1. Stratified Diagonal Layered SpaceTime Architecture Mathini Sellathurai* and Gerard J. Foschini** *Communications Research Centre, Canada (This work has been carried out at Bell Labs during an internship) ** Bell Laboratories, Crawford-Hill, NJ DMACS Workshop on Signal Processing for Wireless Transmission

2. (M,N) communication systems A communication structure uses multiple transmit (M) and multiple receive (N) antennas and provides enormous capacity in rich scattering environments. Heavy Scattering M transmit antennasN receive antennas Known as BLAST systems

3. Primitive bit stream Complex no. stream 1-Dim encoder * SPACETIME CODES DESIGNED USING 1-D CODER(S) (1(2(3(4(5(1(2(3(4(5 Transmit anennas... SPACE TIME   symbol duration **HOW ARE 1-D CODED STREAM(S) ARRAYED IN SPACETIME TO ENABLE 1-D RECEIVER SIGNAL PROCESSING? *** CAN WE ACHIEVE NEAR CAPACITY IN A QUASI- STATIC FADING ENVIRONMENT? LAYERED SPACETIME CODES

4. BLAST: Prior Work *Introduction by Foschini (’96) –Diagonal-Layered Spacetime : Foschini (’96) Wrapped spacetime coding: Caire and Colavolpe (‘2001) Trellis coding for D-BLAST: Wesel and Matache (‘2002) –Horizontal Layered Spacetime (also known as V-BLAST): Foschini and colleagues (’98) Codes designed for AWGN channels –Iterative detection for BLAST Hammons & El Gamal (’99), Sellathurai & Haykin (’99), Ariyavastikul (’00) **Stratified Diagonal Layered SpaceTime (SD-BLAST)

5. NARROWBAND, QUASI-STATIC LONG BURST PERSPECTIVE CHANNEL MATRIX, G, UNKNOWN TO TRANS, KNOWN TO REC EQUATION FOR N-D RECEIVE VECTOR, r (M,N) CHANNEL ENVIRONMENT s: M-D trans vector, power P/M per transmit antenna : N-D AWGN noise vector, var /component G: M x N matrix Gaussian channel. Entries are independent and identically distributed complex N(0,1) random variables

6. (M,N) Generalization of Shannon’s (1,1) Capacity Formula LogDet Formula Formula used in inventing BLAST Capacity Lower Bound

7. 5:1 Demux Time Primitive data stream Code/mod (1(2(3(4(5(1(2(3(4(5 Space DIAGONAL BLAST Capacity Lower-bound (5,5) system with interference nulling and cancellation... THE RECEIVER BASED ON MAXIMIZING THE SIGNAL TO INTERFERENCE AND NOISE RATIO OF THIS ARCHITECTURE “ATTAINS” LogDet FOR ALL M AND N, AND ALL  AND ALL CHANNELS.

8. DIAGONAL BLAST (D-BLAST) ADVANTAGES - THEORETICAL STANDPOINT LogDet capacity “attained” for all (M,N) and all channels all SNRs with 1-D codecs... Wasted Burst duration Time Five point space............. BURST Payload may not be adequate Wasted spacetime at start and at end of a burst can result in not enough layers (coded blocks). Somewhat problematic to code for periodic SINR DISADVANTAGES Symbol time  ||  THIN: (one symbol) Poor code but little waste

9. 5:1 Demux Primitive data stream... Code/mod (1(2(3(4(5(1(2(3(4(5 Time Performs well and is easy to implement (Extremely Practical) Encoders can be 1-D codes designed for AWGN Receiver uses interference avoidance and cancellation Achieves near capacity for N>>M HORIZONTAL BLAST Capacity lower-bound: Maximum capacity can be achieved by optimizing *the number of antennas used * detection order.

10. CAPACITY CONTRASTS 16 RECEIVE ANTENNAS,  = 18 dB, OPEN LOOP - COMPLEMENTARY DISTRIBUTION FUNCTIONS - Outage capacity Receive diversity

11. ……….. (M,N) BURST PERFECTLY PACKED WITH M “CONTINUOUS DIAGONALS” Burst duration Time Space five transmit antennas EX: M = 5 WITH GREEN AND BLUE “CONTINUOUS DIAGONALS”, THERE ARE TWO OTHERS. - ISSUE: POSSIBILITY OF EFFICIENT COMMUNICATION OVER THE M PARALLEL 1-D “HELICAL LAYERS” - Used with iterative interference cancellation detection We will introduce Stratified Diagonal Layers (SD-BLAST).

12. Stratified Diagonal BLAST Fundamentals of stratification SD-BLAST transmitter-receiver structure Mutual information of SD-BLAST: Highlights of the derivation Outage capacity evaluation of SD-BLAST: Monte-Carlo method

13. STRATIFICATION AND CAPACITY C = log 2 [1+(P/  2 )] bps/Hz Assume infinite signaling intervals SIGNAL OF POWER P + AWGN (  2 ) REPLACE BY K SIGNALS OF TOTAL POWER P WITH SUITABLE ENCODING THE CAPACITY DOESN’T CHANGE. Exploited in multi-level coding schemes to achieve high band efficiency using binary codes Equal powers and various rates Various powers

14. WHY STRATIFY ALL M = 5 TRANSMISSIONS? No stratification. Only purple layer is stratified as shown to left. Stratification is harmless but pointless. Stratification effective when all five are stratified. For example, now suffers less interference. PLY

15. CYCLE EVERY SYMBOL      1:5 1:4 DEMUX 1:4 CODE/MOD PRIMITIVE DATA STREAM DEMUX SD-BLAST: Transmitter Stratification

16. STRATIFIED DIAGONALS: KEY FEATURES... Time Symbol duration  ||  - CONTINUOUS DIAGONALS: Good code, spacetime perfectly packed - no waste, binary alphabet acceptable. *DIAGONALIZATION  TRANSMIT ANTENNAS FARE DIFFERENTLY STRATIFICATION  SELF INTERFERING VECTOR SIGNALS ONION PEELING RECEIVER  EACH STRATUM FACES DIFFERENT MUTUAL INFORMATION

17. Detection Peeling away of successive plies (five strata each) from outside-in. White Noise Each onion ring (ply) is coded using 1-D Codes (for example Turbo or LDPC codes ) Equal powers and various rates Various powers and various rates r2r2 r3r3 r4r4 r1r1

18. Mutual Information of SD- BLAST: Highlights Power into a transmit antenna in an (M,N) system Power into a stratum in an (M,N) system with n stratification

19. Each helix winding on the i th strata has constituents: s mi (t), signal from m th ant. on i th strata with interf. + noise N-D vector signal g m s mi, plus spatially colored “noise” PEELING OFF ONE PLY  PEELING OFF ITS M HELICAL CONSTITUENTS r m =

20. Whitening Matched Filter Receiver Identify “additive noise”,  mi, and convert impaired signal to form for Max Ratio Combining (MRC) Additive Noise: Whitening and MRC: Whitened Noise Noise covariance

21. Mutual Information Signal-to-Interference plus noise ratio over a symbol-stratum interval Mutual information of a stratum is

22. Mutual Information (MI) of a Ply MI added by the ith strata PLY M Asymptotic MI as Small approximation Sum goes over to an integral

23. Asymptotic sum capacity As, asymptotic sum capacity Achieves LogDet Capacity

24. Outage capacity Unlike traditional outage capacity definition, in SD- BLAST we need to define outage capacities for –n-strata capacities C i, i=1,2,….,n. –For comparison purposes, we also define outage capacity based on sum capacity. How much outage capacity can we achieve using finite strata (n)?

25. Outage Capacity: Upper-bound The traditional outage capacity is based on the sum capacity However, we have additional demands to satisfy: if any one of the strata fails to satisfy its corresponding capacity demand, then the message will fail. Thus, we need to relax the capacity demand. Say, the sum capacity demand at outage level is

26. Outage capacity-lower bound We relax (reduce) the demand on the outage capacity Then minimize the relaxation A tighter capacity bound can be found by minimizing

27. SD-BLAST and Eigenvalue Hardening SD-BLAST achieves capacity –All (M,1) and (N,1) case (single eigenvalue) –As set of eigenvalues of (M,N) systems harden, The capacity accumulated through each stratum is a function of eigenvalues of the channel matrix

28. DEALING WITH UNCERTAINTY IN BITS/SEC/PLY - Monte-Carlo Method - Q%: OUTAGE REQUIRED WITH K PLIES COMPUTE WEAKER TOTAL CAPACITY DEMANDS FOR OUTAGE q% < Q% CALCULATE Q% OF CHANNELS NOT MEETING WEAKER DEMANDS FOR STRATA CAPACITIES (CLEARLY Q% > q%) ITERATE ON (q,Q) UNTIL Q%  Q% Required Q  Outage (%) 0 QqQq  Total capacity  Strata capacity iterate

29. Numerical Results Finite M, N

30. (8,1) SD-BLAST, SUM CAPACITY vs AVERAGE SNR at 10% outage M=8, N=1 -50510152025 0 1 2 3 4 5 6 7 8 n=1 2 4 8 16 32 64 Average SNR (dB) Capacity (bits/sec/Hz) SD-BLAST D-BLAST V-BLAST Power optimization (M,1) case

31. (8,3) SD-BLAST, SUM CAPACITY vs AVERAGE SNR at 10% outage -50510152025 0 5 10 15 20 n=1 2 4 8 16 32 64 Average SNR ( dB) Capacity (bits/sec/Hz) M=8, N=3 SD-BLAST D-BLAST V-BLAST SD-BLAST (Upper Bound) SD-BLAST (1-64 Strata True Capacity) M > N case

32. SUM CAPACITY CONTRASTS (4,2), (8,3) and (16,5) systems  = 10 dB, OPEN LOOP - COMPLEMENTARY DISTRIBUTION FUNCTIONS - 02468101214161820 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Probability {Capacity > Rate}= Rate (bps/Hz/dim) (16,5) (8,3) (4,2) SD-BLAST, n=64 SD-BLAST, upper-bound D-BLAST V-BLAST M > N case

33. SUM CAPACITY CONTRASTS (2,4) systems  = 10, 18 dB, OPEN LOOP - COMPLEMENTARY DISTRIBUTION FUNCTIONS - CCDF for (2,4) system M < N case

34. Conclusions SD-BLAST ARCHITECTRE OFFERS ENORMOUS CAPACITY (NEAR-CAPACITY) -HIGH BAND WIDTH EFFICIENCY -AVOIDS WASTE OF SPACETIME RESOURCE -IMPLEMENTATION WITH 1-D CODECS -BINARY CODES - CAPACITY CAN BE APPROACHED FOR- –( M,1) systems –As set of eigenvalues harden

35. Acknowledgement – Dr. Reinaldo Valenzuela Director Wireless Communications Research Department, Bell Labs, Lucent Technologies, Holmdel, NJ, for financial assistance –Members of wireless research group, Holmdel, NJ, for many interesting discussions. Thank you