Space-time Diversity Codes for Fading Channels

Space-time Diversity Codes for Fading Channels

by Professor R. A. CarrascoSchool of ComputingSchool of Computing

Staffordshire UniversityStaffordshire [email protected]@staffs.ac.uk

Summary

1. Introduction2. Diversity: frequency, time and space3. Space diversity and MIMO channels4. Maximum likelihood sequence detection for ST codes5. Space-time block coding6. Capacity of MIMO systems on fading channels7. Space-time trellis codes8. Code design9. System block diagram10. Example11. BER performance12. Conclusions

1. Introduction

• Migration to 3G standards high data rate communications required

high quality transmission and bandwidth efficient communications low decoding complexity

• Major obstacles to be solved: Multipath fading: signal is scattered among several paths, each path has

a different time delay. Interference: + ISI in case of channels with memory

+ multi-user interference

2 Mbps indoors

144 kbps outdoors

2. Diversity

• Solution of the multipath fading problem, by transmission of several redundant replicas that undergo different multipath profiles

• Types: Frequency diversity: same information is transmitted on different

frequency carriers, which will face different multipath fading.

Time diversity: replicas of the signal are provided in the form of redundancy in the time domain by the use of an error control code together with a proper interleaver

s1 s2 s3 … s1 time

f1 f2

f [Hz]

S(f)

redundant of

2. Diversity

• Types: (cont.) Space diversity: redundancy is provided by employing an array of

antennas, with a minimum separation of λ/2 between neighbouring antennas. Differently polarized antennas can also be used.

2/

3. Space Diversity and MIMO channels

• where:– ci(l) is the modulation symbol transmitted by antenna i at the time instant l. It is

generated by a space-time encoder. – gij is the path gain from Tx antenna i to Rx antenna j.

– ηj(t) is an independent Gaussian random variable (AWGN channel)

n

iii ttctgtr

1111 )()()()(

...

g11

g22

g12g21

g1m

gnm

gn1...

Tx 1

Tx 2

Tx n

Rx 1

Rx 2

Rx m

)3()2()1( 111 ccc

)3()2()1( 222 ccc

)3()2()1( nnn ccc

n

iii ttctgtr

1222 )()()()(

n

imiimm ttctgtr

1

)()()()(

…

3. Space diversity and MIMO channels

• Therefore we can create the MIMO channel matrix

mxnnmmmm

n

n

n

tgtgtgtg

tgtgtgtgtgtgtgtgtgtgtgtg

tG

)(...)()()(|||||

)(...)()()()(...)()()()(...)()()(

)(

321

3332313

2322212

1312111

4. Maximum likelihood sequence detection for ST codes

The probability of receiving a sequence if the code matrix

Taking the likelihood function as the logarithm:

)(...)2()1(||||

)(...)2()1()(...)2()1(

222

111

Lrrr

LrrrLrrr

R

mmm

)(...)2()1(||||

)(...)2()1()(...)2()1(

222

111

Lccc

LcccLccc

C

nnn

has been transmitted is:

L

l

m

j

n

iiijj

N

lclglr

NGCRp

1 1

2

0

1

0 2

)()()(exp1),|(

L

l

m

j

n

iiijj

N

lclglr

NGCRp

1 1

2

0

1

0 2

)()()(1ln),|(ln

L

l

m

j

n

iiijj

N

lclglrNmL

1 12

2

10

04

)()()(ln

2

4. Maximum likelihood sequence detection for ST codes

When maximising the log-likelihood we eliminate the constant term. After this, the problem is equivalent to minimise the following expression:

This can be easily done with the Viterbi algorithm, using the above expressionas a metric and computing the maximum likelihood path through the trellis.

L

l

m

j

n

iiijjijij lclglrLlmjnilglclrm

1 1

2

1

)()()()...1;,...1;,...,1);(|)(),((

5. Space-time block coding

RECEIVER

x1

xk

gij

1x̂

2x̂

kx̂

1tC

ntC

1tr

mtr

1t

mt

At time t, the signal , received at antenna j is given byjtr

jt

n

i

itij

jt Cgr

1

5. Space-time block codingwhere the noise samples j

t are independent samples of a zero-mean complex Gaussian random variable with variance n/(2*SNR) per sample dimension.

The average energy of symbols transmitted from each antenna is normalised to be one.

Assuming perfect channel state information is available, the receiver computes the decision metric

over all codewords

and decides in favour of the codeword that minimizes the sum.

2

1 1 1

l

t

m

j

n

i

itij

jt Cgr

nlll

nn CCCCCCCCC ............ 212

22

121

21

11

5. Space-time block coding Encoding algorithm

A space-time block code is defined by a p x n transmission matrix H. The entries of the matrix H are linear combinations of the variable x1, x2, …, xk and their conjugates. The number of Transmission antennas is n.

We assume that transmission at the baseband employs a signal constellation A,

with 2b elements. At time slot 1, Kb bits arrive at the encoder and select constellation signals s1,……,sK, setting xi = si for i = 1,2,….,K in H, we arrive at a matrix C with entries linear combinations of s1,s2,…..,sK and their conjugates. So, while H contains indeterminates x1,x2,….,xK C contains specific constellation symbols.

5. Space-time block coding Encoding algorithm

Examples: H2 represents a code that utilizes two antennas, H3 represents a code that utilizes three antennas and H4 represents a code that utilizes four antennas.

*1

*2

*3

*4

*2

*1

*4

*3

*3

*4

*1

*2

*4

*3

*2

*1

1234

2143

3412

4321

4

*2

*3

*4

*1

*4

*3

*4

*1

*2

*3

*2

*1

234

143

412

321

3

*1

*2

212

xxxxxxxx

xxxxxxxxxxxxxxxx

xxxxxxxx

H

xxxxxxxxx

xxxxxxxxxxxx

xxx

H

xxxx

H

5. Space-time block coding Decoding algorithm

Maximum likelihood decoding of any space-time block code can be achieved using linear processing at the receiver. Then maximum likelihood detection amounts to minimizing the decision metric

over all possible values of s1 and s2.

We expand the above metric and delete the terms that are independent of the code words and observe that the above minimization is equivalent to minimizing

The above metric decomposes in two parts, one of which

m

jjj

jjj

j sgsgrsgsgr1

2*1,1

*2,12

2

2,21,11 (1)

2

1

2

1,

22

21

*1,2

*21

*,22

*2,1

*22,

*122,2

*1

*2

*,211,1

*1

*1

1

*,11

m

j ijij

jj

j

jj

jj

jj

jj

jj

jj

j

gsssgrsgr

sgrsgrsgrsgrsgrsgr

m

j

n

iji

m

jj

jj

jj

jj

j gssgrsgrsgrsgr1 1

2

,2

11

*1,2

*21

*,221,1

*1

*1

*,11


is only a function of s1, and the other one

is only a function of s2. Thus the minimization of (1) is equivalent to minimizing these two parts separately. This in turn is equivalent to minimizing the decision metric

for detecting s1, and the decision metric

for detecting s2.

Similarly, the decoders for H3 and H4 can be derived.

2

1

2

1,

22

1

*2,1

*22

*,122,2

*1

*2

*,22

m

j iji

m

jj

jj

jj

jj

j ssgrsgrsgrsgr

21

1

2

1

2

,

2

11

,2*

2*,11 1 sgsgrgr

m

j iji

m

jj

jj

j

22

1

2

1

2

,

2

21

,1*

2*

,21 1 sgsgrgrm

j iji

m

jj

jj

j

5. Space-time block codingThe decoder for H3 minimizes the decision metric

for decoding s1. The decision metric


for decoding s3, and the decision metric

for decoding s4.

21

1

3

1

2

,

2

11

,3*

7,2*

6,1*

5*,33

*,22

*,11 21 sgsgrgrgrgrgrgr

m

j iji

m

jj

jj

jj

jj

jj

jj

j

22

1

3

1

2

,

2

21

,3*

8,1*

6,2*

5*,34

*,12


m

j iji

m

jj

jj

jj

jj

jj

jj

j

23

1

3

1

2

,

2

31

,2*

8,1*

7,3*

5*

,24*,13


m

j iji

m

jj

jj

jj

jj

jj

jj

j

24

1

3

1

2

,

2

41

,1*

8,2*

7,3*

6*,14

*,23


m

j iji

m

jj

jj

jj

jj

jj

jj

j

5. Space-time block codingFor decoding H4, the decoder minimizes the decision metric



for decoding s3, and the decision metric

for decoding s4.

21

1

4

1

2

,

2

11

,4*

8,3*

7,2*

6,1*

5*

,44*,33

*,22

*,11 21 sgsgrgrgrgrgrgrgrgr

m

j iji

m

jj

jj

jj

jj

jj

jj

jj

jj

j

22

1

4

1

2

,

2

21

,3*

8,4*

7,1*

6,2*

5*,34

*,43

*,12


m

j iji

m

jj

jj

jj

jj

jj

jj

jj

jj

j

23

1

4

1

2

,

2

31

,2*

8,1*

7,4*

6,3*

5*

,24*,13

*,42


m

j iji

m

jj

jj

jj

jj

jj

jj

jj

jj

j

24

1

4

1

2

,

2

41

,1*

8,2*

7,3*

6,4*

5*,14

*,23

*,32


m

j iji

m

jj

jj

jj

jj

jj

jj

jj

jj

j


There are two attractions in providing transmit diversity via orthogonal designs.

• There is no loss in bandwidth, in the sense that orthogonal designs provide the maximum possible transmission rate at full diversity.

• There is an extremely simple maximum-likelihood decoding algorithm which only uses linear combining at the receiver. The simplicity of the algorithm comes from the orthogonality of the columns of the orthogonal design.

6. Capacity of MIMO systems on fading channels• For the single Tx/Rx channel the capacity is given by Shannon’s classical formula:

where B is the bandwidth g is the fading gain (the realization of a complex Gaussian random variable)

• For a MIMO channel of n inputs and m outputs, the capacity is now given by:

where Im is the identity matrix of order m

snr is the signal-to-noise ratio per receive antennaG is the MIMO channel matrix* denotes the transpose conjugate

)1(log 22 gsnrBC bits/sec

*

2 detlog GGn

snrIBC m bits/sec

6. Capacity of MIMO systems on fading channels

• A particular case is when m = n and G = In (completely uncorrelated parallel sub-channels), then:

• Conclusion:o Capacity can scale linearly with increasing snro Capacity can increase in almost n more bits/cycle for every 3 dB increase

in the snr.

nI

nsnrBC 1detlog2 bits/sec

nsnrnC 1log2 bits/sec/Hz


• Average capacity of a MIMO Rayleigh fading channel

0

5

10

15

20

25

30

35

40

45

50

55

60

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

SNR [dB]

Ave

rage

Cap

acity

[bits

/sec

/Hz]

N=1 M=1 N=2 M=1 N=1 M=2 N=2 M=2 N=2 M=4 N=2 M=6 N=4 M=4 N=8 M=8


Channel correlation influence in the MIMO channel capacityAssume that all the received powers are equal. In this case we define:

where R is the normalized channel correlation matrix ( )whose components are

Therefore

, Where r = correlation coefficient

12

i

ijj g

R

nsnrIC detlog2

1ijr

**1kjki

kkjki

jiij ggggr

22

22

11

1log11logr

nsnr

nsnrrn

snrnC

6. Capacity of MIMO systems on fading channels• In the case of n >> 1 and r < 1, we finally obtain

When n → ∞

When r = 0 (H = I)

and

)1(1log 2

2 rn

snrnC

)1(2ln

2rsnrC

nsnrnC 1log2

2lnsnrC

Channel Capacity for 3dB & 7dB

3 dB & 7 dB SNR

0

1

2

3

4

5

6

7

8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Correlation Coefficient (r)

Cap

acity

(bit/

sec/

Hz)

4 Antennas

10 Antennas

20 Antennas

50 Antennas

4 Antennas

10 Antennas

20 Antennas

50 Antennas

7 dB SNR

3 dB SNR


5 dB & 9 dB SNR

0

2

4

6

8

10

12

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


Cap

acity

(bit/

sec/

Hz)

4 Antennas

10 Antennas

20 Antennas

50 Antennas

4 Antennas

10 Antennas

20 Antennas

50 Antennas

5 dB SNR

9 dB SNR


11 dB & 30 dB SNR

0

50

100

150

200

250

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


Cap

acity

(bit/

sec/

Hz)

4 Antennas

10 Antennas

20 Antennas

50 Antennas

4 Antennas

10 Antennas

20 Antennas

50 Antennas

11 dB SNR

30 dB SNR

7. Space-time trellis codes

The matrix C is called the code matrix, whose element ci(l) is the symbol transmitted by antenna i at the instant l. and l = 1, …, L

The system model is:

..… )()....2()1( 111 LcccAnt 1

..… )()....2()1( 222 LcccAnt 2

..… )()....2()1( Lccc nnnAnt n

⇒….

….

….

)(...)3()2()1(||||

)(...)3()2()1()(...)3()2()1(

2222

1111

Lcccc

LccccLcccc

C

nnnn

)()()(E)(1

ttctgtr j

n

iiijsj

, where Es is the average symbol energy

ηj(t) is an independent sample of a complex Gaussian random variable with varianceNo/2 per dimension.gij(t) is a complex Gaussian random variable with variance 0.5 per dimension.

Signal to noise ratio per receive antenna:

0

ENnsnr s

7. Space-time trellis codesThe probability of decoding erroneously the code matrix C and choosing instead another code matrix E, assuming ideal channel state information, is given by:

where

and the distance between codewords C and E is given by

after some manipulation we rewrite the distance as:

0

2

N2E),()|( sECdQGECP

x

x dxexQ 2/2

21)(

L

l

m

j

n

iiiij lelclgECd

1 1

2

1

2 )()()(),(

)()()(),(1 1

2 lvlAlvECd j

L

l

m

jj

)(|

)()(

)( 2

1

lg

lglg

lv

nj

j

j

j

))()(())()((...))()(())()((||||...))()(())()((

))()(())()((...))()(())()((

)(

11

1122

111111

lelclelclelclelc

lelclelclelclelclelclelc

lA

nnnnnn

nn

with: and

7. Space-time trellis codes• A(l) is an Hermitian matrix, therefore there exists a unitary matrix U and a diagonal

matrix D such that

Let , so

Considering the Chernoff bound of the error probability:

Now we must distinguish between two cases.

)(0.

0)()(

1*

l

lDUlUA

n

*2

1

)(

)(|

)()(

Ulv

lh

lhlh

j

nj

j

j

2

1

)()()()()(

n

iijijj lhllvlAlv

0

2

N4E),(exp)|( sECdGECP

lj

n

iiji

s lhl, 1

2

0

)()(N4Eexp

7. Space-time trellis codesA. Quasi-static fadinggij(l) is constant within a frame of length L and changes randomly from one frame to another.

where r(A) is the rank of matrix A.

B. Time-varying fading

where Ν(C,E) is the set of indexes of the all zero columns of the difference matrix C-E.

where D = C-E is the difference matrix r(D) is its rank

Ω(D) is the set of column indexes that differ from zero.

mAr

s

mAr

iiGECP

)(

0

)(

1 N4E)|(

),( 01

2

N4E)()()|(

ECl

m

sn

iii lelcGECP

)(

)( 01

2

N4E)()()|(

Dmr

Dl

smn

iii lelcGECP

• Error Probability for fading channels.Single Input/Single Output (SISO)

Multi-antenna (MIMO), from the Chernoff bound of the error probability.

041

NEP

bb (coherent binary PSK, Rayleigh fading)

021

NEP

bb (coherent orthogonal, Rayleigh fading)

021

NEP

bb

(orthogonal, noncoherent, Rayleigh fading)

'

1 2

0

),(1

1),(L

kk

se

cadNEcaP

The output noise power of the branch K can be written as:

Assume that the total energy of a block is limited to Etot = K.E0 (E0 is the transmitting energy for each source symbol).

Using the orthogonal martix H to transmit the sequence (x1,x2,…..,xK)

01 1

2

,01 1

2

,

2

, NgNggPm

j

n

iji

m

j

P

t

jkKt

jKtn

Where

Signal-to-Noise (SNR)

2

1,

1

2

,

2

,

n

iji

P

t

jkKt

jKt ggg

0

1 1

2

,

2

1 1

2

,

01 1

2

,

2

Ng

Eg

Ng

yEPESNR m

j

n

iji

S

m

j

n

iji

m

j

n

iji

K

N

Rrev

01 1

2

, NEgSNR S

m

j

n

ijirev

The total energy can also be expressed as:

Assume the constellation at the receiver satisfies ER = .dR2 , where dR is the minimum

distance of the constellation, and is a constant depending on the different constallations.

Using minimum distance sphere bound, the instant symbol error rate bound is:

n

t

P

t

it

P

t

n

t

ittot CECEE

1 1

2

1 1

2

Thus

S

n

t

K

kk EknXE ..

1 1

2

nEES

0

2

1 1

2

,

m

j

n

tjiR gE

nEgE

m

j

n

tjiS

0

2

1 1

2

,

N

R

N

Rjie P

EP

dginstP4

exp4

exp,2

,

We assume i,j are independent samples of zero mean complex Gaussian random variables having a variance of 0.5 per dimension. Thus are independent Rayleigh distributed with a PDF of:

Thus, the average symbol error rate bound is given by

mnn

i

m

j

n

i

m

jij

e AAEnN

gP

expexp4

exp1 1

00

1 1

2

nm

be

nNEP

041

1

;4

exp0

m

NEP b

e )( n

AeA 1mn

e AP

11

2

,

2

,,, expexp2 jijijiji ggggP

mn

e AP

11

0

0

4 nNEA

or

Where

Probability of error for different numbers of Tx & Rx antennas

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1 2 3 4 5 6 7 8 9 10 11 12

SNR (dB)

Pe

Pe (2 Tx 2 Rx)

Pe (2 Tx 4 Rx)

Pe (4 Tx 2 Rx)

Pe (4 Tx 4 Rx)

Pe (4 Tx 8 Rx)

Pe (8 Tx 4 Rx)

Pe (8 Tx 8 Rx)

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1 2 3 4 5 6 7 8 9 10 11 12

SNR (dB)

Pe

Pe (2 Rx)

Pe (4 Rx)

Pe (6 Rx)

Pe (8 Rx)

Pe (10 Rx)

Pe (15 Rx)

Pe (20 Rx)

Pe (25 Rx)

Pe (30 Rx)

Pe (40 Rx)

Pe (50 Rx)

Probability of error for large number of Tx antennas (n)

8. Code designA. Diversity advantage (DA) is the exponent in the error probability bound.

In order to improve the performance of ST codes, the diversity advantage must be maximised by maximising the rank of the difference matrix.

1st Design criteria: the minimum of the ranks of all possible matrices D = C-E must be maximised. To achieve the full rank n all matrices D must have full rank.

B. Coding gain (CG) is the term independent of SNR in the upper error bound.It is the product of eigenvalues of the difference matrix or of euclidean distances. 2nd Design criteria: in order to maximise the coding gain, the minimum of the

products of euclidean distance (or equivalently the eigenvalues) taken over all pairs of codes C and E must be maximised.

mArDA )( for quasi-static fadingmDrDA )( for time-varying fading

9. System block diagramI. Encoder

Sn(t)

S2(t)

Inf

MapSpace-Time Ring

Encoderb0

binary

x

InterleaverFrame

Building Modulator

c22

~c

S1(t)

InterleaverFrame

Building Modulator

c11

~c

1~cP1

nc~Pn

InterleaverFrame

Building Modulator

cn nc~

… … …

…

9. System block diagramII. Decoder

Space-Time Viterbi Decoder

(Sequence Detection)

Detection & Demodulation

Channel Estimation

Deinterleaver

Detection & Demodulation

Deinterleaver

Inv Map

rj

binaryoutput

r1(k)

r2(k)Detection & Demodulation

rm(k)

jig ,~

r1(t)

r2(t)

rm(t)

4ˆ Zx

0̂b

In case of iterative channel estimation

10. Example

1. Delay diversity code QPSK modulation code rate ½ 2 Tx antennasEncoder structure:

Example: x = 1 3 2 0 1 c1 = 1 3 2 0 1

c2 = 0 1 3 2 0

x ∈ ℤ4 outputantenna 1

outputantenna 2

input

D

c1

c2

1 symbol delay

State0 0

1

2

0/00

1/10

2/20

3/30

0/00

1/10

2/20

3

3/30

0/01

1/11

2/21

3/31

0/02

1/12

2/22

3/32

0/03

1/13

2/23

3/33

transition label: x /c 1c2

3

2

1

0

3

2

1

0

3

2

1

0

3

2

1

0

2/23

0/02

1/10

10. Example2. Space-time ring TCM code

4-state trellis code rate ½ (and n=2)

Encoder structure:

Trellis diagram:

c1x

D c2+ +

3

1 2 3 1 1 input seq.

output seq.1 2 3 1 1

1 3 0 3 0t0 t1

2 1

0 0

1

2

0/00

1/11

2/22

3/33

0/00

1/11

2/22

3

3/33

1/12

2/233/30

0/01

2/20

3/31

0/02

1/13

3/32

0/03

1/10

2/21 3

2

1

0

3

2

1

0

3

2

1

0

3

2

1

0

3/30

1/13 1/10

0 0 0 0 0

0 1 3 0 1

0 2 2 0 2

0 3 1 0 3

1 0 1 1 1

1 1 0 1 2

1 2 3 1 3

1 3 2 1 0

2 0 2 2 2

2 1 1 2 3

2 2 0 2 0

2 3 3 2 1

3 0 3 3 3

3 1 2 3 0

3 2 1 3 1

3 3 0 3 2

input

St St+1

c1 c2input

St St+1

c1 c2

10. ExampleFor the 2 Tx antennas system the metric is reduced to:

L

l

m

jjjjijij lclglclglrlglclrm

1

2

12211 )()()()()())(|)(),((

State

survivor path

0 0

1

2

3

0/00

1/11

2/22

3/33

22.378

1.0917

21.437

41.135

0/00

1/12

0

1

2

Transitionmetrics

12.071+22.378=34.449

22.952+1.0917=24.044

Accumulatedmetrics

2/23

3

7.8235+21.437=29.2602

4.5824+41.137=45.7176

1.0917+10.1054=11.1971 survivor

22221122

22211111 )1()1()1()1()1()1()1()1()1()1( cgcgrcgcgr

received1st antenna

CSI transitionsignals

received2nd antenna

CSI transitionsignals =

branchlabels

11. BER performance on time-varying Rayleigh fading channels

BER Performance of 4-state ST-RTCM codes with one, two and four receive antennas

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

SNR [dB]

BER

Uncoded QPSK Delay Diversity m=1 Baro et al. m=1 ST-RTCM m=1 Delay Diversity m=2Baro et al. m=2 ST-RTCM m=2 Delay Diversity m=4 Baro et al. m=4 ST-RTCM m=4

100

10

10

10

10

10

10

-1

-2

-3

-4

-5

-6

10-7

, (Tx=2), QPSK, ideal CSI

12. Conclusions

• The use of space-time diversity techniques for transmission over fading channels offers a promising increase in the capacity. The capacity of MIMO channels can even increase linearly with the number of transmit antennas.

• In particular, space-time codes provide a significantly better performance than single antenna systems.

• The design criteria for space-time codes has been presented. These takes into account two factors: the diversity advantage and the coding gain.

• Maximum-likelihood detection techniques can be applied in the decoding process without an increase on the decoder complexity.

• Computer simulations avail these statements by showing a smaller BER at a fixed SNR.

Thank you

Date post:	13-Apr-2017
Category:	Technology
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Space-time Diversity Codes for Fading Channels

Technology