Capacity and Coding for Multi-Antenna Broadcast Channels Wei Yu Electrical Engineering Department Stanford University February 20, 2002 Wei Yu
Introduction • Consider a communication situation involving mutliple transmitters and receivers: z 1 x 1 y 1 z 2 x 2 y 2 Multiuser z m − 1 x n − 1 y m − 1 Channel z m x n y m – What is the value of cooperation? Wei Yu 1
Motivation: Multiuser DSL Environment • DSL environment is interference-limited. FEXT user 1 user 2 user n central downstream NEXT office upstream – Explore the benefit of cooperation. Wei Yu 2
Gaussian Vector Channel • Capacity: C = max I ( X ; Y ) . Z n ˆ X n W ( Y n ) W ∈ 2 nC Y n H • Optimum Spectrum: 2 log | HK xx H T + K zz | 1 maximize | K zz | subject to tr( K xx ) ≤ P, K xx ≥ 0 . Wei Yu 3
Gaussian Vector Broadcast Channel • Capacity Region: { ( R 1 , · · · , R K ) : Pr( W k � = ˆ W k ) → 0 , k = 1 , · · · K } . Z n ˆ W 1 ∈ 2 nR 1 Y n W 1 ( Y n 1 ) 1 X n H ˆ Y n W K ( Y n K ) W K ∈ 2 nR K K • Capacity is known only in special cases. – This talk focuses on sum capacity: C = max { R 1 + · · · + R K } . Wei Yu 4
Broadcast Channel: Prior Work • Introduced by Cover (’72) – Superposition coding: Cover (’72). – Degraded broadcast channel: Bergman (’74), Gallager (’74) – Coding using binning: Marton (’79), El Gamal, van der Meulen (’81) – Sum and product channels: El Gamal (’80) – Gaussian vector channel, 2 × 2 case: Caire, Shamai (’00) • General capacity region remains unknown. Wei Yu 5
Degraded Broadcast Channel Z 1 ∼ N (0 , σ 2 Z 2 ∼ N (0 , σ 2 2 − σ 2 1 ) 1 ) X 1 ∼ N (0 , P 1 ) X Y 1 Y 2 X 2 ∼ N (0 , P 2 ) • Superposition and successive decoding achieve capacity (Cover ’72) � � I ( X 1 ; Y 1 | X 2 ) = 1 1 + P 1 R 1 = 2 log σ 2 1 � � = 1 P 2 R 2 = I ( X 2 ; Y 2 ) 2 log 1 + P 1 + σ 2 2 Wei Yu 6
Gaussian Vector Broadcast Channel Z 1 X 1 ∼ N (0 , K 1 ) H 1 Y 1 X Z 2 X 2 ∼ N (0 , K 2 ) H 2 Y 2 • Superposition coding gives: 2 log | H 1 K 1 H T 1 + H 1 K 2 H T I ( X 1 ; Y 1 ) = 1 1 + K z 1 z 1 | R 1 = | H 1 K 2 H T 1 + K z 1 z 1 | 2 log | H 2 K 2 H T 2 + H 2 K 1 H T I ( X 2 ; Y 2 ) = 1 2 + K z 2 z 2 | R 2 = | H 2 K 1 H T 2 + K z 2 z 2 | Wei Yu 7
Channel with Transmitter Side Information Gaussian Channel ... with Transmitter Side Information Z ∼ N (0 , N ) S ∼ N (0 , Q ) Z ∼ N (0 , N ) X Y X Y P P � � C = 1 1 + P 2 log C =? N Wei Yu 8
Writing on Dirty Paper • A surprising result due to Costa (’83): Sn ∼ N (0 , Q ) Zn ∼ N (0 , N ) W ∈ 2 nR Xn ( W, Sn ) Y n W ( Y n ) ˆ � � C = 1 1 + P 2 log N • This inspired Caire and Shamai’s work on 2x2 broadcast channel (’01). Wei Yu 9
Channel with Side Information Sn W ∈ 2 nR Xn ( W, Sn ) Y n W ( Y n ) ˆ p ( y | x, s ) • Gel’fand and Pinsker (’80), Heegard and El Gamal (’83): p ( u,x | s ) { I ( U ; Y ) − I ( U ; S ) } , C = max • Key: What is the appropriate auxiliary random variable U ? Wei Yu 10
Random Binning and Joint Typicality U X Q P + α 2 Q S • Randomly choose u n ( i ) , i ∈ 2 nI ( U ; Y ) . Binning using B : 2 nI → 2 nC . • Encode: Given s n and message W , find i such that ( u n ( i ) , s n ) is jointly typical, and B ( i ) = W . Send: x n = u n ( i ) − αs n . i )) jointly typical. Recover ˆ • Decode: Find ( y n , u n (ˆ W = B (ˆ i ) . Wei Yu 11
Costa’s Choice for U Zn ∼ N (0 , N ) Sn ∼ N (0 , Q ) W ∈ 2 nR Xn ( W, Sn ) Y n W ( Y n ) ˆ • For i.i.d. S and Z : – Let U = X + αS , where α = P/ ( P + N ) . – Let X be independent of S . – This gives the optimal joint distribution on ( S, X, U, Y, Z ) . � � C = I ( U ; Y ) − I ( U ; S ) = 1 1 + P 2 log N . Wei Yu 12
Colored Gaussian Channel with Side Information Sn ∼ N (0 , Kss ) Zn ∼ N (0 , Kzz ) W ∈ 2 nR Xn ( W, Sn ) Y n W ( Y n ) ˆ • For colored S and Z : – Let U = X + FS , where F = K xx ( K xx + K zz ) − 1 . – Let X be independent of S . 2 log | K xx + K zz | C = I ( U ; Y ) − I ( U ; S ) = 1 | K zz | Wei Yu 13
Wiener Filtering • The optimal non-causal estimate of X given X + Z is ˆ X = F ( X + Z ) , where F = K xx ( K xx + K zz ) − 1 . • The optimal auxiliary random variable for channel with non-causal transmitter side information is U = X + FS , where F = K xx ( K xx + K zz ) − 1 . • Curiously, the two filters are the same. Wei Yu 14
Writing on Colored Paper Gaussian Channel ... with Transmitter Side Information Z ∼ N (0 , Kzz ) S ∼ N (0 , Kss ) Z ∼ N (0 , Kzz ) X Y X Y 2 log | K xx + K zz | 2 log | K xx + K zz | C = 1 C = 1 | K zz | | K zz | • Capacities are the same if S is known non-causally at the transmitter. – Several other proofs have been found by Cohen and Lapidoth (’01), and Zamir, Shamai and Erez (’01) under different assumptions Wei Yu 15
New Achievable Region Z n 1 ˆ X n 1 ( W 1 , X n W 1 ( Y n W 1 ∈ 2 nR 1 Y n 2 ) H 1 1 ) 1 X n Z n 2 ˆ W 2 ∈ 2 nR 2 X n Y n W 2 ( Y n 2 ( W 2 ) H 2 2 ) 2 2 log | H 1 K 1 H T 1 + K z 1 z 1 | I ( X 1 ; Y 1 | X 2 ) = 1 R 1 = | K z 1 z 1 | 2 log | H 2 K 2 H T 2 + H 2 K 1 H T = 1 2 + K z 2 z 2 | R 2 = I ( X 2 ; Y 2 ) | H 2 K 1 H T 2 + K z 2 z 2 | Wei Yu 16
Converse • Broadcast capacity does not depend on noise correlation: Sato (’78). z ′ z ′ z 1 1 1 x 1 y 1 x 1 y 1 x 1 y 1 = z ′ z ′ ≤ z 2 2 2 x 2 y 2 x 2 y 2 x 2 y 2 � �� � � p ( z 1 ) = p ( z ′ 1 ) 2 ) , not necessarily p ( z 1 , z 2 ) = p ( z ′ 1 , z ′ if 2 ) . p ( z 2 ) = p ( z ′ • Thus, sum-capacity C ≤ min K nn max K xx I ( X ; Y ) . Wei Yu 17
Strategy for Proving Achievability 1. Find the worst-case noise correlation z ∼ N (0 , K zz ) . 2. Design an optimal receiver for the vector channel with worst-case noise: y = H x + z 3. Precode x so that receiver coordination is not necessary. • Tools: – Convex optimization – Generalized Decision-Feedback Equalization (GDFE) Cioffi, Forney (’95), Varanasi, Guess (’97) Wei Yu 18
Least Favorable Noise • Fix Gaussian input K xx : 2 log | HK xx H T + K zz | 1 minimize | K zz | � K z 1 z 1 � ⋆ subject to K zz = ⋆ K z 2 z 2 K zz ≥ 0 • Minimizing a convex function over convex constraints. � � Ψ 1 0 zz − ( HK xx H T + K zz ) − 1 = • Optimality condition: K − 1 . 0 Ψ 2 Wei Yu 19
Generalized Decision Feedback Equalizer • Key idea: MMSE estimation is capacity-lossless z e y x ˆ x x ( H T H + K − 1 xx ) − 1 H T H xx ) − 1 = G − 1 ∆ − 1 G − T . • Channel can be triangularized: ( H T H + K − 1 z � x 1 � � � y x 1 ˆ ∆ − 1 G − T H T Decision H x 2 x 2 ˆ I − G Wei Yu 20
GDFE with Transmit Filter z ∼ N (0 , Q Λ Q T ) x ∼ N (0 , K xx ) u y H T + I ) − 1 ˆ u ˜ ( ˜ H ˜ 1 H T Λ Q F H √ � �� � � �� � ˜ 1 H = Λ QHF MMSE estimation √ • Set z ∼ N (0 , K zz ) to be the least favorable noise. • Fix x ∼ N (0 , K xx ) , and u ∼ N (0 , I ) . Choose a transmit filter F . Wei Yu 21
GDFE Precoder ˜ z � ˆ � u 1 ˜ ˜ ∆ − 1 G − T H T Decision u H u 2 ˆ � �� � feedforward filter I − G • Decision-feedback may be moved to the transmitter by precoding. • Least Favorable Noise ⇐ ⇒ Feedforward/whitening filter is diagonal! C = min K nn I ( X ; Y ) (i.e. with least favorable noise) is achievable. Wei Yu 22
Gaussian Broadcast Channel Sum Capacity • Achievability: C ≥ max K xx min K zz I ( X ; Y ) . • Converse (Sato): C ≤ min K zz max K xx I ( X ; Y ) . • (Diggavi, Cover ’98): min K zz max K xx I ( X ; Y ) = max K xx min K zz I ( X ; Y ) . Theorem 1. Gaussian vector broadcast channel sum capacity is: 2 log | HK xx H T + K zz | 1 C = max K xx min | K zz | K zz Wei Yu 23
Gaussian Mutual Information Game Z ∼ N (0 , K zz ) X ∼ N (0 , K xx ) H Y Strategy Objective { K xx : trace( K xx ) ≤ P } Signal Player max I ( X ; Y ) � K z 1 z 1 � � � Fictitious ⋆ K zz : K zz = ≥ 0 min I ( X ; Y ) Noise Player ⋆ K z 2 z 2 Nash equilibrium exists. Wei Yu 24
Saddle-Point is the Broadcast Capacity C ( K xx , K zz ) • The optimum K ∗ xx is a water- filling covariance against K ∗ zz . • The optimum K ∗ zz is a least- favorable noise for K ∗ xx . K xx ( K ∗ xx , K ∗ zz ) K zz Broadcast Channel Sum Capacity = Nash Equilibrium Wei Yu 25
The Value of Cooperation z 1 z 1 z 1 x 1 y 1 x 1 y 1 x 1 y 1 z 2 z 2 z 2 x 2 y 2 x 2 y 2 x 2 y 2 max K xx I ( X ; X + Z ) max K xx I ( X ; X + Z ) min K zz max K xx I ( X ; X + Z ) � � � � K 1 0 K z 1 z 1 ⋆ K xx = K zz = 0 K 2 ⋆ K z 2 z 2 s . t . trace( K xx ) ≤ P s . t . s . t . trace( K i ) ≤ P i , trace( K xx ) ≤ P Wei Yu 26
Application: Vector Transmission in DSL Z X 1 Y 1 − − X 2 Y 2 X 3 Y 3 • If interference is known in advance, it can be pre-subtracted: – Send X ′ 1 = X 1 − X 2 − X 3 . 1 || 2 = || X 1 || 2 + || X 2 || 2 + || X 3 || 2 . • Problem: energy enhancement || X ′ Wei Yu 27
Reducing Energy Enhancement: Tomlinson Precoder 3 M 2 S Z M Equivalent 2 − X Y Points ˆ Mod- M Mod- M U U − M 2 − 3 M 2 • Key idea: Use modulo operation to reduce energy enhancement – X is uniformly distributed in [ − M 2 , M 2 ] . • Capacity loss due to shaping: 1.53dB. (Erez, Shamai, Zamir ’00) Wei Yu 28
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