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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
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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
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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
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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
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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
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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
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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
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S/Ptemporal
encoder
s(t)
coded modulation
coded modulation
VBeamformer
Figure 3. Second generation space-time coding with beamforming
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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
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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)
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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
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−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
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−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
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−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
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−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
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−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
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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)
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S/Ptemporal
encoder
s(t)
Modulation
Space-Time
Figure 10. Third generation space-time coding.
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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
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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
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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)
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EU (R)
Ro
CRoRc R
Figure 11. Error exponent EU (R) and cut-off rate bound
y(R) = Ro �R.22
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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)
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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
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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)
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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.
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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
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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 �
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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�
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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
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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
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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.
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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
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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
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> 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
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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 .
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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
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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
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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.
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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.
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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
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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.
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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.
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