Broadcasting over Fading Channelswith Mixed Delay Constraints

Transcription

1 Broadcasting over Fading Channels with Mixed Delay Constraints Shlomo Shamai (Shitz) Department of Electrical Engineering, Technion - Israel Institute of Technology Joint work with Kfir M. Cohen and Avi Steiner Information Theory and Applications Workshop San Deigo, USA February 10-15, / 27

2 1 Preliminaries 2 Non Cooperative Encoders (NCE) NCE scheme Outage and Broadcast Approach Joint Optimality of DC-Layering NDC Layering 3 DC Writing on Paper with NDC Dirt (DPC) Model DPC with Multi Layer 4 NDC Writing on Notes with DC Dirt (DNC) Reversing Encoders Order Numerical Results for Rayleigh 5 Conclusions 2/ 27

3 Background and Some Related Work Gaussian Broadcast channel - (Cover 72),(Bergmans 74) Broadcast approach for fading channels - (Shamai 97),(Steiner-Shamai 03) Mixed traffic settings - (Liang-Zhang-Cioffi 06), (Zhang 08) Coding with CSIT Non causal - (Gel fand-pinsker 80), Dirty Paper Coding (DPC) (Costa 83) Casual - (Shannon 58), Dirty Tape Coding (DTC) - (Erez-Shamai-Zamir 05) Fading DPC, Linear Assignment Capacity - (Zhang-Kotagiri-Laneman 07),(Bennatan-Burshtein 08) This talk is follow up on Cohen, K.M.; Steiner, A.; Shamai, S.;, The broadcast approach under mixed delay constraints, Information Theory Proceedings (ISIT), 2012 IEEE International Symposium on, vol., no., pp , 1-6 July / 27

4 Fading Channel Model Single user, SISO real-value flat fading AWGN channel Y i = S i X i + N i E X i 2 P. S i R +. Distributed by f S (s). CSIR only. E[S] = 1 σ 2 N = 1 = SNR = P. Slow fading L is large enough for reliable communication yet much shorter than the dynamics of the channel K is large enough to capture the empirical distribution of {S k } 4/ 27

5 Mixed Traffic Data classification 1 Delay Constraint (DC) - Delay sensitive traffic (L channel-uses). Outage approach broadcast approach 2 Non Delay Constraint (NDC) - Has relaxed delay requirements. K L channel-uses. Experiences all fading levels. Dominated by the Ergodic Capacity is C erg = 1 2 E S[log(1 + S P)] R DC Average throughput for DC traffic (reliable instantaneous rate can not always be guaranteed). R NDC Fixed rate for the NDC traffic. 5/ 27

6 Goal Goal of Work Characterizing mixed traffic (DC-NDC) joint achievable rate region under three cases, as abbreviated below and augmented on the upper bar. 6/ 27

7 NCE scheme NCE Communication scheme Y = S (W + Z) + N. E W 2 = βp, E Z 2 = (1 β)p, 0 β 1. Encoders cooperation - Not allowed for this scheme. Decoders cooperation - DC decoder informs NDC decoder on the decoded W DE. 7/ 27

8 Outage and Broadcast Approach Outage and Broadcast Approach for DC traffic ρ(s) - DC transmit power distribution per fading I (S) - DC residual interference per fading, I (s) = s ρ(u) du Superposition - w L (m 1, m 2,..., m ) = j=1 w j L(m j) (a) Single Layer (b) Multi layering (c) Continuous Layering 8/ 27

9 Joint Optimality of DC-Layering Defining an Optimization Problem P is fixed β is an independent variable defining the angle I (s) (and thus ρ(s) by ρ(s) = d ds I (s)) will be set upon the former parameters. Optimality at the Sum-Rate sense was studied. subject to I (0) = βp and I ( ) = 0. I (s) = arg max {R DC + R NDC } I (s) Outage Approach - I (s) is fully characterized by a single scalar - s 1 Broadcast Approach - I (s) is right continuous non increasing function over a continuum 9/ 27

10 Joint Optimality of DC-Layering NCE Numerical Results for Rayleigh fading channel NCE1L Achievability Regions 1L and P=25dB 2.5 R NDC [Nats/Channel use] NCE1L1L NCE1Lbc Ergodic Bound R DC [Nats/Channel use] Ergodic bound: R DC + R NDC C erg 10/ 27

11 NDC Layering NDC Dual-Layer Y = S (W + Z 1 + Z 2 ) + N Encoding Fixed γ [0, 1]. E W 2 = βp. E Z 1 2 = (1 β)γp. E Z 2 2 = (1 β)(1 γ)p. Decoding order 1 DC layers. 2 First NDC layer. 3 Additional DC layers (to reduce interference). 4 Second NDC layer. 11/ 27

12 NDC Layering NDC Multi-Layer Y = S (W + Z q ) + N, q=1 Proposition (NDC Continuous Layering) E Z q 2 = (1 β)p q=1 Any expected NDC rate for multilayer NDC part of the form R NDC 0 dsf S (s) 1 + si 0 ( sλ(a) da s 1 s[(1 β)p Λ(a)] ) + Λ(a)s is achievable, for any λ(a) = d daλ(a), Λ(0) = (1 β)p and lim a Λ(a) = 0. DC throughput stays the same Optimization over I (s) and Λ(a) may be performed 12/ 27

13 NDC Layering Numerical Results for NDC Multi-Layering NCE2L1L and NCE2Lbc Achievability P=25dB R NDC [Nats/Channel use] Ergodic Bound NCE,1L,bc* best γ NCE, L,bc Bound** R DC [Nats/Channel use] 13/ 27

14 Model DPC channel model Y = S (W + Z) + N ( ) X KL {m DC (k)} K k=1, m NDC = ( ( ) )} kl K {W L m DC (k), Z KL (m NDC ) (k 1)L+1 k=1 + Z KL (m NDC ) 14/ 27

15 Model DPC with Single Layer Theorem (DPC Expected Rates under Single DC Layering) The following is an achievable region R DPC,1L,1L = 1 F R DC S (s 1) 1+s 2 log 1P 1+(1 β)s 1P η(α,β,s 1) ( ) 1 s1 R NDC 2 0 f sp+1 S(s) log βsp+1 ds f S (s) log(1+(1 β)sp η(α, β, s)) ds s 1 (α,β,s 1) Costa s linear assignment U = W + αz. η(α, β, s) concave quadratic function of α. α = 0 degenerates to NCE. 15/ 27

16 Model Numerical Results for Rayleigh Channel NCE1L1L vs. P=25dB, β= NCE1L1L DPC1L1L α<0 R NDC [Nats/Channel use] α= 0.5 α=0 α=0.5 DPC1L1L α=0 DPC1L1L α>0 Ergodic Bound R DC [Nats/Channel use] Conjecture (Observations based) For Rayleigh distribution, DPC single layering has no advantage over the NCE single layering R DPC,1L,1L = R NCE,1L,1L. 16/ 27

17 DPC with Multi Layer DPC with dual layers Y = S (W 1 + W 2 + Z) + N Encoding order 1 Z with power β 0 P. 2 W 1 (Z) with power β 1 P suited for S = s 1. 3 W 2 (Z, W 1 ) with power β 2 P suited for S = s 2. 17/ 27

18 DPC with Multi Layer DPC with Multi Layer Theorem (DPC Expected Rates under Multi DC Layering) The following is an achievable region for M DC-layers { R DC } M m=1 R DPC,1L,ML = (1 F S(s m )) R DC,m (s m ) R NDC M sm+1 m=0 f S (s) R NDC,m (s) ds β M 0,sM 1, p(u M 1,w M 1 s,z) R DC,m (S) = I(U m ; Y, U m 1 S) I(U m ; Z, W m 1 ) R NDC,m (S) = I(Z; Y, U m S). s 0 0 = s 0 s 1 < s 2 <... < s M < s M+1 =. β 0 + β β M = 1. 18/ 27

19 DPC with Multi Layer DPC with continuous broadcasting Conjecture For Rayleigh distribution, DPC with continuous layering has no advantage over the NCE continuous layering R DPC,1L,bc = R NCE,1L,bc. Leads to the question Why does not DPC outperform NCE? Decoder s incapability of reconstructing the signal W, has to console with the auxiliary RV U. Successive Cancellation for Y = S (W + Z) + N NCE : Y SW = S Z + N. DPC : Y SU = S (1 α)z + N. Worsen SNR! Troublesome in multi layering. 19/ 27

20 Reversing Encoders Order Dirty Notes Channel model Y = S (W + Z) + N X KL ( {m DC (k)} K k=1, m NDC ) = { W L (m DC (k)) } K k=1 { ( + (Z k ) L m NDC, { W L (m DC (j)) } k j=1 )} K. k=1 20/ 27

21 Reversing Encoders Order Definition A (K, L) Dirty Notes Coding (DNC) is defined over the AWGN Gaussian interference channel Y = X + S + N where S is CSIT known progressively by L blocks and coding is done over K such blocks. Horizontal: Time slot DTC Vertical: Available CSIT time slots DNC DPC available CSIT time slots available CSIT time slots available CSIT time slots DTC real time DNC real time DPC real time time slot time slot time slot C DTC C DNC C DPC. 21/ 27

22 Reversing Encoders Order NDC Capacity under Dirty Notes Coding Our settings are Fading Dirty Notes Coding. RDC DNC isindifferent. Lemma: DNC Capacity For some fixed setting (β, P, I (S)) the NDC capacity is C DNC NDC = 1 KL K k=1 max P U L,Z L W kl 1 L E Z L 2 (1 β)p Assuming non cross-notes coding {I(U L ;Y kl 1,Sk 1,(W DE ) kl 1 ) I(U L ;W kl 1 )}. CNDC DNC = max {I (U; Y, S, W DE ) I (U; W )}. P U,Z W E Z 2 (1 β)p Still no close form solution (W DE and W are correlated). 22/ 27

23 Reversing Encoders Order Bounds Inner Bound - The NCE (poorer domain) Loosen Outer Bound - NDC decoder knows the dirt (DC codeword) [ ] 1 I FS (s) s f S (s) βp (s) = s 2 f (1 β)p. S (s) 0 Tighter Outer Bound - NDC encoder accesses non causally the dirt (DC codeword). DPC note-wise U = Z + α W. { RDC 1 sρ(s) R DNC,GA = 2 (1 F 0 S (s)) 1+sI (s)+(1 β)ps ds } R NDC 1 (1 β)sp+si (s)+1 2 f 0 S (s) log η(α,s)+si (s)+1 ds 0 β 1 I I(βP) 0 α 1 η quadratic convex function of α. 23/ 27

24 Numerical Results for Rayleigh Dirty Notes Coding Numerical Results for Rayleigh mdnc3 achievable P=25dB 2.5 Ergodic Bound NCE,1L,bc DNC outer bound modified DNC version Ergodic Bound NCE,1L,bc DNC outer bound Genie Aided DNC R NDC [Nats/Channel use] R NDC [Nats/Channel use] R DC [Nats/Channel use] R [Nats/Channel use] DC 24/ 27

25 Conclusions Conclusions Dirty Paper Coding Intrinsic incapability to reconstruct codewords. Conjectured (by observations) to have inferior region w.r.t. NCE (intuition: DPC can not help as compared to layering over a degraded broadcast channel). Dirty Notes Coding Has a richer domain than NCE. Fading DNC faces too much uncertainty. No simple coding that outperforms NCE. Even with genie aid, moderate surpassing the NCE. The bottom line The simple layered NCE scheme demonstrates reasonable performance compared to cooperative schemes. 25/ 27

26 Thank You. 26/ 27

27 Broadcasting over Fading Channels with Mixed Delay Constraints Kfir Cohen, Avi Steiner and Shlomo Shamai Department of Electrical Engineering Technion-Israel Institute of Technology Haifa 32000, ISRAEL. Abstract Reliable communications over a block fading channel is considered, where the channel state information is available to the receiver only. We consider reliable communications of two data streams, the first is subject to a stringent delay constraint (DC) where the transmission must be completed within a single fading block. The other stream is not required to meet any delay demands, and is called the non-delay constrained (NDC) stream, and hence can enjoy the ergodic features of the channel. We study different settings of layered communications (the broadcast approach), where the encoders of the different layers act in a non-cooperative or cooperative manner employing dirty-paper strategies. Bounds on the rate region for the DC vs. NDC streams are evaluated and discussed. 27/ 27