Minimax Duality of Gaussian Vector Broadcast Channels

Minimax Duality of Gaussian Vector Broadcast 

Channels 

Wei Yu and Tian Lan 

Electrical and Computer Engineering Department 

University of Toronto 

June 29, 2004 

Wei Yu and Tian Lan, University of Toronto

Reciprocity in Gaussian Vector Channels 

Z 1 ∼ N (0, I) Z 2 ∼ N (0, I) 

X 

H 

Y 

Y 

H T 

X 

E[X T X] ≤ P 

E[Y T Y ] ≤ P 

• Capacities of the channels H and H T are the same 

– under the same power constraint; 

– even if H is not square. 

• Proof: H and H T have the same singular values. 

Wei Yu and Tian Lan, University of Toronto 1

Reciprocity for Multi-User Channels 

Z 1 

X 

P 

H Y X ′ 1 

1 

1 

H T 1 

Z 2 

H 2 

Y 2 

Y ′ 

X ′ 2 

P 

H T 2 

Z 

• Duality exists between Gaussian vector MAC and BC Channels 

(Vishwanath, Jindal, Goldsmith ’03, Viswanath, Tse ’03). 

• Goal of this talk: Generalization and re-interpretation of duality. 


Uplink-Downlink Duality 

The multiple-access channel and the broadcast channel are duals. 

Z 1 

X 

P 

H Y X ′ 1 

1 

1 

H T 1 

Z 2 

H 2 

Y 2 

Y ′ 

X ′ 2 

P 

H T 2 

Z 

Previous proofs of duality depend critically on the power constraint. 

Why is there duality? 


Main Results of This Talk 

1. Uplink-Downlink Duality is equivalent to Lagrangian Duality 

2. Duality generalizes beyond the sum power constraint 

Z 1 

X 

H Y X ′ 1 

1 

1 

H T 1 

Z 2 

H 2 

Y 2 

Y ′ 

X ′ 2 

H T 2 

Z 

Broadcast Channel with ⇐⇒ Multiple Access Channel with 

Arbitrary input constraints 

Uncertain noise 


Multiple Antenna Broadcast Channel 

• Non-degraded Gaussian vector broadcast channel: 

Z n 

W 1 ∈ 2 nR 1 

W K ∈ 2 nR K 

X n 

H 

Y n 

1 

Y n K 

Ŵ 1 (Y n 

1 ) 

Ŵ K (Y n K ) 

• Capacity region is solved recently. 

– This talk focuses on sum capacity C = max{R 1 + · · · + R K }. 


Achievability: Writing on Dirty Paper 

Gaussian Channel 

... with Transmitter Side Information 

Z ∼ N (0, Sz) 

S ∼ N (0, Ss) 

Z ∼ N (0, Sz) 

X 

Y 

X 

Y 

C = 1 2 log |S x + S z | 

|S z | 

C = 1 2 log |S x + S z | 

|S z | 

• Capacities are the same if S is known non-causally at the transmitter. 


Converse: Sato’s Outer Bound 

• Broadcast capacity does not depend on noise correlation: Sato (’78). 

z 1 

z ′ z ′ 1 

1 

x 1 

y 1 x 1 

z 2 

x 1 y 1 y 1 

= z ′ 2 

≤ 

z ′ 2 

x 2 y 2 x 2 

y 2 x 2 

y 2 

} {{ } 

{ 

p(z1 ) = p(z 

if 

1) 

′ 

p(z 2 ) = p(z 2) ′ , not necessarily p(z 1, z 2 ) = p(z 1, ′ z 2). 

′ 

• So, sum capacity C ≤ min 

S z 

max 

S x 

I(X; Y). 


Three Achievability Proofs 

1. Decision-Feedback Equalization approach (Yu, Cioffi) 

• Worst-noise diagonalizes the feedforward matrix of a DFE. 

2. Uplink-Downlink duality approach (Viswanath, Tse) 

• Noise covariance is equivalent to the input constraint in dual channel. 

• Worst-noise decouples the inputs in the dual channel. 

3. Convex duality approach (Vishwanath, Jindal, Goldsmith) 

• Channel flipping between multiple access channel and broadcast 

channel. 

This talk: A new derivation of duality. 


Gaussian Broadcast Channel Sum Capacity 

• Achievability: C ≥ max 

S x 

min 

S z 

I(X; Y). 

• Converse: 

C ≤ min 

S z 

max 

S x 

I(X; Y). 

• Gaussian vector broadcast channel sum capacity is therefore exactly: 

. 

C = max 

S x 

1 

min 

S z 2 log |HS xH T + S z | 

|S z | 

Duality can be derived directly from the minimax expression! 


Minimax Optimization 

• Gaussian vector broadcast channel sum capacity is the solution of 

1 

max min 

S x S z 2 log |HS xH T + S z | 

|S z | 

subject to tr(S x ) ≤ P 

[ ] I ⋆ 

S z = 

⋆ I 

S x , S z ≥ 0 

• The minimax problem is convex in S z , concave in S x . 

– How to solve this minimax problem? 


Duality through Minimax 

• Two KKT conditions must be satisfied simultaneously: 

H T (HS x H T + S z ) −1 H = λI 

S −1 

z − (HS x H T + S z ) −1 = 

• For the moment, assume that H is invertible. 

[ ] 

Ψ1 0 

0 Ψ 2 

⇒ 

⇒ 

H T Sz 

−1 H − λI = H T ΨH 

H(H T ΨH + λI) −1 H T = S z 

• Further manipulation: ⇒ (λI) −1 − (H T ΨH + λI) −1 = S x . 


A New Minimax Duality 

KKT conditions lead to a new minimax problem: 

max 

S x 

1 

min 


|S z | 

min 

λ 

max 

Ψ 

1 

2 log |HT ΨH + λI| 

|λI| 

s.t. tr(S x ) ≤ P s.t. tr(Ψ) ≤ λP 

[ ] 

I ⋆ 

S z = 

Ψ is diagonal 

⋆ I 

S x , S z ≥ 0 Ψ ≥ 0, λ ≥ 0 

Minimax duality is equivalent to Lagrangian duality. 


Construct the Dual Channel 

KKT condition: H(H T ΨH + λI) −1 H T = S z 

• Define the diagonal matrix: D = Ψ/λ. trace(D) = ∑ i Ψ i/λ = P . 

• S z = 

[ 

I ⋆ 

⋆ I 

] 

. Thus, constraint on D: trace(D 1 ) + trace(D 2 ) ≤ P . 

E[X ′ 1 X′T 1 ] = D 1 

E[X ′ 2 X′T 2 ] = D 2 

trace(D 1 ) + trace(D 2 ) ≤ P 

X ′ 1 

X ′ 2 

H T 1 

H T 2 

Z 

Y ′ 


Uplink-Downlink Duality 

Z 1 

X 

P 

H Y X ′ 1 

1 

1 

H T 1 

Z 2 

H 2 

Y 2 

Y ′ 

X ′ 2 

P 

H T 2 

Z 

Theorem 1. Under the same power constraint, the Gaussian vector 

multiple-access channel and the Gaussian vector broadcast channel have 

the same sum capacity. 

New Proof: By re-defining D = Ψ/λ, the minimization part of the 

minimax dual problem disappears. 


Generalized Minimax Duality 

Theorem 1 may be generalized to arbitrary linear constraints: 

max 

S x 

1 

min 


|S z | 

max 

Σ z 

1 

min 

Σ x 2 log |HT Σ z H + Σ x | 

|Σ x | 

s.t. tr(S x Q x ) ≤ 1 s.t. tr(Σ z Ψ z ) ≤ 1 

tr(S z Q z ) ≤ 1 tr(Σ x Ψ x ) ≤ 1 

S x , S z ≥ 0 Σ x , Σ z ≥ 0 

Relation: S x = λ x Ψ x S z = λ z Ψ z Σ x = λ x Q x Σ z = λ z Q z 


Generalized Uplink-Downlink Duality 

Z 1 

X 

H Y X ′ 1 

Z ′ 

1 

1 

Z 2 

H 2 

H T 1 

Y 2 

Y ′ 

X ′ 2 

H T 2 

tr(S x Q 1 ) ≤ P 

S z ∼ N (0, Q 2 ) tr(S x ′Q 2 ) ≤ P S z ′ ∼ N (0, Q 1 ) 

Q 1 : Input constraint in BC and Noise covariance in MAC. 

Q 2 : Worst noise covariance in BC and Input constraint in MAC. 


Per-Antenna Power Constrained Broadcast Channel 

max 

S x 

1 

min 


|S z | 

max 

Ψ 

min 

Λ 

1 

2 log |HT ΨH + Λ| 

|Λ| 

s.t. S x (i, i) ≤ P i s.t. tr(Ψ) ≤ ∑ i 

P i 

S z = 

[ I ⋆ 

⋆ I 

] 

∑ 

Λ(i, i)P i ≤ 1 

S x , S z ≥ 0 Λ, Ψ ≥ 0, and diagonal 

Theorem 2. The dual of a Gaussian vector broadcast channel with 

individual per-antenna power constraint is a multiple-access channel with a 

diagonal and linearly constrained uncertain noise. 

i 


Per-Antenna Power Constrained Broadcast Channel 

Z 1 

X 

S x (i) ≤ P i 

H Y X ′ 1 

Z ′ 

1 

1 

Z 2 

H 2 

H T 1 

Y 2 

Y ′ 

X ′ 2 

H T 2 

≤∑ ∑ 

S z (i) ∼ N (0, I) tr(S x ′) Pi Sz ′(i)P i ≤ 1 

In addition, this duality applies not only to the sum capacity but also 

the entire region. 


Summary and Conclusions 

• Sum capacity of a Gaussian vector broadcast channel is: 

C = max 

S x 

1 

min 


|S z | 

• Lagrangian duality leads to uplink-downlink duality. 

• The dual of a broadcast channel with individual power constraint is a 

multiple access channel with unknown noise. 

• Duality can be very useful from a computational perspective.

Minimax Duality of Gaussian Vector Broadcast Channels

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