[PPT] - Feedback based distributed adaptive transmit beamforming PowerPoint Presentation

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Feedback based distributed adaptive transmit beamforming

Algorithmic considerations

Stephan Sigg

Informatik Kolloquium, 31.01.2011, TU Braunschweig

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

Outline

Introduction An upper bound on the expected optimisation time A lower bound on the expected optimisation time An asymptotically optimal optimisation scheme An adaptive protocol for distributed adaptive beamforming Conclusion

Stephan Sigg | Feedback based distributed adaptive beamforming | 2

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

Introduction

Distributed adaptive transmit beamforming

Distributed nodes synchronise the carrier frequency and phase offset of transmit signals Low power and processing devices Non-synchronised local oscillators

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

Introduction

Distributed synchronisation schemes

Closed loop carrier synchronisation1

Receiver Transmitter Receiver Transmitter Receiver Transmitter

Source Source Source Source Source

common master beacon to all source nodes Receive node broadcasts Receiver Transmitter Receive nodes bounce the beacon back on distinct CDMA channels phase offset of each node on these CDMA channels Receiver transmits the relative Synchronised nodes transmit as a distributed beamformer to the receiver

1Y. Tu and G. Pottie, Coherent Cooperative Transmission from Multiple Adjacent Antennas to a Distant Stationary

Antenna Through AWGN Channels, Proceedings of the IEEE VTC, 2002 Stephan Sigg | Feedback based distributed adaptive beamforming | 4

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

Introduction

Distributed synchronisation schemes

Open loop carrier synchronisation2

frequency and local oscillators in a closed−loop synchronisation Transmit nodes synchronise their The receiver broadcasts a sinusoidal signal for open−loop synchronisation to the transmit nodes The synchronised nodes transmit as a distributed beamformer to the receiver Receiver Transmitter

Master Source Source Source Source

Receiver Transmitter Receiver Transmitter

2R. Mudumbai, G. Barriac and U. Madhow, On the feasibility of distributed beamforming in wireless networks, IEEE

Transactions on Wireless Communications, Vol 6, May 2007 Stephan Sigg | Feedback based distributed adaptive beamforming | 5

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

Introduction

f2t+ γ2 2π f1t+ γ1 2 π

i

fn 2π

it+γn

i+1

fn 2π

i+1

t+γn

Iteration i+1 Frequency Time Mutation Iteration i

1 2 3 4 1

Superimposed received sum signal Receiver feedback

Distributed synchronisation schemes

Cosed loop feedback based carrier synchronisationa

aR. Mudumbai, J. Hespanha, U. Madhow, G. Barriac,

Distributed transmit beamforming using feedback control, IEEE Transactions on Information Theory 56(1), volume 56, January 2010 Stephan Sigg | Feedback based distributed adaptive beamforming | 6

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

Introduction

Receiver feedback

Cosed loop feedback based carrier synchronisation

Algorithm always converges to the optimum a Expected optimisation time O(n) when in each iteration the optimum Probability distribution is chosen a Optimisation time can be improved by factor 2 when erroneous decisions are not discarded but inverted b Phase and frequency synchronisation feasiblec

aR. Mudumbai, J. Hespanha, U. Madhow, G. Barriac, Distributed transmit

beamforming using feedback control, IEEE Transactions on Information Theory 56(1), volume 56, January 2010

bJ. Bucklew, W. Sethares, Convergence of a class of decentralised beamforming

algorithms, IEEE Transactions on Signal Processing 56(6), volume 56, 2008

cM. Seo, M. Rodwell, U. Madhow, A Feedback-Based Distributed phased array

technique and its application to 60-GHz wireless sensor network, IEEE MTT-S International Microwave Symposium Digest, 2008 Stephan Sigg | Feedback based distributed adaptive beamforming | 7

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

Outline

Introduction An upper bound on the expected optimisation time A lower bound on the expected optimisation time An asymptotically optimal optimisation scheme An adaptive protocol for distributed adaptive beamforming Conclusion

Stephan Sigg | Feedback based distributed adaptive beamforming | 8

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

An upper bound on the expected optimisation time

Observations

Iterative approach similar to evolutionary random search

New search points are requested by altering the carrier phases Fitness function implemented by receiver feedback Selection of individuals based on feedback values Population size and offspring population size: µ = ν = 1

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

An upper bound on the expected optimisation time

Individual representation

Here: Binary representation of phase/frequency offsets

log(k) bits to represent k phase offsets log(ϕ) bits to represent ϕ frequency offsets Configurations for all nodes concatenated

Phase and frequency offsets enumerated in ascending order Neighbourhood: Gray encoded bit sequence to respect neighbourhood similarities

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

An upper bound on the expected optimisation time

Assumptions : Network of n nodes Each node changes the phase of its carrier signal with probability 1

n

Carrier phase altered uniformly at random from [0, 2π]

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

An upper bound on the expected optimisation time

Optimisation aim : Achieve maximum relative phase offset of 2π

k

Between any two carrier signals For arbitrary k Divide phase space into k intervals of width 2π

k

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

An upper bound on the expected optimisation time

Alter 1 carrier and keep n − 1 signals This happens with probability n − i 1

· 1

n · 1 k ·

1 − 1

n n−1 = n − i n · k

·
1 − 1

n n−1

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

An upper bound on the expected optimisation time

Since

1 − 1

n n < 1 e <

1 − 1

n n−1 Probability that Li is left for partition j, j > i: P[Li] ≥ n − i n · e · k

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

An upper bound on the expected optimisation time

Expected number of iterations to change layer bounded from above by P[Li]−1: E[TP] ≤

n−1

i=0

e · n · k n − i = e · n · k ·

n

i=1

1 i < e · n · k · (ln(n) + 1) = O (n · k · log n)

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

Outline

Introduction An upper bound on the expected optimisation time A lower bound on the expected optimisation time An asymptotically optimal optimisation scheme An adaptive protocol for distributed adaptive beamforming Conclusion

Stephan Sigg | Feedback based distributed adaptive beamforming | 16

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

A lower bound on the expected optimisation time

A lower bound on the synchronisation performance

We utilise the method of the expected progress After initialisation, phases of carrier signals are identically and independently distributed. Each bit in the binary representation of search point sζ has equal probability to be 1 or 0.

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

A lower bound on the expected optimisation time

Probability to start with hamming distance h(sopt, sζ) ≤ l; l ≪ n · log(k) to global optima sopt at most P[h(sopt, sζ) ≤ l] =

l

i=0
n · log(k)

n · log(k) − i

·

k 2n·log(k)−i ≤ (n · log(k))l+2 2n·log(k)−l

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

A lower bound on the expected optimisation time

P[h(sopt, sζ) ≤ l] =

l

i=0
n · log(k)

n · log(k) − i

·

k 2n·log(k)−i ≤ (n · log(k))l+2 2n·log(k)−l This means that with high probability (w.h.p.) the hamming distance to the nearest global optimum is at least l.

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

A lower bound on the expected optimisation time

Use method of expected progress to calculate lower bound: (sζ, t) denotes that sζ is achieved after t iterations Assume progress measure Λ : Bn·log(k) → R+ Λ(sζ, t) < ∆: Global optimum not found in first t iterations For every t ∈ N we have E[TP] ≥ t · P[TP > t] = t · P[Λ(sζ, t) < ∆] = t · (1 − P[Λ(sζ, t) ≥ ∆])

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

A lower bound on the expected optimisation time

E[TP] ≥ t · (1 − P[Λ(sζ, t) ≥ ∆]) With the help of the Markov-inequality we obtain P[Λ(sζ, t) ≥ ∆] ≤ E[Λ(sζ, t)] ∆ and therefore E[TP] ≥ t ·

1 − E[Λ(sζ, t)]

∆

Obtain lower bound by providing expected progress after t iterations

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

A lower bound on the expected optimisation time

Probability for l bits to correctly flip at most

1 −

1 n · log(k) n·log(k)−l ·

1

n · log(k) l ≤ 1 (n · log(k))l Expected progress in one iteration: E[Λ(sζ, t), Λ(sζ′, t + 1)] ≤

l

i=1

i (n · log(k))i < 2 n · log(k) Expected progress in t iterations: ≤

2t n·log(k)

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

A lower bound on the expected optimisation time

Choose t = n·log(k)·∆

4

− 1 Double of expected progress still smaller than ∆. With Markov inequality: Progress not achieved with prob. 1

2.

Expected optimisation time bounded from below by E[TP] ≥ t ·

1 − E[Λ(sζ, t)]

∆

≥

n · log(k) · ∆ 4 ·  1 −

2·n·log(k) 4·n·log(k) · ∆

∆   = Ω(n · log(k) · ∆) With ∆ = k · log(n)

log(k): Same order as upper bound:

E[TP] = Θ (n · k · log(n))

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

Outline

Introduction An upper bound on the expected optimisation time A lower bound on the expected optimisation time An asymptotically optimal optimisation scheme An adaptive protocol for distributed adaptive beamforming Conclusion

Stephan Sigg | Feedback based distributed adaptive beamforming | 24

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

An asymptotically optimal optimisation scheme

Reduce the amount of randomness in the optimisation Improve the synchronisation performance Improve the synchronisation quality

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

An asymptotically optimal optimisation scheme

Global or local optima?

Weak multimodal fitness function

e

j(2π f t +γi)

i

cos( ) ϕ

i

i i

e

j ( +γi) 2π f t γ i 1 ϕ

i

−δ δi

i

ϕ

e

j ( 2π f +γ t ) j ϕ cos( ) G a i n Stephan Sigg | Feedback based distributed adaptive beamforming | 26

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

An asymptotically optimal optimisation scheme

Fitness function observed by single node Constant carrier phase

ffset for n − 1 nodes

Fitness function: F(Φi) = A sin(Φi + φ) + c

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

An asymptotically optimal optimisation scheme

Approach: Measure feedback at 3 points Solve multivariable equations Apply optimum phase

ffset calculated

F(Φi) = A sin(Φi + φ) + c

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

An asymptotically optimal optimisation scheme

Problem:

Calculation not accurate when two or more nodes alter the phase of their transmit signals

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

An asymptotically optimal optimisation scheme

Node estimates the quality of the function estimation itself Transmit with optimum phase offset and measure channel again When Expected fitness deviates significantly from measured fitness, discard altered phase offset Deviation: 1 node: ≈ 0.6% 2 nodes: ≈ 1.5% 3 nodes: > 3%

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

An asymptotically optimal optimisation scheme

1 Transmit with phase offsets γ1 = γ2 = γ3; measure feedback 2 Estimate feedback function and calculate γ∗

i

3 Transmit with γ4 = γ∗

i

4 If deviation smaller 1% finished, otherwise discard γ∗

i

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

An asymptotically optimal optimisation scheme

Asymptotic synchronisation time: O(n) Classic approach:3 Θ(n · k · log(n))

3Sigg, El Masri and Beigl, A sharp asymptotic bound for feedback based closed-loop distributed adaptive beamforming in wireless sensor networks (Accepted for Transactions on Mobile Computing) Stephan Sigg | Feedback based distributed adaptive beamforming | 32

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

An asymptotically optimal optimisation scheme

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

An asymptotically optimal optimisation scheme

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

An asymptotically optimal optimisation scheme

Phase offset of distinct nodes is within +/ − 0.05π for up to 99%

f all nodes.

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

An asymptotically optimal optimisation scheme

Asymptotically optimal synchronisation time Simulations: ≈ 12n Further improvement:

3 iterations per turn Utilise last transmission from previous iteration

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

Outline

Introduction An upper bound on the expected optimisation time A lower bound on the expected optimisation time An asymptotically optimal optimisation scheme An adaptive protocol for distributed adaptive beamforming Conclusion

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

An adaptive protocol for distributed adaptive beamforming

0.5 1 1.5 2 2.5 3 −2 2 x 10

−10

Time [ms]

Modulated transmit signal for device 1

0.5 1 1.5 2 2.5 3 0.5 1 Time [ms]

Transmitted bit sequence

0.5 1 1.5 2 2.5 3 −2 2 x 10

−10

Time [ms]

Modulated transmit signal for device n

0.5 1 1.5 2 2.5 3 −2 2 x 10

−9

Time [ms]

Received superimposed sum signal

0.5 1 1.5 2 2.5 3 −5 5 x 10

−9

Time [ms]

Demodulated received sum signal

Shift in the phase offset

f transmit signals

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

An adaptive protocol for distributed adaptive beamforming

0.5 1 1.5 2 2.5 3 −1 1x 10

−11

Time [ms]

Modulated transmit signal for device 1

0.5 1 1.5 2 2.5 3 0.5 1 Time [ms]

Transmitted bit sequence

0.5 1 1.5 2 2.5 3 −1 1x 10

−11

Time [ms]

Modulated transmit signal for device n

0.5 1 1.5 2 2.5 3 −5 5x 10

−10

Time [ms]

Received superimposed sum signal

0.5 1 1.5 2 2.5 3 −5 5x 10

−10

Time [ms]

Demodulated received sum signal Stephan Sigg | Feedback based distributed adaptive beamforming | 39

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

An adaptive protocol for distributed adaptive beamforming

0.5 1 1.5 2 2.5 3 −2 2 x 10

−10

Time [ms]

Modulated transmit signal for device 1

0.5 1 1.5 2 2.5 3 0.5 1 Time [ms]

Transmitted bit sequence

0.5 1 1.5 2 2.5 3 −2 2 x 10

−10

Time [ms]

Modulated transmit signal for device n

0.5 1 1.5 2 2.5 3 −2 2 x 10

−9

Time [ms]

Received superimposed sum signal

0.5 1 1.5 2 2.5 3 −2 2 x 10

−9

Time [ms]

Demodulated received sum signal Stephan Sigg | Feedback based distributed adaptive beamforming | 40

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

An adaptive protocol for distributed adaptive beamforming

Stephan Sigg | Feedback based distributed adaptive beamforming | 41

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

An adaptive protocol for distributed adaptive beamforming

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

An adaptive protocol for distributed adaptive beamforming

7 0.90625

1.151049 e−09

8 0.85938

1.182819 e−09

9 0.89062

1.209551 e−09

6 5 0.9375 4 0.875 3 0.75 2 0.25 1 0.5

1.438299 e−09 1.198927 e−09 1.191585 e−09

0.8375

1.139293 e−09 1.155027 e−09 1.230101 e−09

RMSE Nr prob

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

An adaptive protocol for distributed adaptive beamforming

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Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion

An adaptive protocol for distributed adaptive beamforming

Situation mean median σ Door state (opened/closed) 0.952 0.9513 0.0099 Presence of individual 0.817 0.8238 0.0455 Phone call (gsm) 0.9 1.0 0.32 Door opened (cond.: Empty room) 1.0 1.0 0.0 Door closed (cond.: Empty room) 1.0 1.0 0.0 Door closed (cond.: Room occupied) 0.832 0.83 0.041 Door opened (cond.: Room occupied) 0.976 0.98 0.0184 Room occupied (cond.: Door closed) 0.673 0.66 0.1143 Room occupied (cond.: Door open) 0.595 0.54 0.1247 Empty room (cond.: Door closed) 1.0 1.0 0.0 Empty room (cond.: Door open) 1.0 1.0 0.0

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Questions?

Stephan Sigg sigg@ibr.cs.tu-bs.de

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