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On the SNR Exponent of Hybrid Digital Analog Space Time Coding

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. s. s. s. K. 1. 2. ;. ;. :. :. :. ;. ^. ^. s. s. K. 1. ;. :. :. :. ;.

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On the SNR Exponent of Hybrid Digital Analog Space Time Coding

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  1. On the SNR Exponent ofHybrid 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. s s s K 1 2 ; ; : : : ; ^ ^ s s K 1 ; : : : ; What this talk is about • Transmit an analog source over a MIMO channel T uses of the channel Quasi-static (Block) Fading Encoder Receiver • Performance criterion : End to End quadratic distortion • How to best use multiple antennas to minimize distortion? Wireless Communications Lab, TAMU

  4. s s s K 1 2 ; ; : : : ; ^ ^ s s K 1 ; : : : ; 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 Quasi-static (Block) Fading Encoder Receiver Wireless Communications Lab, TAMU

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

  6. N M £ C ( ) h h H C N 0 1 2 w e r e » i j H ( ) E [ ] ( [ ] ) h f l l l M N M N , l ; d h h i i X X X M T i i · t t · t t t > ( ) ; h d h l T i i i e c a s e o o w s a s a s m p e e x e n s o n x x s n o r m a z e s u c a r t t = ( ) T 1 d h l S N R S N R i i i s e u r a o n n c a n n e u s e s t t t t ; : : : ; , ½ e n o e s e g n a - o - o s e a o p ½ H T 1 t + y x w = = t t t M ; ; : : : ; Block Fading Channel Model Wireless Communications Lab, TAMU

  7. Diversity Multiplexing Tradeoff • Channel has zero capacity, we talk of outage capacity • For Pe() ! 0, we need SNR !1 • Makes sense to look at the exponent of SNR • Diversity gain : Pe() /-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 Wireless Communications Lab, TAMU

  8. f ( ) g C d f l f d h C i i i i t l R o n s e r a a m y o s p a c e - m e c o n g s c e m e s ½ r o g ½ = r ( ) d ¡ c c c F l P r c ( ) ( ) o r a r g e ½ / ½ O l d d ¤ h h l i h l l i i i i i i t t t t t t o u a g e p m a e x p o n e n w e r e a s r a e n c r e a s e s r w s u a p s r s e m u p e x n g g a n ½ r r o g ½ r , = = c c c c Optimal D-M Tradeoff (Zheng and Tse) Wireless Communications Lab, TAMU

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

  10. ( ) N 0 1 s » i ; ¢ 1 2 E ( ) [ j j ] e D ( ) ( ) s s s ¡ l D ¡ K ½ 2 K 1 2 s s ½ R f d f M S E h S N R i D a ´ = o g ½ ( ) t t ; ; : : : ; S N R E l i a e o e c a y o w : f ( ) g K / ½ ¡ t F l f h l d h S C i i ´ x p o n e n : = a ´ m = ^ ^ a m y o s o u r c e - c a n n e c o n g s c e m e s ½ T ½ 1 l ( ) ( ) ! ? s s ´ o g ½ K 1 a ´ s u p a ´ = ; : : : ; Precise Problem Statement Fading Channel Source Channel Encoder Receiver SNR = Wireless Communications Lab, TAMU

  11. l l b b R R i i t t r r o o g g ½ ½ s s = = c s c s E h l T h i i 1 t t t t t q u a n g e x p o n e n s 2 w e g e e r e s u n e o r e m ¤ ( ) ¡ d ¡ ( ) ( ) ( ) S h R K R T R R i ( ) r D R D R P , D R · r · + + c n c e w e a v e ´ c a e = = ½ a ½ ´ Q s c c s s e p s s , s e p s Q Exponent for Separation Scheme MIMO Channel Sp-Time Code Not in outage In outage Wireless Communications Lab, TAMU

  12. Wireless Communications Lab, TAMU

  13. [ ] h h i b l b i T E 1 t t e o r e m x p o n e n a c e v a e y s e p a r a o n h ´ ? ? ( ( ) ( ) ( ) ) ( ) d d j j j j j 2 1 1 2 1 ¡ ¡ ¡ ¡ j 2 ( ) f M j 1 2 o r a ´ ´ = = ( ( ) ( ) ) ( ) ( ) s e p d d d d j j j j 2 1 1 ; ; : : : ; + ? ¡ ¡ ? ? ¡ ? , ´ h b l b d h l d h i i i ¤ t s a c e v a e y a a n e m s o u r c e - c a n n e c o n g s c e m e Main Results – Separation based approach • Comments – Separation is not optimal • Can we outperform the optimized separated scheme ? • Yes, we will see that later Wireless Communications Lab, TAMU

  14. h i 1 E ( ) D ¸ ½ d = 2 t H ( ) H ´ I H H ¡ ¢ + e ( ) l d H I H H R ½ · t + o g e ½ c 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) Wireless Communications Lab, TAMU

  15. [ ] h f d i b d T I 2 t t t e o r e m n o r m e r a n s m e r u p p e r o u n ( ) T h l d S N R b d d b ? i i i i t t t t e o p m a s o r o n e x p o n e n s u p p e r o u n e y a ´ M ½ ¾ 2 X ( ) j j ( ) M N i i 2 1 1 ¡ + ¡ a ´ m n = b u ; ´ i 1 = ¤ Upper bound Wireless Communications Lab, TAMU

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

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

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

  19. 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 Wireless Communications Lab, TAMU

  20. K ( ( ) ) d M M T £ £ ^ l ^ b X X K C C a i t d 2 2 M 2 e s s r o g e s s ½ s K K K 1 1 1 s ; ; ; : : : : : : : : : ; ; ; 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 Space-Time Encoder Quantizer - Reconstruct Spatial Multiplexer Wireless Communications Lab, TAMU

  21. QAM Wireless Communications Lab, TAMU

  22. [ ] h b i d h l b d T H 3 T h b h h d i i e o r e m y r s c e m e o w e r o u n t t t t t t s s e e r a n e s e p a r a e e x p o n e n ( ) F B W h h b d d l l H D A d h i i i i i i i t t t o r e x p a n s o n e y r g a - a n a o g s p a c e - m e c o n g a c e v e s , µ ¶ ( ) ( ) ( ) d d ? ? j j j j 2 1 1 1 1 ¡ ¡ ¡ ¡ ( ) ( ) 1 1 + ¡ a ´ = h b d i y r 2 1 ( ) ( ) M d d j j 1 ? ? ¡ + ¡ ¡ ´ M ´ Exponent of Hybrid Digital Analog Coding Schemes Wireless Communications Lab, TAMU

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

  24. Wireless Communications Lab, TAMU

  25. [ ] h b i d h l b d T H 3 e o r e m y r s c e m e o w e r o u n B d d h C i i t a n w o m p r e s s o n : M 2 ( ) M 2 ¸ a ´ ´ = h b d i y r ; ´ Exponent of HDA Schemes • It is remarkable that this is equal to the upper bound • Hence it is optimal Wireless Communications Lab, TAMU

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

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

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

  29. 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 Wireless Communications Lab, TAMU

  30. 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 Wireless Communications Lab, TAMU

  31. 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 Wireless Communications Lab, TAMU

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