The Structure of the Worst Noise in Gaussian Vector Broadcast - PowerPoint PPT Presentation

The Structure of the Worst Noise in Gaussian Vector Broadcast Channels Wei Yu University of Toronto March 19, 2003 DIMACS Workshop on Network Information Theory

Outline • Sum capacity of Gaussian vector broadcast channels. • Complete characterization of the worst-noise. • Efficient numerical solution for the dual channel. • Does duality extend beyond the power constrained channels? DIMACS Workshop on Network Information Theory 1

Gaussian Vector Broadcast Channel • Non-degraded broadcast channel: Z n ˆ W 1 ∈ 2 nR 1 W 1 ( Y n Y n 1 ) 1 X n H ˆ Y n W K ( Y n K ) W K ∈ 2 nR K K • Capacity region is still unknown. – Sum capacity C = max { R 1 + · · · + R K } is recently solved. DIMACS Workshop on Network Information Theory 2

Marton’s Achievability Region • For a broadcast channel p ( y 1 , y 2 | x ) : R 1 ≤ I ( U 1 ; Y 1 ) R 2 ≤ I ( U 2 ; Y 2 ) R 1 + R 2 ≤ I ( U 1 ; Y 1 ) + I ( U 2 ; Y 2 ) − I ( U 1 ; U 2 ) for some auxiliary random variables p ( u 1 , u 2 ) p ( x | u 1 , u 2 ) . • For the Gaussian broadcast channel: I ( U 2 ; Y 2 ) − I ( U 1 ; U 2 ) is achieved with precoding. DIMACS Workshop on Network Information Theory 3

Writing on Dirty Paper Gaussian Channel ... with Transmitter Side Information Z ∼ N (0 , Szz ) S ∼ N (0 , Sss ) Z ∼ N (0 , Szz ) X Y X Y C = 1 2 log | S xx + S zz | C = 1 2 log | S xx + S zz | | S zz | | S zz | • Capacities are the same if S is known non-causally at the transmitter. C = max p ( u,x | s ) I ( U ; Y ) − I ( U ; S ) = max p ( x ) I ( X ; Y | S ) DIMACS Workshop on Network Information Theory 4

Precoding for the Broadcast Channel 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 S 1 H T I ( X 1 ; Y 1 | X 2 ) = 1 1 + S z 1 z 1 | R 1 = | S z 1 z 1 | 2 log | H 2 S 2 H T 2 + H 2 S 1 H T = 1 2 + S z 2 z 2 | R 2 = I ( X 2 ; Y 2 ) | H 2 S 1 H T 2 + S z 2 z 2 | DIMACS Workshop on Network Information Theory 5

Converse: Sato’s Outer Bound • 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 ′ • So, sum capacity C ≤ min S zz max S xx I ( X ; Y ) . DIMACS Workshop on Network Information Theory 6

Three Proofs of the Sum Capacity Result 1. Decision-Feedback Equalization approach (Yu, Cioffi) 2. Uplink-Downlink duality approach (Viswanath, Tse) 3. Convex duality approach (Jindal, Vishwanath, Goldsmith) DIMACS Workshop on Network Information Theory 7

DFE Approach z ′ ∆ − 1 G − T H T Decision x H � �� � feedforward filter I − G • Decision-feedback at the receiver is equivalent to transmitter precoding. • (Non-Singular) Worst Noise ⇐ ⇒ Diagonal feedforward filter Fix S xx , min S zz I ( X ; Y ) is achievable. DIMACS Workshop on Network Information Theory 8

Uplink-Downlink Duality Approach Z 1 ∼ N (0 , Q ) Z 2 ∼ N (0 , I ) H T X 1 H Y 1 X 2 Y 2 E [ X T E [ X T 1 X 1 ] ≤ P 2 QX 2 ] ≤ P • Uplink and downlink channels are duals. • The noise covariance and input constraint are duals. • Worst-noise gives an input constraint that decouples the inputs. C = max S xx min S zz I ( X ; Y ) DIMACS Workshop on Network Information Theory 9

Convex Duality Approach Z 1 H T Z X ′ H 1 Y 1 1 1 Z 2 Y ′ X X ′ H T H 2 Y 2 2 2 P P • Sato’s bound: C ≤ min S zz max S xx I ( X ; Y ) . S x ′ x ′ I ( X ′ ; Y ′ ) . • Broadcast/Multiple-Access duality: C ≥ max S x ′ x ′ I ( X ′ ; Y ′ ) . • Convex duality: max S xx min S zz I ( X ; Y ) = max DIMACS Workshop on Network Information Theory 10

Objective • Completely characterize the worst-noise. – Duality through minimax. – Worst-noise through duality. • Efficient numerical solution for the dual channel. • Does duality extend beyond the power constrained channel? DIMACS Workshop on Network Information Theory 11

Minimax Capacity • Gaussian vector broadcast channel sum capacity is the solution of 2 log | HS xx H T + S zz | 1 max S xx min | S zz | S zz subject to tr( S xx ) ≤ P � I � ⋆ S zz = ⋆ I S xx , S zz ≥ 0 • The minimax problem is convex in S zz , concave in S xx . – How to solve this minimax problem? DIMACS Workshop on Network Information Theory 12

Duality through Minimax • Two KKT conditions must be satisfied simultaneously: H T ( HS xx H T + S zz ) − 1 H = λI � � Ψ 1 0 zz − ( HS xx H T + S zz ) − 1 = S − 1 0 Ψ 2 • For the moment, assume that H is invertible. H T S − 1 zz H − λI = H T Ψ H ⇒ H ( H T Ψ H + λI ) − 1 H T = S zz ⇒ This is a “water-filling” condition for the dual channel. DIMACS Workshop on Network Information Theory 13

Power Constraint in the Dual Channel • Interpretation of dual variable: λ = ∂C ∂P , Ψ i = − ∂C . ∂S z i z i �� � – Thus, capacity is preserved if λ ∆ P = Ψ i ∆ S z i z i i 2 log | HS xx H T + S zz | • Capacity C = min max 1 . | S zz | – Thus, capacity is preserved if ∆ P = ∆ S z i z i . P 1 � i Ψ i Therefore, = P . λ DIMACS Workshop on Network Information Theory 14

Construct the Dual Channel KKT condition: H ( H T DH + I ) − 1 H T = 1 λ S zz • where D = Ψ /λ is diagonal, trace( D ) = � i Ψ i /λ = P . � � I ⋆ • S zz = . Thus, constraint on D : trace( D 1 ) + trace( D 2 ) ≤ P . ⋆ I Z X ′ H T 1 1 E [ X ′ 1 X ′ T 1 ] = D 1 E [ X ′ 2 X ′ T Y ′ 2 ] = D 2 trace( D 1 ) + trace( D 2 ) ≤ P X ′ H T 2 2 DIMACS Workshop on Network Information Theory 15

Yet Another Derivation for Duality The duality between broadcast channel and multiple-access channel: 2 log | HS xx H T + S zz | 2 log | H T DH + I | 1 1 max S xx min max | S zz | | I | S zz D s . t . tr( S xx ) ≤ P s . t . tr( D ) ≤ P � � I ⋆ S zz = D is diagonal ⋆ I S xx , S zz ≥ 0 D ≥ 0 KKT conditions for minimax = ⇒ KKT condition for max. DIMACS Workshop on Network Information Theory 16

Worst-Noise Through Minimax • Solve the dual multiple access channel problem with power constraint P . Obtain (Ψ , λ ) . Then: H ( H T Ψ H + λI ) − 1 H T S zz = ( λI ) − 1 − ( H T Ψ H + λI ) − 1 S xx = • What if H is not invertible, or S zz is singular? DIMACS Workshop on Network Information Theory 17

Decision-Feedback Equalization with Singular Noise � � Ψ 1 0 zz − ( HS xx H T + S zz ) − 1 = • With non-singular noise: S − 1 . 0 Ψ 2 • If H is low-rank, S zz can be singular. Z Linear Estimation/DFE X H is not unique if | Sz | = 0 . m × n m -dimensional n > m DIMACS Workshop on Network Information Theory 18

Necessary and Sufficient Condition for Diagonalization • Suppose that the worst-noise | S zz | = 0 , let z U T , S zz = US ˜ z ˜ where S zz is n × n , S ˜ z is m × m , m < n . z ˜ • It is always possible to write H = U ˜ H . • There exists a DFE with diagonal feedforward filter if and only if � Ψ 1 � 0 H T + S ˜ z ) − 1 = U T z − ( ˜ HS xx ˜ S − 1 U z ˜ z ˜ ˜ 0 Ψ 2 DIMACS Workshop on Network Information Theory 19

Singular Worst-Noise • It can be verified that the diagonalization condition is satisfied by: S (0) H ( H T Ψ H + λI ) − 1 H T = zz ( λI ) − 1 − ( H T Ψ H + λI ) − 1 S xx = • However: S (0) zz does not necessarily have 1 ’s on the diagonal.   I ⋆ ⋆ S (0)  . zz = ⋆ I ⋆  ⋆ ⋆ ⋆ DIMACS Workshop on Network Information Theory 20

Characterization of the Worst-Noise Theorem 1. The following steps solve the worst noise in y = Hx + z : 1. Find the optimal (Ψ , λ ) in the dual multiple access channel. 2. Form S (0) zz = H ( H T Ψ H + λI ) − 1 H T , S xx = ( λI ) − 1 − ( H T Ψ H + λI ) − 1 . 3. If S xx is not full rank, reduce the rank of H , and repeat 1-2. 4. The class of worst-noise is precisely S (0) zz + S ′ zz .       I ⋆ ⋆ 0 0 0 I ⋆ ⋆  +  =  . ⋆ I ⋆ 0 0 0 ⋆ I ⋆    ⋆ ⋆ ⋆ 0 0 ⋆ ⋆ ⋆ I DIMACS Workshop on Network Information Theory 21

Worst-Noise is Not Unique • The same S xx water-fills the entire class of S (0) zz + S ′ zz . � � �� � � �� S ′ S ′ 0 0 S ˜ 0 • S (0) z ˜ z = [ U | U ′ ] 11 12 [ U | U ′ ] T , zz + + S ′ S ′ S ′ 0 0 0 21 22 zz 12 S ′− 1 – where S ′ 11 − S ′ 22 S ′ 21 = 0 . – The entire class of worst-noise is related by linear estimation: z + z ′ 1 | z ′ E [˜ 2 ] = ˜ z. • The class of ( S xx , S zz ) that satisfies the KKT condition is precisely: ( S xx , S (0) zz + S ′ zz ) DIMACS Workshop on Network Information Theory 22

Outline • Complete characterization of the worst-noise. – Duality through minimax. – Worst-noise through duality. • Efficient numerical solution for the dual channel. • Does duality extend beyond the power constrained channel? DIMACS Workshop on Network Information Theory 23