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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

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SLIDE 1

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

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SLIDE 2

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

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SLIDE 3

Gaussian Vector Broadcast Channel

  • Non-degraded broadcast channel:

Zn Xn Y n

1

Y n

K

H W1 ∈ 2nR1 WK ∈ 2nRK ˆ W1(Y n

1 )

ˆ WK(Y n

K)

  • Capacity region is still unknown.

– Sum capacity C = max{R1 + · · · + RK} is recently solved.

DIMACS Workshop on Network Information Theory 2

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SLIDE 4

Marton’s Achievability Region

  • For a broadcast channel p(y1, y2|x):

R1 ≤ I(U1; Y1) R2 ≤ I(U2; Y2) R1 + R2 ≤ I(U1; Y1) + I(U2; Y2) − I(U1; U2) for some auxiliary random variables p(u1, u2)p(x|u1, u2).

  • For the Gaussian broadcast channel:

I(U2; Y2) − I(U1; U2) is achieved with precoding.

DIMACS Workshop on Network Information Theory 3

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SLIDE 5

Writing on Dirty Paper

Gaussian Channel ... with Transmitter Side Information

Z ∼ N(0, Szz) Z ∼ N(0, Szz) S ∼ N(0, Sss) X X Y Y

C = 1 2 log |Sxx + Szz| |Szz| C = 1 2 log |Sxx + Szz| |Szz|

  • 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

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SLIDE 6

Precoding for the Broadcast Channel

W1 ∈ 2nR1 W2 ∈ 2nR2 H1 H2 Xn

1 (W1, Xn 2 )

Xn

2 (W2)

Xn Zn

1

Zn

2

Y n

1

Y n

2

ˆ W1(Y n

1 )

ˆ W2(Y n

2 )

R1 = I(X1; Y1|X2) = 1 2 log |H1S1HT

1 + Sz1z1|

|Sz1z1| R2 = I(X2; Y2) = 1 2 log |H2S2HT

2 + H2S1HT 2 + Sz2z2|

|H2S1HT

2 + Sz2z2|

DIMACS Workshop on Network Information Theory 5

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SLIDE 7

Converse: Sato’s Outer Bound

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

x1 x1 x1 x2 x2 x2 y1 y1 y1 y2 y2 y2 z1 z2 z′

1

z′

1

z′

2

z′

2

=

≤

  • if
  • p(z1) = p(z′

1)

p(z2) = p(z′

2) , not necessarily p(z1, z2) = p(z′ 1, z′ 2).

  • So, sum capacity C ≤ min

Szz max Sxx I(X; Y).

DIMACS Workshop on Network Information Theory 6

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SLIDE 8

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

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SLIDE 9

DFE Approach

x z′ H HT

  • feedforward filter

∆−1G−T Decision I − G

  • Decision-feedback at the receiver is equivalent to transmitter precoding.
  • (Non-Singular) Worst Noise ⇐

⇒ Diagonal feedforward filter Fix Sxx, min

Szz I(X; Y) is achievable.

DIMACS Workshop on Network Information Theory 8

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SLIDE 10

Uplink-Downlink Duality Approach

X1 X2 Y1 Y2 Z1 ∼ N (0, Q) Z2 ∼ N (0, I) E[XT

1 X1] ≤ P

E[XT

2 QX2] ≤ P

H HT

  • 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

Sxx min Szz I(X; Y)

DIMACS Workshop on Network Information Theory 9

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SLIDE 11

Convex Duality Approach

X X′

1

X′

2

Y1 Y2 Y ′ Z1 Z2 Z P P H1 H2 HT

1

HT

2

  • Sato’s bound: C ≤ min

Szz max Sxx I(X; Y).

  • Broadcast/Multiple-Access duality: C ≥ max

Sx′x′ I(X′; Y′).

  • Convex duality: max

Sxx min Szz I(X; Y) = max Sx′x′ I(X′; Y′).

DIMACS Workshop on Network Information Theory 10

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SLIDE 12

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

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SLIDE 13

Minimax Capacity

  • Gaussian vector broadcast channel sum capacity is the solution of

max

Sxx min Szz

1 2 log |HSxxHT + Szz| |Szz| subject to tr(Sxx) ≤ P Szz = I ⋆ ⋆ I

  • Sxx, Szz ≥ 0
  • The minimax problem is convex in Szz, concave in Sxx.

– How to solve this minimax problem?

DIMACS Workshop on Network Information Theory 12

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SLIDE 14

Duality through Minimax

  • Two KKT conditions must be satisfied simultaneously:

HT(HSxxHT + Szz)−1H = λI S−1

zz − (HSxxHT + Szz)−1 =

  • Ψ1

Ψ2

  • For the moment, assume that H is invertible.

⇒ HTS−1

zz H − λI = HTΨH

⇒ H(HTΨH + λI)−1HT = Szz This is a “water-filling” condition for the dual channel.

DIMACS Workshop on Network Information Theory 13

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SLIDE 15

Power Constraint in the Dual Channel

  • Interpretation of dual variable: λ = ∂C

∂P , Ψi = − ∂C ∂Szizi . – Thus, capacity is preserved if λ∆P =

  • i

Ψi

  • ∆Szizi
  • Capacity C = min max 1

2 log |HSxxHT + Szz| |Szz| . – Thus, capacity is preserved if ∆P P = ∆Szizi 1 . Therefore,

  • i Ψi

λ = P.

DIMACS Workshop on Network Information Theory 14

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SLIDE 16

Construct the Dual Channel

KKT condition: H(HTDH + I)−1HT = 1

λSzz

  • where D = Ψ/λ is diagonal, trace(D) =

i Ψi/λ = P.

  • Szz =
  • I

⋆ ⋆ I

  • . Thus, constraint on D: trace(D1) + trace(D2) ≤ P.

X′

1

X′

2

Y ′ Z E[X′

1X′T 1 ] = D1

E[X′

2X′T 2 ] = D2

trace(D1) + trace(D2) ≤ P HT

1

HT

2 DIMACS Workshop on Network Information Theory 15

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SLIDE 17

Yet Another Derivation for Duality

The duality between broadcast channel and multiple-access channel: max

Sxx min Szz

1 2 log |HSxxHT + Szz| |Szz| max

D

1 2 log |HTDH + I| |I| s.t. tr(Sxx) ≤ P s.t. tr(D) ≤ P Szz =

  • I

⋆ ⋆ I

  • D is diagonal

Sxx, Szz ≥ 0 D ≥ 0 KKT conditions for minimax = ⇒ KKT condition for max.

DIMACS Workshop on Network Information Theory 16

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SLIDE 18

Worst-Noise Through Minimax

  • Solve the dual multiple access channel problem with power constraint P.

Obtain (Ψ, λ). Then: Szz = H(HTΨH + λI)−1HT Sxx = (λI)−1 − (HTΨH + λI)−1

  • What if H is not invertible, or Szz is singular?

DIMACS Workshop on Network Information Theory 17

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SLIDE 19

Decision-Feedback Equalization with Singular Noise

  • With non-singular noise: S−1

zz − (HSxxHT + Szz)−1 =

  • Ψ1

Ψ2

  • .
  • If H is low-rank, Szz can be singular.

X H Z m-dimensional m × n n > m Linear Estimation/DFE is not unique if |Sz| = 0.

DIMACS Workshop on Network Information Theory 18

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SLIDE 20

Necessary and Sufficient Condition for Diagonalization

  • Suppose that the worst-noise |Szz| = 0, let

Szz = US˜

z˜ zU T,

where Szz is n × n, S˜

z˜ z is m × m, m < n.

  • It is always possible to write H = U ˜

H.

  • There exists a DFE with diagonal feedforward filter if and only if

S−1

˜ z˜ z − ( ˜

HSxx ˜ HT + S˜

z˜ z)−1 = U T

Ψ1 Ψ2

  • U

DIMACS Workshop on Network Information Theory 19

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SLIDE 21

Singular Worst-Noise

  • It can be verified that the diagonalization condition is satisfied by:

S(0)

zz

= H(HTΨH + λI)−1HT Sxx = (λI)−1 − (HTΨH + λI)−1

  • However: S(0)

zz does not necessarily have 1’s on the diagonal.

S(0)

zz =

  I ⋆ ⋆ ⋆ I ⋆ ⋆ ⋆ ⋆   .

DIMACS Workshop on Network Information Theory 20

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SLIDE 22

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(HTΨH + λI)−1HT,

Sxx = (λI)−1 − (HTΨH + λI)−1.

  • 3. If Sxx 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 ⋆ ⋆ ⋆ I ⋆ ⋆ ⋆ ⋆   +   ⋆   =   I ⋆ ⋆ ⋆ I ⋆ ⋆ ⋆ I   .

DIMACS Workshop on Network Information Theory 21

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SLIDE 23

Worst-Noise is Not Unique

  • The same Sxx water-fills the entire class of S(0)

zz + S′ zz.

  • S(0)

zz +

  • S′

zz

  • = [U|U ′]
  • S˜

z˜ z

  • +
  • S′

11

S′

12

S′

21

S′

22

  • [U|U ′]T,

– where S′

11 − S′ 12S′−1 22 S′ 21 = 0.

– The entire class of worst-noise is related by linear estimation: E[˜ z + z′

1|z′ 2] = ˜

z.

  • The class of (Sxx, Szz) that satisfies the KKT condition is precisely:

(Sxx, S(0)

zz + S′ zz)

DIMACS Workshop on Network Information Theory 22

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SLIDE 24

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

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SLIDE 25

Sum Power Gaussian Vector Multiple Access Channel

X1 X2 H1 H2 Z Y P

max

Sxx

1 2 log |HTSxxH + I| s.t. tr(Sxx) ≤ P Sxx is diagonal Sxx ≥ 0

  • An efficient way to find the worst-noise is to solve the dual problem.

– Previous numerical solution: Jindal, Jafar, Vishwanath, Goldsmith.

DIMACS Workshop on Network Information Theory 24

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SLIDE 26

Iterative Water-filling

  • Iterative water-filling: Optimize each of Si while fixing all others.

max

Si

1 2 log

  • i

HiSiHT

i + I

  • max

Si

1 2 log

  • i

HiSiHT

i + I

  • s.t.

tr(Si) ≤ Pi s.t.

  • i

tr(Si) ≤ P Si ≥ 0 Si ≥ 0 Individual Constraints Coupled Constraint

  • Iterative water-filling only works with the individual power constraints.

DIMACS Workshop on Network Information Theory 25

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SLIDE 27

Dual Decomposition for the Sum-Power Problem

Take Lagrangian dual with respect to the coupled constraint only: max 1 2 log

  • i

HiSiHT

i + I

  • g(ν) = max

1 2 log

  • i

HiSiHT

i + I

  • s.t.
  • i

Pi ≤ P − ν

  • i

Pi − P

  • tr(Si) ≤ Pi

s.t. tr(Si) ≤ Pi Si ≥ 0 Si ≥ 0 Sum Power Capacity = min

ν>0 g(ν)

DIMACS Workshop on Network Information Theory 26

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SLIDE 28

Iterative Water-filling for the Dual Problem

  • By introducing a Lagrange multiplier ν, constraints are decoupled:

g(ν) = max

Si

1 2 log

  • i

HiSiHT

i + I

  • − ν
  • i

Pi − P

  • s.t.

tr(Si) ≤ Pi Si ≥ 0 – To solve g(ν): Iteratively optimize each of (Si, Pi). – To find min g(ν) over ν > 0: Decrease ν if

i Pi < P. Increase ν if i Pi > P.

DIMACS Workshop on Network Information Theory 27

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SLIDE 29

Convergence of the Dual Decomposition Algorithm

  • 3 transmit antennas
  • 50 receivers each with

a single antenna – typically 3-6 active

  • i.i.d. Gaussian channel
  • Bisection on ν.

20 40 60 80 100 120 2.5 3 3.5 4 4.5 5 iterations sum capacity (bits/transmission)

DIMACS Workshop on Network Information Theory 28

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SLIDE 30

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 29

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SLIDE 31

Broadcast Channel under Linear Covariance Constraint

  • The DFE achievability result works with any fixed Sxx.
  • The capacity of the broadcast channel under covariance constraint:

max

Sxx min Szz

1 2 log |HSxxHT + Szz| |Szz| subject to tr(QSxx) ≤ P Szz =

  • I

⋆ ⋆ I

  • Sxx, Szz ≥ 0
  • What is the duality result in this case?

DIMACS Workshop on Network Information Theory 30

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SLIDE 32

KKT Condition for Minimax

  • Two KKT conditions must be satisfied simultaneously:

HT(HSxxHT + Szz)−1H = λQ S−1

zz − (HSxxHT + Szz)−1 =

  • Ψ1

Ψ2

  • For simplicity, assume invertible H.

H(HTΨH + λQ)−1HT = Szz with

  • i tr(Ψi)

λ = P

DIMACS Workshop on Network Information Theory 31

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SLIDE 33

Duality under Linear Covariance Constraint

The duality between broadcast channel and multiple-access channel: max

Sxx min Szz

1 2 log |HSxxHT + Szz| |Szz| max

D

1 2 log |HTDH + Q| |Q| s.t. tr(QSxx) ≤ P s.t. tr(D) ≤ P Szz =

  • I

⋆ ⋆ I

  • D is diagonal

Sxx, Szz ≥ 0 D ≥ 0 The above two problems have the same KKT conditions.

DIMACS Workshop on Network Information Theory 32

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SLIDE 34

Generalized Duality

X X′

1

X′

2

Y1 Y2 Y ′ Z1 Z2 Z′ tr(SxxQ1) ≤ P tr(Sx′x′Q2) ≤ P Szz ∼ N (0, Q2) Sz′z′ ∼ N (0, Q1) H1 H2 HT

1

HT

2

Q1: Input constraint in BC and Noise covariance in MAC. Q2: Worst noise covariance in BC and Input constraint in MAC.

DIMACS Workshop on Network Information Theory 33

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SLIDE 35

Broadcast Channel under Convex Covariance Constraint

  • Under arbitrary convex constraint, DFE still works.

max

Sxx min Szz

1 2 log |HSxxHT + Szz| |Szz| subject to f(Sxx) ≤ P Szz =

  • I

⋆ ⋆ I

  • Sxx, Szz ≥ 0

Does duality exist in this case?

DIMACS Workshop on Network Information Theory 34

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SLIDE 36

Duality under Convex Covariance Constraint

Duality still exists, but the values of the dual variables are not known: max

Sxx min Szz

1 2 log |HSxxHT + Szz| |Szz| max

D

1 2 log |HTΨH + λQ| |λQ| s.t. f(Sxx) ≤ P s.t. tr(Ψ) ≤ P ′ Szz =

  • I

⋆ ⋆ I

  • D is diagonal

Sxx, Szz ≥ 0 D ≥ 0 Q = f ′(·). But if f(·) is non-linear, tr(Ψ) = λP.

DIMACS Workshop on Network Information Theory 35

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SLIDE 37

Peak Power Constrained Broadcast Channel

  • Duality exists, but not computationally useful. Need to solve minimax.

max

Sxx min Szz

1 2 log |HSxxHT + Szz| |Szz| max

D

1 2 log |HTΨH + Q| |Q| s.t. Sxx(i, i) ≤ Pi s.t. tr(Ψ) ≤ P ′ Szz =

  • I

⋆ ⋆ I

  • D is diagonal

Sxx, Szz ≥ 0 D ≥ 0

  • Here, Q =

  µ1 ... µn  . But, µi, P ′ are not known.

DIMACS Workshop on Network Information Theory 36

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SLIDE 38

Concluding Remarks

  • Sum capacity of a Gaussian vector broadcast channel is:

C = max

Sxx min Szz

1 2 log |HSxxHT + Szz| |Szz|

  • If the input constraint is a linear covariance constraint:

C = max

D

1 2 log |HTDH + Q| |Q|

  • Minimax is a more fundamental expression than duality.
  • Duality, when exists, has computational advantage.

DIMACS Workshop on Network Information Theory 37