1

Noncooperative Optimization of Space-Time Signals in

Ad hoc Networks

Ronald A. Iltis and Duong Hoang

Dept. of Electrical and Computer Engineering

University of California

Santa Barbara, CA 93106

{iltis,hoang}@ece.ucsb.edu

http://stnlabs.ece.ucsb.edu

Supported in part by NSF CCF-0429596 and the International Foundation for

Telemetering

Presented at ITA 2006 UCSD

2

Outline

• Ad hoc network definition and space-time waveforms.

Time-reversal relationships.

• Optimization problem: Minimize sum power under

decoupled QoS constraint.

• Network duality and weak duality.

• Network Lagrangian: Greedy algorithms vs. taxation.

• IMMSE-TR as a noncooperative game with taxation.

• Extension to ST eigencoding. Waterfilling with

generalized eigenvalues. IMMSE-TR for eigencoding.

• Simulation results.

3

Ad hoc Quasi-Synchronous Network

1

l(1) = 2

2

3

5 l(3) = 5

l(5) = 3

4

6

4

Space-Time QS Waveforms

il(i)

gl(i)

5

QS Channel Model

Space-Time Channels are Block Toeplitz

Proposition: ST channel transpose is reciprocal time-reverse.

6

Time-Reverse Relationships

7

Optimization for QoS

• Minimize network sum power

Subject to SNR constraint

Equivalent to decoupled capacity constraint.

log(1+Γl(i)

)= rl(i)

8

Summary of Ad hoc QS Network

1. Each node i transmits a space-time waveform

2. Each node i = 1,2,…,N has a unique link node l(i)

≠i.

3. Spatial channels are symmetric,

4. Transmission is asynchronous, half-duplex.

5. Node i defined by (receive, transmit) S-T

beamformer pair and transmit power Pi.

6. SNR at node l(i) defined by

9

S-T MMSE Receive Beamformer

Optimal receive solution – easily implemented using LMS/RLS algorithm and

preamble, ACK, CTS, RTS training sequences (e.g. in 802.11b/g)

i

Training

+

-

c.c.

10

Optimization Using MMSE Beamformer

Optimal MMSE S-T receive beamformer

Lagrangian for optimization problem – Equivalent to capacity QoS constraint.

Definition of MAI plus noise covariance

11

Selected Results on Beamforming and Eigencoding

• Point-to-Point MIMO: OFDM with beamforming is optimal in absence of MAI. (Raleigh and Jones 99)

• Spatial-only eigencoding in Ad hoc networks – greedy optimization performs poorly (Ye and Blum 03.)

• Iterative MMSE beamforming approximately minimizes sum power under decoupled capacity QoS constraint (Bromberg and Agee 03.)

• IMMSE yields minimum power solution under SNR constraints for multi-access channel (duality) (Viswanath/Tse03, Visotsky/Madhow 99, Rashid-Farrokhi et. al. 98)

• Noncooperative beamforming games (Iltis 04, Oteri and Paulraj 05.)

12

• For TDD networks with symmetric channels, the optimum

transmit beamformer is the conjugate of the MMSE

receive beamformer (duality – Bromberg and Agee 03)

• For MISO multiuser networks, duality also holds and

iterative MMSE beamforming yields optimal solutions

(Rashid-Farrokhi et. al. 98)

• For Ad hoc MIMO networks (non-TDD), weak duality

holds (Iltis/Kim/Hoang 05.) IMMSE satisfies the FONC

for optimization using Lagrangian analysis.

Summary of Network Duality Results

gi = wr,∗i

13

FONC Theorem for Optimization

Theorem

Corollary: A greedy noncooperative algorithm cannot satisfy the FONCs.

The optimal normalized transmit ST vectors gi, in either a QS or synchro-nous network, are given by the eigenvector equation

gi = λiQ−1i H

Hl(i)R

−1l(i)Hl(i),igi. (1)

These vectors gi are stationary points of the Lagrangian and satisfy the KKTconditions, and λi are the Lagrange multipliers. The pseudo-covariance matrixQi ∈ C

MNs×MNs is given by

Qi = I+∑

k �=i,l(i)

2∑

q=0

ρl(k)(Hql(k),i)

Hwl(k)wHl(k)H

ql(k),i. (2)

14

Optimality of FDMA in Ad hoc Networks

Let the network defined by the matrices Hpl(i),j be synchronous, so that

Hpl(i),j = 0 for p �= 0. Assume that the waveforms gi are augmented by a

cyclic prefix so that the H0l(i),j matrices become circulant. Then the optimal

ST waveforms in the absence of MAI are given by

gopti =

√Picki ⊗ ai, (1)

where ck = [1 exp(i2πk/Ns) . . . exp(i2πk(Ns − 1)/NS)]T is an FFT vector with

frequency k. The ST waveform is normalized to ||cki ⊗ai||2 = 1. The minimum

power Pi yielding an SNR of γ0 is Pi = γ0/λmax(HHl(i),iHl(i),i), where λmax(A)

is the maximum eigenvalue of matrix A. The optimal frequency ki and spatialsignature ai without MAI satisfy

ki,ai = argmaxk,a

aHHHl(i),i(k)Hl(i),i(k)a, (2)

where ||ai||2 = 1/Ns. Furthermore, when the decoupled optimum frequencies

ki are unique so that ki �= kj for j �= i, l(i), then the FDMA solution is thenetwork optimum of the problem.

15

IMMSE Algorithm for S-T Beamforming

1 2

Step 1: Transmit training sequence from 1, use LMS/RLS at

node 2 = l(1) to approximate MMSE/MVDR S-T beamformer.

(Assume zero misadjustment)

Step 2: Set transmit beamformer at l(1) to c.c. time-reverse of

normalized MVDR receive beamformer.

16

IMMSE S-T

1 2

Step 3: Transmit training sequence from 2 to 1, use LMS/RLS and

repeat MVDR beamformer computation.

Step 4: Replace g1 by (w1*) r. Substitution reveals power alg.

17

Noncooperative Game Interpretation

Matrix pencil interpretation of IMMSE.

Use Gauss-Seidel iterations with gin+1 updated sequentially.

Power update achieves γ0 immediately.

18

Noncooperative Game Interpretation

Equivalent utility function: Pin+1, gi

n+1 maximizes un() (Best

response.)

Normalized SNR for link

i to l(i), γi(gi,g-i).Interference tax

19

Total Inference Function for Spatial-Only Beamforming

Decomposition of TIF after Gauss-Seidel update of gin.

20

Relation of IMMSE-TR to Lagrangian

Let {gi} be the IMMSE-TR solution for the ST waveforms and assume thenetwork is synchronous. Then the {gi} correspond to a stationary point of theLagrangian where each λi is minimized and satisfies the KKT conditions. Theresulting dual of the Lagrangian for any set of transmit beamformers satisfyinggi = w

r,∗i ∀i is

Psum =N∑

i=1

||gi||2 =

N∑

i=1

λiγ0 −N∑

i=1

∑

j �=i,l(i)

|gr,Ti Hi,jgj |2. (1)

Hence, IMMSE-TR, where each λi is minimized can only correspond to theminimum of Psum in the absence of MAI, i.e. when gr,Ti Hi,jgj = 0.

Interpretation: IMMSE-TR satisfies FONC, but in general is not

optimum.

Theorem

21

IMMSE-TR and FONC

• Recall Theorem giving FONC.

IMMSE-TR matrix pencil interpretation.

Satisfies FONC for smallest Lagrange multiplier satisfying KKT

conditions.

µigi = R−1i HH

l(i)R−1l(i)Hl(i),igi.

gi = λiQ−1i HH

l(i)R−1l(i)Hl(i),igi.

22

Power Efficiency

• Power efficiency: Power required to maintain power on

link i to l(i) in absence of multiuser interference/ Actual

power used in IMMSE.

• Similar to CDMA asymptotic efficiency for Gaussian

MAI (Verdú 86, 98).Maximum normalized SNR on link i to l(i).

Power efficiency:

23

Simulation of IMMSE – Exact Eigenvector Solution

M = 4 elements, Ns = 14 samples, SNR target = 10 dB, three path fading

channel with 1/r2 loss.

100 200 300 400 500 600 700 800 900 1000

0

100

200

300

400

500

600

700

800

900

Meters

Me

ters

Ns =14 M

r,t =4

0 2 4 6 8 10 12 140

2

4

6

8

10

12

Nyquist Sample

Use

r N

um

be

r

24

Power Efficiency for Space-Time Waveforms

0 50 100 150 200 250 300 350 400 450 5000.8

0.82

0.84

0.86

0.88

0.9

Iteration

Po

we

r E

ffic

ien

cy η

Greedy Algorithm

IMMSE-TR Efficiency η

Cooperative Augmented Lagrangian Efficiency

N = 10 NodesN

s = 14, M = 4

25

Power Efficiency

Number of nodes 10 14 18 22 26 30

IMMSE-TR η 0.88 0.87 0.83 0.82 0.68 0.67Greedy η 0.84 0.83 0.73 0.73 0.58 0.57Centralized η 0.89 0.87 0.84 0.83 0.70 0.70

Table 1: Power efficiency of 3 algorithms averaged over 50 network instances

Cooperative augemented Lagrangian (centralized) always yields best

performance.

IMMSE-TR always outperforms greedy optimization, very close to

cooperative solution.

26

Extension to Space-Time Eigencoding

• Global optimization problem. Minimize sum power

under assumptions (a) MAI is Gaussian (b)

Decoupled capacity constraint.

minimize

N∑

i=1

tr{GH

i Gi}

subject to for all l(i)

log |I+ GHi H

Hl(i),iR

−1l(i)Hl(i),iGi| = rl(i),

27

Noncooperative ST Eigencoding (Taxation)

Interference taxation due to minimization objective.

Minimize tr{GH

i R∗,ri Gi}

Subject to

log |I+ GHi H

Hl(i),iR

−1l(i)(G−i)Hl(i),iGi| = rl(i)

with G−i fixed.

Tr{GH

i R∗,ri GH

i

}=

Tr{GHi G

Hi } + Tr

GHi

∑

k �=i

Hr,∗I,kG

r,∗k G

r,Tk H

r,Tl,k G

Hi }

28

Waterfilling using Generalized Eigenvalues

Noncooperative optimum.

Powers give in terms of generalized eigenvalues.

Pi,k =(

µiΛIi,k

− 1ΛSi,k

)+

Properties of generalized eigenvalues and eigenmatrices.

HHl(i),iR

−1l(i)Hl(i),iGi = (R∗,r

i )GiΛGi

GHi R

∗,ri Gi = ΛIi

GHi H

Hl(i),iR

−1l(i)Hl(i),iGi = ΛSi

ΛGi = ΛSi (ΛIi )−1,

29

IMMSE-TR for ST Eigencoding

Consider matrix ST MMSE detector.

MMSE,i = argminWE{||bl(i) −WHi ri||

2}

MMSE matrix form of solution

Wi = R−1i Hi,l(i)Gl(i)Mi

IMMSE-TR – column-wise conjugation and time-reversal

Gi =Wr,∗i

Mi =[I+ GH

l(i)HHi,l(i)R

−1i Hi,l(i)Gl(i)

]−1

30

Relationship of IMMSE-TR to Noncooperative Optimum

Theorem:

Consider the IMMSE-TR algorithm where the transmit matrices satisfyGi =W

r,∗i for all nodes i. Then at least one fixed point of IMMSE-TR corre-

sponds to the generalized eigenmatrix solution rewritten as

Gi = (Rr,∗i )−1HH

l(i),iR−1l(i)Hl(i),iGi(Λ

gi )−1. (1)

Outline of proof: From definition of MMSE detector

Wi = R−1i Hi,l(i)Gl(i)P

1/2l(i)MiDi

Use IMMSE-TR relationship

Gi = (Rr,∗i )−1HH

l(i),iR−1l(i) ×Hl(i),iGiP

1/2i Ml(i)Dl(i)P

1/2l(i)M

r,∗i D

r,∗i

But Di becomes diagonal if Gi is a generalized eigenmatrix. Leads to (1) above.

31

Simulation Results

0 200 400 600 800 10000

100

200

300

400

500

600

700

800

900

1000

x, meters

y, m

ete

rs

Exact channel/covariance knowledge – eigenmatrix

solution. Ns = 6, M = 4, r = 3 b/s/Hz, 14 nodes.

32

Power Efficiency – ST Eigencoding vs. OFDM

0 20 40 60 80 100 120 140 160 180 2000.3

0.4

0.5

0.6

0.7

0.8

0.9

1P

ow

er

effic

ien

cy

Iterations

M = 4 N

s = 6

r = 3 b/s/Hz

ST Generalized Eigencoding

ST Greedy

OFDM-Generalized Eigencoding

OFDM - Greedy

Centralized

33

Santa Barbara ZAPR Scenario

Zero-infrastructure Access Point Router – Rank-1 dominant beampattern approximations

for noncooperative ST eigencoding.

34

Sum Power Consumption – SB ZAPR Network

Ns = 64 (OFDM-type) M = 6 antennas, 10 b/s/Hz.

50 100 150 200 250 3000

20

40

60

80

100

120

Iteration

To

tal p

ow

er,

W

Ns=64

M=6

10 b/s/Hz

Generalized Eigencoding

Greedy

35

Conclusions

• Lagrangian optimization, noncooperative game theory and taxation closely related to IMMSE-TR.

• Implicit interference taxation in IMMSE-TR yields consistently better solutions than greedy optimization.

• ST eigencoding with taxation – leads to waterfilling algorithm based on generalized eigenvalues.

• Questions:

– Can taxation be optimized? Use game theory concepts? (Problems with lack of ordered action set, supermodularity.)

– Can IMMSE-TR be modified to yield consistent eigenmatrixsolutions for ST eigencoding?

– Robust algorithms for channel/covariance estimation?

– Implement in reconfigurable hardware?