1. On the SNR Exponent of Hybrid Digital Analog Space Time Coding Krishna R. Narayanan Texas A&M University http://ee.tamu.edu/~krn Joint work with Prof. Giuseppe Caire University of Southern California

2. Wireless Communications Lab, TAMU

3. What this talk is about  Transmit an analog source over a MIMO channel Encoder s 1 ; s 2 ; ::: ; s K ^ s 1 ; ::: ; ^ s K Receiver Quasi-static (Block) Fading  Performance criterion : End to End quadratic distortion  How to best use multiple antennas to minimize distortion? T uses of the channel

4. Wireless Communications Lab, TAMU Example 1: Streaming real-time video  Due to stringent delay constraints, we cannot code over many realizations of the channel  Channel changes from one block to another Encoder s 1 ; s 2 ; ::: ; s K ^ s 1 ; ::: ; ^ s K Receiver Quasi-static (Block) Fading

5. Wireless Communications Lab, TAMU Example 2: Broadcasting an analog source  Empirical distribution of H l converges to some ensemble distribution as l ! 1 Encoder s 1 ; s 2 ; ::: ; s K ^ s 1 ; ::: ; ^ s K User 1 ^ s 1 ; ::: ; ^ s K User 2 ^ s 1 ; ::: ; ^ s K User 3 H 1 H 2 H 3

6. Wireless Communications Lab, TAMU Block Fading Channel Model y t = p ½ M H x t + w t ; t = 1 ; ::: ; T ½ d eno t es t h e S i g na l - t o- N o i se R a t i o ( SNR ) X = [ x 1 ; ::: ; x T ], i snorma l i ze d suc h t h a tt r ( E [ X H X ]) · MT T i s t h e d ura t i on ( i nc h anne l uses ) M · N ( t h ecase M > N f o ll owsasas i m p l eex t ens i on ) H 2 C N £ M, w h ere h i ; j » CN ( 0 ; 1 )

7. Wireless Communications Lab, TAMU Diversity Multiplexing Tradeoff  Channel has zero capacity, we talk of outage capacity  For P e (  ) ! 0, we need SNR ! 1  Makes sense to look at the exponent of SNR  Diversity gain : P e (  ) /  -d  Multiplexing gain: Use many antennas to transmit many data streams  Zheng and Tse: Both are simultaneously achievable, there is a fundamental trade off of how much of each we can achieve

8. Wireless Communications Lab, TAMU Optimal D-M Tradeoff (Zheng and Tse)     C ons i d era f am i l y o f s p ace- t i meco d i n g sc h emes f C r c ( ½ ) g R c = r c l o g½ r c i s t h emu l i p l ex i n gg a i n O p t i ma l ex p onen t d ¤ ( r c ) = su p d ( r c ) w h ereasra t e i ncreasesw i t h ½ as r = r c l o g½ F or l ar g e ½, P ou t a g e / ½ ¡ d ( r c )

9. Wireless Communications Lab, TAMU d * (r) for Rayleigh Fading Channels d ¤ ( r ) = ( M ¡ r )( N ¡ r )

10. Wireless Communications Lab, TAMU Precise Problem Statement Source Channel Encoder s 1 ; s 2 ; ::: ; s K ^ s 1 ; ::: ; ^ s K Receiver Fading s i » N ( 0 ; 1 ) D ( ½ ) ¢ = 1 K E [ j s ¡ e s j 2 ] F am i l y o f source-c h anne l co d i n g sc h emes f SC ´ ( ½ ) g R a t eo fd eca y o fMSE w i t hSNR : D / ½ ¡ a ( ´ ) SNRE x p onen t : a ( ´ ) = ¡ l i m ½ ! 1 l o g D ( ½ ) l o g ½ a ? ( ´ ) = su p a ( ´ ) Channel SNR = ½ ´ = 2 K T

11. Wireless Communications Lab, TAMU Exponent for Separation Scheme Sp-Time Code Q MIMO Channel R s = r s l og½ b i t s R c = r c l og½ b i t s S i nce R s K = R c T, we h ave R c = ´ R s E q ua t i n g ex p onen t s, we g e tt h eresu l t i n Th eorem 1 D se p ( R s ) · D Q ( R s ) + a P ( e ) D se p ( R s ) · ½ ¡ 2 ´ r c + a ½ ¡ d ¤ ( r c ) Not in outageIn outage

12. Wireless Communications Lab, TAMU

13. Main Results – Separation based approach  Comments – Separation is not optimal  Can we outperform the optimized separated scheme ? Yes, we will see that later

14. Wireless Communications Lab, TAMU Upper bound on the exponent  An upper bound is obtained by assuming that the channel is instantaneously known at the transmitter  Coding rate is chosen according to  Large SNR behavior can be analyzed using techniques similar to those in Zheng and Tse (Wishart distribution, Varadhan’s lemma) R c ( H ) · l o g d e t ¡ I + ½ HH H ¢ D ( ½ ) ¸ E h 1 d e t ( I + ½ HH H ) 2 = ´ i

15. Wireless Communications Lab, TAMU Upper bound

16. Wireless Communications Lab, TAMU The case for analog coding schemes – known SNR n 1 ; ::: ; n k Channel code Q + Decoder s 1 ; ::: ; s k OR s 1 ; ::: ; s k + MMSE estimate n 1 ; ::: ; n k r s = 1 2 l og ( 1 + ½ ) D ( ½ ) = 1 1 + ½ D ( ½ ) = 1 1 + ½

17. Wireless Communications Lab, TAMU Separation scheme with Fading n 1 ; ::: ; n k Channel code Q + Decoder s 1 ; ::: ; s k  Separation based scheme is optimal only when ||h i || 2 = 1  Can be quite bad otherwise  Cannot exploit higher instaneous channel gain r s = 1 2 l og ( 1 + ½ ) h i

18. Wireless Communications Lab, TAMU Analog scheme with Fading s 1 ; ::: ; s k + MMSE estimate n 1 ; ::: ; n k  Analog scheme is simultaneously optimal for all SNRs  Graceful degradation of MSE with SNR h i D ( ½ ; h i ) = 1 1 + jj h i jj 2 ½

19. Wireless Communications Lab, TAMU Why Hybrid then?  Alas! things are never that easy  The optimality is valid only for the SISO channel with T = K  If more bandwidth is available, it is difficult to take advantage  Same with lesser bandwidth also  Hence, we need to look for hybrid digital and analog solutions

20. Wireless Communications Lab, TAMU HDA Solution for T > K (Bandwidth expansion)  Involves some math, but the exponent can be analyzed  Express MSE as a function of the Eigen values, use the Wishart distribution and Varadhan’s lemma Quantizer K r s l og½ b i t s - Reconstruct Space-Time Encoder X ( d ) 2 C M £ T d Spatial Multiplexer X ( a ) 2 C M £ K 2 M s 1 ;:::; s K ^ s 1 ;:::; ^ s K e 1 ;:::; e K

21. Wireless Communications Lab, TAMU QAM

22. Wireless Communications Lab, TAMU Exponent of Hybrid Digital Analog Coding Schemes Th i s i s b e tt er t h an t h esepara t e d exponen t

23. Wireless Communications Lab, TAMU HDA Scheme for T < K (Bandwidth Compression) Quantizer Space-Time Encoder Spatial Multiplexer + X ( D ) 2 C M £ T X ( a ) 2 C M £ T bits M a i n t r i c k i s i nc h os i n g ¯ t o b e d e p en d en t on SNR as ½ ¡ ° an d o p t i m i z i n g °

24. Wireless Communications Lab, TAMU

25. Exponent of HDA Schemes  It is remarkable that this is equal to the upper bound  Hence it is optimal T h eorem 3 [ H y b r id sc h eme l ower b oun d ] B an d w i d t hC ompress i on: a h y b r i d ( ´ ) = 2 M ´ ; ´ ¸ 2 M

26. Wireless Communications Lab, TAMU Upper bound for the Parallel Channel F or M p ara ll e l c h anne l s a u b = m i n ( M ; M 2 ´ )

27. Wireless Communications Lab, TAMU Scalar channel M = N = 1

28. Wireless Communications Lab, TAMU MIMO M = N = 2

29. Wireless Communications Lab, TAMU Comments  SISO Channel with fading Even if T > K, the best exponent is 1 More bandwidth does not buy us anything It is the degrees of freedom, not the bandwidth that is important For this case, the exponent for the entire region is fully known Gunduz-Erkip - infinite layers of superposition coding is optimal Our approach is much simpler

30. Wireless Communications Lab, TAMU MIMO Case  However, in the MIMO case things are different More bandwidth helps us buy diversity Antennas can be used to compress It is very easy to find practical schemes to get the correct exponent For example, uniform scalar quantization is optimal at high SNRs! We may require very large SNR for the asymptotics to kick in

31. Wireless Communications Lab, TAMU Outlook  Our conjecture is that the upper bound is loose (it is not entirely clear) for the bandwidth expansion case  Better constructive schemes to improve the achievable part  In some sense, the key problem is to find a better exponent for the parallel channel