Optimal Signal Constellations for Fading Space-Time Channels

Transcription

1 Optimal Signal Constellations for Fading Space-Time Channels 1. Space-time channels Alfred O. Hero University of Michigan - Ann Arbor Outline 2. Random coding exponent and cutoff rate 3. Discrete K-dimensional constellations 4. Bound on minimum distance 5. Low dimensional constellations. Conclusions 1

2 s t1 s t2 s t3 ; t = 1;:::;T 2 3 h12 h11 H = h22 Transmitter h21 h32 4 h31 5 x t1 x t2 ; t = 1;:::;T T =coherent fade interval M=number of transmit antennas N=number of receive antennas = receiver SNR Receiver Figure 1. Narrowband space time channel for M = 3, N = 2 2

3 [x l t1 ;:::;xl tn ] = p [s l t1 ;:::;s l tm ] X l = p S l H l + W l h l m1 h l mn Received signal in l-th frame (t = 1;:::;T) 2 l 11 h l 1n h [wl t1 ;:::;wl tn ]; or, equivalently 4 ffl X l : T N received signal matrices ffl S l : T M transmitted signal matrices ffl H l : i.i.d. M N channel matrices ο Cl N (0;I M N IN ) ffl W l : i.i.d. T N noise matrices ο Cl N (0;I T N IN ) 3

4 Block coding over L frames produces blocks of L symbols [S 1 ;:::;S L ] where S = S l is selected from a symbol alphabet S Random Block Coding: select S l at random from S according to probability distribution P 2 P. ffl Objective: Find optimal distribution P (S) over P ffl Optimality criteria: capacity, outage capacity, random coding error exponent, cut-off rate Transmitter constraints: ffl average power E[kSk constraint: ] = R ksk 2 dp» TM 2 peak power constraint: ksk 2» TM,forallS 2 S where ksk 2 = trfs H Sg 4

5 cos(w t) o X s(t) temporal encoder S/P Z -1 X cos(w t) o Figure 2. First generation space-time coding: Seshadri&Winters FDD transmitter with delay diversity (1994) 5

6 max E[I(S; XjH)] = (SjH) P Capacity Results: (avg. power constraint - Telatar, BLTM 95): 1. Channel H l Known to Txmt and Rcv: Capacity: (bits/sec/hz) or( ) bits=channel use T C = max E[H(XjH) H(XjS; H)] (SjH) P ff-outage Capacity: C = fc o : P (C(H) > C o ) = ffg

7 + Since H(XjH)» ln(ji N + H H R s Hj); and H(XjS; H) = H(N) = max C(H) s : trfr s g»m ln(ji N + HR s H H j) R where, for H = UDV = ln(ji + HR o s Hj) 1 μ R o s = V H diag V and μ is such that (water-filling) d 2 i trfr o sg = M

8 coded modulation s(t) temporal encoder S/P V Beamformer coded modulation Figure 3. Second generation space-time coding with beamforming 8

9 2. Channel H l known only to Rcv: = C max = max E[H(XjH) E[I(X; H(XjS; SjH)] H)] (S) P (S) P ) C = E log fi fi IN + M HH H fi fi Λ Capacity achieving distribution: ffl S Gaussian with orthogonal rows and columns of identical energy ) BLAST (Foschini, BLTJ 199) ) Space time 4-PSK/4-TCM (Tarokh&etal IT 98, Tarokh&etal COM 99) In practice must transmit training within each frame to learn H 9

10 Capacity bounds: (Hochwald&Marzetta SPIE99, Driesen&Foschini COM99) + MN) z } log(1» C(H)» min(m;n) log + MN 1 min(m;n) = 00 when rank(h)=1 z } 00 when rank(h)=min(m;n) = 10

11 coded modulation s(t) temporal encoder S/P coded modulation 4-PSK/4-TCM: 2 bits/sec/hz (simulation), M=N=2 ffl BLAST: 1.2 Mbps over 30kHz (40 bits/sec/hz) in 800MHz band, M=8, N=12 ffl Figure 4. Second generation space-time coding 11

12 Coherent Transmission and Reception T/R know channel M=32 M=1 M= 4 M= 1 b/s/hz SNR (db) 12

13 Incoherent Transmission R knows channel M=32 M=1 M= 4 M= 1 b/s/hz SNR (db) 13

14 1 Effect of Incoherent Transmission C/C o M=32 M=1 M= 4 M= SNR (db) 14

15 b/s/hz Coherent Transmission and Reception with Training Errors: T train = M=32 M=1 M= 4 M= SNR (db) 15

16 1 Effect of Training Errors (coherent transmission): T train = C/C o M=32 M=1 M= 4 M= SNR (db) 1

17 3. Slow fading Rayleigh channel: H unknown Capacity? (Marzetta&Hochwald, BL TM 98, IT 99) ffl C = E [log P (XjS)=P (X)] (bits/channel-use) Capacity achieving distribution? ffl S = ΦV where Φ and V are mutually independent matrices Φ: T M unitary: Φ H Φ = I M ffl V : M M real diagonal ffl V! ci M 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) 1

18 s(t) temporal encoder S/P Space-Time Modulation Figure 10. Third generation space-time coding. 18

19 1 = 1 p Φ 8 S = fφ 1 ;:::;Φ K g; Φ k = k Φ 1 1 e j 2ß 8 5 e j 2ß 8 1 e j 2ß 8 2 e j 2ß e j 2ß 8 e j 2ß e j 2ß 8 1 e j 2ß 8 1 e j 2ß 8 e j 2ß e j 2ß 8 3 e j 2ß e j 2ß 8 () Example: Unitary space-time constellation (Hochwald&etal BLTM 98) ffl T = 8, M = 3, K = 25 unitary signal matrices e j 2ß 8 (0) 0 ; = 0 1 e j 2ß

20 1 Random Coding Error Exponent The minimum error probability of any decoder of a block code over L frames satisfies (Fano 1) min P e» e LE U (R) ; R < C where R: symbol rate (nats/symbol) ffl C: channel capacity (nats/symbol) ffl E U (R): error exponent ffl 20

21 E U (R) =»Z ( Z ln μr 1=(1+μ) ) 1+μ dx ; nats=sym (S) dp (p(xjs)) max max P 2P μ2[0;1] X2X S2S E U (R) has been studied under avg. power constraint for ) Known H (Telatar BL TM 9) ) Unknown H (Abou-Faycal & Hochwald BL TM 99) 21

22 R o R c R o E U (R) C R Figure 11. Error exponent E U (R) and cut-off rate bound y(r) = R o R. 22

23 R» R o p p(xjs)dp (S) 2 dx; R o computational cut-off rate lower bound (Gallager IT 4) E U (R) R o R; Z»Z o = max R 2P P ln nats=symbol X2X S2S where P are suitably constrained distributions over Cl T M ) Cut-off rate analysis has been used to evaluate practical coding limits (Wang&Costello COM 95, Hagenauer&etal ffl IT 9) ffl different coding and modulation schemes (Massey 4) ffl signal design for optical fiber links (Snyder&Rhodes IT 80) ffl signaling over multiple access channels (Narayan&Snyder IT81) 23

24 R» R o FACTS: R o» C ffl R o is highest practical rate for sequential decoders (Savage 5) ffl E U (R) ß R o R when R ß R c, the critical rate ffl R o specifies upper bound on optimal decoder error ffl P e» e L(R o R) ; 24

25 2 Integral Representation for R o Z Z ln dp (S 1 ) dp (S 2 ) e ND(S 1kS 2 ) : o = max R 2P P where S 1 2S S 2 2S D(S 1 ks 2 ) Low SNR approximation: def 1 2 ln j I T + 2 (S 1S H 1 +S 2S H 2 ) j 2 = T + S 1 S H 1 jji T + S 2 S H 2 j : ji D(S 1 ks 2 ) = 2 =8kS 1 S H 1 S 2 S H 2 k 2 + o( 2 ) 25

26 3 h Λ 5 The following parallels Theorems 1 and 2 of Marzetta&Hochwald IT 99 Proposition 1 Assume that the transmitted signal S is constrained to satisfy the peak power constraint ksk 2» MT. There is no advantage to using M > T transmit antennas. Furthermore, for M» T the signal matrices achieving R o can be expressed as 2 4 S = i where Φ ffl Φ is T M unitary matrix V H V = I M ffl Λ is M M non-negative diagonal matrix. 2

27 max ;S g K ln i=1 i i fp KX ffl P = [P 1 ;:::;P K ] T min ;S g P E T K P i=1 K i i fp 3 Case of Discrete K-dimensional Constellations Specialize P to the discrete distributions over Cl T M Then R o = ~ R o (K) is given by P j e ND(S iks j ) = ln where i=1 P i K X j=1 E k = ((D(S i js j )) K i;j=1 : dissimilarity (distance) matrix ffl 2

28 o (K) = ln min ~R i g K i=1 fs minimized for equalizer probability P = P Λ min g P E T K P i=1 K i fp Under peak power constraint, ks i k» TM,! ψ Inner maximization: min : 1 T K P =1 Φ T P E K P Ψ >0 P Lagrangian J(P ) = P T E K P 2c(1 T KP 1) 28

29 KX Fact: optimal constellation E satisfies 1 1 K 0 K o (K) = ln min ~R i g K i=1 fs P j e ND(S iks j ) = c E K P Λ = c1 K ) j=1 Λ = ce 1 K 1 K ; and c = 1 P and T K E 1 K 1 K 1 1 T KE 1 1 max = g K ln i=1 1 T KE 1 K 1 K i fs K 1 K 29

30 j TM 2 p 1 j ψ! TM 2 j j ψ! TM=2 K 2=T 4 Bound on minimum distance To o( 2 ) we have bounds ΛΛ min = D 2 max K min i=1 i=j D(S iks j ) 8 g fs i 2 2 (TM) 1) 2 > (TM) 2 p (2 128 TM TM TM (0; 0) TM (0; 0) TM 2 2 p 1 (a) (b) Figure 12. Constellations of signal matrix singular values 30

31 5 Finite K cutoff rate curve: M=2, eta=2, T=4*M N= 1 N= 2 N= 3 N= 4 4 R o log (k) 2 31

32 3 Finite K cutoff rate curve: M=2, N=1, eta=2, T=2*M R o K o K c log 2 (k) Figure 13. Cutoff-rate curve as function of size K. 32

33 ) diminished returns by increasing K beyond K c Define: K o = bt=mc: orthogonal size for which closed form expression ~ R o exists =maxvaluek and K c : logk transition point = knee of ~ R o 33

34 o (K) max ~R i g fp min = min D D(S iks j ) > flk 2=T i=j 1 P ip j e ND(S iks j ) Pi;j 1 P ip j + P i=j P ip j e ND min Pi;j 1 + (K 1)=K e ND min 1=K 5 Bound on logk transition point of constellation Pick test constellation fs i g K i=1 for which = (TM) fl! ψ log! ψ log max i g fp = log 34

35 > log 1 + (K 1)=K e NflK 2=T 1=K 1 + (K 1) e NflK 2=T = log(k) log This gives lower bound on K o ß log(k); (K 1) e NflK 2=T» 1 K o n K 2=T o ln K = fln : K 35

36 M=2,N=2 M=3,N=3 M=4,N=4 M=5,N=5 K c as a function of T for eta=1 10 log 2 (K c ) T Figure 14. Corner sizes for equal M and N. 3

37 ρ = o argmax m ln m2f1;:::;mg M (1 + TM=(2m))2 S 1 = TM Φ 1 ; S 2 = TM Φ 2 Low Dimensional Constellations K» K o For given, T and M define the integer M o ff : + TM=m 1 First a result on max attainable distance under peak power constraint Proposition 2 Let 2M» T.Then D max def max = 2 K 1 ks 2 D(S ) = M o ln (1 + TM=(2M o)) 2 : (1) 1 + TM=M o peak 2S ;S 1 S Furthermore, the optimal signal matrices which attain D max can be taken as scaled rank M o mutually orthogonal unitary T M matrices of the form 3

38 » = ~ S H 2 ~ S 1 ~S i = 2 S i [I M + 2 SH i S i ] 1 where, for j = 1; 2, Φ H j Φ j = I Mo ; and Φ H i Φ j = 0; i = j: Proof is based on alternative representation for D(S 1 ks 2 ) 1 ks 2 ) = 1 2 ln j I M + 2 SH 1 S 1j 2 ji M + 2 SH 2 S 2j 2 D(S M + S H 1 S 1jjI M + S H 2 S 2j ji fi fi» H» fi fi 2 ; IM where» is a M M multiple signal correlation matrix 38

39 M o M o SNR Figure 15. Top M panel: o as a function of the SNR parameter TM. Bottom panel: blow up of first panel over a reduced range of SNR. 39

40 ~R o (K) = ln K + (K 1)e ND max 1 Proposition 3 Let 2M» T and let M o be as defined in (1). Suppose that M o» minfm;t=kg. Then the peak constrained K dimensional cut-off rate is D and max is given by (1). Furthermore, the optimal constellation attaining R o (K) is the set of K rank M o mutually orthogonal unitary matrices and the optimal probability assignment is uniform: ~ Λ i = 1=K, P = 1;:::;K. i 40

41 >< >: >< >: >= >; >= >; Example constellations T M = 4 2 for M o = 1, K = 4: ( 2 TM < 4:8) ffl fs i g K i=1 = ; 5 ; 5 ; ffl M o = 2, K = 2: ( 2 TM 4:8) fs i g K i=1 = ;

42 Conclusions ffl Peak power contrained cut-off rate reduces to minimizing Q-form ffl optimal constellation equalizes the decoder error rates Average distance for optimal K-dim constellation decreases at most ffl K by 2=T Optimal low rate constellation is a set of scaled mutually orthogonal ffl unitary matrices. ffl Rank of the unitary signal matrices decreases in SNR For very low SNR, no diversity advantage: apply power to a single ffl antenna element at a time. 42

43 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, [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, [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 , Jan