The and The Binary Dirty Multiple Access Channel with Common Interference Dept. EE - Systems, Tel Aviv University, Tel Aviv, Israel April 25th, 2010 M.Sc. Presentation
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B/G Model Compound CSI Smart Rx Smart Tx MIMO SI@Tx DPC THP Motivation Compound Compound GP Channels with Side Information at the Transmitter S W Encoder X p(y x, s) Y Decoder Ŵ Channel S - Side information available at Tx. Causal/Non-Causal Side Information Shannon setting: S is known causally at Tx. Gel fand-pinsker setting: S is known non-causally at Tx.
B/G Model Compound CSI Smart Rx Smart Tx MIMO SI@Tx DPC THP Motivation Compound Compound GP Channels with Side Information at the Transmitter Shannon Setting Capacity ( 58) C causal =max p(t) P(T ) I (T ; Y ) T - the set of all functions ( strategies ) from S to X. Equivalent to a DMC whose inputs are all strategies t T. Uses only the current state s i. (though has access to all past states s i 1 ) Gel fand-pinsker Setting Capacity ( 80) C non-causal =max p(u,x s) {I (U; Y ) I (U; S)} U - auxiliary random variable, which satisfies: U (X, S) Y Both results generalize to continuous alphabets.
B/G Model Compound CSI Smart Rx Smart Tx MIMO SI@Tx DPC THP Motivation Compound Compound GP Dirty Paper Channel Y = X + S + N S N W Encoder 1 n n i=1 x2 i P X X Y Ŵ Decoder S - interference, known (causally/non-causally) at Tx with average power P S. N -AWGNwithaveragepowerP N.
B/G Model Compound CSI Smart Rx Smart Tx MIMO SI@Tx DPC THP Motivation Compound Compound GP Dirty Paper Channel Non-Causal Gaussian Side Information (Costa 83) Gaussian interference: S N(0, P S ). Select the auxiliary U = X + αs, X N(0, P X ) independent of S N(0, P S ). Interference does not reduce capacity: C = 1 2 log(1 + SNR) Causal Gaussian Side Information (ESZ 05) Capacity is not known in general. In the limit of high SNR (SNR ): C causal = 1 2πe 2 log (1 + SNR) 12 Extension (Cohen-Lapidoth 02, ZSE 02, ESZ 05) True to arbitrary side information as well.
B/G Model Compound CSI Smart Rx Smart Tx MIMO SI@Tx DPC THP Motivation Compound Compound GP Lattice Strategies / Tomlinson-Harashima Precoding V U αs Transmitter Receiver Encoder + mod-λ X Channel Decoder N + Y α + X =[v αs U] mod Λ; ( mod [ L 2, L 2) ) U Unif[ L/2, L/2) - dither, known at both ends. v - information baring signal. Y =[αy + U] mod Λ = [v + N eff ] mod Λ, N eff =(1 α)u + αn. Remark In this scheme, P S can be arbitrarily large. S mod-λ Y
B/G Model Compound CSI Smart Rx Smart Tx MIMO SI@Tx DPC THP Motivation Compound Compound GP Lattice Strategies / Tomlinson-Harashima Precoding Performances SNR eff P X P Neff SNR α MMSE SNR+1 R 1 2 log(snr eff) 1 2 Shaping Loss 1+SNR. 2πe log 12 = 1 2 log(1 + SNR) Causality Loss {}}{ Reducing the causality/shaping loss is possible using multi-dimensional lattices. Requires non-causal knowledge of S. 1 2πe log 2 12. In the limit of dimension going to infinity, the shaping loss can be avoided. (ZSE 02, ESZ 05) (The Voronoi region approaches a sphere)
B/G Model Compound CSI Smart Rx Smart Tx MIMO SI@Tx DPC THP Motivation Compound Compound GP Applications of Dirty Paper Coding Model serves as an information-theoretic framework for (known) interference cancellation in: ISI channels. Information embedding. MIMO broadcast channels. Assumption: Tx knows channel gains.
B/G Model Compound CSI Smart Rx Smart Tx MIMO SI@Tx DPC THP Motivation Compound Compound GP Imperfect Channel Knowledge Perfect channel knowledge is not always available! Why Imperfect Channel Knowledge? Channel estimation errors. Finite feedback. Quantization errors. Finite accuracy. Compound Channel with Side Information Special case of the compound channel with side information problem.
B/G Model Compound CSI Smart Rx Smart Tx MIMO SI@Tx DPC THP Motivation Compound Compound GP Compound Channel W Encoder X Channel p β (y x, s) β B Y Decoder β is constant and unknown to Tx. Capacity is the same when β is known/unknown at Rx. Compound Achievable rate = worst case over all β values. Capacity (Blackwell-Breiman-Thomasian 59, Dobrushin 59, Wolfowitz 60) C = max inf I β(x ; Y ) p(x) P(X ) β B Remark Generalizes to continuous alphabets. Ŵ
B/G Model Compound CSI Smart Rx Smart Tx MIMO SI@Tx DPC THP Motivation Compound Compound GP Compound Channels with Side Information at Tx S W Encoder X p β (y x, s) Y Decoder Ŵ Channel Capacity is not known in general. Extending the proof of Gel fand and Pinsker gives upper and lower bounds. Bounds do not coincide due to the presence of feedback Y1 i 1 in U.
B/G Model Compound CSI Smart Rx Smart Tx MIMO SI@Tx DPC THP Motivation Compound Compound GP Compound Channels with Side Information at Tx Upper and Lower Bounds for Non-Causal Side Information (Mitran-Devroye-Tarokh 06) C l C C u C l = max inf [I β(u; Y W ) I (U; S W )] p(u x,s,w)p(x s,w)p(w) β B C u = max inf max [I β(u; Y W ) I β (U; S W )] p(x s,w)p(w) β B p β (u x,s,w)
B/G Model Compound CSI Smart Rx Smart Tx MIMO Perfect DP Compound DP Dirty Paper Channel Dirty paper channel with perfect channel knowledge: Y = X + S + N S β S N W Encoder 1 n n i=1 x2 i P X X Y Ŵ Decoder SNR P X P N β [1 Δ, 1 + Δ] is known to Rx but not to Tx. SIR P X P S β constant (non-ergodic) for whole transmission. Constant (non-ergodic) for whole transmission.
B/G Model Compound CSI Smart Rx Smart Tx MIMO Perfect DP Compound DP Compound Dirty Paper Channel Dirty paper channel with imperfect channel knowledge: Y = X + S β + N S N 1 β W Encoder X Y Decoder Ŵ 1 n n i=1 x2 i P X β [1 Δ, 1 + Δ] - Compound. Constant (non-ergodic) for whole transmission. β is known to Rx but not to Tx. Compound Achievable rate = worst case over all β values.
B/G Model Compound CSI Smart Rx Smart Tx MIMO THP Compound Channel with Causal Side Information Non-Causal Side Information Gives upper and lower bounds (Mitran et al. 06). Causal Side Information Extension of Shannon s technique works! Capacity: C casual = max inf I β(t ; Y ) p(t) P(T ) β B Strategies from the current state only (s i ) need to be considered. Non Single-Letter Expression for Non-Causal Side Information 1 k I β(t; Y) C non casual = lim sup k max inf p(t) β
B/G Model Compound CSI Smart Rx Smart Tx MIMO THP Back to the Compound Dirty Paper Channel... Scalar Lattice Strategies We focus first on scalar (one-dim.) lattice strategies. Remark α is used at both ends (Tx and Rx). Transmitter Knows β The problem reduces to classical DPC case (β 1). Transmitter Does NOT know β What is the best strategy? Work as before? ( Naïve Approach ) Can receiver do better? ( Smart Rx ) Can transmitter do better? ( Smart Tx )
B/G Model Compound CSI Smart Rx Smart Tx MIMO Naïve Approach Work as Before Transmitter X =[v αs U] mod Λ. Receiver Remark Y =[αy + U] mod Λ = [v + N β eff ] mod Λ, 1) =(1 α)u + α(β S + αn. β N β eff P S R =0 even when SNR and for any value of α! What can we do?
B/G Model Compound CSI Smart Rx Smart Tx MIMO Scheme Performances Moral Smart Receiver Distinguish between α R (α @Rx)andα T (α @Tx) Transmitter X =[v α T S U] mod Λ. Receiver Y =[α R Y + U] mod Λ = [v + N β eff ] mod Λ, N β eff =(1 α R)U +(α T β α R ) S β + α RN. Remarks α R = α T - Bad! For time being, we optimize w.r.t. SNR wc eff.
B/G Model Compound CSI Smart Rx Smart Tx MIMO Scheme Performances Moral Naïve Approach SNR eff = λ Naïve (β)(1 + SNR) 1 λ Naïve (β) 1+ 1. (1 β)2 Smart Rx (MMSE Estimation) SIR + SNR SIR SNR eff = λ(β)(1 + SNR) λ MMSE (β) α MMSE R = 1+ 1 SIR + 1 SNR 1+ 1 SIR + 1 SNR + SNR SIR (1 β)2 αmmse T β 1+ SIR 1+ 1 SIR + 1 SNR
B/G Model Compound CSI Smart Rx Smart Tx MIMO Scheme Performances Moral High SNR Limit α T 1. α MMSE R = 1+ β SIR 1+ 1 SIR = SIR+β SIR+1. Optimized α R SNR wc eff = 1+SIR (1 β) 2. Non-Optimized α R SNR wc eff = SIR (1 β) 2. Strong interference: (SIR 0) Smart Rx achieves positive rates!
B/G Model Compound CSI Smart Rx Smart Tx MIMO Scheme Performances Moral Example: SIR = 1, Δ=1/3 25 20 Ignoring β MMSE Estimation SINR [db] 15 10 5 0 0 5 10 15 20 25 30 SNR [db] Remark Even for SIR = 1, the gain is 3dB, in the limit of high SNR.
B/G Model Compound CSI Smart Rx Smart Tx MIMO Scheme Performances Moral Moral Conclusions In the high SNR regime, the scheme is interference limited due to β. But... Ignoring β at Rx is suboptimal and could lead to significant rate losses. For large S, Rx partially compensates for ignorance of Tx. max SNR wc eff criterion max MI criterion Achievable Rate of (Deterministic) THP Strategies R d THP =max α T 1 min max β α R ( ε(β,α T,α R ) h N β eff,g 2 log(snr eff)+ε(β,α T,α R ) 1 2 log ) ( ) h N β eff and N β eff,g is Gaussian with the same power as Nβ eff. ( ) 2πe, 12
B/G Model Compound CSI Smart Rx Smart Tx MIMO Motivation Criteria Rates Noisy Non-Causal Smart Transmitter So far - Smart Rx. Tx does not know β, but is aware of its ignorance... Can Tx do better? How about Guessing β? α T varies from symbol to symbol. Common Randomness: for any α T,Rxusesoptimalα R as previously. Increases (worsens) the MSE. Improves the mutual-information.
B/G Model Compound CSI Smart Rx Smart Tx MIMO Motivation Criteria Rates Noisy Non-Causal Motivation Effective Noise Expression (Reminder) Setting N β eff =(1 α R)U +(α T β α R ) S β + α RN Concentrate on SIR 0andSNR. Model as a noiseless channel (N = 0). All noise is self noise : Y =[v + N β eff ] mod Λ N β eff = α β α U Remarks True for multi-dimensional lattices as well. Capacity would be infinite if β were known at Tx!
B/G Model Compound CSI Smart Rx Smart Tx MIMO Motivation Criteria Rates Noisy Non-Causal Motivation Example (B =2) β {1 ± Δ} α T 1 minimizes the MSE but achieves a finite rate. P(α T =1 Δ) = P(α T =1+Δ)=0.5 obtains a larger MSE, but achieves infinite rate. Corollary Optimizing MSE Optimizing Rate
B/G Model Compound CSI Smart Rx Smart Tx MIMO Motivation Criteria Rates Noisy Non-Causal MMSE Max.SNR Max. Rate MMSE Criterion SNR eff =max min f (α) β P X E α (N β eff ) 2. Maximal SNR Criterion SNR eff =max min E α f (α) β P X ( N β eff ) 2. Maximal Rate Criterion RTHP r =max f (α) Rr THP (f )=max min I β(v ; Y α). f (α) β
B/G Model Compound CSI Smart Rx Smart Tx MIMO Motivation Criteria Rates Noisy Non-Causal Achievable Rates Using Random THP strategies Maximal Rate R r THP = for Δ 1/3. max f (α): Supp{f (α)} [ Δ,Δ] [ min E α β [ Δ,Δ] Remark True for multi-dimensional lattices as well. Finding f (α) Finding optimal f (α) is analytically hard. log Instead, we evaluate for several f (α) choices. α β α ],
B/G Model Compound CSI Smart Rx Smart Tx MIMO Motivation Criteria Rates Noisy Non-Causal Robust DPC Performance 7 6 f (α) P(α=1)=1 5 R [nats] 4 1 Δ 1 1+Δ α 3 2 1 0 0.05 0.1 0.15 0.2 0.25 0.3 Δ
B/G Model Compound CSI Smart Rx Smart Tx MIMO Motivation Criteria Rates Noisy Non-Causal Robust DPC Performance 7 6 f (α) Unif[1 Δ,1+Δ] P(α=1)=1 5 R [nats] 4 1 Δ 1 1+Δ α 3 2 1 0 0.05 0.1 0.15 0.2 0.25 0.3 Δ
B/G Model Compound CSI Smart Rx Smart Tx MIMO Motivation Criteria Rates Noisy Non-Causal Robust DPC Performance 7 6 f (α) V like Unif[1 Δ,1+Δ] P(α=1)=1 5 R [nats] 4 1 Δ 1 1+Δ α 3 2 1 0 0.05 0.1 0.15 0.2 0.25 0.3 Δ
B/G Model Compound CSI Smart Rx Smart Tx MIMO Motivation Criteria Rates Noisy Non-Causal Robust DPC Performance 7 6 Upper Bound V like Unif[1 Δ,1+Δ] P(α=1)=1 5 R [nats] 4 3 2 1 0 0.05 0.1 0.15 0.2 0.25 0.3 Δ
B/G Model Compound CSI Smart Rx Smart Tx MIMO Motivation Criteria Rates Noisy Non-Causal Noisy Case Effective Noise Expression ( N β eff =(1 α R)U + α R α) β S β + α RN. Strong Interference: N β eff = α β α U + β α N. Self noise component is dominant Use random α. Gaussian component is dominant Use deterministic α.
B/G Model Compound CSI Smart Rx Smart Tx MIMO Motivation Criteria Rates Noisy Non-Causal Rates for SNR = 17dB 1.6 1.4 α=1 α=α MMSE Unif[1 Δ,1+Δ] Unif[α MMSE (1 Δ),α MMSE (1+Δ)] R [nats] 1.2 1 0.8 0.6 0.4 0 0.05 0.1 0.15 0.2 0.25 0.3 Δ
B/G Model Compound CSI Smart Rx Smart Tx MIMO Motivation Criteria Rates Noisy Non-Causal Non-Causal Case and Multi-Dimensional Lattices High SNR, Low SIR Results stay exactly the same. Only self noise component is present. Noise and information signal gain shaping together. Corollary: No need to increase dimension! General Case Gaining some shaping is possible. Costa Random Binning for Gaussian S Non-Compound Case: U = X + αs is optimal. Compound Case: Selecting Costa s α like α T in the THP scheme, gives identical results!
B/G Model Compound CSI Smart Rx Smart Tx MIMO Model Linear-ZF DPC Rates Robustness Implications for MIMO Broadcast Channels Model and Transmission Scheme Y i = h T i X + N i i =1, 2 X = X 1 t 1 + X 2 t 2 X i - Information signal to user i =1, 2 t i - Unit vector in Tx direction of user i =1, 2 Channel Estimation Inaccuracies h i - Actual channel vectors. Transmitter has estimation h i h i. Assume small inaccuracies angular inaccuracies are dominant.
B/G Model Compound CSI Smart Rx Smart Tx MIMO Model Linear-ZF DPC Rates Robustness Linear Zero-Forcing x 1 x 2 h 1 h 1 ε 1 θ ε2 h 2 h 2 h i - true channel vectors h i - estimated channels @Tx ε i - angular errors
B/G Model Compound CSI Smart Rx Smart Tx MIMO Model Linear-ZF DPC Rates Robustness Dirty Paper Coding x 1 h 1 h 1 ε 1 θ ε2 x 2 h 2 h 2 h i - true channel vectors h i - estimated channels @Tx ε i - angular errors
B/G Model Compound CSI Smart Rx Smart Tx MIMO Model Linear-ZF DPC Rates Robustness Achievable Rates Linear Zero-Forcing Dirty-Paper Coding ( ) R 1 1 2 log 1+ SNR 1 h 1 2 sin 2 (θ) SNR 2 h 1 2 o(1) Δ 2 +1 ( ) R 2 1 2 log 1+ SNR 2 h 2 2 sin 2 (θ) SNR 1 h 2 2 o(1) Δ 2 +1 ) R 1 (1+ 1 2 log SNR 1 h 1 2 sin 2 (θ) SNR 1 h 1 2 o(1) sec 2 (θ)δ 2 +1 ) R 2 (1+ 1 2 log SNR 2 h 2 2 SNR 1 h 2 2 o(1). Δ 2 +1
B/G Model Compound CSI Smart Rx Smart Tx MIMO Model Linear-ZF DPC Rates Robustness Robustness to Channel Uncertainty Which Scheme is More Robust? When SNR 2 sec 2 (θ) > SNR 1 : Dirty paper coding is more robust than linear ZF! Improved Schemes Both schemes can be improved using: randomization @Tx ( Smart Tx )
Model and Sum-Capacity Rate Region Binary Dirty Multiple-Access with Common Interference Binary Dirty MAC with Common Interference
Channel Model Model and Sum-Capacity Rate Region N Bernoulli(ε) -Noise. Y = X 1 X 2 S N S Bernoulli(1/2) - Known to both encoders. 1 Input ( power ) constraints: n w H(x i ) q i. Sum-Capacity R1+R2 = u.c.e. max { H b (q 1 + q 2 ) H b (ε), 0 } Time-sharing is essential. (In contrast to the Gaussian case) Cooperation between encoders does not increase capacity. (In contrast to the Gaussian case) Binary Dirty MAC with Common Interference
Model and Sum-Capacity Rate Region Clean MAC Dirty MAC Clean MAC Rate Regions for q 1 =1/6, q 2 =1/10 0.5 0.45 0.4 0.35 R2 [bits] 0.3 0.25 0.2 0.15 0.1 0.05 Capacity Region Sum Capacity Improved Onion Peeling Onion Peeling Time Sharing (one Tx at a time) h b (q1*q2) h b (q2) h (q2*q1) h (q1) b b 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 R1 [bits] Binary Dirty MAC with Common Interference
Model and Sum-Capacity Rate Region Clean MAC Dirty MAC Dirty MAC Rate Regions for q 1 =1/6, q 2 =1/10 0.5 0.45 0.4 0.35 R2 [bits] 0.3 0.25 0.2 0.15 0.1 0.05 Clean Sum Capacity (UB) Clean MAC Capacity (UB) TS between OP Improved Onion Peeling Time Sharing (one Tx at a time) Onion Peeling h b (q1*q2) h b (q2) h (q2*q1) h (q1) b b 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 R1 [bits] Binary Dirty MAC with Common Interference
Smart Tx DMAC Supplementary Supplementary
Smart Tx DMAC Rates UB Achievable Rates Using Random THP strategies Deterministic Selection α 1 RTHP r (f Deter) = log Δ = log 1 Δ Uniform Distribution α [ Δ, Δ] RTHP r (f Unif) = 1 [ (1 + Δ) log(1 + Δ) 2Δ (1 Δ) log(1 Δ) 2Δ log(2δ) ] RTHP r (f Deter) V-like Distribution f V like (α) = α 1 Δ 2, α 1 Δ R r THP (f V like) = 1 2Δ 2 [ (1 Δ 2 )log(1 Δ 2 )+Δ 2 log(δ 2 ) ]. Supplementary
Upper Bound Smart Tx DMAC Rates UB Remarks Optimal f (α) I β (V ; Y )thesameforallβ. Optimal f (α) is not symmetric around 1. Becomes more symmetric as Δ decreases. Upper Bound R r THP log(1 + Δ) log(δ) + 1 Proof: I β (V ; Y )=min{e α [log α] E α [log α β ]} β min {log(1 + Δ) E α [log( α β mod Λ)]} β log(1 + Δ) 1 Δ log x dx 2Δ Δ =log(1+δ) log(δ) + 1. Supplementary
Smart Tx DMAC Binary Dirty Multiple Access Channel Doubly-Dirty MAC (PZE 09) S = S 1 S 2. S 1, S 2 Bernoulli(1/2). Capacity Region: { } C(q 1, q 2 )= (R 1, R 2 ):R 1 + R 2 uch [H b (q min ) H b (ε)] q min min{q 1, q 2 } Single-Informed User (PZE 09) Only user 1 knows S. Capacity Region: C(q 1, q 2 ) = cl conv { (R 1, R 2 ): R 2 H b (q 2 ε) H b (ε) R 1 + R 2 H b (q 1 ) H b (ε) }. Supplementary