Optimal Signal Constellations for Fading Space-TimeChannels

Alfred O. Hero

University of Michigan - Ann Arbor

Outline

1. Space-time channels

2. Random coding exponent and cutoff rate

3. DiscreteK-dimensional constellations

4. Bound on minimum distance

5. Low dimensional constellations

6. Conclusions

1

Receiver

Transmitter

H =2

664h11 h12

h21 h22

h31 h323

775

st1 st2 st3; t = 1; : : : ; T

xt1 xt2; t = 1; : : : ; T

T=coherent fade interval

M=number of transmit antennas

N=number of receive antennas

�= receiver SNR

Figure 1. Narrowband space time channel for M = 3,N = 2

2

Received signal inl-th frame (t = 1; : : : ; T )

[xlt1; : : : ; xl

tn] =

p�[slt1; : : : ; sl

tm]2

6664hl11 � � � hl1n

......

...

hlm1 � � � hlmn3

7775+ [wlt1; : : : ; wl

tn];

or, equivalentlyX l =p

�SlH l +W l

� X l: T �N received signal matrices

� Sl: T �M transmitted signal matrices

� H l: i.i.d. M �N channel matrices� Cl N (0; IMN

IN )

� W l: i.i.d. T �N noise matrices� Cl N (0; ITN

IN )3

Block codingoverL frames produces blocks ofL symbols

[S1; : : : ; SL]

whereS = Sl is selected from a symbol alphabetS

Random Block Coding: selectSl at random fromS according toprobability distributionP 2 P.

� Objective: Find optimal distributionP (S) overP

� Optimality criteria: capacity, outage capacity, random coding errorexponent, cut-off rate

� Transmitter constraints:

– average power constraint:E[kSk2] = R kSk2dP � TM

– peak power constraint:kSk2 � TM , for all S 2 S

where

kSk2 = trfSHSg

4

Z-1

cos(w t)o

cos(w t)o

S/Ptemporal

encoder

X

X

s(t)

Figure 2. First generation space-time coding: Seshadri&Winters FDD

transmitter with delay diversity (1994)5

Capacity Results: (avg. power constraint - Telatar, BLTM 95):

1. ChannelH l Known to Txmt and Rcv:

Capacity: (bits/sec/hz) or (bits=channel�use

T

)

C = max

P (SjH)E[I(S;XjH)] = max

P (SjH)E[H(XjH)�H(XjS;H)]

�-Outage Capacity: C = fCo : P (C(H) > Co) = �g

6

SinceH(XjH) � ln(jIN + �HHRsHj); and H(XjS;H) = H(N)

C(H) = max

Rs : trfRsg�M

ln(jIN + �HRsHH j)

= ln(jI + �HRosHj)

where, forH = UDVRos = V Hdiag

��� 1

�d2i�+

V

and� is such that (water-filling)

trfRosg =M

7

S/Ptemporal

encoder

s(t)

coded modulation

coded modulation

VBeamformer

Figure 3. Second generation space-time coding with beamforming

8

2. ChannelH l known only to Rcv:C = max

P (S)E[I(X;SjH)] = max

P (S)E[H(XjH)�H(XjS;H)]

) C = E�

log��IN + �

M HHH���

Capacity achieving distribution:

� S Gaussian with orthogonal rows and columns of identical energy

) BLAST (Foschini, BLTJ 1996)

) Space time 4-PSK/4-TCM (Tarokh&etal IT 98, Tarokh&etal COM 99)

In practice must transmit training within each frame to learnH9

Capacity bounds: (Hochwald&Marzetta SPIE99, Driesen&Foschini

COM99)

log(1 + �MN)| {z }

\=00when rank(H)=1� C(H) � min(M;N) log�

1 +

�MN

min(M;N)�

| {z }

\=00when rank(H)=min(M;N)

10

S/Ptemporal

encoder

s(t)

coded modulation

coded modulation

� 4-PSK/4-TCM: 2 bits/sec/Hz (simulation), M=N=2

� BLAST: 1.2 Mbps over 30kHz (40 bits/sec/Hz) in 800MHz band, M=8, N=12

Figure 4. Second generation space-time coding11

−10 −5 0 5 10 15 200

5

10

15

20

25

30

35

40

45Coherent Transmission and Reception − T/R know channel

SNR (dB)

b/s/

hzM=32M=16M= 4M= 1

12

−10 −5 0 5 10 15 200

5

10

15

20

25

30

35

40Incoherent Transmission − R knows channel

SNR (dB)

b/s/

hzM=32M=16M= 4M= 1

13

−10 −5 0 5 10 15 200.3

0.4

0.5

0.6

0.7

0.8

0.9

1Effect of Incoherent Transmission

SNR (dB)

C/C

o

M=32M=16M= 4M= 1

14

−10 −5 0 5 10 15 200

5

10

15

20

25

30

35

40

45

Coherent Transmission and Reception with Training Errors: Ttrain

=128

SNR (dB)

b/s/

hzM=32M=16M= 4M= 1

15

−10 −5 0 5 10 15 200.4

0.5

0.6

0.7

0.8

0.9

1

Effect of Training Errors (coherent transmission): Ttrain

=128

SNR (dB)

C/C

o

M=32M=16M= 4M= 1

16

3. Slow fading Rayleigh channel:H unknown

Capacity? (Marzetta&Hochwald, BL TM 98, IT 99)

� C = E [logP (XjS)=P (X)] (bits/channel-use)

Capacity achieving distribution?

� S = �V

where� andV are mutually independent matrices

� �: T �M unitary:�H� = IM

� V : M �M real diagonal

V ! cIM as� !1 or T !1.

) Unitary space-time modulation (Hochwald&etal BL TM 1998)

) Differential space-time modulation (Hochwald&Sweldens COM99)

) Space-time group codes (Hughes SAM 00, Hassibi&etal BLTM 00)

17

S/Ptemporal

encoder

s(t)

Modulation

Space-Time

Figure 10. Third generation space-time coding.

18

Example: Unitary space-time constellation (Hochwald&etal BLTM 98)� T = 8, M = 3, K = 256 unitary signal matrices

S = f�1; : : : ;�Kg; �k = �k�1

�1 =

1p8

2666666666666666641 1 1

1 ej2�

8

5 ej2�

8

6

1 ej2�

8

2 ej2�

8

4

1 ej2�

8

7 ej2�

8

2

1 ej2�

8

4 1

1 ej2�

8

1 ej2�

8

6

1 ej2�

8

6 ej2�

8

4

1 ej2�

8

3 ej2�

8

2

377777777777777775

; � =2

6664ej2�

8

(0) 0 � � �

0

...

0 � � � ej2�

8

(7)3

7775

19

1 Random Coding Error Exponent

The minimum error probability of any decoder of a block code overL

frames satisfies (Fano 61)

minPe � e�LEU (R); R < C

where

� R: symbol rate (nats/symbol)

� C: channel capacity (nats/symbol)

� EU (R): error exponent

20

EU (R) =

max

�2[0;1]max

P2P(

��R� lnZ

X2X

�ZS2S(p(XjS))1=(1+�) dP (S)�1+�

dX)

; nats=sym

EU (R) has been studied underavg. power constraintfor

) KnownH (Telatar BL TM 96)

) UnknownH (Abou-Faycal & Hochwald BL TM 99)

21

EU (R)

Ro

CRoRc R

Figure 11. Error exponent EU (R) and cut-off rate bound

y(R) = Ro �R.22

Ro computational cut-off ratelower bound (Gallager IT 64)

EU (R) � Ro �R; R � Ro

Ro = max

P2P� lnZ

X2X

�ZS2S

pp(XjS)dP (S)�2

dX; nats=symbol

whereP are suitably constrained distributions overCl T�M

) Cut-off rate analysis has been used to evaluate

� practical coding limits (Wang&Costello COM 95, Hagenauer&etalIT 96)

� different coding and modulation schemes (Massey 74)

� signal design for optical fiber links (Snyder&Rhodes IT 80)

� signaling over multiple access channels (Narayan&Snyder IT81)

23

FACTS:

� Ro � C

� Ro is highest practical rate for sequential decoders (Savage 65)

� EU (R) � Ro �R whenR � Rc, the critical rate

� Ro specifies upper bound on optimal decoder error

Pe � e�L(Ro�R); R � Ro

24

2 Integral Representation forRo

Ro = max

P2P� lnZ

S12SdP (S1)Z

S22SdP (S2) e�ND(S1kS2):

where

D(S1kS2) def

= 12lnjIT+�2(S1SH

1 +S2SH

2 )j2

jIT+�S1SH1 jjIT+�S2SH2 j :

Low SNR approximation:

D(S1kS2) = �2=8kS1SH1 � S2SH

2 k2 + o(�2)

25

The following parallels Theorems 1 and 2 of Marzetta&Hochwald IT 99

Proposition 1 Assume that the transmitted signalS is constrained to

satisfy the peak power constraintkSk2 �MT . There is no advantage to

usingM > T transmit antennas. Furthermore, forM � T the signal

matrices achievingRo can be expressed as

S =2

4�

35h �i

where

� � is T �M unitary matrixV HV = IM

� � isM �M non-negative diagonal matrix.

26

3 Case of DiscreteK-dimensional Constellations

SpecializeP to the discrete distributions overCl T�M

ThenRo = ~Ro(K) is given by

max

fPi;SigKi=1� ln

KXi=1Pi

KXj=1Pj e�ND(SikSj) = � ln min

fPi;SigKi=1PTEKP

where

� Ek = ((D(SijSj))Ki;j=1: dissimilarity (distance) matrix

� P = [P1; : : : ; PK ]T

27

Under peak power constraint,kSik � TM ,

~Ro(K) = � ln min

fSigKi=1

min

fPigKi=1PTEKP

!

Inner maximization:

min

P>0 : 1TK

P=1�

PTEKP

Lagrangian

J(P ) = PTEKP � 2c(1TKP � 1)minimized forequalizer probabilityP = P �

28

EKP� = c1K )

KXj=1Pj e�ND(SikSj) = c

Fact: optimal constellation satisfiesE�1K 1K � 0

P � = cE�1K 1K ; and c =

1

1TKE�1

K 1K

and

~Ro(K) = � ln min

fSigKi=1

1

1TKE�1

K 1K= max

fSigKi=1ln�

1TKE�1

K 1K�

29

4 Bound on minimum distance

To o(�2) we have bounds

D��min = max

fSigKi=1min

i6=jD(SikSj) � �2

8

(TM)2

(2p � 1)2>�2(TM)2

128

K�2=T

(a)

(0; 0) TM

TM

! j TM2p�1

j

(b)(0; 0) TM

TM

TM2

TM2

! j j

TM=2

2p�1

Figure 12. Constellations of signal matrix singular values

30

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

log2(k)

Ro

Finite K cutoff rate curve: M=2, eta=2, T=4*M

N= 1N= 2N= 3N= 4

31

0 1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

3

log2(k)

Ro

Finite K cutoff rate curve: M=2, N=1, eta=2, T=2*M

Kc

Ko

Figure 13. Cutoff-rate curve as function of size K.

32

Define:

Ko = bT=Mc: “orthogonal size”

= max valueK for which closed form expression~Ro exists

and

Kc: “logK” transition point

= knee of ~Ro

) diminished returns by increasingK beyondKc

33

5 Bound on logK transition point of constellation

Pick “test constellation”fSigKi=1 for which

Dmin = mini6=jD(SikSj) > K�2=T

= �2(TM)2

128 .

~Ro(K) � maxfPiglog

1Pi;j PiPje�ND(SikSj)

!

� maxfPiglog

1Pi;j PiPj +P

i 6=j PiPje�NDmin

!

= log�

1

1=K + (K � 1)=K e�NDmin

�34

> log�

1

1=K + (K � 1)=K e�N K�2=T

�

= log(K)� log�

1 + (K � 1) e�N K�2=T�

� log(K); (K � 1) e�N K�2=T � 1

This gives lower bound onKo

Ko �n

K : K2=T lnK = No

35

5 10 150

2

4

6

8

10

12

14

16

log

2(Kc)

T

Kc as a function of T for eta=1

M=2,N=2M=3,N=3M=4,N=4M=5,N=5

Figure 14. Corner sizes for equal M and N .

36

6 Low Dimensional ConstellationsK � Ko

For given�, T andM define the integerMo

Mo = argmaxm2f1;:::;Mg�

m ln(1 + �TM=(2m))2

1 + �TM=m

�:

First a result on max attainable distance under peak power constraint

Proposition 2 Let2M � T . Then

Dmax

def

= max

S1;S22SKpeakD(S1kS2) =Mo ln(1 + �TM=(2Mo))2

1 + �TM=Mo

: (1)

Furthermore, the optimal signal matrices which attainDmax can be taken

as scaled rankMo mutually orthogonal unitaryT �M matrices of the

form

S1 = �TM �1; S2 = �TM �2

37

where, forj = 1; 2,

�Hj �j = IMo ; and �Hi �j = 0; i 6= j:

Proof is based on alternative representation forD(S1kS2)

D(S1kS2) = 12lnjIM+�

2SH1 S1j2jIM+�

2SH2 S2j2

jIM+�SH1

S1jjIM+�SH2

S2j��IM � �H���2 ;

where� is aM �M multiple signal correlation matrix� = ~SH2 ~S1

~Si =�

2Si [IM +�

2SHi Si]�1

38

0 20 40 60 80 100 1200

10

20

30

40

Mo

0 5 10 15 200

2

4

6

8

Mo

SNR

Figure 15. Top panel: Mo as a function of the SNR param-eter �TM . Bottom panel: blow up of first panel over areduced range of SNR.

39

Proposition 3 Let2M � T and letMo be as defined in (1). Suppose that

Mo � minfM;T=Kg. Then the peak constrainedK dimensional cut-off

rate is~Ro(K) = ln�

K

1 + (K � 1)e�NDmax

�

andDmax is given by (1). Furthermore, the optimal constellation

attaining ~Ro(K) is the set ofK rankMo mutually orthogonal unitary

matrices and the optimal probability assignment is uniform:P �i = 1=K,

i = 1; : : : ;K.

40

Example constellations forT �M = 4� 2

�Mo = 1, K = 4: (�2TM < 4:8)fSigKi=1 =

8>>>>><>>>>>:

26666641 0

0 0

0 0

0 03

777775 ;2

6666640 0

1 0

0 0

0 03

777775 ;2

6666640 0

0 0

1 0

0 03

777775 ;2

6666640 0

0 0

0 0

1 03

7777759>>>>>=

>>>>>;

�Mo = 2, K = 2: (�2TM � 4:8)

fSigKi=1 =8>>>>><

>>>>>:2

6666641 0

0 1

0 0

0 03

777775 ;2

6666640 1

1 0

0 0

0 03

7777759>>>>>=

>>>>>;41

7 Conclusions� Peak power contrained cut-off rate reduces to minimizing Q-form

� optimal constellation equalizes the decoder error rates

� Average distance for optimalK-dim constellation decreases at most

byK�2=T

� Optimal low rate constellation is a set of scaled mutually orthogonal

unitary matrices.

� Rank of the unitary signal matrices decreases in SNR

� For very low SNR, no diversity advantage: apply power to a single

antenna element at a time.

42

References

[1] A. O. Hero and T. L. Marzetta, “On computational cut-off rate for space time coding,”

Technical Memorandum, Bell Laboratories, Lucent Technologies, Murray Hill, NJ,

2000.

[2] I. Abou-Faycal and B. M. Hochwald, “Coding requirements for multiple-antenna

channels with unknown Rayleigh fading,” Technical Memorandum, Bell Laboratories,

Lucent Technologies, Murray Hill, NJ, 1999.

[3] T. L. Marzetta and B. M. Hochwald, “Capacity of a mobile multiple-antenna

communication link in Rayleigh fading,”IEEE Trans. on Inform. Theory, vol. IT-45, pp.

139–158, Jan. 1999.

43