feedback based distributed adaptive transmit beamforming ... · m. seo, m. rodwell, u. madhow, a...

Feedback based distributed adaptive transmitbeamformingAlgorithmic considerations

Stephan Sigg

Informatik Kolloquium, 31.01.2011, TU Braunschweig

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


Introduction

Distributed adaptive transmit beamformingDistributed nodes synchronise the carrier frequency and phase offset oftransmit signalsLow power and processing devicesNon-synchronised local oscillators



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



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



Introduction

f2t + γ22 π

f1t + γ12 π

ifn2 π

it+γ

n i+1fn2 π

i+1t+γ

n

Iteration i+1

Fre

quen

cy

Time

Mutation

Iteration i1

2

3

4

1

Superimposed received sum signal

Receiv

er f

eed

back

Distributed synchronisationschemes

Cosed loop feedback basedcarrier synchronisationa

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

Distributed transmit beamforming using feedback control, IEEETransactions on Information Theory 56(1), volume 56, January2010



Introduction

Receiv

er

feed

back

Cosed loop feedback based carriersynchronisation

Algorithm always converges to the optimum a

Expected optimisation time O(n) when ineach iteration the optimum Probabilitydistribution is chosen a

Optimisation time can be improved by factor2 when erroneous decisions are not discardedbut 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 Theory56(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, 2008c

M. Seo, M. Rodwell, U. Madhow, A Feedback-Based Distributed phased arraytechnique and its application to 60-GHz wireless sensor network, IEEE MTT-SInternational Microwave Symposium Digest, 2008



Outline

Introduction





Conclusion




ObservationsIterative approach similar to evolutionary random search

New search points are requested by altering the carrier phasesFitness function implemented by receiver feedbackSelection of individuals based on feedback valuesPopulation size and offspring population size: µ = ν = 1




Individual representationHere: Binary representation of phase/frequency offsets

log(k) bits to represent k phase offsetslog(ϕ) bits to represent ϕ frequency offsetsConfigurations for all nodes concatenated

Phase and frequency offsets enumerated in ascending orderNeighbourhood: Gray encoded bit sequence to respect neighbourhoodsimilarities




Assumptions :

Network of n nodesEach node changes the phase of its carrier signal withprobability 1

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




Optimisation aim :Achieve maximum relative phase offset of 2π

kBetween any two carrier signalsFor arbitrary kDivide phase space into k intervals of width 2π

k




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




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




Expected number of iterations to change layer bounded from aboveby P[Li ]

−1:

E [TP ] ≤n−1∑i=0

e · n · kn − i

= e · n · k ·n∑

i=1

1

i

< e · n · k · (ln(n) + 1)

= O (n · k · log n)



Outline

Introduction





Conclusion




A lower bound on the synchronisation performance

We utilise the method of the expected progressAfter initialisation, phases of carrier signals are identically andindependently distributed.Each bit in the binary representation of search point sζ has equalprobability to be 1 or 0.




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




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 hammingdistance to the nearest global optimum is at least l .




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+0

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) ≥ ∆])




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




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: ≤ 2tn·log(k)




Choose t = n·log(k)·∆4 − 1

Double of expected progress still smaller than ∆.

With Markov inequality: Progress not achieved with prob. 12 .

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))



Outline

Introduction





Conclusion




Reduce the amount of randomness in the optimisation

Improve the synchronisation performance

Improve the synchronisation quality




Global or local optima?

Weak multimodal fitnessfunction

e j(2π f t +γi )

ico

s(

)ϕ

i

i

i

ej

( +γi)

2π f t γi

1

ϕi

−δ

δi

i

ϕ

ej( 2

πf

+γ

t

)j

ϕco

s(

)

Gain




Fitness function observedby single node

Constant carrier phaseoffset for n − 1 nodes

Fitness function:

F(Φi ) = A sin(Φi + φ) + c




Approach:

Measure feedback at 3points

Solve multivariableequations

Apply optimum phaseoffset calculated

F(Φi ) = A sin(Φi + φ) + c




Problem:

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




Node estimates the quality of thefunction estimation itself

Transmit with optimum phase offsetand measure channel again

When Expected fitness deviatessignificantly from measured fitness,discard altered phase offset

Deviation:

1 node: ≈ 0.6%2 nodes: ≈ 1.5%3 nodes: > 3%




1 Transmit with phase offsets γ1 6= γ2 6= γ3; measure feedback

2 Estimate feedback function and calculate γ∗i3 Transmit with γ4 = γ∗i4 If deviation smaller 1% finished, otherwise discard γ∗i




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)




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




Asymptotically optimal synchronisation time

Simulations: ≈ 12n

Further improvement:

3 iterations per turnUtilise last transmission from previous iteration



Outline

Introduction





Conclusion




0 0.5 1 1.5 2 2.5 3−2

0

2x 10

−10

Time [ms]

Modulated transmit signal for device 1

0 0.5 1 1.5 2 2.5 30

0.5

1

Time [ms]

Transmitted bit sequence

0 0.5 1 1.5 2 2.5 3−2

0

2x 10

−10

Time [ms]

Modulated transmit signal for device n

0 0.5 1 1.5 2 2.5 3−2

0

2x 10

−9

Time [ms]

Received superimposed sum signal

0 0.5 1 1.5 2 2.5 3−5

0

5x 10

−9

Time [ms]

Demodulated received sum signal

Shift in the phase offset of transmit signals




0 0.5 1 1.5 2 2.5 3−1

0

1x 10

−11

Time [ms]

Modulated transmit signal for device 10 0.5 1 1.5 2 2.5 3

0.5

1

Time [ms]


0 0.5 1 1.5 2 2.5 3−1

0

1x 10

−11

Time [ms]


0 0.5 1 1.5 2 2.5 3−5

0

5x 10

−10

Time [ms]


0 0.5 1 1.5 2 2.5 3−5

0

5x 10

−10

Time [ms]





0 0.5 1 1.5 2 2.5 3−2

0

2x 10

−10

Time [ms]

Modulated transmit signal for device 10 0.5 1 1.5 2 2.5 3

0

0.5

1

Time [ms]


0 0.5 1 1.5 2 2.5 3−2

0

2x 10

−10

Time [ms]


0 0.5 1 1.5 2 2.5 3−2

0

2x 10

−9

Time [ms]


0 0.5 1 1.5 2 2.5 3−2

0

2x 10

−9

Time [ms]





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 0.752 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




Situation mean median σ

Door state (opened/closed) 0.952 0.9513 0.0099Presence of individual 0.817 0.8238 0.0455Phone call (gsm) 0.9 1.0 0.32

Door opened (cond.: Empty room) 1.0 1.0 0.0Door closed (cond.: Empty room) 1.0 1.0 0.0Door closed (cond.: Room occupied) 0.832 0.83 0.041Door opened (cond.: Room occupied) 0.976 0.98 0.0184Room occupied (cond.: Door closed) 0.673 0.66 0.1143Room occupied (cond.: Door open) 0.595 0.54 0.1247Empty room (cond.: Door closed) 1.0 1.0 0.0Empty room (cond.: Door open) 1.0 1.0 0.0


Questions?

Stephan [email protected]


feedback based distributed adaptive transmit beamforming ... · m. seo, m. rodwell, u. madhow, a...

Documents