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UNIVERSITT STUTTGART
INSTITUT FR NACHRICHTENBERTRAGUNG
Prof. Dr.-Ing. J. Speidel
www.inue.uni-stuttgart.de
Supplementary Material
for the LectureSpace-Time WirelessCommunications
(to be enhanced during lecture)
S T
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INSTITUT FR NACHRICHTENBERTRAGUNG UNIVERSITTProf. Dr.-Ing. Joachim Speidel STUTTGART
Space-Time Wireless Communications
Overview
1.)Multiple Input Multiple Output (MIMO) channel
2.)Spatial multiplex, diversity, beamforming principles
3.)Linear flat fading and frequency selective fading wireless MIMO channel
4.)MIMO receiver: Zero Forcing, Minimum Mean Square Error (MMSE),Maximum Likelihood (ML)
5.)MIMO channel capacity
6.)Space-time coding methods
7.)Convolutional coding, Turbo coding
8.)Decoding principles, iterative receivers
9.)Applications
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INSTITUT FR NACHRICHTENBERTRAGUNG UNIVERSITTProf. Dr.-Ing. Joachim Speidel STUTTGART
Space-Time Wireless Communications
Literature
[1] J. Speidel:Multiple Input Multiple Output (MIMO) - Drahtlose Nachrichtenbertragung hoherBitrate und Qualitt mit Mehrfachantennen. Telekommunikation Aktuell, vol. 59, issue 7-10/05,July-Oct. 2005, pp. 1-63.
[2] B. Vucetic et al.: Space-Time Coding. John Wiley Publisher, 2003.
[3] A. Paulraj et al.:Introduction to Space-Time Wireless Communications. Cambridge UniversityPress, 2003.
[4] E. Larsson; P. Stoica: Space-Time Block Coding for Wireless Communications. Cambridge Uni-
versity Press, 2003.
[5] S. Alamouti:A simple transmit diversity technique for wireless communications. IEEE Transac-tions on Selected Areas of Communications, vol. 16, Oct. 1998.
[6] V. Tarokh; N. Seshardi; A.R. Calderbank: Space-time codes for high data rates wireless commu-nications: Performance criterion and code construction. IEEE Transactions on InformationTheory, vol. 44, 1998.
[7] J. Proakis: Digital Communications. Mc Graw-Hill Book Company, 4th edn. 2008.
[8] E. Lee; D. Messerschmitt; J. Barry: Digital Communication. Springer, 2004.
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INSTITUT FR NACHRICHTENBERTRAGUNG UNIVERSITTProf. Dr.-Ing. Joachim Speidel STUTTGART
Space-Time Wireless Communications
Reprint prohibited - all rights reserved 3
Block diagram wireless MIMO transmission (details)
rN k
bandpass lowpasse
j 0t
g
r t r k b n
detector
mapper
impulseshaper
Re
u t s k
ej0t
sM k
uM t
g
r1 t r1 k
t0 kT+
MIMO receiver
bit sequence
b n
bit ratevB
P
S
serial-parallelconverter
a t
symbol ratevs 1 T= rN t
s1 k
MIMO channel
u1 t
MIMO transmitter
equivalent analog MIMO lowpass channel h t
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INSTITUT FR NACHRICHTENBERTRAGUNG UNIVERSITTProf. Dr.-Ing. Joachim Speidel STUTTGART
Space-Time Wireless Communications
complex time-variant impulse response, ( ; )
Block diagram of wireless MIMO transmission (model)
Definition
MIMO Multiple Input Multiple Output : ,
SISO Single Input Single Output :
t0 kT+
s1
k
s k
sM k
b n b n
r1 k
r k
rN k
h t r t
r1 t
rN t
equivalent analogMIMO lowpass channel
MIMO Tx MIMO Rx
equivalent discrete-timeMIMO lowpass channel h m k
h t 1 M = 1 N =
M 1 N 1
M 1= N 1=
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INSTITUT FR NACHRICHTENBERTRAGUNG UNIVERSITTProf. Dr.-Ing. Joachim Speidel STUTTGART
Space-Time Wireless Communications
MIMO noise vector
a) Gaussian , real , zero mean , variance
pdf
b) complex Gaussian
with property (1)
statistically independent
variance of
of course, has zero mean
c) random vector
all as in b), i. e. zero mean, variance , pdf (2)
all , statistically independent
x 2
p x 1
2
---------------- e
1
2---
x2
2-------
=
n x jy+= n
x y
x y
pnn p x p y
1
2----------------
2 e
1
2---
x2
y2
+
2---------------------
1
22-------------- e
1
2---
n
n
2-------------
1
n2
--------- e
n
n
n2
-------------
= = = =
22 n2
= n
n
n n1nN T
=
n n2
n n
pn n pn n 1=
N
1 n
2-----------
N
e
nH
n
n2---------
1
n2
----------- N
e
n2
n2---------
= = =(2)
nH
n n2
=
(1)
(2)
(3)
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INSTITUT FR NACHRICHTENBERTRAGUNG UNIVERSITTProf. Dr.-Ing. Joachim Speidel STUTTGART
Space-Time Wireless Communications
Minimum mean squared error (MMSE) receiver
Given:
tx signal with autocorrelation matrix (1)
noise with autocorrelation matrix (2)
and statistically independent
Error ; squared error (3a,b)
MMSE: (4a,b)
Result: (5)
Proof of (5):
Introduce error autocorrelation matrix (6)
Then (7)
With (3a):
MIMO channel MMSE rx
H W
s H s
n
r y W r=
H N M
s Rss E ssH =
n Rnn E nn
H
=
s n
e s y s W r= = e2
eH
e=
J E eH
e mi n W min Jarg= = =
W RssHH
H RssHH
Rnn+ 1
=
Ree E eeH =
J trRee trE eeH = = trA sum of main diagonal elements=
J tr E s W r sH rHWH tr E ssH srHWH W rsH W rrHWH+ = =
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INSTITUT FR NACHRICHTENBERTRAGUNG UNIVERSITTProf. Dr.-Ing. Joachim Speidel STUTTGART
Space-Time Wireless Communications
Eigenvalue decomposition of :
(9)
is called "square root" of
diagonal matrix with eigenvalues of
According to Lemma 1, decomposition (9) is always possible
In (10), 2nd and 3rd term are artificially introduced and cancel out.
With from (10)
(11a)
and with Lemma 2
(11b)
Due to (4a) for all also for , for which . Thus, the first term in(11b) is also . Consequently, results for
=
=
= (10)
-
=
=
=
Rrr
Rrr UH U UH1 2 1 2 U 1 2 U
H1 2 U AHA= = = =
A Rrr
diag 1 N = Rrr
Rrr AH
A andA1A I into (8) yields=
J tr Rss RsrA1AW
H W A
HA
H 1Rrs W A
HAW
H+
tr Rss RsrA1
AH
1Rrs RsrA
1A
H 1Rrs+
RsrA1 A WH W AH AH 1 Rrs W AHA WH+
RsrH
Rrs=
J tr Rss RsrA1
AH
1Rrs W A
HRsrA
1 W AH RsrA
1
H+ =
J tr Rss RsrA1
AH
1Rrs tr W A
HRsrA
1 W AH RsrA
1
H +=
J 0 W W W AH RsrA1
0=0 J min=
tr W AH
RsrA1
W AH RsrA1
H
W AH RsrA1
F2
0= =
W AH RsrA1
0
W AH
RsrA1
W RsrA1
AH
1Rsr A
HA
1RsrRrr
1= =
(9)
1
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INSTITUT FR NACHRICHTENBERTRAGUNG UNIVERSITTProf. Dr.-Ing. Joachim Speidel STUTTGART
Space-Time Wireless Communications
As and are statistically independent, they are also uncorrelated, is of zero mean.
Thus , (16)
(16) into (13): (17)
(16) into (15): (18)
(17), (18) into (12): (19)
and (5) is proven.
Important case: ,
From (5) follows after some manipulation
; (20)
Lemma 3: Matrix inversion
With Lemma 3 follows from (20)
(21)
Proof of (21):
s n n
Rsn 0= Rns 0=
Rrr H RssHH
Rnn+=
Rsr RssHH
=
W RssHH
H RssHH
Rnn+ 1
=
Rss EsIM= Rnn n2
IN=
W HH
H HH IN+
1= n
2Es=
A BC D+
1A
1A
1B C
1D A
1B+
1D A
1=
W HH
H IM+ 1
HH=
HH
H HH IN+
1H
H IN HIMHH
+ 1
= =
A B C D
H 1 1 H 1 1 H 1 H 1 H 1 H 1 1 H 1
Lemma 3
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INSTITUT FR NACHRICHTENBERTRAGUNG UNIVERSITTProf. Dr.-Ing. Joachim Speidel STUTTGART
Space-Time Wireless Communications
Proof of some statements of linear algebra
Let be Hermiteian matrix with eigenvalues .
A) is a quadratic form.
for any (22)
Proof: q. e. d.
B) A matrix with property (22) is called positive semi-definite.
(all are real)
Proof:
Let be the eigenvector associated to , i. e.
(23)
We calculate
q. e. d.
; (24)
Q N N 1 N =
Q aaH
=
zH
Qz
zH
Qz 0 z 0
zH
Qz zH
aaH
z zH
a zH
a H
zH
a2
0= = =
Q
Q with property (22) 0 1 N =
u Q u u=
uH
Q u uH u u
Hu u
20= = =
(23) (22)
0
Rrr E rrH = Rrr
HE rr
H H
E rrH Rrr= = =
0
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INSTITUT FR NACHRICHTENBERTRAGUNG UNIVERSITTProf. Dr.-Ing. Joachim Speidel STUTTGART
Space-Time Wireless Communications
(27)
As , , and we conclude q. e. d.
Note, that (25) holds also for non-normalized vectors, i. e.
If the eigenvectors are normalized, i. e. , then they are orthonormal.
Matrix of orthonormal eigenvectors
(28)
due to orthonormality of (29)
with property (29) is called an unitary matrix. As can be seen column vectors
of an unitary matrix are orthogonal and normalized. (30)
Then (31)
(32)
From (31): =
=
With (29) = (33)
Q u2 2 u2=
1 u1 H
u2 Q u1 H
= u2 u1H
QH
u2 u1H
Q u2= =
u1H
= u22(27)
1 0 2 0 1 2 u1H
u2 0=
1u1H 2u2 0= 1 2 0
u u 2 1=
U u1 uN =
U1
UH
= u
U
UH
QU =
diag 1 N =
UH
Q U 1
Q UH
1U 1
Q UUH
1
u1H
u2 2
u1H
u2=
(23)
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INSTITUT FR NACHRICHTENBERTRAGUNG UNIVERSITTProf. Dr.-Ing. Joachim Speidel STUTTGART
Space-Time Wireless Communications
Singular Value Decomposition (SVD):
SVD Lemma:
Given the matrix
An eigenvalue decomposition of does not always exist. In this case an SVD is feasible.
(1)
is called SVD of , where
(2)
and , are unitary matrices, i. e.
, (3a,b)
are called singular values of . (4)
are the positive (non-zero) eigenvalues of the Hermiteian matrix
(5)
where *) (6)
and (7)
The remaining eigenvalues of are zero.
The eigenvalue decomposition of is
H CN M
H
H UD VH
=
H
D
1 00
0 P0 0
RN M=
U CN N V CM M
U1
UH
= V1
VH
=
1 P = H 1 P =
QN H HH
CN N=
rankH rankQN P= =
P min M N
P 1 N += QNQN
H
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INSTITUT FR NACHRICHTENBERTRAGUNG UNIVERSITTProf. Dr.-Ing. Joachim Speidel STUTTGART
Space-Time Wireless Communications
One method to find is the eigenvalue decomposition of as follows
(10)
with (11)
and (12)
Note,
Thus, is the matrix of eigenvectors with respect to the eigenvalues of .
SVD (1) exists for all cases and . (13)
Proof of SVD Lemma:
a) Proof of(1) - (9):
(8) exists with (9), (6) and (7).
With and (5) follows from (8)
; (14a)
We decompose
(14b)
V QM
VH
QMV M=
QM HH
H CM M=
M1 0
P0
0 0
RM M=
rankQM rankQN P= =
V 1 M QM
M N M N
QN QNH
is Hermiteian =
V VH
IM=(3b)
UH
H V VH
HH
U N= A CN M
N =1 0
00
1 00
0 DDH
=
A= A
H
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INSTITUT FR NACHRICHTENBERTRAGUNG UNIVERSITTProf. Dr.-Ing. Joachim Speidel STUTTGART
Space-Time Wireless Communications
Comparing (14a) and (14b) we conclude
(15)
From (15)
and (1) is proven.
Note, that the only requirement on is (unitary).
b) We now prove, that can be obtained by eigenvalue decomposition of as outlined
by (10) - (12).
From (1) follows using Hermiteian operation
(16)
Multiplying (1) and (16) results in
(17)
With and in (14b) follows
(18)
with in (12)
Comparing and we conclude that and have the same non-
zero eigenvalues . Of course, the remaining eigenvalues are zero. Note, that in general
the total number and of eigenvalues is different for and , respectively. Also the
associated matrices and of eigenvectors are different in general.
UH
HV D=
H UD VH
=
V CM M V 1 VH=
V QM HH
H=
HH
V DH
UH
=
HH
H V DH
UH
UD VH
V DH
DVH
= =
DH
D
DH
D M=
M
N M QN H HH= QM H
HH=
1 P N M QN QM
U V
IN=(3a)
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INSTITUT FR NACHRICHTENBERTRAGUNG UNIVERSITTProf. Dr.-Ing. Joachim Speidel STUTTGART
Space-Time Wireless Communications
MIMO channel capacity
Derivation of
Given (1)
Eigenvalue decomposition of yields
; (2)
(2) --> (1) (3)
Matrix calculus:
, also (4)
(4) --> (3)
q. e. d.
Derivation of
C ld IN H HHES
n2
------+=
C ld IN NES
n2
------+=
H HH
UH
H HH
U NP 0
0 0N N = = P diag 1 P =
C ld IN UHH HHU ES
n2
------+=
IM AB+ IN BA+ A M N B N M = M N=
C = ld IN H HH
U UHES
n2
------+
= ld IN H HHES
n2
------+
C ld IM HH
HES
n2
------+=
IN
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Space-Time Wireless Communications
Reprint prohibited - all rights reserved 15
Max. channel capacity in bit/channel use
Prerequisites: with full rank, no fading, . Total tx mean power .
MIMO
Remark
for large SNR
MIMO larger than forN=M
MIMO same asN=M
SISO
SIMOlarger than for SISOgain of for large SNR
MISO same as SISO
Cmax
NM
rxtx
M N= Nld 1Estot
n2
----------+ N
ldEstot
n2
----------
M N Mld 1 NM-----
Estot
n2
----------+
M N Nld 1 Estotn
2----------+
M N 1= = ld 1Estot
n2
----------+
M 1=
N 1ld 1 NE
stot
n2
----------+ ldN
N 1=
M 1ld 1
Estot
n2
----------+
H h 1= 1 N ; 1 M = = Estot
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INSTITUT FR NACHRICHTENBERTRAGUNG UNIVERSITTProf. Dr.-Ing. Joachim Speidel STUTTGART
Space-Time Wireless Communications
Orthogonal space-time block code matrices
a) For real symbols , tx antennas, spatial code rate 1
b) For complex symbols , tx antennas, spatial code rate 1/2
c) For complex symbols , tx antennas, spatial code rate 1: Alamouti ST code
S
s k M 4=
S
s1 s2 s3 s4
s2
s1
s4
s3
s3 s4 s1 s2
s4 s3 s2 s1
=
s k M 3=
S
s1 s2 s3 s4 s1* s2* s3* s4*
s2 s1 s4 s3 s2* s1
* s4* s3
*
s3 s4 s1 s2 s3* s4
* s1* s2
*
=
s k M 2=
Ss1 s2
*
s2 s1*
=
space
time
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INSTITUT FR NACHRICHTENBERTRAGUNG UNIVERSITTProf. Dr.-Ing. Joachim Speidel STUTTGART
Space-Time Wireless Communications
Reprint prohibited - all rights reserved 17
Space-Time Trellis Coding
Input-output relation given by state transition diagram or trellis diagram (which includes time axis)
Example: Transmit delay diversity with M=2 antennas
c k b n
s1
k c k =
4 PS K
Tc k 1
s2
k c k 1 =
state machine
c1c2
c3 c4
Im c k
Re c k c k c1 c4 =
state 1
state 2
state 3
state 4
Trellis Diagram
transitions areindicated by
c c
input outputof state machine
c1 c1
c4 c4
c2 c2
c3 c3 12
3 4
k
c2 c
1
c1 c4
c
4
c
1
c 1
c 4
c
4c
2
not all transitions
are plotted
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Space-Time Wireless Communications
Vertical Encoding (VE)
V-BLAST Vertical-Bell Labs Layered Architecture for Space-Time Coding
c k b'' n' b' n'
bitsequence
temporalencoder
b n demux
bitsequence
spatialdemux
symbolsequence
bitsequence
1 :M
interleaver QAM-mapper
1
M
c k c k L-1+
vector symbolsequence
c k M 1+ c k L-1-M+1+
y1
mux
1
ZF/MMSE-
receiver
QAM-
demapperdecision
device
de-
interleaver
temporal
decoder
1
1
r1
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INSTITUT FR NACHRICHTENBERTRAGUNG UNIVERSITTProf. Dr.-Ing. Joachim Speidel STUTTGART
Space-Time Wireless Communications
Reprint prohibited - all rights reserved 19
Horizontal Encoding (HE)
H-BLAST Horizontal Bell Labs Layered Architecture for Space-Time Coding
cM
k b''M n'' b'M n'' bM
n' M
c1 k b''1 n'' b'1 n'' b1 n'
bitsequence
temporalencoder
b n demux
bitsequence
interleaverQAM
mapper
symbolsequence
c1 k
cM
k
c1 k L-1+
cM
k L-1+
vector symbolsequence
antenna 1
antennaM
Transmitter
1 :M
1
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INSTITUT FR NACHRICHTENBERTRAGUNG UNIVERSITTProf. Dr.-Ing. Joachim Speidel STUTTGART
Space-Time Wireless Communications
Reprint prohibited - all rights reserved 20
One solution for H-BLAST receiver
Note:rx is rather simple, because it can operate just in temporal direction ("horizontally"), except for ZF or MMSE part, which operates in space direction.
cM k cM k bM n'' b' M n'' bM n' M
1
1 1
MN
c1 k c1 k b'' 1 n'' b1' n'' b1 n'
ZF orMMSEreceiver
QAMdemapper
QAMdecisiondevice
de-interleaver
temporaldecoder mux
b n
estimates
M: 1
r1 k
rN k
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INSTITUT FR NACHRICHTENBERTRAGUNG UNIVERSITTProf. Dr.-Ing. Joachim Speidel STUTTGART
Space-Time Wireless Communications
The Q-Function
, where variance of with
, where
(erf: error function; erfc: error function complement)
Q 12
---------- e
1
2--- u
2
du ; Q 1 Q =
=
Q 1 ; Q 0 12--- ; Q 0===
P X Q ---
= 2 X p x 12
--------------- e
1
2---
x
---
2
=
P A X B Q A---
Q B---
=
Q 1
2--- 1 erf
2-------
1
2---erfc
2-------
= = erf z
2
------- eu
2
du0
z
=
erfc z 1 erf z =
1
103
4
Q
1--------------- 1
1------
e1
2--- 2
1
2--- e
1
2--- 2
1
2 --------------- e
1
2--- 2
Chernov bound
0.1
0.01
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Space-Time Wireless Communications
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Bit error ratio (BER) as a function of SNR for MxN MIMO system, with ZF receiver, 4-PSK,
Gray mapping. Encoded with convolutional code or uncoded.
Bit error ratio (BER) as a function of SNR for 4x4 MIMO system with ZF receiver, 4-PSK,
Gray mapping. Parallel detection or sequential detection with interferencecancellation (SAL). Encoded with convolutional code or uncoded.
Bit error ratio (BER) as a function of SNR for 6x6 MIMO system. ZF or MMSE receiver, 16-QAM,
Gray mapping, parallel detection. Encoded with convolutional code or uncoded.
SAL (Symbolauslschung) meansordered successive interference cancellation (OSIC)
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Space-Time Wireless Communications
Reprint prohibited - all rights reserved 23
Iterative V-BLAST receiver
Bit error ratio (BER) as a function of SNR for 6x6 V-BLAST MIMO system with regular
16-QAM, Gray mapping, convolutional encoding and iterative V-BLAST receiver.
Bit error ratio (BER) as a function of SNR for 6x6 V-BLAST MIMO system
with convolutional encoding, Anti-Gray mapping, and iterative V-BLAST MMSE receiver.A1 regular 16-QAM, A2 is non-regular 16-QAM.
Non-regular 16-QAM
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Space-Time Wireless Communications
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Ergodic MIMO channel capacity for a 4x4 MIMO system as a function of transmit
correlation coefficient, no receive correlation. Exponential correlation matrix model,
uncorrelated fading. Large transmit correlation coefficients have a catastrophic impact.
Ergodic capacity for a 4x4 MIMO system as a function of SNR and
various transmit correlation coefficients. Receive correlation 0,5.
Exponential correlation model, uncorrelated fading.
Relative water-filling gain for a 4x4 MIMO system as a function of SNR
and various transmit correlation coefficients, receive correlation 0.7.
Exponential correlation matrix model, uncorrelated fading.
Bit error ratio (BER) as a function of SNR for uncorrelated MIMO channel
with T=4 transmit and R receive antennas. Rank of channel matrix is L=4.
The slope of the asymptotics depend on R, which is clearly visible.
T - number or tx antennas
R - number of rx antennas