi HijVjV
Hj HH
ijUi
], (5.17)
under the same constraints.
Here we observe that (5.13) and (5.17) are two equivalently different ways to obtain
the optimum Ui. Since Hij is known at the receiver perfectly, (5.17) is a well-known
trace optimization problem with orthonormal constraints and the solution is given by the
94
MIMO Interference Channels
eigenvectors corresponding to the d maximum eigenvalues of∑K
j=1,j 6=i HijVjVHj HH
ij . Thus,
the local projectors are obtained as Pi = UiUHi . However, the independence of the interference
signal from the desired signal is still not guaranteed.
In the downlink, each transmitter then updates the corresponding precoding matrices in
such a way that the interference power observed in the unwanted receivers is minimized as
much as possible. This guarantees that the aligned interfering signals are not contained in
the subspace orthogonal to the one spanned by Ui.
Optimization of Vj
The d dimensional independent signals from any transmitter j to the unwanted receivers
i, ∀i 6= j is HijVj . Please note here that the channel Hij in the uplink is from all transmitters
j 6= i to the receiver i while the channel Hij in the downlink is from the transmitter j to all
the receivers i 6= j and Vj is obtained by minimizing the following :
minVj
K∑i=1,i 6=j
||P⊥i HijVj||2F (5.18)
minVj
K∑i=1,i 6=j
trace[P⊥j HijVjV
Hj HH
ij
], (5.19)
minVj
trace[ K∑i=1,i 6=j
VHj (HH
ijP⊥i Hij)Vj
], (5.20)
under the constraints
UHi Ui = Id, VH
j Vj = Id. (5.21)
Equation (5.19) and (5.20) follow from (5.18) by the trace norm relationship and the
cyclical invariant properties of the trace. From (5.20), it is clear that Vj is determined if
we know P⊥i and the channel states Hij exactly. At that time, it is again a well-known
trace optimization problem and the optimum Vj is given by the eigenvectors corresponding
to the d minimum eigenvalues of∑
i=1,i 6=j HHijP
⊥i Hij, which is the basis of most of the IA
algorithms such as [2, 32]. However, channel state information at the transmitter (CSIT) is
always difficult to determine in the time-varying wireless channels and also the knowledge of
optimum value of d is not available.
95
MIMO Interference Channels
Grouping of Receivers and Reformulation of Optimization of Vj
As discussed in the previous section, we would like to form a group among the receivers and
collectively align the interference onto a common subspace. Hence, we imagine a system
where each receiver has some common channel parts and other independent channel parts
and transmitter knows the common channel and independent channels from each receiver
locally. Such system model can be used in future wired plus wireless communication systems,
such as in trains and buses as we discussed before.
The topological information about the common and independent channels are presented
in the 0-1 selection matrices Si ∈ RN×NT , ∀i = {1, 2, · · · , K}. If Hj ∈ CNT×M is the
channel matrix available at the transmitter j, the elements of which are the channel states
corresponding to the available spatial dimensions (antenna elements ) in all the interfering
receivers, then the channel matrix at each receiver i corresponding to transmitter j can be
written as:
Hij = SiHj.
The optimization problem in (5.20) is then expressed as:
minVj
trace[VHj HH
j
( K∑i=1,i 6=j
SHi P⊥i Si)
HjVj
], (5.22)
under the same constraints, where the term Φ =∑K
i=1,i 6=j SHi P⊥i Si is the alignment matrix
as defined in non-linear manifold learning [47,82–84].
Expression (5.22) is particularly significant because we can express HjVj as a single
variable, we can play around with another parameter Si (that provides flexibility in topology)
and we can easily analyze the error if P⊥i is not known perfectly. Mathematically, HjVj is
the d dimensional precoded interference signal from the transmitter j and we can clearly
observe that the trace in (5.22) is minimum when HjVj ∈ N (Φ), where N (Φ) represents
the null space of Φ. This fact motivates us to ask the following questions: a) Does N (Φ)
exist? b) If it does exist, what is the dimension of N (Φ) ?
We observe that the N (Φ) exists as long as there is overlapping channels or physically the
overlapping spatial dimensions between any two receivers. But such overlapping is not always
optimal. Our aim is to determine the optimal overlapping between them. Next, we discuss
the concept of optimal overlapping between any two sub-spaces created by the projection of
any two sub-matrices.
96
MIMO Interference Channels
Z1
Z2
S1
S2
ZΦf(S1,S2,Z1,Z2)
ΦZ = 0 for any S1 and S2.
Figure 5.5: Relationship between Φ and Z.
Optimal Overlapping
Consider Zk and Z`, ∀k 6= ` are two sub-matrices of Z and the corresponding projection
matrices PZk and PZ` are the projection onto any two subspaces M and N , then the two
subspaces are said to be optimally overlapped if one of the sub-matrix has full column rank
and
min[
rank(Zk), rank(Z`)]
= rank(Zk`), (5.23)
where Zk` = Zk ∩ Z` represents the matrix of the common rows between Zk and Z`.
When they are optimally overlapped, it is no longer possible to find two independent
vectors wk and w` such that Zk`wk = Zk`w`. Thus, a unique solution exists that satisfies
this condition. The study of the uniqueness of the solutions of a merging or the realization
problem is called rigidity [85]. In other words, the subspaces are optimally overlapped when
the rigid solution exists and in that case the total space is exactly equal to the overlapped
subspace, which is exactly equal to the null space of the alignment matrix obtained from two
sub-matrices as given by
R(Z) = R(Zk) ∩R(Z`) = N (Φ). (5.24)
The relationship between Z and Φ is clearly depicted in Figure 5.5 for any two sub-matrices
of Z named as Z1 and Z2. N (Φ) and Z are exactly same when optimal overlap exists between
Z1 and Z2.
How do we obtain the optimal overlapping subspace?
One idea is to determine the optimally overlapped sub-matrices of the given matrix Z. In
order to do so, we define the selection matrices Si such that Zi = SiZ. Now, we can tune
97
MIMO Interference Channels
Sis optimally to determine the optimal Φ. The optimal tuning of Sis help to optimally
overlap the rows of Zis, which allows more flexibility in the system in terms of the topological
arrangement of the receivers. In fact, each rows of Zis represent the number of antenna
elements in the receiver i and hence the overlapping of the rows can be mathematically
viewed as the concept of virtual antennas overlapping or the antennas sharing. We call this
virtual sharing because the sharing is only mathematically observed and not physically.
If rank(Zk) = rank(Z`) = d, then when Zk and Z` are optimally overlapped, rank(Zk`) =
dim(N (Φ)) = d is satisfied. So, we can choose any arbitrary d and determine the optimal
Φ such that dim(N (Φ)) = d by properly tuning the selection matrices. In this regard, we
require a scheme to determine the rank of N (Φ) for different Sis.
5.2.2 Determining the Rank of N (Φ)
Theorem 5.1. Consider that Z1 and Z2 are two sub-matrices of Z ∈ CNT×M with some
overlapping rows, expressed as
Z1 = S1Z =
[Z11
Z12
], Z2 = S2Z =
[Z21
Z22
], (5.25)
with Z11 ∈ CN11×M and Z22 ∈ CN22×M are the non overlapped parts and Z12 = Z21 ∈ CN12×M
are the overlapped parts, and Φ = SH1 (IN11+N12−Z1Z†1)S1 +SH2 (IN22+N12−Z2Z
†2)S2 as defined
previously then
rank(Φ) = N11 +N12 +N22 − (r1 + r2 + r3), (5.26)
rank(N (Φ)) = r1 + r2 + r3, (5.27)
where
r1 = rank{Z12}, (5.28)
r2 = rank{(Z11R1)(:,r1+1:M)}, (5.29)
r3 = rank{(Z22R1)(:,r1+1:M)}. (5.30)
R1 ∈ CM×M is the right singular vector obtained from singular value decomposition (SVD) of
Z12.
Proof. The proof is provided in the appendix A.
The rank ofN (Φ) for different values of M and the same value of NT for varying overlapping
dimensions is plotted in Figure 5.6. One of the interesting things we observe here is the case
when there is no overlapping. In that case, r1 + r2 + r3 = 2M but we expected all r1, r2
98
MIMO Interference Channels
and r3 to be zero since they depend on R1, the right singular vector of an empty matrix.
However, we observe that though Z12 is an empty matrix, the right singular vector R1 is an
identity matrix IM . Thus, r1 = 0, r2 = M and r3 = M . Intuitively if we observe the rank
of matrix Φ, this makes sense because Φ is the embedding of the orthogonal projections of
dimensions N11 −M and N22 −M in this case and hence the rank of N (Φ) is 2M .
As the overlapping increases, r2 and r3 depends on r1 and the rank of N (Φ) is r1 + 2(M −r1) = 2M − r1. r2 and r3, in fact, represent the number of independent dimensions at the two
receivers and r1 represents the overlapping dimensions between them. For all r1 > M , the
rank is always M . The N (Φ) does not exist when 2M − r1 ≥ 2N because the complement of
the projection matrix becomes close to zero in that case. In the simulation shown in 5.6, the
rank of N (Φ) is not defined for the cases when (M = 10, r1 = 0) and (M = 12, r1 = 0, 2, 4).
The optimal overlapping is observed when r1= number of transmit antennas=M . Since we
assume that each receiver observes equal overlapping all the time, the number of overlapping
rows is always considered to be even in the simulations.
The rank of N (Φ) is observed for two overlapping receivers that share common antennas
(spatial dimensions) between them. As we discussed before, the Sis provide flexibility to
define different topology of antennas sharing and in the next section, we discuss the approach
to determine the rank of N (Φ) for the ring topological sharing and the star topological
sharing in the case of three user MIMO-IC.
5.2.3 Ring Topology
System Model
If the overlapping of certain number of antennas is observed between the adjacent receivers,
we call this scheme the ‘ring topology’. Symmetric antennas overlapping for ring topology of
three user MIMO-IC is depicted in Figure 5.7, where the antennas in each receiver are indexed
with alphabets. Different antennas overlap between the adjacent receivers. For example, the
antennas indexed ‘d’ and ‘e’ overlap between RX1 and RX2, the antennas indexed ‘h’ and ‘i’
overlap between RX2 and RX3 and the antennas indexed ‘l’ and ‘a’ overlap between RX1 and
RX3 as shown in Figure 5.7. In that case, RX1 observes all the information in antennas ‘l, a,
b ,c ,d, e’, RX2 observes all the information in antennas ‘d, e, f, g, h, i’ and RX3 observes all
the information in antennas ‘h, i, j, k, l, a’.
Thus, for the ring topology each receiver observes three parts of channel information- the
part corresponding to the common channel with the previous receiver, the part corresponding
99
MIMO Interference Channels
0 2 4 6 8 10 12 14 16 18 202
4
6
8
10
12
14
16
18
20
Rank of N (Φ)
(N = 10, K = 2)
Ran
k(r
1+r2+r3)
Overlapping Dimensions(r1 = M1)
M = 2M = 4M = 6M = 8M = 10M = 12
Figure 5.6: Interference dimension for different M
100
MIMO Interference Channels
Rx1
Rx3
Rx2
a b c d
d e
e f g h
i j k l l a h i
Figure 5.7: An example of ring topology for antennas overlapping. Each antenna is indexedwith the alphabets. ‘a,b,c,d’ are the antennas in RX1, ‘e,f,g,h’ are the antennas in RX2, ‘i,j,k,l’are the antennas in RX3. All receivers share different antennas with the adjacent receivers.The antenna elements in the dashed box are the antennas shared between two receivers.
101
MIMO Interference Channels
to the independent channel and the part corresponding to the common channel with the next
receiver. The signal received by any receiver i, transmitted from all the transmitters j, that
observes overlap of antennas with the adjacent receivers i+ 1 and i− 1 is given by
yi =
H(i)i(i−1)
H(i)ii
H(i)i(i+1)
Visi +K∑
j=1,j 6=i
H(j)i(i−1)
H(j)ii
H(j)i(i+1)
Vjsj + ni, (5.31)
where H(j)i(i−1) ∈ C2Nc×M is the common channel between the receivers i and (i−1) observed
at receiver i corresponding to the transmitter j, H(j)ii ∈ C(N−Nc)×M is the independent channel
observed at receiver i corresponding to transmitter j, and H(j)i(i+1) ∈ C2Nc×M is the common
channel between the receivers i and (i + 1) observed at receiver i corresponding to the
transmitter j; and ni ∈ CNk×1 is the additive white Gaussian noise (AWGN) with zero mean
and unit variance. Nc is the number of overlapped antennas per user between the adjacent
receivers and Nk = N + 2Nc; sj ∈ Cd×1 are the d dimensional data streams transmitted from
any transmitter j and Vj ∈ CM×d are the precoding matrix corresponding to transmitter j.
Now, the channel from any transmitter j to all the receivers is Hj ∈ CKN×M and each
channel is selected from Hj by the selection matrices Si ∈ R(N+2Nc)×M corresponding to
receiver i. Thus SiHjVj is the sub-matrix of HjVj. We use the same MAC-BC duality
approach as described before to determine all Uis and the corresponding Φ, only differing
in the dimensions. The aim is again to obtain optimum Vj which is contained in N (Φ)
from (5.22). Hence, we obtain the optimal dimensions of N (Φ) for ring topology next.
Optimal Dimensions of N (Φ) for Ring Topology
Theorem 5.2. For K = 3 user IC with ring topological antennas overlap, consider any matrix
Z = HjVj ∈ CKN×M due to any transmitter j, the selection matrices Si ∈ R(N+2Nc)×KN
observed at any receiver i and the sub-matrices Zi = SiHjVj ∈ C(N+2Nc)×M , where N is the
number of receive antennas, Nc is the number of overlap antennas per user between adjacent
receivers and M is the number of transmit antennas; then the alignment matrix formed by
the receivers due to any transmitter j Φj is defined as Φj = SHj PZjSj +∑K
i=1,i 6=j SHi (P⊥Zi)Si
and the rank of null space of Φj is given by
rank(N (Φj)) =
N +M − (K + 1)Nc, 0 ≤ Nc ≤ M
(K−1)
N − (K − 1)Nc,M
(K−1)≤ Nc ≤ N
(K−1)
0, Nc ≥ N(K−1)
. (5.32)
102
MIMO Interference Channels
Proof. The proof is provided in appendix B.
Intuitively, we observe that the N (Φj) is the possible space created by the interfered
receivers where the observed interference is aligned, plus the available free dimensions in
the intended receiver. If d = M streams of data is transmitted from any transmitter Tx1,
then the interference occupies M dimensions in Rx2 and M dimensions in Rx3 and there are
N −M free dimensions in Rx1. Hence, the dimension of N (Φj) is N −M +M +M = N +M .
When the antennas overlap, the overlap dimension belongs to the null space. If there is
overlap of 1 antenna per user, then two interference streams from each user belong to that
space, thus reducing the dimension of null space by (K − 1)1 = 2. Also, due to ring topology
2 ∗Nc = 2× 1 = 2 free dimensions at the desired receiver overlap with other receivers and
available free dimensions is N −M − 2Nc. Hence, for Nc overlap, the dimension of the null
space is reduced by (K + 1)Nc = 4Nc for the case when Nc ≤ M2
as given by theorem 5.2.
For M2≤ Nc ≤ N
2, the two interfered receivers completely overlap all the interference
signal, thus contributing only M dimensions. The overlap, however varies the free dimensions
available in the desired receiver as N −M − 2Nc. Hence the rank of null space is M +N −M − 2Nc = N − 2Nc
The number of optimal overlapping rows is achieved when all the interference observed by
the interfered receivers are aligned, i.e, when only M dimensions are occupied by interference
in two receivers. In that case, the rank(N (Φj)) = N −M + M − 2Nc = N − 2Nc, where
N −M − 2Nc is due to the free dimensions in the first receiver and M is due to other two
receivers. Hence, the optimal number of overlapping rows Nc is achieved when
N − 2Nc = N +M − (K + 1)Nc ⇒ Nc =M
K − 1. (5.33)
The rank of the N (Φj) for different number of transmit antennas and N = 10 receive
antennas with varying overlapping dimensions is plotted in Figure 5.8 for three users that
observe common channels over different antennas (ring topology). These results are plotted
for the Φj obtained from the designed post processing matrices Uis. We also consider that
each transmitter transmits d = M streams of data and the number of antennas in the receiver
is always greater than the number of transmit antennas.
The plot in Figure 5.8 clearly shows the two region, the optimal overlapping rows is given
by the number of overlapping rows corresponding to that point where the two regions are
separated. This is also shown in the graph by drawing a dotted line from the the point where
the two regions are separated in the graph. For example, the number of optimal overlapping
rows when M = 3 is 1, which is bM2c as obtained in (5.33). This makes sense because it
103
MIMO Interference Channels
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
2
4
6
8
10
12
14
16
18
Rank of N (Φ) vs Overlap for Ring topology
(N = 10, K = 3)
Overlapping Rows
Ran
kof
N(Φ
)
M = d = 3M = d = 4M = d = 5M = d = 6M = d = 7M = d = 8
Figure 5.8: Rank of N (Φj) for different M , when different antennas overlap between theadjacent receivers. The null space is decreasing as the overlapping rows increase. The graphshows two regions and the point where the graph changes as shown by the dashed line givesthe optimal number of overlapping rows.
104
MIMO Interference Channels
requires at least one overlapping rows between the users that receive interference in order to
create the optimal interference overlap space for three independent interference streams. If
two rows per user overlap, it uses four channels to align three independent streams, which
does not improve more than using one rows per user to overlap.
5.2.4 Star Topology
System Model
When the overlapping of the group of same antennas is observed in all the receivers, we call
such overlapping the ‘star topology’. An example scenario for the star topology is shown in
Figure 5.9, where all the antennas in each receiver are indexed with certain alphabets and all
the receivers have the same overlapping antennas ‘a, e, i’. Thus, RX1 observes ‘a, b, c, d, e,
i’, RX2 observes ‘a, e, f, g, h, i’ and RX3 observes ‘a, e, i, j, k, l’ antennas.
The concept is similar to the ‘ring topology’ except that the overlapping antennas are
same in all receivers. This approach of antennas overlapping may have a lot of practical
applications in future networks, where both wired and wireless system serve the same user
at the same time. All the common channel from the wired server includes all the common
control information and all other private channels from the wireless server includes the private
data, which allows better system performance.
The signal received by any receiver i from all the transmitters j = {1, 2, · · · , K} for the
star topology is expressed as :
yi =
[H
(p)ii
H(c)i
]Visi +
K∑j=1,j 6=i
[H
(p)ij
H(c)j
]Vjsj + ni, (5.34)
where H(p)ij ∈ C(N−Nc/K)×M is the private channel states observed at receiver i from the
transmitter j and H(c)j is the common channel states at all receivers due to transmitter j.
Nc/K is the common rows per receiver, assuming that the overlapping is symmetric.
Optimal Dimensions of N (Φ) for Star Topology
For this channel setup with star topology antennas overlapping, the selection matrix Sis
are defined accordingly and then the alignment matrix Φ is obtained by estimating suitable
Uis as described before and then the optimal dimensions for N (Φ) is determined. The
rank(N (Φ)) for star topological antennas overlapping is given by the following theorem:
105
MIMO Interference Channels
Rx1
a b c d
Rx2
e f g h
Rx3
i j k l
a e i
a e i a e i
Figure 5.9: An example of star topology for antennas overlapping in three user MIMO IC.Each antenna is indexed with the alphabets. Same antennas overlap for all the receivers. Forexample, the antennas in the dashed box ‘a’,’e’,’i’ overlap in all three receivers.
106
MIMO Interference Channels
Theorem 5.3. For K = 3, consider any matrix Z = HjVj ∈ CKN×M with star topo-
logical setup having Nc overlapping antennas per user, such that selection matrices Si ∈C(N+(K−1)Nc)×KN and the sub-matrices Zi = SiHjVj ∈ C(N+(K−1)Nc)×M for which the align-
ment matrix due to any transmitter j is defined as Φj = SHj PZjSj +∑K
i=1,i 6=j SHi (P⊥Zi)Si,
then the rank of the null space of Φj is given by:
rank(N (Φj)) =
N + (K − 2)M − (K − 1)2Nc, 0 ≤ Nc ≤ M
K
N −Nc,MK≤ Nc ≤ N
0 Nc ≥ N
, (5.35)
Proof. The proof is provided in appendix C.
Similar to the ring topology, the null space of Φj in the star topology is also the available
possible interference space observed by the receivers which aim to align interference and
the space of free dimensions in the receiver which does not receive any interference. When
there is no overlap, the star topology and the ring topology behaves similarly and hence the
rank(N (Φj)) = N +M .
When the overlap is 0 ≤ Nc ≤ MK
, the total interference dimensions in the interfered users
is (K − 1)(M −KNc
)+KNc because of the K overlap unlike K − 1 overlap in ring topology,
while the total free dimensions is N −M −Nc. Hence,
rank(N (Φj)) = (K − 1)(M −KNc
)+KNc +N −M −Nc,
= (K − 2)M −K2Nc + 2KNc −Nc +N,
= (K − 2)M − (K2 − 2K + 1)Nc +N,
= N + (K − 2)M − (K − 1)2Nc.
When Nc ≥ MK
, all interfered receivers only contribute M dimensions and total free
dimensions in desired receiver is N −M −Nc. So
rank(N (Φj)) = N −M −Nc +M = N −Nc.
The rank of rank(N (Φj)) for star topology with 10 receive antennas and varying transmit
antennas such that each transmitter transmits d = M streams of data is plotted in the
Figure 5.10. The simulation is obtained by designing the suitable selection matrices Sis and
hence alignment matrix Φj. In the graph, we clearly see the two regions as given by the
two conditions and the number of optimal overlapping rows is given by the overlapping rows
corresponding to the point where the two regions are separated as indicated by the dashed
lines in the graph in Figure 5.10. We observe that the number of optimal overlapping rows is
107
MIMO Interference Channels
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
2
4
6
8
10
12
14
16
18
Rank of N (Φ) vs Overlap for Star topology
(N = 10, K = 3)
Overlapping Rows
Ran
kof
N(Φ
)
M = d = 3M = d = 4M = d = 5M = d = 6M = d = 7M = d = 8
Figure 5.10: Rank of N (Φ) for different transmit antennas varying with the number ofoverlapping antennas in the ring topological overlap structure. The graph shows two regionsand the point where the region changes shows the optimal overlapping rows as shown by thedashed line.
108
Conclusions
given by
N∗c = bMKc.
This makes sense because the overlap is formed by the K users unlike K − 1 users in ring
topology. We also observe that the graph does not go to zero at just Nc = 5 overlap unlike
the ring topology. This is because in ring topology there occurs 5 overlap from two users and
no free dimensions are available at the desired receiver. However, in the star topology there
are still 10− 5 = 5 free dimensions.
5.3 Conclusions
In this chapter, we introduced the concept of vertical alignment by forming a group amongst
the receivers, which can collectively align the interference onto a common overlap space.
These receivers can only form a group when there is enough correlation or overlap between
them. Thus, we assume that these receivers have common channels and independent channels.
This is a reasonable assumption for future communication networks in crowded trains, buses
and stadiums where we can obtain common channels from wired media and the independent
channels from the wireless media. We introduced the alignment matrices for different
topological antennas overlapping and determine the optimal overlapping condition for each of
the topology. This new idea of collective interference management can provide new directions
in the future research.
109
Conclusions
110
Chapter 6
DOF Analysis of Three User MIMO
IC via Receiver Chain Alignment
6.1 Introduction
Demand for higher data rates and the better quality of service in wireless systems continue to
increase, unconcerned with the fact that transmit power, bandwidth and design complexity
are limited and expensive resources. Multi-user communications in a single input single
output (SISO) setting and/or in a MIMO setting is believed to be an effective solution to
these exponentially increasing demands. However, multi-user communications are interference
limited due to multiple users transmitting and receiving at the same time with the same
frequency and majority of today’s research is focused on designing the effective interference
management and mitigation techniques, both academically [1, 38,64,86,87] and industrially
[88–90].
One of such interference management techniques is the IA, which received significant
interests amongst the researchers, because of the fact that this is proved to be the degrees
of freedom (DOF) optimal [1, 22]. DOF is an important metric to study the interference
problem as it is viewed as a first order capacity approximation at high signal to noise ratio
(SNR) [91,92].
The DOF of a variety of interfering networks with different channel assumptions are studied
under the prism of IA in a number of recent articles such as [2, 3, 27–29, 68, 69, 75] but an
effective and generalized IA algorithm for all networks and for all channel conditions still fail
and is an open problem.
The DOF of the three user MIMO-IC is of special interest to the researchers because it is
the smallest MIMO network where the effect of IA is distinctly visible in terms of spatial
extensions, which is achieved by proper precoding in the absence of infinite channel uses in
time or frequency [2]. Further, due to less number of unknown parameters this channel acts
as a gateway towards the generalization to large number of users.
111
Receiver Chain Alignment
The DOF of interference channels are well studied for almost all channel realizations
when the transmitters and receivers are equipped with same number of antennas, i.e, for
a symmetric system with M = N , where M is the number of transmit antennas and N
is the number of receive antennas. However, when the transmitters and the receivers are
equipped with different number of transmit and receive antennas, the generalized scheme for
the characterization of DOF for more than two users IC is not still well studied. Some of the
recent works by Wang et al. [4] proposed that the DOF of three user IC can be characterized
by the notion of subspace alignment chains (SACs) which identifies the extra dimensions
required for the optimal IA schemes.
From the last chapter, we learned that the optimal overlapping is always possible when
there is certain collective cooperative operation between any two receivers. Motivated by
this fact, in this chapter, we describe a new notion of cooperative receiver chain alignment
scheme to characterize the DOF of three user IC with different number of transmit and
receive antennas.
In this regard, we explain a simple method in terms of overlapping spaces observed by the
adjacent receivers as a chain diagram. Such overlapping chain diagrams among the receivers is
termed as receiver chain alignment (RCA) [93]. Note here that the DOF analysis of three-user
MIMO-IC with arbitrary number of transmit and receive antennas is still unsolved and by
considering the symmetry, we reduce the number of parameters for ease of analysis. Unlike
the SAC described in [4], RCA is achieved by the knowledge of optimal overlap dimensions
as obtained from the null space of Φ observed by all the receivers as defined in the previous
chapter. In the next section, we describe the RCA scheme in detail.
6.2 Receiver Chain Alignment
Receiver chain alignment is a scheme at the receiver side, where the adjacent receivers form
an optimal overlap space to align the observed interference. The optimal space is allocated
as long as the spatial dimensions available in each receiver are enough.
In order to better understand the RCA, consider an example with M = N = 4 and K = 3,
where the adjacent receivers RX2, RX3; RX3, RX1 and RX1, RX2 optimally allocate an
overlapping space for the interference represented in the form of chain as shown in Figure 6.1.
Here, each overlap space is of dimension M , with the contribution of M2
dimensions from
each receiver. We observed in chapter 5 that this is the optimal number of overlapping rows
when two receivers collectively align the interference. The total overlap dimensions between
112
Receiver Chain Alignment
any two adjacent receivers are denoted by the dotted lines and the remaining free dimensions
are denoted by the solid lines in the Venn-diagram like structure as shown in Figure 6.1.
Let us assume that each optimal overlap is of length 1. Since there are no more free
dimensions after three optimal overlaps for M = N = 4, the total length of the chain (L) is
3. We also observe that for any M = N = p(even), L is always K (3 here) as shown in the
lower part of the Figure 6.1.
M = N = 4
M = N = p (even)
RX2 RX3 RX3 RX1 RX1 RX2
RX2 RX3 RX3 RX1 RX1 RX2
p2
p2
p2
p pp
−Free dimensions −Overlapped dimensions
Figure 6.1: RCA for M = N = 4 and M = N = p (general case) in 3-user IC with optimaloverlapping.
When N > M , we observe that there are still N −M free dimensions per user after the
chain length K. Hence, the total chain length is
L = K +KN −MM
. (6.1)
The fraction N−MM
comes from the fact that the overlap space of dimension M is assumed
to be of length 1, which means the N −M dimensions has length N−MM
. The RCA for
N = 5,M = 4 is depicted in Figure 6.2 and this is same for all M and N with N −M = 1
and K ≤M .
113
DoF Analysis and Achievability
M = 4, N = 5
RX2 RX3 RX3 RX1 RX1 RX2
RX2RX3RX1
−Free dimensions −Overlapped dimensions
Chain I -
Chain II-
Figure 6.2: RCA for M = 4, N = 5 with optimal overlapping.
6.3 DoF Analysis and Achievability
Theorem 6.1. For symmetric K = 3 user IC with M transmit and N receive antennas
(N ≥M , M is even), the achievable DOF per user when optimal (M) overlapping rows exist
between the adjacent receivers, is given by
d =
2M3, ∀ 1
2< N−M
M≤ 1
N1+(1+min(N−M
M,1)), otherwise,
(6.2)
where 1 + N−MM
is the length of RCA normalized per user.
Proof. Clearly, we observe that the achievable DOF is dependent on N−MM
, which constitutes
the length of RCA. Thus, we prove the theorem by constructing RCA for the following cases,
6.3.1 Case 1 : 0 ≤ N−MM ≤ 1
2
Assume that d0 is the spatially normalized achievable DOF, i.e., the DOF per transmit
antenna, then the total desired streams transmitted by K users with M antennas per user is
K × (Md0). Since the chain length is L, there are also L overlapped interference space, each
of dimension M that contribute L× (Md0) interference streams. In order to decode all the
desired streams independent of the interference streams, the total (desired + interference)
114
DoF Analysis and Achievability
streams can not exceed the total number of antennas in all the receivers. Hence,
(K + L)Md0 ≤ KN ⇒Md0 ≤KN
K + L. (6.3)
The RCA for N = M was discussed in the previous section and depicted in Figure 6.1,
where L = K. The achievable DOF per user in that case is
d = Md0 ≤KN
2K=N
2. (6.4)
The RCA for N−MM
= 14
is depicted in Figure 6.2. The total length of the chain in that
case is given by (6.1) and also holds true for all values of M and N , when N−MM≤ 1
2. Hence
the achievable DOF per user is
d = Md0 ≤KN
2K +K N−MM
=NM
N +M, (6.5)
In fact (6.4) is obtained from (6.5) with M = N . We observe that similar result is observed
in [3] and [4] for this case.
6.3.2 Case 2 : 12 <
N−MM ≤ 1
The RCA for one of the cases that satisfies 12< N−M
M≤ 1 is depicted in Figure 6.3, where
M = 4 and N = 7. It is observed that after the first K chain lengths, each receiver has
N −M free dimensions and one more overlap of M dimensions is still possible. This means
that K + 1 complete overlaps are possible. But, after K + 1 overlaps, K − 2 receivers (here
RX1) has N −M free dimensions, while K − 1 receivers (here RX2 and RX3) have N − 3M2
free dimension each.
Since there are enough available free dimensions, the next chain (RX3, RX1) is still possible.
However, in order to create the optimum overlap space, RX3 and RX1 both require equal free
dimensions (M2
each). Since RX1 has (N −M) dimensions, and RX3 is in short of 2M −Ndimensions, RX3 borrows the required dimensions from the RX1 to form the optimum chain
(RX3, RX1). This borrowing phenomenon acts like adding one virtual antenna at the RX3
and the total antennas at all receivers is KN + (2M −N) = (K − 1)N + 2M instead of KN .
The first K + 1 chain length always exist and still there are (K − 1)(N − 3M2
) + (K −2)(N −M) = 3N − 4M free dimensions for K = 3. After borrowing of required dimensions,
another chain of dimension M is possible, leaving 3N − 4M −M = 3N − 5M free dimensions.
Hence, total chain length is
L1 = K + 2 +3N − 5M
M. (6.6)
Let us consider another example with M = 6 and N = 10 as shown in Figure 6.4. Here,
115
DoF Analysis and Achievability
M = 4, N = 7
RX2 RX3 RX3 RX1 RX1 RX2
RX2RX3RX3RX1RX2
−Free dimensions −Overlapped dimensions
Chain I -
Chain II-
Figure 6.3: RCA for M = 4, N = 7 with optimal overlapping.
RX3 is in short of two more dimensions in order to create the optimum overlap space of
dimension M and to complete the chain (RX3, RX1). But, RX3 can borrow one extra
dimension from RX1 and the other one from RX2, equivalently adding two virtual antennas.
In both the examples, 2M − N virtual antennas are added and chain length is given
by (6.6). Hence,
(K + L1)Md0 ≤ (K − 1)N + 2M, (6.7)
d = Md0 ≤(K − 1)N + 2M
K +K + 2 + 3N−5MM
=2M
3,∀K = 3. (6.8)
6.3.3 Case 3 : 1 < N−MM ≤ 2
The RCA for one of the example scenarios that satisfies 1 < N−MM
< 2 is shown in Figure 6.5,
for M = 2 and N = 5. In this case, we clearly observe that there are 2K complete overlaps
and still (N −M)−M = N − 2M free dimensions per user. Hence, the total length of the
chain is
L2 = 2K +K(N − 2M)
M, (6.9)
116
DoF Analysis and Achievability
M = 6, N = 10
RX2 RX3 RX3 RX1 RX1 RX2
RX2RX3RX3RX1RX2
−Free dimensions −Overlapped dimensions
Chain I -
Chain II-
Figure 6.4: RCA for M=6 and N=10
and the total data streams (desired + interference) is given by
Ts = KMd0 + (2K +K(N − 2M)
M)Md0. (6.10)
However, for the K user IC, each receiver only observes the maximum of (K − 1)Md0
interference streams and Md0 desired streams. For K = 3, the maximum possible total
(desired + interference) streams is 9Md0 and not Ts as given by (6.10). Hence,
9Md0 ≤ 3N ⇒ d = Md0 ≤N
3. (6.11)
Hence for case 1 and case 3, d is expressed in general as :
d ≤ N
1 + (1 + min(N−MM
, 1)). (6.12)
One of the interesting features of RCA is the pictorial representation of the ratio R = N−MM
.
When R = 0, the chain length (L) is K and the RCA terminates in chain I, while for
0 < R ≤ 1, we have K < L ≤ 2K and the RCA terminates anywhere in the chain II.
Similarly, for 1 < R ≤ 2, we have 2K < L ≤ 3K as shown in Figure 6.5. Also, each
intermediate value of L, e.g., L = K + α, ∀α ≤ K represent all the intermediate values of R,
e.g., β : 0 < β ≤ 1.
For all the cases when N−MM≥ 2, no IA is required and the achievable DOF is always M
117
DoF Analysis and Achievability
M = 2, N = 5
RX2 RX3 RX3 RX1 RX1 RX2
RX2RX3RX3RX1RX1RX2
RX2 RX3 RX1
−Free dimensions −Overlapped dimensions
Chain I -
Chain II -
Chain III-
Figure 6.5: RCA for M = 2, N = 5 in 3-user IC for optimal overlapping.
118
Information theoretic outer bound for DOF
(N > M). No RCA is created.
Although the RCA evaluates the DOF of IC with N ≥M and M is even, the achievable
DOF is completely symmetric over the ICs and the DOF evaluated for the case N > M holds
true also when N and M are interchanged. The DOF is also the linear function of the ratio
of M and N , and hence the DOF for odd values of M are easily determined by evaluating
the DOF for 2M and 2N and dividing the obtained value by 2. For example, the achievable
DOF for M = 3, N = 4 is obtained by evaluating the achievable DOF for M = 6, N = 8 and
dividing the result by 2. In this regard, RCA is useful to evaluate the DOF of three user
MIMO IC for any values of M and N .
The variation of DOF per user for different transmit antennas (M) with respect to different
receive antennas (N) antennas is plotted in the Figure 6.6, where the variation of DOF clearly
shows specific patterns in certain region, mainly four; the first linear region is observed for
N ≤ M/3, the second non-linear region exists for M/3 < N ≤ 2M , the third linear region
exists for 2M < N ≤ 3M and the fourth constant region for N ≥ 3M . The second non-linear
is especially important because it is the region where MIMO processing and IA is both
possible. This is the region where improvement is still possible and the problem is still open.
6.4 Information theoretic outer bound for DOF
Let Wk be the independent transmitted messages from TXk to RXk and Rk(ρ) be the
achievable rate as a function of SNR, denoted by ρ, corresponding to user k, ∀k = {1, 2, 3}.The rate is achieved by choosing appropriately large n, where n can be considered as the
number of channel uses [1, 14]. If dk is the DOF corresponding to user k then
dk = limρ→∞
Rk(ρ)
log(ρ). (6.13)
Let Rsum =∑3
k=1Rk and Yk = (YkD, YkI) is the signal received at receiver k, where YkD is
the desired signal and YkI is the interference signal. Also, let h(X) be the differential entropy,
I(X, Y ) be the differential mutual information, h(X|Y ) be the differential conditional entropy,
I(X;Y |Z) represents the differential conditional mutual information and W = {W1,W2,W3}
119
Information theoretic outer bound for DOF
0 5 10 15 20 25 300
1
2
3
4
5
6
7
8
9
10
DoF of three user (M ×N) MIMO IC.
Receive Antennas(N)
Degrees
ofFreedom
DoF RCA
DoF JafarDoF Wang
M = 4
M = 6
M = 10
M = 15
Figure 6.6: DOF plot for different transmit and receive antennas under different DOFachievable schemes. The ‘*’ line represents the DOF achieved using the proposed RCAscheme, the ‘o’ line represents the DOF achieved with the scheme proposed by Jafar et. al [3]and the ‘square’ line represents the DOF achieved with SAC scheme proposed by Wang et.al [4].
120
Information theoretic outer bound for DOF
[14] then from definitions, we have
nRsum = h(W ) = I(W ;Y nk ) + h(W |Y n
k ), (6.14)
≤ I(W ;Y nk ) + nεn, (6.15)
= I(W ;Y nkD, Y
nkI) + nεn, (6.16)
= I(W ;Y nkD) + I(W ;Y n
kI |Y nkD) + nεn, (6.17)
≤ nN log ρ+ n o(log ρ) + I(W ;Y nkI |Y n
kD) + nεn. (6.18)
Equation (6.15) follows from the Fano’s inequality [14], while (6.17) follows from the chain
rule of mutual information [14] and (6.18) follows from the fact that the DOF can not be
more than the number of antennas at the receiver (antenna bound) and o(log ρ) approaches
zero as ρ approaches infinity. From definitions,
I(W ;Y nkI |Y n
kD) = h(Y nkI |Y n
kD)− h(Y nkI |Y n
kD,W ),
≤ h(Y nkI |Y n
kD) + o(n). (6.19)
Equation (6.19) follows from the fact that given the input messages and the received signal
at particular receiver, the interference signal is decoded with small noise distortion [ [4]
Lemma 3 ]. Hence from (6.18) and (6.19),
nRsum ≤ nN log ρ+ h(Y nkI |Y n
kD) + n o(log ρ) + o(n). (6.20)
Consider the example scheme where M = 2, N = 3, with optimum overlap as shown in the
Figure 6.7, where the interference and desired signals observed at the receiver k after the
first overlap are: YkI = (Y(k+1)a, Y(k−1)c), YkD = (Yka, Ykc), ∀k = {1, 2, 3} and k is cyclic.
Hence, we can further express (6.20) as
nRsum ≤ nN log ρ+ h(Y n(k+1)a, Y
n(k−1)c|Y n
kc) + n o(log ρ) + o(n), (6.21)
and also as
nRsum ≤ nN log ρ+ h(Y n(k+1)a, Y
n(k−1)c|Y n
ka) + n o(log ρ) + o(n). (6.22)
Using the chain rule for conditional differential entropy and expressing Yka and Ykc as
a function of X(k+1)a and X(k−1)c respectively as shown in Figure 6.7, where Xka, Xkb,
X(k+1)a, X(k+1)b, X(k−1)a, X(k−1)b are the signals transmitted from all the transmitters and (6.21)
is further simplified to,
nRsum ≤ nN log ρ+ h(Xn(k−1)a|Xn
(k−1)b) + h(Xn(k+1)b|Xn
(k−1)b, Xn(k−1)a) + n o(log ρ) + o(n),
= nN log ρ+ nR(k−1) − h(Xn(k−1)b) + n o(log ρ) + o(n), (6.23)
where (6.23) follows from the fact that the achievable rate can be expressed in terms of
the joint entropy nRk = h(Xka, Xkb) and the corresponding chain rule of joint differential
121
Information theoretic outer bound for DOF
.
.
.
.
.
.
.
.
.
.
.
.
X1a
X1b
X2a
X2b
X3a
X3b
TX1
TX2
TX3
RX1
RX2
RX3
Y1a = f(X2a)
Y1b
Y1c = f(X3b)
Y2a = f(X3a)
Y2b
Y2c = f(X1b)
Y3a = f(X1a)
Y3b
Y3c = f(X2b)
Figure 6.7: Overlap interference and transmission in K = 3 user IC with M = 2 and N = 3.
122
Information theoretic outer bound for DOF
entropy. Similarly, (6.22) is simplified to
nRsum ≤ nN log ρ+ nR(k+1) − h(Xn(k+1)a) + n o(log ρ) + o(n). (6.24)
Adding (6.23) and (6.24) and summing over all k = {1, 2, 3},3∑
k=1
2nRsum ≤3∑
k=1
(2nN log ρ+ nR(k−1) + nR(k+1) − h(Xn(k−1)b)− h(Xn
(k+1)a) + n o(log ρ) + o(n)).
(6.25)
Further simplifying, we obtain
4nR ≤ 2nN log ρ− 1
3
[h(Xn
a ) + h(Xnb )]
+ n o(log ρ) + o(n), (6.26)
where h(Xna ) =
∑3k=1 h(Xn
ka), h(Xnb ) =
∑3k=1 h(Xn
kb) and R = Rsum
3, the rate per user.
Similarly, in the next overlap, (chain II in RCA), there is only one free antenna per receiver,
which satisfies
nRk ≤ h(Xnka) + h(Xn
kb). (6.27)
Summing over all k = {1, 2, 3}, (6.27) is further expressed as
nRsum ≤ h(Xna ) + h(Xn
b )⇒ nR ≤ 1
3
[h(Xn
a ) + h(Xnb )]. (6.28)
Now, adding (6.26) and (6.28),
5nR ≤ 2nN log ρ+ n o(log ρ) + o(n), (6.29)
⇒ 5R
log ρ≤ 2N +
o(log ρ)
log ρ+ o(n). (6.30)
As n→∞ and ρ→∞, we getR
log ρ≤ 2N
5=
6
5, (6.31)
which is the per user achievable DOF for M = 2, N = 3 and K = 3.
For general case with M transmit antennas and N receive antennas, with 0 < N−MM≤ 1
2, we
follow the similar approach but there are M inputs instead of two and total of 3M conditions
in the first overlap (chain I in RCA). Adding all the conditions and simplifying, we obtain:
2MnR ≤MNn log ρ− 1
3
[h(Xn
a ) + h(Xnb ) + · · ·+ h(Xn
m)]
+ n o(log ρ) + o(n). (6.32)
Also, for next overlap (chain II in RCA) there are N −M free antennas. Thus,
n(N −M)R ≤ 1
3
[h(Xn
a ) + h(Xnb ) + · · ·+ h(Xn
m)]. (6.33)
Adding (6.32) and (6.33), we get:
(M +N)nR ≤MNn log ρ+ n o(log ρ) + o(n), (6.34)
⇒ (M +N)R
log ρ≤MN +
o(log ρ)
log ρ+ o(n). (6.35)
123
Information theoretic outer bound for DOF
As n→∞ and ρ→∞,R
log ρ≤ MN
M +N= d. (6.36)
Thus, we show that the DOF is information theoretically achievable for the the case when
0 ≤ N−MM≤ 1
2.
When N ≥ 3M , no alignment is required and M DOF is easily achievable. Let us consider
the case when 2M < N ≤ 3M . As given by RCA, the DOF per user in this case is outer
bounded by N3
. We validate this result using information theoretic approach.
Consider an example scenario with M = 2 and N = 5, that satisfies 2M < N ≤ 3M . Our
approach is based on optimum overlapping between the receivers as depicted in the figure 6.8,
where we observe that two layers of alignment are possible unlike the previous case.
By using the same information theoretic approaches and same notational definitions
explained for the previous case, we obtain the expression similar to (6.26) for the case of
M = 2 and N = 3 corresponding to the chain I of RCA. This is true because each transmitter
transmits M = 2 streams of data, which is same as in the previous case. Thus,
4nR ≤ 2nN log ρ− 1
3
[h(Xn
a ) + h(Xnb )]
+ n o(log ρ) + o(n). (6.37)
Each receiver still has two antennas for optimal overlapping the interfering signals, hence
they perform second layer of overlapping as shown in Figure 6.8. The second layer contains
two antennas more than the first layer, but still behaves as if there are only N2
antennas
due to complete overlapping of the first layer over second layer. The signals observed at
interfering antennas of RX1 is Y1I = (Y3d, Y3e, Y2a, Y2b), where Y3e is overlapped with Y3d
and Y2a is overlapped with Y2b; similarly, the signals observed at desired antennas of RX1
is Y1D = (Y1a, Y1b, Y1d, Y1e), where Y1a is overlapped with Y1b and Y1e is overlapped with Y1d.
Similar analysis is true for the signals observed at all the other receivers.
The conditional differential entropy corresponding to the signal at interfering antennas
given the signals at desired antennas for n channel uses are expressed as
h(Y n1I |Y n
1D) = h(Y n3d, Y
n2b|Y n
1b) ≤ h(Xn2a, X
n2b, X
n3a, X
n3b|Xn
2a, Xn2b),
= h(Xn3a, X
n3b|Xn
2a, Xn2b). (6.38)
Since (Xn3a, X
n3b) and (Xn
2a, Xn2b) are independent, we can further express (6.38) as
h(Y n1I |Y n
1D) ≤ h(Xn3a, X
n3b) = nR3. (6.39)
Similar analysis for the conditional entropy is valid for the other received signals at desired
antennas, which gives h(Y n1I |Y n
1D) ≤ nR2. Also, we perform similar analyses on the signals
received at the interfering and desired antennas corresponding to the RX2 and RX3, for which
124
Information theoretic outer bound for DOF
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
X1a
X1b
X2a
X2b
X3a
X3b
TX1
TX2
TX3
RX1
RX2
RX3
Y1a = f(X2a)
Y1b = f(X2b)
Y1c
Y1d = f(X3a)
Y1e = f(X3b)
Y2a = f(X3a)
Y2b = f(X3b)
Y2c
Y2d = f(X1a)
Y2e = f(X1b)
Y3a = f(X1a)
Y3b = f(X1b)
Y3c
Y3d = f(X2a)
Y3e = f(X2b)
Figure 6.8: An example analysis of alignment due to antenna sharing.
125
Information theoretic outer bound for DOF
we obtain the conditional differential entropies h(Y n2I |Y n
2D) ≤ nR1, h(Y n2I |Y n
2D) ≤ nR3 and
h(Y n3I |Y n
3D) ≤ nR2, h(Y n3I |Y n
3D) ≤ nR1 respectively. For all the conditional entropies, we can
obtain the corresponding expression for Rsum as given by expressions (6.18) and (6.19) and
sum them all to obtain:
6nRsum ≤ 6nN
2log ρ+ 2nRsum + n o(log ρ) + o(n),
4nRsum ≤ 3nN log ρ+ n o(log ρ) + o(n),
4nR ≤ nN log ρ+ n o(log ρ) + o(n). (6.40)
imilarly, for the third chain, there is still one antenna remaining in each receiver, and
similar to the case of previous example, we obtain the expression
nR ≤ 1
3
[h(Xn
a ) + h(Xnb )]. (6.41)
Adding up (6.37), (6.40) and (6.41),
9nR ≤ 3nN log ρ+ n o(log ρ) + o(n), (6.42)
9R
log ρ≤ 3N +
o(log ρ)
log ρ+ o(n). (6.43)
As n→∞ and ρ→∞,R
log ρ≤ 3N
9=N
3=
5
3= D0, (6.44)
which is the per user DoF when M = 2 and N = 3 as obtained from the RCA.
For the general case with M transmit and 2M < N ≤ 3M receive antennas, the general
expression for chain I is given by (6.32). Since there are M more transmit antennas in chain
II, which is effectively halved due to overlapping,
3MnRsum ≤3MN
2log ρ+MnRsum + n o(log ρ) + o(n),
2MnR ≤ MN
2log ρ+ n o(log ρ) + o(n). (6.45)
For the chain III, with M transmit antennas, there are M2
antennas remaining at each
receiver which is exactly enough for optimal overlap and the general expression is given by
Mn
2Rsum ≤ h(Xn
a ) + h(Xnb ) + · · ·+ h(Xn
m),
Mn
2R ≤ 1
3[h(Xn
a ) + h(Xnb ) + · · ·+ h(Xn
m)], (6.46)
Adding (6.32), (6.45) and (6.46), we obtain
(4Mn+Mn
2)R ≤ 3MNn
2+ n o(log ρ) + o(n). (6.47)
126
Achievability of the DOF
As n→∞ and ρ→∞,R
log ρ≤ 3MN
2(9M2
)=N
3= D0. (6.48)
6.5 Achievability of the DOF
The precoding matrices are designed which align all the interferences in the common subspace
between any two interfering receivers. For the random post processing matrices designed at
the receiver, there are always two parts, one due to the common channel states and the other
due to the independent channel states. Thus at any receiver k, the post-processing matrix
Uk is defined as:
Uk =
[Uk1
Uk2
]∈ C(mk1+mk2)×d, (6.49)
where for the symmetric case with optimal overlapping,
mk2 = M and mk1 = N − M
2.
Hence, we consider M is even to ensure that N − M2
is an integer.
We know that for optimal overlapping, M is the number of overlapping antennas between
any two adjacent receivers where the interfering signals are aligned. Consider the case when
the interference signal from TX1 as observed by RX2 and Rx3 is aligned in the common
subspace formed by RX2 and Rx3, which is achieved by designing proper post-processing
matrix. In fact, the post processing matrix Uk at any receiver k has two parts, the first
part zero-forces the interference observed and the second part aligns the interference in the
common subspace formed by two adjacent receivers.
The alignment conditions for RX2 and RX3 in order to align the interference observed from
TX1 are then given by
(I) Aligning the observed interference over the subspace created from common antenna
elements:
UH22H
(c)21 V1 = UH
31H(c)21 V1, (6.50a)
⇒(UH22 −UH
31)H(c)21 V1 = 0, (6.50b)
⇒V1 ∈ N((UH
22 −UH31)H
(c)21
), (6.50c)
(II) Zero forcing the interference observed over the subspace created from independent
127
Achievability of the DOF
antenna elements in RX2:
UH21H
(i)21V1 = 0, (6.51a)
⇒V1 ∈ N(UH
21H(i)21
), (6.51b)
(III) Zero forcing the interference observed over the subspace created from independent
antenna elements in RX3:
UH32H
(i)31V1 = 0, (6.52a)
⇒V1 ∈ N(UH
32H(i)31
). (6.52b)
From these three conditions, we can state that
V1 ∈ {N((UH
22 −UH31)H
(c)21
)∩N
(UH
21H(i)21
)∩N
(UH
32H(i)31
)}. (6.53)
Furthermore, utilizing the property of null space that N (A) = N (AHA) for any given
matrix A [81], we can express (6.53) as:
V1 ∈ {N(CH
1 C1
)∩N
(CH
2 C2
)∩N
(CH
3 C3
)}, (6.54)
where C1 = (UH22 −UH
31)H(c)21 , C2 = UH
21H(i)21 and C3 = UH
32H(i)31 . Then,
CH1 C1 = H
(c)21
H(U22U
H22 + U31U
H31 −U22U
H31 −U31U
H22
)H
(c)21 ,
= H(c)21
H(U22U
H22 + U31U
H31 −U22U
H31 − (U22U
H31)H
)H
(c)21 , (6.55a)
CH2 C2 = H
(i)21
HU21U
H21H
(i)21 , (6.55b)
CH3 C3 = H
(i)31
HU32U
H32H
(i)31 . (6.55c)
Now, realizing the fact that N (A + B) = N (A) ∩ N (B for any positive semidefinite
matrices A and B [81, 84], we can express (6.54) as:
V1 ∈ {N(CH
1 C1 + CH2 C2 + CH
3 C3
)}. (6.56)
Since N (AH) = R(A), we have N (AH) ∩N (A) = {φ}, an empty set. Hence,
N (CH1 C1) = N
(H
(c)21
HU22U
H22H
(c)21
)∩N
(H
(c)21
HU31U
H31H
(c)21
),
= N(H
(c)21
H(U22U
H22 + U31U
H31)H
(c)21
). (6.57)
Hence from equations (6.55b), (6.55c), (6.56) and (6.57), the alignment is achieved when
V1 ∈ N(H
(c)21
H(U22U
H22 + U31U
H31
)H
(c)21 + H
(i)21
HU21U
H21H
(i)21 + H
(i)31
HU32U
H32H
(i)31
). (6.58)
Since H(i)21 , H
(c)21 , H
(i)31 and H
(c)31 are obtained from the channel matrices H21 and H31, let us
128
Achievability of the DOF
assume that,
H(i)21 = T11H21, H
(c)21 = T12H21 (6.59)
H(i)31 = T22H31, H
(c)31 = T21H31, (6.60)
where T11,T12,T21 and T22 are the 0− 1 selection matrices of relevant dimensions, that
depends on the dimensions of H21,H31 and H(i)21 , H
(c)21 , H
(i)31 , H
(c)31 .
Further, let us consider the summation terms in (6.58) be denoted by Σ such that
Σ =H(c)21
HU22U
H22H
(c)21 + H
(c)21
HU31U
H31H
(c)21 + H
(i)21
HU21U
H21H
(i)21 + H
(i)31
HU32U
H32H
(i)31 , (6.61)
=(T12H21)HU22UH22T12H21 + (T21H31)HU31U
H31T21H31 + (T11H21)HU21U
H21T11H21
+ (T22H31)HU32UH32T22H31, (6.62)
=HH21(TH
12U22UH22T12 + TH11U21U
H21T11)H21 + HH
31(TH21U31U
H31T21 + TH
22U32UH32T22)H31,
(6.63)
=HH21
{[0 0
0 U22UH22
]+
[U21U
H21 0
0 0
]}H21 + HH
31
{[U31UH31 0
0 0
]+
[0 0
0 U32UH32
]}H31,
(6.64)
=HH21
{[U21UH21 0
0 U22UH22
]}H21 + HH
31
{[U31UH31 0
0 U32UH32
]}H31. (6.65)
Now realize that H21 and H31 are obtained as sub-matrices of some matrix H1 such that
H21 = S1H1 and H31 = S2H1, then Σ is expressed as:
Σ = HH1
(SH1
[U21U
H21 0
0 U22UH22
]S1 + SH2
[U31U
H31 0
0 U32UH32
]S2
)H1. (6.66)
Hence, for any V1 which has the same dimensions as the dimensions of the null space of Σ,
V1 ∈ N (Σ)⇒ ΣV1 = 0⇒ VH1 ΣV1 = 0. (6.67)
However, the N (Σ) may not exist and V1 can have only trivial solution. But when we
substitute the value of Σ, we can express (6.67) as:
VH1 HH
1 ΦH1V1 = 0⇒ H1V1 ∈ N (Φ), (6.68)
where
Φ = SH1
[U21U
H21 0
0 U22UH22
]S1 + SH2
[U31U
H31 0
0 U32UH32
]S2. (6.69)
As we discussed previously, Φ is the higher dimensional embedding of the lower dimensional
subspace and the N (Φ) always exist and the achievable DOF with optimal overlap is the
dimensions of the N (Φ).
129
Conclusions
Since
H1V1 ∈ N(SH1 U2U
H2 S1 + SH2 U3U
H3 S2
), (6.70)
we can determine V1 for M dimensional overlapping Rx2 and Rx3. Similarly, we can
determine V2 and V3 for the M dimensional overlapping Rx3, Rx1 and Rx1, Rx2. This shows
that the precoder always exist as long as the optimal overlap is possible. Hence the chain
stops when the optimal overlap is not possible. This proofs the achievability of the scheme.
6.6 Conclusions
The DOF characterization of three user MIMO IC with arbitrary M transit and N receive
antennas is achieved via a scheme based on creating an overlapping space by the adjacent
receivers, called receiver chain alignment. The DOF is expressed in terms of the length of
the alignment chain. We provide a proof based on information theory and also presented
an achievable scheme in order to create an overlap space and designing a precoder which
achieves the overlap.
130
Chapter 7
Conclusions and Future Works
7.1 Concluding Remarks
Interference management is one of the challenging issues in the future multi-user wireless
communication systems and interference alignment (IA) is the promising scheme to improve
the degrees of freedom (DOF) in multi-user systems. In this thesis, we incorporated the
IA technique in the multi-user interference channels (ICs) and the interference broadcast
channels (IBCs).
A novel IA achieving algorithm is presented in the first part to design the precoder and
the zero-forcing matrix for any K-user IC. We showed that by using the hybrid scheme
where the receivers cooperate and the transmitters transmit greedily, we can design precoders
and zero-forcing matrices that achieve better results than the existing ones in the literature.
This scheme minimizes the interference power cooperatively and maximizes the signal power
greedily while most of the existing literatures such as in [2] only minimizes interference power
or maximizes signal power.
Instantaneous channel state information at the transmitter (CSIT) is difficult to obtain in
a time varying wireless channels and this limits the performance of IA. Existing literature
suggests that no benefits of IA can be achieved in the absence of instantaneous CSIT. However,
we presented schemes that achieve DOF gain with the help of delayed CSIT. Our main
contribution was to present a space time based transmission scheme to achieve better than
1 total DOF in a two-cell multiple input single output (MISO) IBC. In fact, we showed
that the total achievable DOF converges to 85
as the number of users approaches infinity.
Interference management in multi-cell MISO IBC is not well studied in literature and our
results which showed that with proper transmission scheme and proper knowledge of past
channel states, inter cell interference (ICI) and inter user interference (IUI) can be mitigated
wisely to improve the DOF is a strong contribution.
We showed that receivers can collectively align the received interference from any transmitter
onto a common subspace when there is enough overlapping between received signal dimensions.
131
Discussion and Future Works
In this regard, we considered multiple input multiple output (MIMO) IBC and three user
MIMO IC to show that optimal overlapping can be achieved. We showed that the interference
signal is contained in the null space of the so called alignment matrix, denoted by Φ and
derived the expression to determine the dimension of these null spaces which is the dimensions
of the overlap space.
The receiver chain alignment (RCA) scheme is presented to determine the DOF of three
user MIMO IC with arbitrary number of transmit and receive antennas by forming the chain
of optimal overlapping space between any two receivers and we showed that the DOF can
be expressed in terms of the length of such alignment chain. We provided an information
theoretic outer bound for such schemes and also showed that such schemes are achievable.
Thus, overall this thesis is a contribution in the interference management and DOF analysis
of the multi-user interference channels and interference broadcast channels and we present a
number of directions for the future research.
7.2 Discussion and Future Works
Most of the interference management works in multi-user communication are open. Especially,
the works regarding the DOF in multi-user (more than two) in different network scenarios
are not well studied. Hence, we list here some of the works where we can further work in this
thesis:
More than two cell IBC
We observed that even with the two cells, it is a challenging task to manage both IUI and
ICI. However, we also observed that the users can cooperatively align the received ICI onto a
common subspace. With more than two cells, precoders can be properly designed so as to
align the ICI observed from all the cells and the users in each cell can cooperatively create a
subspace to place these interference. However, the problem is still challenging due to more
variables.
Rank Deficient MIMO Channels
The concept of alignment matrix can be useful to solve number of MIMO channels and one
of them could be a rank-deficient MIMO IC [94]. We can easily create an alignment matrix
in such case and determine the overlap space by obtaining the null space of such matrix.
132
Discussion and Future Works
Alignment Matrix With Error
Most of the analysis we performed in chapter 5 are by assuming that the alignment matrix
were known perfectly but this is not always true. So we can further study on the alignment
matrix with error and design better practical approaches to create the overlap space.
By some abuse of notation, let us represent that Φ by Φ and the actual Φ obtained
trace[(HjVj)
H(Φ−Φ∗
)(HjVj)
], (7.1)
= trace[(HjVj)
H( K∑i=1,i 6=j
SHi (P⊥i −P∗i⊥)Si
)(HjVj)
], (7.2)
Since P∗i⊥ is not known, we would like to obtain the difference between the observed and
the exact value, which is measured as an error given by
||E||2 =K∑
i=1,i 6=j
||SHi(P⊥i −P∗i
⊥)Si||2, (7.3)
≤K∑
i=1,i 6=j
||P⊥i −P∗i⊥||2, (7.4)
≤K∑
i=1,i 6=j
||P⊥i P∗i ||2, (7.5)
=K∑
i=1,i 6=j
||(IN −UiUHi )SiHjVj(SiHjVj)
†||2, (7.6)
≤K∑
i=1,i 6=j
||(IN −UiUHi )SiHjVj||2||(SiHjVj)
†||2, (7.7)
=K∑
i=1,i 6=j
||Ei||2||SiHjVj||2
κ(SiHjVj) ≤K∑
i=1,i 6=j
||Ei||2, (7.8)
where κ(SiHjVj) = ||SiHjVj|| ||(SiHjVj)†|| is the condition number for matrix SiHjVj,
||Ei||2 = ||(IN −UiUHi )SiHjVj||2 is the local projection error observed at any receiver i. The
inequality (7.5) follows from the difference and product relationship of any two projection
matrices given by Krein-Krasnoselskii-Milman equality [95] and stated as
||P−Q||2 = max{||P(I−Q)||2, ||Q(I−P)||2}, (7.9)
for any two projection matrices P and Q and (7.7) follows directly from (7.6) by the property
of 2-norm.
By designing Ui and Vj iteratively, we aim to minimize the alignment error as much as
possible. We observed that the alignment error is decreasing iteratively and it is minimum
133
Discussion and Future Works
when the overlap is more as shown in Figure 7.1.
0 10 20 30 40 50 60 70 80 90 100
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Alignment Error for Varying Overlap
(N = 10,M = 8)
No. of iterations
Align
mentError
Overlap=8Overlap=6Overlap=4Overlap=2Overlap=0
Figure 7.1: Alignment error for different total overlap dimensions due to two receivers. Theerror is decreasing as the overlap is increasing.
134
Appendices
135
Appendix A
Proof of Theorem 5.1
We consider all the symbols and terms as defined in theorem 5.1. Using the singular value
decomposition (SVD), we decompose Z12 as:
Z12 = Z21 = Q1
[Σ1 0
0 0
]RH
1 , (A.1)
and further Z11 and Z22 as
Z11 = Q2
[Γ1 Σ2 0
Γ1 0 0
][I 0
0 RH2
]RH
1 , (A.2)
Z22 = Q3
[Γ2 Σ3 0
Γ2 0 0
][I 0
0 RH3
]RH
1 . (A.3)
where Q1 ∈ Cm12×m12 ,Q2 ∈ Cm11×m11 ,Q3 ∈ Cm22×m22 , R1 ∈ CM×M ,R2 ∈ CM−r1×M−r1 and
R3 ∈ CM−r2×M−r2 are the the unitary matrices; Σ1 ∈ Cr1×r1 ,Σ2 ∈ Cr2×r2 and Σ3 ∈ Cr3×r3are the diagonal matrices. Also, Γ1 and Γ1 are the top r2 rows and bottom m11 − r2 rows of
QH2 (Z11R1)(:,1:r1) respectively and Γ2 and Γ2 are the top r3 rows and bottom m22 − r3 rows
of QH3 (Z22R1)(:,1:r1) respectively, and
r1 = rank{Z12}, (A.4)
r2 = rank{(Z11R1)(:,r1+1:M)}, (A.5)
r3 = rank{(Z22R1)(:,r1+1:M)}. (A.6)
For the further simplification, consider Z1 = QH21Z1, for
Q21 =
[Q2 0
0 Q1
]∈ C(m11+m12)×(m11+m12) (A.7)
is the unitary matrix and based on the decompositions (A.1), (A.2) and (A.3), we obtain
span{Z1
}= span
{QH
21Z1
}= span
{Y1
}, where
Y1 =
0 I
W1 0
I 0
0 0
[R1 0
0 I
]∈ C(m11+m12)×(r1+r2), (A.8)
137
Proof of Theorem 5.1
W1 = Γ1Σ−11 and R1 = (I + WH
1 W1)−12 . We are interested in the orthogonal complement of
span{Y1
}, the basis of which is given by
Y⊥1 =
0 0
I 0
−WH1 0
0 I
[D1 0
0 I
]∈ C(m11+m12)×(m11−r2+m12−r1), (A.9)
for D1 = (I + W1WH1 )−
12 .
Similarly, considering Z2 = QH13Z2 for the unitary matrix
Q13 =
[Q1 0
0 Q3
]∈ C(m12+m22)×(m12+m22). (A.10)
Also, based on the decompositions (A.1), (A.2) and (A.3) span{Z2
}= span
{QH
13Z2
}=
span{Y2
}, where
Y2 =
I 0
0 0
0 I
W2 0
[R2 0
0 I
]∈ C(m12+m22)×(r1+r3), (A.11)
W2 = Γ2Σ−11 and R2 = (I + WH
2 W2)−12 . The orthogonal complement of span
{Y2
}is given
by
Y⊥2 =
−WH
2 0
0 I
0 0
I 0
[D2 0
0 I
]∈ C(m12+m22)×(m22−r3+m12−r1), (A.12)
for D2 = (I + W2WH2 )−
12 .
By embedding the projection matrix due to Y⊥1 and the projection matrix due to Y⊥2 onto
m11 +m12 +m22 dimensional space and adding them , we obtain the alignment matrix Φ as
given by
Φ = ST1 P⊥Y1S1 + ST2 P⊥Y2
S2, (A.13)
for the given selection matrices S1 and S2. Since Y⊥1 and Y⊥2 are known, Φ is determined by
obtaining the row-wise embedding E1 and E2
E1 =
[Y⊥1
0
], E2 =
[0
Y⊥2
], (A.14)
138
Proof of Theorem 5.1
and expressing as Φ = E1EH1 + E2E
H2 = EEH , where
E =[E1 E2
]=
0 0 0 0
D1 0 0 0
−WH1 D1 0 −WH
2 D2 0
0 I 0 I
0 0 0 0
0 0 D2 0
. (A.15)
Since E spans the same space as EEH , dim(Φ) = dim(E) = m11 +m12 +m22−(r1 +r2 +r3),
which is clear from the fact that m12 − r1 columns of E are the same.
The alignment matrix due to Z1 and Z2, Φ is related to the alignment matrix due to Z1
and Z2, Φ by the expression
Φ = QHΦQ, (A.16)
for Unitary matrix Q =
Q2 0 0
0 Q1 0
0 0 Q3
.
Hence rank(Φ) = rank(Φ) = m11 +m12 +m22−(r1 +r2 +r3) and rank(N (Φ)) = r1 +r2 +r3.
For the fully overlapped case, m11 + m12 + m22 = 2N, r1 = M, r2 = 0, r3 = 0, hence
rank(Φ) = 2N −M and rank(N (Φ)) = M = r1.
139
Proof of Theorem 5.1
140
Appendix B
Proof of Theorem 5.2
For the given ring topological setup with M transmit and N receive antennas, where M < N ,
we define the alignment matrix collectively formed by all the receivers due to any transmitter
j, i.e, Φj as:
Φj = SHj PZjSj +K∑
i=1,i 6=j
SHi P⊥ZiSi, (B.1)
as defined in theorem 5.2, where PZj = UjUHj are the projection observed due to the basis
matrix Uj at any receiver j and P⊥Zi = Ir −UiUHi are the projection observed due to the
orthogonal complement of the basis matrix Ui.
Now we can express N (Φj) as:
N (Φj) = N(SHj PZjSj +
K∑i=1,i 6=j
SHi P⊥ZiSi
), (B.2)
Using the property of the null space, i.e, the null space of the sum of any two positive
definite matrices A and B is the intersection of their null spaces as [81]
N (A + B) = N (A) ∩N (B), (B.3)
we can express
N (Φj) = N(SHj PZjSj
)⋂N( K∑i=1,i 6=j
SHi P⊥ZiSi
). (B.4)
Let us look at N(SHj PZjSj
)and N
(∑Ki=1,i 6=j SHi P⊥ZiSi
)separately. We have assumed
that M data streams are transmitted from any transmitter. Hence N(SHj PZjSj
)contains
all the space in KN dimensional region due to K receivers each with N antennas except the
M dimensional space because PZj is M dimensional.
Similarly N(∑K
i=1,i 6=j SHi P⊥ZiSi
)is the intersection of the null spaces of each K − 1 terms.
If K = 3, there are two terms N(SH2 P⊥Z2
S2
)and N
(SH3 P⊥Z3
S3
). Due to the projection onto
the orthogonal complement, N(SH2 P⊥Z2
S2
)contains only M dimensional space corresponding
to Rx2 and all other spaces corresponding to Rx1 and Rx3. Also, N(SH3 P⊥Z3
S3
)contains
141
Proof of Theorem 5.2
only M dimensional space corresponding to Rx3 and all other spaces corresponding to Rx1
and Rx2. Hence when there is no overlap, the intersection between these two null spaces
is M dimensional spaces corresponding to each of the receiver and N dimensional space
corresponding to Rx1.
The total intersection space observed by all the receivers is thus N −M dimensions in Rx1,
M dimensions in Rx2 and M dimensions in Rx3. Since they are independent, the rank of the
null space when there is no overlapping is N −M +M +M = N +M .
When there areNc overlap between the adjacent receivers, then the null spaceN(SH1 PZ1S1
)still contains all the available space except M dimensional space in Rx1. However the
N(SH2 P⊥Z2
S2
)contains Nc less dimensions from Rx1 and another Nc less from Rx3. Similarly,
N(SH3 P⊥Z3
S3
)contains Nc dimensions less from Rx1 and Nc dimensions less from Rx2. Hence
the intersection space decreases by 4Nc and the rank of the intersection space is N +M −4Nc
as stated in the theorem.
After Nc = M2
overlapping, the total intersection space depends only on the space observed
at Rx1 because then Rx2 and Rx3 completely overlap all the received interference signal and
increasing Nc does not affect the intersection space at Rx2 and Rx3. However increasing Nc
decreases the intersection space observed by Rx1 by two times due to the overlap of Rx1 with
Rx2 and Rx3. Hence the dimension of the N (Φj) varies as N − 2Nc.
142
Appendix C
Proof of Theorem 5.3
The proof follows similarly as in the ring topology by designing the suitable alignment matrix
due to any transmitter j collectively observed by all the receivers and given by:
Φj = SHj PZjSj +K∑
i=1,i 6=j
SHi P⊥ZiSi. (C.1)
Similarly, we can express N (Φj) as
N (Φj) = N(SHj PZjSj
)⋂N( K∑i=1,i 6=j
SHi P⊥ZiSi
). (C.2)
When there is no overlap, the rank of N (Φj) is same as ring topology and is given by
N +M . When there is overlap of Nc rows per user, then total of KNc common dimensions
always decrease and Nc dimensions decrease from Rx1 due to being in the overlap. Hence
the total intersection dimension is again N +M − (K + 1)Nc. This is valid as long as the
number of overlapping rows is MK
. When overlapping rows per user is greater than MK
, the
total number of overlaps is greater than M and the dimensions of N (Φj) no longer varies
with M but only varies linearly with N −Nc.
143
144
References
[1] V. R. Cadambe and S. A. Jafar, “Interference alignment and degrees of freedom of the
K-user interference channel,” IEEE Transactions on Information Theory, vol. 54, no. 8,
pp. 3425–3441, 2008.
[2] K. S. Gomadam, V. R. Cadambe, and S. A. Jafar, “Approaching the capacity of
wireless networks through distributed interference alignment,” in GLOBECOM, 2008,
pp. 4260–4265.
[3] T. Gou and S. A. Jafar, “Degrees of freedom of the K-user M ×N MIMO interference
channel,” IEEE Transactions on Information Theory, vol. 56, no. 12, pp. 6040–6057,
2010.
[4] C. Wang, T. Gou, and S. Jafar, “Subspace alignment chains and the degrees of freedom
of the three-user MIMO interference channel,” Information Theory, IEEE Transactions
on, vol. 60, no. 5, pp. 2432–2479, May 2014.
[5] D. Tse and P. Viswanath, Fundamentals of wireless communication. New York, NY,
USA: Cambridge University Press, 2005.
[6] A. Goldsmith, Wireless Communications. New York, NY, USA: Cambridge University
Press, 2005.
[7] T. Rappaport, Wireless Communications: Principles and Practice, 2nd ed. Upper
Saddle River, NJ, USA: Prentice Hall PTR, 2001.
[8] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” European Transactions
on Telecommunications, vol. 10, pp. 585–595, 1999.
[9] C. Shannon, “Two-way communication channels,” Proc. 4th Berkeley Symp. Math. Stat.
Prob., vol. 1, pp. 611–644, 1961.
[10] T. M. Cover, “Comments on broadcast channels,” IEEE Transactions on Information
Theory, vol. 44, no. 6, pp. 2524–2530, 1998.
[11] H. Liao, “Multiple access channels,” Ph.D. dissertation, Department of Electrical Engi-
neering, University of Hawaii, Honolulu, 1972.
145
References
[12] A. B. Carleial, “A case where interference does not reduce capacity,” IEEE Transactions
on Information Theory, vol. 21, no. 5, pp. 569––570, Sep. 1975.
[13] ——, “Interference channels,” IEEE Transaction on Information Theory, vol. 24, no. 1,
pp. 60––70, Jan. 1978.
[14] T. M. Cover and J. A. Thomas, Elements of information theory. New York, NY, USA:
Wiley-Interscience, 1991.
[15] A. Goldsmith, S. A. Jafar, N. Jindal, and S. Vishwanath, “Fundamental capacity of
MIMO channels,” IEEE Journal on Selected Areas in Communications, Special Issue on
MIMO systems, vol. 21, 2003.
[16] M. Costa, “Writing on dirty paper (corresp.),” IEEE Transactions on Information
Theory, vol. 29, no. 3, pp. 439–441, May 1983.
[17] N. Jindal, S. Vishwanath, and A. Goldsmith, “On the duality of Gaussian multiple-access
and broadcast channels,” IEEE Transactions on Information Theory, vol. 50, no. 5, pp.
768–783, May 2004.
[18] T. S. Han and K. Kobayashi, “A new achievable rate region for the interference channel,”
IEEE Transactions on Information Theory, vol. 27, no. 1, pp. 49–60, 1981.
[19] R. H. Etkin, D. N. C. Tse, and H. Wang, “Gaussian interference channel capacity
to within one bit,” IEEE Transactions on Information Theory, vol. 54, no. 12, pp.
5534–5562, 2008.
[20] C. E. Shannon, “Communication in the presence of noise,” Proc. Institute of Radio
Engineers, vol. 37, no. 1, pp. 10–21, 1949.
[21] A. Host-Madsen and A. Nosratinia, “The multiplexing gain of wireless networks,” in
Proceedings of International Symposium on Information Theory, 2005., Sept 2005, pp.
2065–2069.
[22] M. A. Maddah-Ali, A. S. Motahari, and A. K. Khandani, “Communication over MIMO
X channels: Interference alignment, decomposition, and performance analysis,” IEEE
Transactions on Information Theory, vol. 54, no. 8, pp. 3457–3470, 2008.
146
References
[23] V. R. Cadambe, S. A. Jafar, and S. Shamai, “Interference alignment on the deterministic
channel and application to fully connected Gaussian interference networks,” IEEE
Transactions on Information Theory, vol. 55, no. 1, pp. 269–274, 2009.
[24] S. A. Jafar and S. Shamai, “Degrees of freedom region of the MIMO X channel,” IEEE
Transactions on Information Theory, vol. 54, no. 1, pp. 151–170, 2008.
[25] S. Akhlaghi, M. A. Maddah-Ali, and E. Rahimi, “A fixed precoding approach to achieve
the degrees of freedom in X channel,” CoRR, vol. abs/1006.3385, 2010.
[26] B. Nazer, M. Gastpar, S. A. Jafar, and S. Vishwanath, “Ergodic interference alignment,”
IEEE Transactions on Information Theory, vol. 58, no. 10, pp. 6355–6371, Oct 2012.
[27] S. A. Jafar, “Exploiting channel correlations - simple interference alignment schemes
with no CSIT,” in GLOBECOM, 2010, pp. 1–5.
[28] H. Maleki, S. A. Jafar, and S. Shamai, “Retrospective interference alignment,” in
International Symposium on Information Theory, 2011, pp. 2756–2760.
[29] H. Bolcskei and J. Thukral, “Interference alignment with limited feedback,” in ISIT,
2009, pp. 1759–1763.
[30] A. S. Motahari, S. Oveis-Gharan, M. A. Maddah-Ali, and A. K. Khandani, “Real
interference alignment: Exploiting the potential of single antenna systems,” IEEE
Transactions on Information Theory, vol. 60, no. 8, pp. 4799–4810, Aug 2014.
[31] D. S. Papailiopoulos and A. G. Dimakis, “Interference alignment as a rank constrained
rank minimization,” IEEE Transactions on Signal Processing, vol. 60, no. 8, pp. 4278–
4288, 2012.
[32] S. W. Peters and R. W. H. Jr., “Interference alignment via alternating minimization,”
in ICASSP, 2009, pp. 2445–2448.
[33] B. Nosrat-Makouei, J. G. Andrews, and R. W. H. Jr., “A simple SINR characterization
for linear interference alignment over uncertain MIMO channels,” in ISIT, 2010, pp.
2288–2292.
[34] C. M. Yetis, T. Gou, S. A. Jafar, and A. H. Kayran, “On feasibility of interference
alignment in MIMO interference networks,” IEEE Transactions on Signal Processing,
vol. 58, no. 9, pp. 4771–4782, 2010.
147
References
[35] B. Niu and A. M. Haimovich, “Interference subspace tracking for network interference
alignment in cellular systems,” in GLOBECOM, 2009, pp. 1–5.
[36] S. A. Jafar, “On asymptotic interference alignment,” plenary talk, International confer-
ence on signal processing and communication(SPCOM), 2010.
[37] H. Shen and B. Li, “A novel iterative interference alignment scheme via convex opti-
mization for the MIMO interference channel,” in VTC Fall, 2010, pp. 1–5.
[38] F. Pantisano, M. Bennis, W. Saad, and M. Debbah, “Cooperative interference alignment
in femtocell networks,” in GLOBECOM, 2011, pp. 1–6.
[39] N. Lee and R. W. Heath, “Space-time interference alignment and degree-of-freedom
regions for the miso broadcast channel with periodic csi feedback,” IEEE Transactions
on Information Theory, vol. 60, no. 1, pp. 515–528, Jan 2014.
[40] ——, “Not too delayed csit achieves the optimal degrees of freedom,” in Communication,
Control, and Computing (Allerton), 2012 50th Annual Allerton Conference on, Oct 2012,
pp. 1262–1269.
[41] M. A. Maddah-Ali and D. Tse, “Completely stale transmitter channel state information
is still very useful,” IEEE Transactions on Information Theory, vol. 58, no. 7, pp.
4418–4431, July 2012.
[42] R. Tandon, S. Mohajer, H. V. Poor, and S. Shamai, “Degrees of freedom region of the
MIMO interference channel with output feedback and delayed CSIT,” IEEE Transactions
on Information Theory, vol. 59, no. 3, pp. 1444–1457, 2013.
[43] S. Yang, M. Kobayashi, D. Gesbert, and X. Yi, “Degrees of freedom of time correlated
MISO broadcast channel with delayed CSIT,” IEEE Transactions on Information Theory,
vol. 59, no. 1, pp. 315–328, Jan 2013.
[44] P. d. Kerret and D. Gesbert, “Interference alignment with incomplete CSIT sharing,”
IEEE Transactions on Wireless Communications, vol. 13, no. 5, pp. 2563–2573, May
2014.
[45] S. A. Jafar, “Topological interference management through index coding,” IEEE Trans-
actions on Information Theory, vol. 60, no. 1, pp. 529–568, Jan 2014.
148
References
[46] A. S. Motahari, S. O. Gharan, M. A. Maddah-Ali, and A. K. Khandani, “Real interference
alignment,” CoRR, vol. abs/1001.3403, 2010.
[47] Z. Zhang and H. Zha, “Principal manifolds and nonlinear dimension reduction via local
tangent space alignment,” SIAM Journal of Scientific Computing, vol. 26, pp. 313–338,
2004.
[48] S. A. Jafar, “Interference alignment: A new look at signal dimensions in a communication
network,” Foundations and Trends in Communications and Information Theory, vol. 7,
no. 1, pp. 1–136, 2011.
[49] G. Bresler, A. Parekh, and D. N. C. Tse, “The approximate capacity of the many-to-one
and one-to-many gaussian interference channels,” IEEE Transactions on Information
Theory, vol. 56, no. 9, pp. 4566–4592, Sept 2010.
[50] S. A. Jafar and S. Vishwanath, “Generalized degrees of freedom of the symmetric
Gaussian K-user interference channel,” IEEE Transactions on Information Theory,
vol. 56, no. 7, pp. 3297–3303, 2010.
[51] C. Huang, V. R. Cadambe, and S. A. Jafar, “On the capacity and generalized degrees of
freedom of the X channel,” CoRR, vol. abs/0810.4741, 2008.
[52] S. Karmakar and M. K. Varanasi, “The generalized degrees of freedom of the MIMO
interference channel,” in ISIT, 2011, pp. 2198–2202.
[53] H. Huang, V. K. N. Lau, Y. Du, and S. Liu, “Robust lattice alignment for k-user MIMO
interference channels with imperfect channel knowledge,” CoRR, vol. abs/1103.4525,
2011.
[54] B. Nosrat-Makouei, J. G. Andrews, and R. W. Heath, “MIMO interference alignment
over correlated channels with imperfect CSI,” IEEE Transactions on Signal Processing,
vol. 59, no. 6, pp. 2783–2794, June 2011.
[55] C. S. Vaze and M. K. Varanasi, “The degrees of freedom region and interference
alignment for the MIMO interference channel with delayed CSIT,” IEEE Transactions
on Information Theory, vol. 58, no. 7, pp. 4396–4417, 2012.
[56] ——, “A new outer-bound via interference localization and the degrees of freedom regions
of MIMO interference networks with no CSIT,” CoRR, vol. abs/1105.6033, 2011.
149
References
[57] Y. Birk and T. Kol, “Informed-source coding-on-demand (ISCOD) over broadcast
channels,” in INFOCOM, 1998, pp. 1257–1264.
[58] S. Y. E. Rouayheb, A. Sprintson, and C. N. Georghiades, “On the index coding prob-
lem and its relation to network coding and matroid theory.” IEEE Transactions on
Information Theory, vol. 56, no. 7, pp. 3187–3195.
[59] Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, “Zero-forcing methods for down-
link spatial multiplexing in multiuser mimo channels,” IEEE Transactions on Signal
Processing, vol. 52, no. 2, pp. 461–471, Feb 2004.
[60] J. Parajuli and G. Abreu, “Interference alignment using alignment matrix,” in 2015 49th
Asilomar Conference on Signals, Systems and Computers, Nov 2015, pp. 1092–1096.
[61] G. Bresler, D. Cartwright, and D. Tse, “Settling the feasibility of interference align-
ment for the MIMO interference channel: the symmetric square case,” CoRR, vol.
abs/1104.0888, 2011.
[62] R. Tresch, M. Guillaud, and E. Riegler, “On the achievability of interference alignment
in the K-User constant MIMO interference channel,” CoRR, vol. abs/0904.4343, 2009.
[63] M. Razaviyayn, M. Sanjabi, and Z.-Q. Luo, “Linear transceiver design for interference
alignment: Complexity and computation,” CoRR, vol. abs/1009.3481, 2010.
[64] I.-H. Wang and D. N. C. Tse, “Interference mitigation through limited receiver coop-
eration,” IEEE Transactions on Information Theory, vol. 57, no. 5, pp. 2913–2940,
2011.
[65] S. Mathur, L. Sankar, and N. B. Mandayam, “Coalitions in cooperative wireless networks,”
IEEE Journal on Selected Areas in Communications, vol. 26, no. 7, pp. 1104–1115, 2008.
[66] M. J. Osborne and A. Rubinstein, A course in game theory. MIT Press, 1994.
[67] C. D. Meyer, Ed., Matrix Analysis and Applied Linear Algebra. Philadelphia, PA, USA:
Society for Industrial and Applied Mathematics, 2000.
[68] J. Parajuli and G. Abreu, “Interference alignment with hybrid optimization and receiver
cooperation,” in 2013 IEEE 14th Workshop on Signal Processing Advances in Wireless
Communications (SPAWC), June 2013, pp. 300–304.
150
References
[69] M. A. Maddah-Ali and D. Tse, “On the degrees of freedom of MISO broadcast channels
with delayed feedback,” EECS Department, University of California, Berkeley, Tech.
Rep., Sep 2010.
[70] T. Gou and S. A. Jafar, “Optimal use of current and outdated channel state information:
Degrees of freedom of the MISO BC with mixed CSIT,” IEEE Communications Letters,
vol. 16, no. 7, pp. 1084–1087, July 2012.
[71] S. Park and I. Lee, “Analysis of degrees of freedom of interfering MISO broadcast
channels,” in GLOBECOM, 2009, pp. 1–6.
[72] C. Suh and D. Tse, “Interference alignment for cellular networks,” in in Communication,
Control, and Computing, 2008 46th Annual Allerton Conference, 2008, pp. 1037–1044.
[73] T. Liu and C. Yang, “On the degrees of freedom of asymmetric MIMO interference
broadcast channels,” CoRR, vol. abs/1310.7311, 2013.
[74] G. Sridharan and W. Yu, “Degrees of freedom of MIMO cellular networks: Decomposition
and linear beamforming design,” CoRR, vol. abs/1312.2681, 2013.
[75] J. Parajuli and G. Abreu, “A space-time Tx scheme for two-cell MISO-BC with delayed
CSIT,” in 2014 IEEE International Symposium on Information Theory, June 2014, pp.
1912–1916.
[76] N. Lee, W. Shin, Y.-J. Hong, and B. Clerckx, “Two-cell miso interfering broadcast
channel with limited feedback: Adaptive feedback strategy and multiplexing gains,” in
ICC, 2011, pp. 1–5.
[77] J. Tang and S. Lambotharan, “Interference alignment techniques for MIMO multi-cell
interfering broadcast channels,” IEEE Transactions on Communications, vol. 61, no. 1,
pp. 164–175, January 2013.
[78] W. Shin, N. Lee, J. B. Lim, C. Shin, and K. Jang, “On the design of interference
alignment scheme for two-cell MIMO interfering broadcast channels,” IEEE Transactions
on Wireless Communications, vol. 10, no. 2, pp. 437–442, February 2011.
[79] M. Torrellas, A. Agustin, and J. Vidal, “Retrospective interference alignment for the
MIMO interference broadcast channel,” CoRR, vol. abs/1501.04204, 2015.
151
References
[80] X. Yi and D. Gesbert, “Precoding on the broadcast MIMO channel with delayed CSIT:
The finite SNR case,” 2012 IEEE International Conference on Acoustics, Speech and
Signal Processing (ICASSP), pp. 2933–2936, March 2012.
[81] A. Galantai, Projectors and projection methods, ser. Advances in mathematics. Boston,
Dordrecht, London: Kluwer Academic, 2004.
[82] H. Zha and Z. Zhang, “Spectral properties of the alignment matrices in manifold learning,”
SIAM Review, 2008.
[83] Q. Ye and W. Zhi, “Eigenvalue bounds for an alignment matrix in manifold learning.”
Linear Algebra and its Applications., vol. 436, no. 8, pp. 2944–2962, 2012.
[84] C.-K. Li, R.-C. Li, and Q. Ye, “Eigenvalues of an alignment matrix in nonlinear manifold
learning.” Communications in Mathematical Sciences., vol. 5, no. 2, pp. 313–329, 2007.
[85] S. J. Gortler, C. Gotsman, L. Liu, and D. Thurston, “On affine rigidity,” Journal of
Computational Geometry, vol. 4, no. 1, pp. 160–181, 2013.
[86] D. Gesbert, S. Kiani, A. Gjendemsjo, and G. Oien, “Adaptation, coordination, and
distributed resource allocation in interference-limited wireless networks,” Proceedings of
the IEEE, vol. 95, no. 12, pp. 2393–2409, Dec 2007.
[87] J. Andrews, W. Choi, and R. Heath, “Overcoming interference in spatial multiplexing
MIMO cellular networks,” Wireless Communications, IEEE, vol. 14, no. 6, pp. 95–104,
December 2007.
[88] G. Boudreau, J. Panicker, N. Guo, R. Chang, N. Wang, and S. Vrzic, “Interference
coordination and cancellation for 4G networks,” Communications Magazine, IEEE,
vol. 47, no. 4, pp. 74–81, April 2009.
[89] K. Yang, “Interference management in LTE wireless networks [industry perspectives],”
Wireless Communications, IEEE, vol. 19, no. 3, pp. 8–9, June 2012.
[90] S. Sun, Q. Gao, Y. Peng, Y. Wang, and L. Song, “Interference management through
CoMP in 3GPP LTE-advanced networks,” Wireless Communications, IEEE, vol. 20,
no. 1, pp. 59–66, February 2013.
152
References
[91] G. Bresler and D. Tse, “3 User interference channel: Degrees of freedom as a function of
channel diversity,” in Communication, Control, and Computing, 2009. Allerton 2009.
47th Annual Allerton Conference on, Sept 2009, pp. 265–271.
[92] Y. Wu, S. S. (Shitz), and S. Verdu, “Information Dimension and the Degrees of Freedom
of the Interference Channel,” IEEE Transactions on Information Theory, vol. 61, no. 1,
pp. 256–279, 2015.
[93] J. Parajuli and G. Abreu, “Degrees of freedom of three-user MIMO-IC via receiver chain
alignment,” in 17th IEEE International Workshop on Signal Processing Advances in
Wireless Communications, SPAWC 2016, Edinburgh, United Kingdom, July 3-6, 2016,
2016, pp. 1–6.
[94] S. Krishnamurthy and S. Jafar, “Degrees of freedom of 2-user and 3-user rank-deficient
MIMO interference channels,” in Global Communications Conference (GLOBECOM),
2012 IEEE, Dec 2012, pp. 2462–2467.
[95] D. P. M. M. G. Krein, M. A. Krasnoselski, “On the defect numbers of the linear operators
in Banach space and on some geometrical questions,” Sb. Tr. Inst. Mat. Akad. Nauk
Ukr. SSR 11, pp. 97–112, 1948 (in Russian).
153
top related