group detection

8/9/2019 Group Detection

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Reduced Complexity Demodulation of MIMO

Bit-Interleaved Coded Modulation using IQ

Group DetectionZak Levi , Dan Raphaeli

Abstract In this paper we propose a novel reduced

complexity technique for the decoding of multiple-input

multiple-output bit interleaved coded modulation (MIMO-

BICM) using IQ Group Detection (GD). It is well known

that the decoding complexity of the MAP detector for

MIMO-BICM increases exponentially in the product of

the number of transmit antennas and number of bits

per modulation symbol, and becomes prohibitive even

for simple schemes. We propose to reduce complexity

by partitioning the signal into disjoint groups at the

receiver and then detecting each group using a MAP

detector. Complexity and performance can be traded off

by the selection of the group size. Group separation and

partitioning is performed such as to maximize the Mutual

Information between the transmitted and received signal.

It is shown that for moderate to high SNR IQ Group

Detection outperforms conventional MMSE per antenna

detection schemes by 1-2[dB] under a fast Rayleigh fading

channel, and by 3-4[dB] under a Quasi static Rayleigh

fading channel with practically no increase in decoding

complexity. It is also shown that higher gains can beachieved with some increase in complexity. We further

propose an ad hoc Iterative Group Cancelation scheme

using hard decision feedback to enhance performance.

Index Terms: Mutual Information, Group Detection, Max-

imum A posteriori, Minimum Mean Square Error, Log

Likelihood Ratio.

I. INTRODUCTION

Bit interleaved coded modulation (BICM) is a well

known transmission technique widely used in practical

single-input single-output (SISO) systems. In SISO sys-tems the BICM approach received a lot of attention due

to its ability to exploit diversity under fading channels

in a simple way [1]. Inspired by such an approach

BICM was proposed as a transmission technique for

multi carrier MIMO systems [7], [2].MAP decoding

of BICM transmission involves the computation of the

Log Likelihood Ratio (LLR) for each transmitted bit.

The LLR computation is performed using a detector,

the detector complexity is exponential in the product

of the number of transmit antennas and number of bits

per modulation symbol. Even for simple scenarios this

complexity becomes overwhelming. Various techniques

have been suggested to reduce the computational burdenof the MAP detector. Most of them can be classified into

either list sphere detector based techniques, or Interfer-

ence Suppression and Cancelation based techniques.

The list sphere detector was proposed by [15], [8]

and is a modification of the sphere decoder. The sphere

decoder is an efficient algorithm for finding the nearest

lattice point to a noisy lattice observation. The vector

space spanned by the MIMO channel matrix is regarded

as a lattice and the received signal as its perturbed

version. The algorithm searches for the nearest latticepoint in a sphere around the perturbed received lattice

point. The list sphere detector is a modification of the

sphere decoder and enables the production of approx-

imate LLR values for coded bits. The complexity and

performance of the list sphere detector greatly depend

on the selection of the sphere radius and the list size.

These depend on the channel matrix and the SNR. An

iterative scheme using the list sphere detector and a

turbo code was reported to closely approach capacity of

the multi antenna fast Rayleigh fading channel [15]. The

complexity of the list sphere detector is generally higher

then that of decoding techniques employing Interference

Suppression and Cancelation.

For decoding techniques employing Interference

Suppression and Cancelation, the MAP detector is ap-

proximated by linear processing of the MIMO channel

outputs followed a per antenna LLR computer. In [10]

the authors proposed a Zero Forcing (ZF) detector

followed by a per antenna LLR computer. The authors

made the simplifying assumption of white post equal-

ization interference. In [11] a Minimum Mean Squared

Error (MMSE) based detector was derived followed by

a per antenna LLR computer without the assumption

of white post equalization interference. It was shown

that such a receiver has the same complexity as the onein [10] but offers superior performance. The authors

in [16] proposed an iterative scheme comprising of

a soft Interference Canceler (IC), an adaptive MMSE

detector followed by a per antenna LLR computer and

a soft output decoder. Soft outputs from the decoder

were used to both reconstruct estimates of the channel

output and to adapt the MMSE detector. The recon-

structed channel output estimates were subtracted from

the true received signal (IC) and the resulting signal was

detected via the adaptive MMSE detector and the LLR

computer. It was shown that when bit reliability at the

output of the soft decoder was high the adaptive MMSE

detector coincided with the Matched Filter. A reducedcomplexity approximation to [16] was proposed in [9]


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where the soft output decoder was replaced by a hard

output Viterbi decoder and the soft Interference Can-

celer by a hard one. After the first decoding stage ,due

to the lack of soft information from the decoder, correct

decisions were assumed. After the initial decoding stage

the MMSE detector was replaced by a Matched Filter.

In this paper we propose a Group Detection (GD) In-

terference Suppression based technique. GD was widely

studied in the context of Multi User Detection (MUD) in

CDMA systems [3]. The idea is to jointly detect a subset

of the transmitted information while treating the rest

of the transmission as noise. Many existing detection

techniques can be regarded as Group Detection based

techniques, namely the per antenna detection techniques

where each antenna can be identified as a single group.

The authors in [4] used Group Detection in the context

of V-BLAST decoding ( [6]) as a remade for error

propagation. In their work a group of the worst p

sub-channels was jointly detected using ML decoding.

A DFE was then used to detect the rest of the sub-channels. In [5] a GD scheme was proposed as a trade

off between diversity gain and spatial multiplexing gain

by partitioning the signal at the transmitter into groups.

Each group was encoded separately and per group rate

adaptation was performed.

In our work group detection was employed only

at the receiver side with no special treatment at the

transmitter. Unlike [4], [5], [11], [9], [16] where a

group was defined as a collection of antennas/sub-

channels, we define a group as a collection of In Phase

and Quadrature (I/Q) components of the transmitted

symbols possibly from different antennas. The smallest

group is defined as a single (I/Q) component of thea transmitted symbol. The GD scheme consists of

group partitioning, group separation and detection. The

proposed GD scheme was derived from an information

theoretic point of view. Group separation was performed

by linear detection. Under a Gaussian assumption on the

transmitted signal, the MMSE detector was identified

as a canonical (information lossless) detector for group

detection. A group partitioning scheme was derived

such as to maximize the sum rate. The selection of

the group size allowed us to tradeoff performance with

complexity. At one end when the number of groups was

set to one, the entire transmission was jointly detected

and the scheme coincided with full MAP, while at the

other end each dimension was decoded separately. An

Iterative group interference canceling technique using

hard outputs from the decoder similar to [9] was also

investigated. Finally, performance was evaluated via

simulations using a rate 1/2 64-state convolutional codewith octal generators (133,171) and random interleav-

ing. The proposed GD scheme was compared to the full

MAP detection scheme and the standard MMSE scheme

[11], [9] for both fast Rayleigh fading and quasi static

Rayleigh fading channels.

The organization of this paper is as follows. In

Section II the system model is presented along witha review of MIMO-BICM MAP detection. Section III

bi

ciRandom

Interleaver

Binary

SourceEncoder

Symbol

Mapper (Gray)

NTX

a

1a

S/P

TX

H

Detector

(Bit LLR)

1y

NRX

y

De

InterleaverDecoder

bi

RX

Fig. 1. MIMO-BICM NRX x NTX System Model.

introduces the concept of Group Detection and deals

with group separation and detection. Group partitioning

is addressed in Section IV. Iterative Group Interfer-ence Cancelation is discussed in Section V. Simulation

results for fast and quasi static Rayleigh fading are

presented in Section VI, and Section VII concludes the

paper.

I I . SYSTEM MODEL AND NOTATIONS

We consider a MIMO-BICM system with NT Xtransmit and NRX receive antennas as illustrated inFig. 1. The information bit sequence b = [b1,...,bNb ] isencoded into coded bits which are then interleaved by

a random interleaver. The interleaved bits, denoted by

c = [c1,...,cNc ], are mapped onto an 2m

QAM signalconstellation using independent I&Q gray mapping. The

block of Nc/m symbols is split into sub-blocks oflength NT X . At each instant n a sub-block a(n) =[a1(n),...,aNTX (n)]

T is transmitted simultaneously by

the NT X antennas. We assume that the transmitted sym-bols are independent with a covariance matrix Raa =2a INTXNTX . The NRX 1 received signal is denotedby y(n) = [y1(n),...,yNRX (n)]

T and is given by

y(n) = H(n) a(n) + z(n) (1)Where H(n) is the NRXNT X complex channel

matrix [hi,j(n)]i=1..NRX ,j=1..NTX and is assumed tobe perfectly known at the receiver (full CSI at thereceiver). z(n) is an NRX1 additive white complexGaussian noise vector z(n) = [z1(n),...,zNRX (n)]

T

with a covariance matrix of Rzz = 2z INRXNRX .

For simplicity we consider the case where

NT X =NRX NT. The extension to an arbitrarynumber of transmit and receive antennas satisfying

NT X NRX is strait forward. The above systemmodel can be used to describe a multi carrier MIMO

system by interpreting the instant index n as afrequency (subcarrier) index. Each block of Nc/msymbols would correspond to a single multi carrier

symbol with NC/(mNT X ) sub carriers,and H(n)would correspond to MIMO channel experienced by


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the nth subcarrier. For the reminder of this paper we

omit the instant index n for clarity of notation.

A. Review of MIMO-BICM MAP detection

The decoding scheme is shown in Fig. 1,the MAP

detector performs soft de-mapping by computing the

conditional Log Likelihood Ratio(LLR) for each codedbit. The conditional LLR of the kth coded bit is the

logarithm of the ratio of the likelihood that the bit

was a one, conditioned by the received signal and

channel state, to the likelihood that the bit was a zero,

conditioned by the received signal and channel state

(full CSI at the receiver). The conditional LLR for the

kth coded bit is given by

LLR( ck| y, H) = log

Pr

ck = 1| y, H

Pr

ck = 0| y, H (2)

For clarity and ease of notation from here on we omit

conditioning on the channel matrix H from our notation.Using Bayes rule and the ideal interleaving assumption,

the conditional LLR of the kth coded bit is given by

LLR

ck| y

= log

aSk,r1

e

12z

yHa2

aSk,r0

e

12z

yHa2

(3)

Where Sk,rl CNT is the set of all complex QAMsymbol vectors whose kth bit in the rth symbol is l {0, 1}. The complexity of the above LLR computationis 2

mNT1

and is thus exponential in the product of theM-QAM constellation size and the number of antennas.

III. GROUP DETECTION

Before presenting the group detection scheme let us

reformulate the system model using real signals only.

Eq. (1) is transformed intoy

Ry

I

=

HR HIHI HR

aRaI

+

zRzI

(4)

The subscripts R or I imply taking the Real

or Imaginary part of the Vector or Matrix it is

associated with. For clarity of notation for therest of this paper all real vectors and matrices

derived from the complex ones described in

Sec. II will inherit the names of their complex

versions. For example from here on the vector a =[real {a1} , ...,real {aNT} , imag {a1} , ...,imag{aNT}]T.Denote the number of real dimensions as N = 2 NT

The transmitted signal a is partitioned into Ng disjoint groups of equal size M, where M = N/Ng.The extension to non equal size groups is trivial. Let

= {1, 2, . . . N } be the set of indexes of entries in thetransmitted vector a. Define the group partitioning of into disjoint groups Gi such that:Gi ,|Gi| = M andNgi=1

Gi = . For any a N, the group aGi |Gi|

y

1Ga

LLR

LLR

LLR

P/S Deinterleaver Decoder

b

Group Separation Group Detection

2Ga

NgG

a

M x 1

M x 1

M x 1

N x 1

N x 1

N x 1

1GW

2GW

NgGW

Fig. 2. Group Detection Scheme.

is obtained from a by striking out ak, k / Gi. Thechannel experienced by the group Gi namely HGi isobtained by striking out the kth column of H k / Gi.

At the receiver the groups are separated by an NNseparation matrix. The sub-matrix WGi of size M N,that filters out the ith group out of the received vector

y, is obtained from W by striking out the kth row ofW

k / Gi. The separate detection of real and imaginaryparts of the transmitted symbols is possible due to the

independent I&Q mapping rule. The separation scheme

is depicted in Fig. 2

A. Group Separation Matrix

Given a group partitioning scheme, we propose to

optimize the separation matrix W such as to maximizethe sum rate

Ng

i=iIWGiy; aGi (5)

Denote the output of the separation matrix corre-

sponding to the group Gi by

aGi WGiy (6)Eq. (5) is maximized by choosing the group sep-

aration matrix WGi such as to maximize the MutualInformation between each transmitted group and the

output of the separation matrix

MN

WoptGi = arg maxWGi{I(aGi; aGi)} (7)

From the data processing inequality [12] follows that

I

y; aGi IWGiy; aGi = IaGi; aGi (8)

Thus if exists a separation matrix WGi that achievesthe equality in Eq. (8) then it clearly maximizes the

mutual information in Eq. (7) and in Eq. (5). It is well

known [22], [13] that the sub-matrix of the MMSE

separation matrix Wmmse

Wmmse = RayR1yy = H

T

HHT +

2z2a

INN

1(9)

corresponding to the group Gi achieves the equalityin Eq. (8). The sub-matrix for group Gi is given by


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WmmseGi = HTGi

HHT +

2z2a

INN

1(10)

The estimation error covariance matrix is given by

Ree = Raa

RayR1

yy

Rya (11)

and the estimation error covariance matrix for the

group Gi is a sub-matrix of Ree and is denoted byRe

GieGi

B. Group LLR Computation

The output of the MMSE separation matrix for group

G is given by

aG = Wmmse

G HGaG + WmmseG

HGaG + z

vG

(12)

Where HG is the N |G| sub matrix of H corre-sponding to group G and HG is the N

G sub matrixofH corresponding to group G,and GG = . Denoteby vG the noise experienced by group G, the covariancematrix of vG is given by

RvGvG = WmmseG

2a2

HGHTG

+2z2

INN

(WmmseG )

T(13)

The conditional LLR for the kth coded bit, where kbelongs to the set of coded bits mapped into one of the

symbols belonging to group G is given by

LLR( ck|

aG) = logPr {ck = 1| aG}Pr {ck = 0| aG} (14)To compute Eq. (14) we need the conditional pdf

f( aG| aG). From Eq. (12) and under the Gaussianassumption on the inter group interference [21] follows

aG| aG N

WmmseG HGaG, RvGvG

(15)

The fact the the noise term in Eq. (12) is colored

complicates the evaluation of Eq. (15). We propose

to whiten the noise in Eq. (12). The noise covariance

matrix is symmetric positive semi-definite and thus has

the following eigen value decomposition

RvGvG = UGGUTG (16)

Where UG is a |G| |G| unitary matrix and G is a|G| |G| diagonal matrix of the eigen values of RvGvG.

UG (UG)T

= I|G||G|

G = diag

G1 , . . . G|G|

(17)The noise whitening matrix is given by

FG = 12G U

TG (18)

The output of the group whitening separation matrixfor group G is given by

aG = FGWmmse

G HGaG + vG (19)

Where

RvG

vG

= I|G||G|

The conditional LLR is then derived by using Bayeslaw and the ideal interleaving assumption. The condi-

tional LLR is given by

LLR( ck| aG) = log

aGS

k,rG,1

e12aGFGWmmseG HGaG2

aGS

k,rG,0

e12aGFGWmmseG HGaG2

(20

Sk,rG,l R|G| is the set of all real |G| dimensionalPAM symbol vectors whose kth bit in the rth symbol

is l {0, 1}. The complexity of the LLR computation

for all groups is Ng2

m2 |G|

and is exponential in theproduct of the group size and the number of bits per real

dimension. We are then able to trade off performance

with complexity by the selection of group size.

C. Simplified LLR Computation for group size of 2

For a group size of Ng = 2 it is possible to derive asimplified closed form approximation for the LLR with-

out computing the noise whitening matrix in Eq. (18).

The derivation is in the spirit of [11] and can be done for

an arbitrary group size. The LLR will be derived given

a zero forcing separation matrix. The MMSE structure

will then emerge from the derivation. As in [11] byusing the log max approximation [1] the conditional

LLR can be expressed as

LLR

ck| y max

dSk,rG,1

{logf ( aZF| aG = d)}max

dSk,rG,0

{log f( aZF| aG = d)} (21)

Where

aZF = H#y =

HTH

1HTy = a + w (22)

H# is the ZF separation matrix given by the Moore

Penrose pseudo inverse of H. The noise covariancematrix is given by

Rww =12

2z

HTH1

(23)

Let the group G = {i, j}. Under the Gaussianassumption on the inter group interference [21] follows

that

f( aZF| aG) = e12Q(aG)

(2)M |G|

Q (aG) =

aZF GT

1G

aZF G

(24)To find the mean and covariance in Eq. (24) we note

that


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aZF = aiei + aj ej +

k /{i,j}

akek + w (25)

Where ei is the N 1 ith unit vector. The mean andvariance are given by

G

= E{ aZF| aG} = aiei + aj ejG = E

aZF G

aZF G

TaG

= 122a

INN VGVTG

+ Rww

VG =

eiej (26)

Substituting Eq. (26) into Eq. (24) gives

Q (aG) = aTZF

1G aZF 2aTZF1G

aiei + aj ej

+

a2i eTi

1G ei + a

2j e

Tj

1G ej

(27)

Substituting Eq. (27) and Eq. (25) into Eq. (21) and

noting that the first term in Eq. (27) is not a function

of aG and thus cancels out we arrive at

LLR

ck| y max

dSk,rG,0

Q (d)

max

dSk,rG,1

Q (d)

(28)

Where

Q (aG) = 2aTZF1G

aiei + aj ej

+ a2i eTi

1G ei

+a2j eTj

1G ej

(29)

In Appendix. (A) we show that

Q (aG) =12

2a

(1pjj )pii(1pii)(1pjj )p2ij

aMMSEipii

ai2

+122a

(1pii)pjj(1pii)(1pjj )p2ij

aMMSEjpjj

aj2

+

12

2a1

(1pii)(1pjj )p2ij

aMMSEi pij aj

2+

12

2a

1(1pii)(1pjj )p2ij

aMMSEj pij ai

2+C

(30)

Where C is not a function of aG and will cancel outin Eq. (28) and pi,j is the i,jth element of P

P = WmmseH (31)

In Appendix. (B) we show that 1/pii is the MMSE

bias compensation factor from [14] and that the bestunbiased linear estimate of aMMSEi from aj is pij aj .Thus in Eq. (30) we identify four euclidian distance

terms. We note that when when i, j are taken as the Iand Q of the same transmit antenna follows that pij = 0and pii = pjj . In Appendix. (B) we show that

pii1 pii = SN RMMSE U,i

By denoting the unbiased euclidian distance by

ij =

aMMSEipii

d1+j

aMMSEj

pii d2

(32)

Eq. (30) coincides with the LLR in [11] namely:

LLR

ck| y SN RMMSE U,i

max

dSk,r{i,j},0

2ij (d)

maxdSk,r

{i,j},1

2ij (d)

(33)

IV. GROUP PARTITIONING

The number of ways to partition the transmitted

signal into groups is a function of the transmitted signal

size N and the groups size M. For example when bothN and M are powers of 2 the number of partitioningpossibilities is given by Eq. (34)

NP =12

N

N/2

12

N/2N/4

. . .

12

2MM

=

12

log2(N/M) N!

M!

log2(N/M)i=1 (N2i)!(34)

Table II summarizes the number of partitioning pos-

sibilities for several values of N and M.

Scheme M NP2 2 (N=4) 2 34 4 (N=8) 2 1054 4 (N=8) 4 35

8 8 (N=16) 2 6756758 8 (N=16) 4 2252258 8 (N=16) 8 6435

TABLE I

NUMBER OF PARTITIONING POSSIBILITIES

We are faced with the problem of choosing a parti-

tioning scheme from amongst all the partitioning pos-

sibilities. A natural selection would be to choose the

partitioning scheme that minimizes some probability

of error measure, this although very intuitive is very

difficult to trace analytically. Instead we propose to

select the partitioning scheme that maximizes the sum

rate of the groups.

By using the chain rule of mutual information the

mutual information of the MIMO channel can be written

as Eq. (35)

Iy; a = Iy; aG1 + Iy; aG2 aG1++I

y; aGNg

aG1 , . . . , aGNg1 (35)When using the GD scheme information is not ex-

changed between groups, and so the mutual information

of Eq. (35) cannot in general be realized. The mutual

information (sum rate) given the GD scheme is given

by Eq. (36)

Ngi=1

I

y; aGi Iy; a (36)

The sum rate is simply the sum of mutual information

since the groups are disjoint. The mutual information inEq. (36) can be written as


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Ngi=1

I

y; aGi

=

Ngi=1

h

aGi Ng

i=1

h

aGi y (37)

Using the fact that the covariance of the MMSE

estimator of aGi from y equals the covariance of theestimation error, namely R

aGiy aGiy = ReGieGi [13]and assuming that the Inter Group Interference is Gaus-

sian [21] follows that aGi y is also Gaussian and that

Ngi=1

I

y; aGi

=

Ngi=1

h

aGi 1

2

Ngi=1

logReGieGi (38)

Under the assumption that transmitted symbols are

i.i.d the first term in Eq. (38) does not depend on the

partitioning scheme, the determinants of the error co-

variance matrices

ReGi

eGi

, i = 1 . . . N g are a functionthe partitioning scheme. Given a partitioning scheme

{G1, G2, . . . GNg } the error covariance matrix for theith group ReGieGi is obtained from the covariancematrix Ree by striking out the kth rows and the kthcolumns k / Gi. Thus finding the partitioning schemethat maximizes the mutual information in Eq. (38)

is equivalent to finding the partitioning scheme that

minimizes the product of the the determinants of the

group error covariance matrices. Thus we need to solve

the following optimization problem

{Gopt1 , Gopt2 , . . . GoptNg } =arg min

{G1, G2, . . . GNg }s.tGi |Gi| = MGi Gj = i = j

Ngi=1

ReGieGi

(39)

The complexity of the above search quickly becomes

overwhelming (Eq. (34)). To reduce the complexity of

Eq. (39) we turn to the structure of Ree. Since Ree is areal covariance matrix it is obviously symmetric positive

semi-definite, however its structure goes deeper due to

the symmetry in the real channel matrix H (Eq. 4). Thisstructure can be utilized to greatly simplify Eq. (39)

for the 2 2 scheme. Intuition form the simplifiedexpressions for the 2

2 scheme will then lead us to

develop simple suboptimal ad hoc approximations to(Eq. (39).

A. Simplified Partitioning for2 2 schemeFor the 2 2 antenna scenario the MMSE error

covariance matrix (Eq. (11)) is given by:

Ree =12

2a

I4x4 HT

HHT +

2z2a

I4x4

1H

(40)

In Appendix. (C) we prove that

Ree =2

2a 1

ad b2 c2

d bb a

0 cc 0

0 cc 0 d bb a

(41)

Where hi denotes the ith column ofH and the scalars,a,b,c,d are given by

=2z2a

a = 1 + hT1 h1 d = 1 + hT2 h2

b = hT1 h2 c = hT1 h4

(42)

For simplicity of notation we drop the scalar mul-

tiplying the matrix in Eq. (41) since it will not affectthe minimization of Eq. (39). For the 2 2 scenariothere are 3 ways to partition the transmitted signal intogroups of size 2,namely

Scheme G1 G21 {1, 2} {3, 4}2 {1, 3} {2, 4}3 {1, 4} {2, 3}

TABLE II

PARTITIONING POSSIBILITIES FOR 2 2 SCENARIO

The group error covariance matrices corresponding

to each partitioning possibility are given in Table. (III).

Scheme ReG1eG1

ReG2eG2

1

d b

b a

d b

b a

2

d 00 d

a 00 a

3

d c

c a

a cc d

TABLE III

ERROR COVARIANCE MATRICES FOR ALL PARTITIONINGPOSSIBILITIES OF 2 2 SCENARIO

Denote the product of the determinants of the group

error covariance matrices of the ith partitioning scheme

by Di, since all the error covariance matrices are posi-tive semi-definite their determinants are nonnegative.

D1=

ad b22 = (ad)2 b2 (2ad 1) 0D2

= (ad)

2 0D3

=

ad c2

2

= (ad)2 c2 (2ad 1) 0(43)

Substituting Eq. (43) into Eq. (39) we obtain

Scheme =Arg mini

(Di) (44)

From Eq. (43) and Eq. (44) follows that partitioning

scheme 2 will be chosen only in cases where 2ad hT1 h4 choose scheme 1 else chosescheme 3.

The above result is simple to compute and very

intuitive. Each one of the columns of the channel matrix

H can be regarded as the channel experienced by asingle element of the transmitted signal. Elements are

thus grouped together based on the correlation between

their corresponding channels. This is intuitive since

grouping correlated elements will result in less noise

enhancement in the group separation process.

B. Ad hoc Simplified Partitioning

In general for NT > 2 obtaining a closed formexpression for the noise covariance matrix as was done

for NT = 2 is hard to tackle. Instead we draw intuitionfrom the 2 2 scenario and propose ad hoc algorithmsfor the partitioning of a general NT NT system intogroups of 2 and 4.

1) Simplified Partitioning into groups of size 2 :The algorithm is a greedy one that partitions the trans-

mitted signal into groups of size 2 such as to maximize

the correlation at the output of the channel. The algo-

rithm stats off with a candidate list consisting of all

the transmitted elements. At each stage, the algorithm

finds the two maximally correlated elements from the

candidate list, groups them together and then erases

them from the candidate list.

Simplified partitioning algorithm for a group size of 2

1) n = 1, n = {(i, j) : i < j}

2) hi,j = hTi hj , (i, j) n3) [in, jn] = arg max(i,j)n (hi,j )

4) Gn = {in, jn}5) n = {(k, jn), (in, k), (jn, k), (k, in) : k,

(k, jn), (in, k), (jn, k), (k, in) n}6)n+1 = n \ n7) if (+ + n) Ng goto 3 else end

Since matrix HTH is a byproduct from the computationof Wmmse (see App ()) it needs not to be recomputedin stage 2. The above algorithm is very simple and its

complexity is that of finding the maximum entry from

a list. At each one of the Ng stages of the algorithmthe list size decreases drastically. This is a tremendous

reduction compared the combinatorial complexity

in Eq. (12). In Sec (IV-C) we show that under a

Gaussian alphabet and Rayleigh channel assumptions,

for NT = 4 the loss of the simplified partitioningalgorithm with respect to optimal partitioning increases

with the SNR. When transmitting 16bit/ChannelUsethe loss is about 0.3[dB] and when transmitting

30bit/ChannelUse the loss is about 0.45[dB]

2) Simplified Partitioning into groups of size 4 :

The algorithm is a greedy one that partitions the

transmitted signal into groups of size 4 namely

Gn = {in, jn, kn, ln}such as to maximize the following heuristical corre-

lation measure

[in, jn] = arg maxi,j hTi hjkn = arg maxk hTinhk + hTjnhk

ln = arg maxlhTinhl + hTjnhl + hTknhl

(47)

The correlation measure is built in the following

fashion. First the pair of elements with maximal cor-

relation is found then the third element is selected

such that maximizes the sum of correlations with the

already selected pair. The fourth element is found using

the same procedure thus selecting the element that

maximizes the sum of correlations with respect to the

pre selected triplet. Using the same notations as in the

previous chapter the partitioning algorithm is given by:

Simplified partitioning algorithm for a group size of 4

1) n = 1, n = {(i, j) : i < j}2) hi,j =

hTi hj , (i, j) n3) [in, jn] = arg max(i,j)n (hi,j)4) kn = arg maxk:(in,k)&(jn,k)n&k=in,jn

(hin,k + hjn,k)5) ln = arg maxl:(in,k)&(jn,k)&(kn,l)n&l=in,jn,kn

(hin,l + hjn,l + hkn,l)6) Gn = {in, jn, kn, ln}7) n = {(t, jn), (jn, t), (t, in), (in, t), (t, kn),

(kn, t), (t, ln), (ln, t) :


8/15

8

{t : (t, jn), (jn, t), (t, in), (in, t), (t, kn), (kn, t), (t, ln), (ln, t) n}

8)n+1 = n \ n9) if + + n Ng goto 3 else end

Since matrix HTH is a byproduct from the com-putation of Wmmse (see App ())it needs not to berecomputed in stage 2. The above algorithm is very

simple and its complexity is that of finding the max-

imum entry from a list. At each one of the Ng stagesof the algorithm the list size decreases drastically. This

is a tremendous reduction compared the combinatorial

complexity in Eq. (12).

In Sec (IV-C) we show that under a Gaussian

alphabet and Rayleigh channel assumptions, for NT = 4the loss of the simplified partitioning algorithm with

respect to optimal partitioning increases with the SNR.

When transmitting 16bit/ChannelUse the loss is about0.25[dB] and when transmitting 30bit/ChannelUsethe loss is about 0.35[dB]

C. Mutual Information for Rayleigh fading channel

The capacity loss resulting from group detection

was computed for the Rayleigh fading channel, thus

the entries of the complex matrix H were indepen-dent Gaussian random variables (Rayleigh Amplitude,

uniform phase) with a variance of 1/NT generatedindependently at each instant. The expectation of the

capacity is given by

C = E

{Iy; a}= E{ 12 log2 2a2z HHT + INN} (48)

The expectation of the sum rate when using group

detection is computed using Eq. (38) and given by:

E{Ngi=1

I

y; aGi} = NgM2 log 122a

12Ngi=1

E{log ReGieGi }(49)

To probe the loss in capacity incurred by using group

detection we turn to simulations. Since the capacity we

are interested in is ergodic we can approximate the

expectation in Eq. (48) and Eq. (49) by the instantaverage. We consider the 2 2 and 4 4 systemsand group sizes of 2 and 4 with Optimal Search (OS)

partitioning, Simplified Search (SS) partitioning and

simple Per Antenna (PA) partitioning.

Fig. 3 summarizes results for a 2 2 system andshows that for medium to high SNR group detection

has a gain of around 1.5[dB] over the simple perantenna partitioning and is only around 0.6[dB] fromthe capacity.

Fig. 5 summarizes results for a 4 4 system forgroups of size 4 and 2.

For medium to high SNR, partitioning into groups

of size 4 with optimal group partitioning losses roughly2[dB] from capacity. Using the simplified partitioning

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

18Rayleigh Fading Capacity (Ntx=Nrx=2)

SNR [dB]

Bits/ChannelU

se

Full Capacity

GD

PA

12 13 14 15 16 17 18 19 20

6.5

7

7.5

8

8.5

9

9.5

SNR[dB]

Bits/ChannelUse

1 6 1 8 2 0 22 24 2 6 2 8

10.5

1 1

11.5

1 2

12.5

1 3

S N R [d B ]

Bits/ChannelUse

0.54[dB]

2[dB]

0.64[dB]

2.09[dB]

Fig. 3. 22 Capacity Loss Per Antenna Vs Group Detection.

0 5 10 15 20 25 300

5

10

15

20

25

30

35


SNR [dB]

Bits/ChannelUse

Full Capacity

2G_OS_GD

2G_PA_GD

4G_OS_GD

4G_PA_GD

8 10 12 14 16 18 20 22

11

12

13

14

15

16

17

18

19

20

21

Rayleigh Fading Capacity (Ntx=Nrx=4)

SNR [dB]

Bits/ChannelUse

16 18 20 22 24 26 28 30

20

21

22

23

24

25

26

27

28

Rayle igh Fad ing Capacity (Ntx=Nrx=4)

SNR [dB]

Bits/ChannelUse

1.7[dB]

2.9[dB]

4.8[dB]

2.2[dB]

3.6[dB]

5.5[dB]

Fig. 4. 44 Capacity Loss Per Antenna Vs Group Detection.

algorithm losses roughly an extra 0.3[dB]. The sim-ple antenna partitioning scheme for a group size of

4 (two antennas per group) losses roughly 3.5[dB]from capacity, thus smart group partitioning shows a

gain of roughly 1 1.5[dB] over simple per antennapartitioning.

For medium to high SNR, partitioning into groups

of size 2 with optimal group partitioning losses roughly3.5[dB] from capacity, using the simplified partitioningalgorithm losses roughly an extra 0.4[dB]. The simpleantenna partitioning scheme (two antennas per group)

losses roughly 5 5.5[dB] from capacity, thus smartgroup partitioning shows a gain of roughly 1 2[dB]over simple per antenna partitioning. It is interesting

to note that partitioning into groups of size 2 withoptimal partitioning achieves just about the same mutual

information as partitioning into groups of size 4 withsimple per antenna partitioning.


9/15

9

0 5 10 15 20 25 300

5

10

15

20

25

30

35


SNR [dB]

Bits/Channel

Use

Full Capacity

2G_OS_GD

2G_SS_GD

4G_OS_GD

4G_SS_GD

20 21 22 23 24 25 26

21

22

23

24

25

26

SNR[dB]

Bits/ChannelUse

13 1 4 1 5 1 6 17 18 19

13

1 3 . 5

14

1 4 . 5

15

1 5 . 5

16

1 6 . 5

17

1 7 . 5

18

SN R [ d B]

Bits/ChannelUse

0.26[dB]

0.3[dB]

0.35[dB]

0.45[dB]

Fig. 5. 4 4 Capacity Loss GD Optimal Search Vs SimpleSearch.

V. ITERATIVE GROUP INTERFERENCE

CANCELATION

The detector in Eq. (3) does not exploit dependencies

between coded bits which leads to degraded perfor-

mance. The detector in Eq. (20) is an approximation

to Eq. (3) and is even more information lossy since

information is not exchanged between groups. An opti-

mal decoder would regard the channel code and MIMO

channel as serially concatenated codes and would de-

code them jointly, such a decoder would have extraor-

dinary complexity. Many authors [18], [15], [16], [17],[9] propose to use iterative schemes since it has been

shown that such schemes are very effective and com-

putationally efficient in other joint detection/decoding

problems [19], [20]. The iterative scheme proposed here

uses hard decisions from the decoder. Using soft outputs

would result in superior performance however hard

output decoders are commonly implemented in many

practical systems and are less complex then soft output

decoders. The iterative scheme proposed here is similar

to the one in [9].

For each group namely group G, hard decoded bitsfrom the decoder are re-encoded, re-interleaved and

used to reconstruct a version of the transmitted MIMOsymbol from all symbols but the ones corresponding

to group G. This reconstructed signal is then passedthrough the effective MIMO channel. Group Interfer-

ence Canceling is performed by subtracting the filtered

reconstructed signal from the true received signal. The

signal after Interference Cancelation is given by:

yiG

= HGaG + HG

aG aiG

eG

+z (50)

The superscript i in Eq. (50) denotes the iteration

number. Assuming correct decisions ai

G = aG theabove expression is further simplified.

yiG

= HGaG + z (51)

The noise after Interference Canceling (assuming

correct decisions) is white and thus a canonical front

end matrix is the Matched Filter HTG

aiG = HTGHGaG + H

TGz (52)

The group noise covariance matrix after matched

filtering is no longer white and is given by

RGG =12

2z HTG HG (53)

A. Group Partitioning For Iterative Group Detection

The partitioning into groups for the iterative stage

introduces a new trade off with respect to the original

group partitioning. In the first part of the decoding

process we traded off decoding complexity with per-formance, where larger groups resulted in better perfor-

mance and higher complexity. After the first decoding

pass we have hard estimates for all bits. If one partitions

the signal into large groups then one is using less

new information and at the extreme not using any

new information when no partitioning is done thus

only one group (MAP decoding). On the other hand

if one partitions the signal into very small groups (at

the extreme groups of 1 bit each) one may be more

susceptible to error propagation since one only has

hard estimates of the decoded bits with no reliability

measure. We propose to partition into groups of size

2. Note that the partitioning scheme in Sec. IV is nolonger relevant since it does not take into account the

new information from the initial stage. We thus propose

to use the simple antenna partitioning. The LLR for

group G can be efficiently computed by Eq. (33) andby setting

P = HTG

HGH

TG +

2z2a

INN

1HG (54)

At the end of each iteration one obtains hard decoded

bits that can be used by the next iteration. Simulation

results in chapter VI suggest that performing two itera-

tions achieves most of the performance gain.

V I . SIMULATION RESULTS

The performance of the GD scheme for MIMO-

BICM was evaluated via Monte-Carlo simulations. At

the transmitter blocks (packets) of 2000 information

bits were encoded and interleaved using a rate 1/264 state convolutional encoder with octal generators

(133, 171) followed by a random per packet inter-leaver. Two antenna configurations were considered,

a 2x2 configuration and a 4x4 configuration. For the

2x2 configuration the detection schemes considered

were full MAP detection, Per Antenna group detec-

tion (conventional MMSE) and optimal search GroupDetection all with zero, one and two hard iterations.


10/15

10

Most of the performance gain due to iterations was

achieved after two iterations. For the 4x4 configuration

two partitioning schemes were considered namely the

partitioning into Ng = 4 groups of size 2 each andthe partitioning into Ng = 2 groups of size 4 each.For both partitioning schemes the detection schemes

considered were full MAP detection (only for fast

fading), Per Antenna group detection (PA GD - con-

ventional MMSE), Optimal Search Group Detection

(OS GD),Simplified Search Group Detection (SS GD)

all with zero,one and two iterations.The complex MIMO

channel matrix entries were drawn from a zero mean

complex Gaussian distribution with variance 1/NT inan iid fashion. Simulation results were summarized via

average Bit Error Rate (BER) and average Packet Error

Rate (PER) versus SNR1 plots.

6 8 10 12 14 16 18 20 22 24 26

10-4

10-3

10-2

10-1

Snr

Ber

Ber(Snr) Mimo (2,2) 64QAM ConvK =7 ,Rate = 1/2

16QAM Coded PA Detection

16QAM Coded GD Detection

16QAM Coded Map Detection

16QAM Coded IPA Detection

16QAM Coded IGD Detection

16QAM Coded IMAP Detection

16QAM Coded I2PA Detection

16QAM Coded I2GD Detection

16QAM Coded I2MAP Detection










16QAM

64QAM

2 Iteration

0 Iteration

1 Iteration

2 Iteration

0 Iteration

1 Iteration

Fig. 6. 2 2 16QAM,64QAM Fast Fading Rayleigh.

Gain [dB] @BER MAP/GD GD/PA

104 105

No Iter 0.2-0.3 0.8-1.7

1 Iter 0.1 0.4-0.8

2 Iter 0.1-0.3 0.35

M AP Iter Gain GD It er Gain

1 Iter 1-1.3 1-1.5

2 Iter 1 0.8-1

PA Iter Gain

1 Iter 2-2.5

2 Iter 1-1.5

TABLE IV

FAS T RAYLEIGH FADING MAP,GD,PA COMPARISON AT

BE R 104 - 105 , 2 2 SCENARIO

A. Fast Fading

For fast fading the MIMO channel was independently

generated at each instant. Fig 6 presents simulation re-

sults for the 2x2 configuration for both 16 and 64QAM.

Table IV summarizes the gain of the MAP scheme

over GD, the gain of GD over PA and the gain due

to iterations for each one of the schemes. The gain

1The SNR is defined as E

Ha2

E

z2

= 1

2z

was measured at a BER of 104 - 105. The gainsin Table IV correspond to both 16 and 64QAM since

they were found to be similar. Performing more than

two iterations did not show much gain. Fig 6 suggests

that the GD gain over PA increases with the SNR.

Without iterations GD shows a substantial gain over

PA. Performing iterations closes the gap between GD

and full MAP as well as the gain of GD over PA.

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 2210

-5

10-4

10-3

10-2

10-1

Snr

Ber

Ber(Snr) Mimo (4,4) 16QAM Conv K=7 ,Rate = 1/2

16QAM Coded Map Detection16QAM Coded 4G_PA_GD Detection16QAM Coded 2G_PA_GD Detection16QAM Coded 4G_OS_GD Detection16QAM Coded 2G_OS_GD Detection16QAM Coded IMAP Detection16QAM Coded 4G_PA_IGD Detection16QAM Coded 2G_PA_IGD Detection16QAM Coded 4G_OS_IGD Detection16QAM Coded 2G_OS_IGD Detection16QAM Coded 4G_PA_I2GD Detection16QAM Coded 2G_PA_I2GD Detection16QAM Coded 4G_OS_I2GD Detection16QAM Coded 2G_OS_I2GD Detection16QAM Coded I2MAP Detection

0 Iteration

1 Iteration

2 Iteration

Fig. 7. 4 4 16QAM Fast Fading Rayleigh.

5 10 15 2010

-5

10-4

10-3

10-2

10-1

Snr

Ber


16QAM Coded 4G_SS_GD Detection

16QAM Coded 4G_OS_GD Detection


16QAM Coded 2G_OS_GD Detection

16QAM Coded 4G_SS_IGD Detection

16QAM Coded 4G_OS_IGD Detection


16QAM Coded 2G_OS_IGD Detection

0 Iteration

1 Iteration

Fig. 8. 4 4 16QAM Fast Fading Rayleigh Optimal SearchVs Simple Search.

Fig 7 summarizes simulation results for the 4x4 con-

figuration for 16QAM. Table V presents a comparison

between the various GD schemes and the MAP scheme

as well as a comparison between GD with group size

of 2 to that of GD with a group size of 4, the gain due

to iteration is also included.

It is interesting to note that GD with a group size

of 2 outperforms PA with a group size of 4, especially

since the detection complexity of the former is much

lower then the later. The results show that performing

iterations reduces the gap between MAP, the various GD

schemes and PA.Fig 8 presents simulation results for the 16QAM


11/15

11

Gain [dB] @BER |G|=2 MAP/GD |G|=2 GD/PA104 105

No Iter 1.5-2 1-2

1 Iter 1 0.5

2 Iter 0.6 0.3

|G|=4 MAP/GD |G|=4 GD/PANo Iter 1 1.5-2

1 Iter 0.4-0.8 0.7

2 Iter 0.3-0.5 0.5

GD,PA |G|=4/2 Iter GainNo Iter 1-2

1 Iter 0.4-0.7 1.5-4.5

2 Iter 0.1-0.3 1-2

TABLE V


BE R 104 - 105 , 4 4 SCENARIO

4x4 configuration for the various GD schemes with

the simplified group partitioning (SS GD) algorithms.

Results were compared to those of the Optimal Search

partitioning (OS GD). The simplified partitioning intogroups of size 2 (See IV-B.1) showed a loss of no

more then 0.2[dB] with respect to optimal partitioning,the loss after one iteration dropped to 0.1[dB]. Thesimplified partitioning into groups of size 4 (See IV-

B.2) showed a loss of no more then 0.35[dB] withrespect to the optimal partitioning, the loss after one

iteration remained around 0.35[dB].

B. Quasi Static Fading

For quasi static fading the MIMO channel remained

constat over a duration of a block and changed indepen-

dently from block to block. Fig 6 presents simulation

results for 16QAM 2x2 configuration. Table VI summa-

rizes the gain of MAP scheme over GD,the gain of GD

scheme with respect to PA scheme as well as the gain

due to iterations for each one of the schemes all at a

PER of 102-103.

Fig 10 and Fig 11 presents simulation results for

16QAM 4x4 configuration. Fig 10 summarizes results

for the partitioning into 4 groups of size 2 each using

the simplified search algorithm, while Fig 11 present

simulation results for partitioning into 2 groups of size

4 each using the simplified search algorithm. Table VII

presents a comparison between the various GD schemesat a PER of102-103, as well as gain due to iterations.

Gain [dB] @PER MAP/GD GD/PA

102 103

No Iter 4-8 3-4

1 Iter 4-5 2-4

2 Iter 2-3 2

MAP Iter Gain GD,PA Iter Gain

1 Iter 1-2 3-4

2 Iter 0.5-1 1

TABLE VI


BE R 104 - 105 , 4 4 SCENARIO

5 7 9 11 13 15 17 19 21 23 25 27 29 31 3310

-3

10-2

10-1

100

Snr

Ber











Fig. 9. 2 2 16QAM Quasi Static Fading Rayleigh.

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 3510

-3

10-2

10-1

100

Snr

PER

PER(Snr) Mimo(4,4) Packet size 2000 infromation bits 16QAM Conv K=7 R=1/2

16QAM Coded 4G_PA_GD Detection


16QAM Coded 4G_PA_IGD Detection


16QAM Coded 4G_PA_I2GD Detection

16QAM Coded 4G_SS_I2GD Detection

Fig. 10. 4 4 16QAM 4 Groups Simple Search Quasi StaticFading Rayleigh.

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 3510

-3

10-2

10

-1

100

Snr

PER

PER(Snr) Mimo(4,4) Packet size 2000 infromation bits 16QAM Conv K=7 R=1/2

16QAM Coded 2G_PA_GD Detection


16QAM Coded 2G_PA_IGD Detection


16QAM Coded 2G_PA_I2GD Detection

16QAM Coded 2G_SS_I2GD Detection

Fig. 11. 4 4 16QAM 2 Groups Simple Search Quasi Static

Fading Rayleigh.


12/15

12

Gain [dB] @BER |G|=2 GD/PA |G|=4 GD/PA104 105

No Iter 3-4 1.5-2

1 Iter 3.5-4 1.5-2

2 Iter 2.5-3 1-1.5

PA |G|=4/2 SS |G|=4/2No Iter 7-10 5-8

1 Iter 4-7 3-5

2 Iter 3-4 2-3

|G|=2 Iter Gain |G|=4 Iter Gain1 Iter 5-6 2.5-3.5

2 Iter 2.5-3.5 1.5-2

TABLE VII

QUASI STATIC RAYLEIGH FADING GD,PA COMPARISON AT

PE R 102 103 , 4 4 SCENARIO

C. Simulation Summary

Simulation results suggest that under a fast Rayleigh

fading at low BER GD achieves gains of 1-2[dB] with

respect to PA with practically no increase in complexity.

An extra gain of 1-2[dB] can be achieved by choosing

larger group sizes with a complexity price. Under for

Quasi static Rayleigh fading at low BER GD achieves

gains of 3-4[dB] with respect to PA with practically

no increase in complexity. An extra gain of 5-10[dB]

can be achieved by choosing larger group sizes with

a complexity price. For both Quasi static and fast

Rayleigh fading performing hard iterations improved

performance of all the schemes as well as reduced the

gaps between them.

VII. CONCLUSIONS

In this paper we proposed a scalable reduced com-plexity detection algorithm for MIMO-BICM. Com-

plexity reduction was achieved by performing detec-

tion in groups instead of joint detection of the entire

MIMO signal. A simple group partitioning algorithm

was derived as well as a approximate expression for

the LLR for group size of 2. Performance and com-

plexity were shown to be traded off by the selection

of the group size. Computer simulations showed that

GD achieves gains of 1-4[dB] with respect to PA with

practically no increase in complexity. Gains of up to

10[dB] were achieved by using larger group size. A

simple hard iterative interference canceling scheme was

further proposed to enhance performance. Performing

hard iterations improved performance of all the schemes

as well as reduced the gaps between them.

APPENDIX

APPENDIX A - CALCULATION OF Q (aG)

To compute Q (aG) we first derive a closed formexpression for 1G . Define

P = INN +2z2a H

TH1

1

(A-1)

And note that

G =12

2a

INN + 2z2a HTH1

P1

+VG (I2x2) VTG

(A-2)

Then make use of the matrix inversion lemma 2

1G =12

2aP

INN + VG I2x2 VTG P VG1

T

VTG P

(A-3)

Noting that

T =

I2x2 VTG P VG1

=

1 pjj pij

pij 1 pii

(1 pii) (1 pjj ) p2ij(A-4)

Where pij is the i,jth element of P in Eq. (A-1).To compute the first term in Eq. (29) we evaluate

aTZF1G ei =

12

2aaTZFP

INN + VGT V

TG P

ei (A-5)

Noting that P converts ZF estimation into MMSEestimation

aTZFP =

PTaZFT

=INN +

2z2a

HTH

11 HTH

1HTy

T=

HTH+

2z

2aIN

1HTy

T=

WmmseyT

= aTMMSE

(A-6)

The forth equality in Eq. (A-6) follows from the

following Eq. (A-7)HTH+

2z2a

I1

HT =2a2z

2a2z

HTH+ I1

HT

=2a2z

I HT

HHT +

2z2a

I1

H

HT

=2a2z

HT

I

HHT +2z2a

I1

HHT

=2a2z

HT

2a2z

HHT + I1

= HT

HHT +2z2a

I1

= Wmmse

(A-7)

The second and forth equalities in Eq. (A-7) follow

from the matrix inversion lemma while the rest are

trivial. Substituting Eq. (A-6) into Eq. (A-5) yields

aTZF1G eiai =

12

2aa

TMMSE

ei + VGT V

TG P ei

ai

= 122a(1pjj )a

MMSEi +pij a

MMSEj

(1pii)(1pjj )p2ijai

aTZF1G ej aj =

12

2aaTMMSE

ej + VGT V

TG P ej

aj

= 122a

pij aMMSEi +(1pii)a

MMSEj

(1pii)(1pjj )p2ijaj

(A-8)

The last two terms in Eq. (29) evaluate to

2

A1

A1

B D1 + CTA1B1 CTA1 =A + BDCT

1


13/15

13

a2i eTi

1G ei =

12

2aeTi P ei +

12

2aeTi P VGT V

TG P ei

= 122a

pii(1pjj )+p2ij

(1pii)(1pjj )p2ija2i

a2j eTj

1G ej =

12

2ae

Tj P ej +

12

2ae

Tj P VGT V

TG P ej

= 122a

pjj (1pii)+p2ij

(1pii)(1pjj )p2ija2j

(A-9)

Substituting Eq. (A-6,A-8) into Eq. (A-3) yields

Q (aG) =12

2a

(1pjj )pii(1pii)(1pjj )p2ij

aMMSEi

pii ai

2+12

2a

(1pii)pjj(1pii)(1pjj )p2ij

aMMSEj

pjj aj

2+

12

2a

1(1pii)(1pjj )p2ij

aMMSEi pij aj

2+

12

2a

1(1pii)(1pjj )p2ij

aMMSEj pij ai

2

+C

(A-10)

APPENDIX B - CONNECTING P WITH MSE

From the matrix inversion lemma follows that

P =

INN +2z2a

HTH

11=

I

2a2z

HTH+ INN

1 (B-1)Again from the matrix inversion lemma follows that

2a2z

HTH+ INN1

=

INN HT HHT + 2z2a INN1 H (B-2)Substituting Eq. (B-2) into Eq. (B-1) gives

INN +

2z2a

HTH

11=

HT

HHT +2z2a

INN

1H

(B-3)

From Eq. (11) and Eq. (A-1) then follows that

Ree =2a2 (INN P) (B-4)

Since Ree is the MMSE error covariance matrix itsdiagonal elements are the MSEMMSE in the estimationof each element of a. The unbiased SNR (See [14]) ofthe ith element ai is given by

SN RMMSE U,i =2a2

MS EMMSE,i 1 = pii1pii (B-5)

The bias compensation scaling factor is given by

2a2

2a2 M SEMMSE,i

=1

pii(B-6)

We next prove that the best unbiased linear estimate

of aMMSEi from aj is pij aj . From Eq. (B-4) follows

that

pij = 22a E

ai aMMSEi

aj aMMSEj

=2

2aE

aMMSEj

ai aMMSEi 22a E{aiaj}

+ 22aE

aMMSEi aj (B-7)

The first term in the second equality is zero from

the orthogonality principle and the second term in the

second equality is zero since transmitted symbols are

statistically independent. Thus

pij =2

2aE

aMMSEi aj

= 22aE

aMMSEj ai

(B-8)

From linear estimation theory follows that

aMMSEi (aj ) =E{aMMSEi aj}

E{a2j} aj =2

2aE

aMMSEi aj

aj

= pijaj(B-9)

Since both aj and aMMSEi are zero mean follows

that the best unbiased linear estimate of aMMSEi from

aj is pij aj . The same proof can be repeated foraMMSEj .

APPENDIX C - COMPUTING Ree FOR THE 2 2SCHEME

Using the matrix inversion lemma follows that

Ree =12

2a

I4x4 HT

HHT +

2z2a

I4x4

1H

=

12

2a

I4x4 +

2a2z

HTH1 (C-1)

Denoting the ith row of H by hTi follows that

HTH =

hT1 h1 hT1 h2 0 h

T1 h4

hT1 h2 hT2 h2 hT1 h4 0

0 hT1 h4 hT1 h1 hT1 h2hT1 h4 0 h

T1 h2 h

T2 h2

(C-2)

Denoting =2z2a

and substituting Eq. (C-2) into

Eq. (C-1)

Ree = 2A

1 0 0 00 1 0 00 0 1 00 0 0 1

+

hT

1 h1 hT

1 h2 0 hT

1 h4hT1 h2 h

T2 h2 hT1 h4 0

0 hT1 h4 hT1 h1 hT1 h2hT1 h4 0 h

T1 h2 h

T2 h2

1 (C-3)

Denote the scalars a,b,c,d

a = 1 + hT1 h1 d = 1 + hT2 h2

b = hT1 h2 c = hT1 h4

(C-4)

The inverse of the matrix in Eq. (C-3) can be

computed in closed form by noting that the matrix in

Eq. (C-3) has the following block symmetry

Ree = 122a A B

BT A1 (C-5)


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14

And then using the block matrix inversion lemma 3

follows

A B

BT A

1=

E1 E1BA1A1BTE1 A1 + A1BTE1BA1 E = A BA1BT

(C-6)

Each one of the sub-matrices can now be computed

in close form thus:

A1 = 1adb2

d bb a

E = A c2adb2 0 1

1 0

d b

b a

0 11 0 =

1 c2adb2

A

E1 =

1adb2c2

d bb a

E1BA1 =c

d bb a

0 1

1 0

d bb a

(adb2)(c2ad+b2)

= cc2ad+b2

0 1

1 0

A1BTE1 = E1BA1T = cc2ad+b2

0 11 0

(C-7)

A1 + A1BT

E1BA1

=

1(adb2)

d b

b a

+

c2

d bb a

0 11 0

0 1

1 0

(adb2)(adb2c2) =

1(adb2) + c2(adb2)(adb2c2) d bb a =1

(adb2c2)

d bb a

= E1

(C-8)

Substituting Eq. (C-7) and Eq. (C-8) into Eq. (C-5)

Ree =2

2a

1

ad b2 c2

d bb a

0 cc 0

0 cc 0

d bb a

(C-9)

3 A BCT D 1 = E1 E1BD1D1BTE1 D1 + D1CTE1BD1 E = A BD1CT

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