Feedback based distributed adaptive transmit beamforming ... · M. Seo, M. Rodwell, U. Madhow, A Feedback-Based Distributed phased array technique and its application to 60-GHz wireless
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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 Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion
Introduction
Distributed adaptive transmit beamformingDistributed nodes synchronise the carrier frequency and phase offset oftransmit signalsLow power and processing devicesNon-synchronised local oscillators
Stephan Sigg | Feedback based distributed adaptive beamforming | 3
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
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
Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion
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
Stephan Sigg | Feedback based distributed adaptive beamforming | 6
Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion
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
Stephan Sigg | Feedback based distributed adaptive beamforming | 7
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
Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion
An upper bound on the expected optimisation time
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
Stephan Sigg | Feedback based distributed adaptive beamforming | 9
Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion
An upper bound on the expected optimisation time
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
Stephan Sigg | Feedback based distributed adaptive beamforming | 10
Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion
An upper bound on the expected optimisation time
Assumptions :
Network of n nodesEach node changes the phase of its carrier signal withprobability 1
nCarrier phase altered uniformly at random from [0, 2π]
Stephan Sigg | Feedback based distributed adaptive beamforming | 11
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
kBetween any two carrier signalsFor arbitrary kDivide phase space into k intervals of width 2π
k
Stephan Sigg | Feedback based distributed adaptive beamforming | 12
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
Stephan Sigg | Feedback based distributed adaptive beamforming | 13
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
Stephan Sigg | Feedback based distributed adaptive beamforming | 14
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 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)
Stephan Sigg | Feedback based distributed adaptive beamforming | 15
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
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 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.
Stephan Sigg | Feedback based distributed adaptive beamforming | 17
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
Stephan Sigg | Feedback based distributed adaptive beamforming | 18
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 hammingdistance to the nearest global optimum is at least l .
Stephan Sigg | Feedback based distributed adaptive beamforming | 19
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+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) ≥ ∆])
Stephan Sigg | Feedback based distributed adaptive beamforming | 20
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
Stephan Sigg | Feedback based distributed adaptive beamforming | 21
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: ≤ 2tn·log(k)
Stephan Sigg | Feedback based distributed adaptive beamforming | 22
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. 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))
Stephan Sigg | Feedback based distributed adaptive beamforming | 23
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
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
Stephan Sigg | Feedback based distributed adaptive beamforming | 25
Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion
An asymptotically optimal optimisation scheme
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
Stephan Sigg | Feedback based distributed adaptive beamforming | 26
Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion
An asymptotically optimal optimisation scheme
Fitness function observedby single node
Constant carrier phaseoffset for n − 1 nodes
Fitness function:
F(Φi ) = A sin(Φi + φ) + c
Stephan Sigg | Feedback based distributed adaptive beamforming | 27
Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion
An asymptotically optimal optimisation scheme
Approach:
Measure feedback at 3points
Solve multivariableequations
Apply optimum phaseoffset calculated
F(Φi ) = A sin(Φi + φ) + c
Stephan Sigg | Feedback based distributed adaptive beamforming | 28
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 oftheir transmit signals
Stephan Sigg | Feedback based distributed adaptive beamforming | 29
Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion
An asymptotically optimal optimisation scheme
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%
Stephan Sigg | Feedback based distributed adaptive beamforming | 30
Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion
An asymptotically optimal optimisation scheme
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
Stephan Sigg | Feedback based distributed adaptive beamforming | 31
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
Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion
An asymptotically optimal optimisation scheme
Stephan Sigg | Feedback based distributed adaptive beamforming | 33
Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion
An asymptotically optimal optimisation scheme
Stephan Sigg | Feedback based distributed adaptive beamforming | 34
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%of all nodes.
Stephan Sigg | Feedback based distributed adaptive beamforming | 35
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 turnUtilise last transmission from previous iteration
Stephan Sigg | Feedback based distributed adaptive beamforming | 36
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 | 37
Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion
An adaptive protocol for distributed adaptive beamforming
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
Stephan Sigg | Feedback based distributed adaptive beamforming | 38
Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion
An adaptive protocol for distributed adaptive beamforming
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]
Transmitted bit sequence
0 0.5 1 1.5 2 2.5 3−1
0
1x 10
−11
Time [ms]
Modulated transmit signal for device n
0 0.5 1 1.5 2 2.5 3−5
0
5x 10
−10
Time [ms]
Received superimposed sum signal
0 0.5 1 1.5 2 2.5 3−5
0
5x 10
−10
Time [ms]
Demodulated received sum signal
Stephan Sigg | Feedback based distributed adaptive beamforming | 39
Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion
An adaptive protocol for distributed adaptive beamforming
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]
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 3−2
0
2x 10
−9
Time [ms]
Demodulated received sum signal
Stephan Sigg | Feedback based distributed adaptive beamforming | 40
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
Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion
An adaptive protocol for distributed adaptive beamforming
Stephan Sigg | Feedback based distributed adaptive beamforming | 42
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 0.752 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
Stephan Sigg | Feedback based distributed adaptive beamforming | 43
Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion
An adaptive protocol for distributed adaptive beamforming
Stephan Sigg | Feedback based distributed adaptive beamforming | 44
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 0.0099Presence 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
Stephan Sigg | Feedback based distributed adaptive beamforming | 45
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