Algorithmic approaches to distributed
adaptive transmit beamforming
5th international conference on Intelligent Sensors, Sensor
Networks and Information Processing
Stephan Sigg, Michael Beigl
Institute of Distributed and Ubiquitous Systems
Technische Universität Braunschweig
Stephan Sigg
ISSNIP 2009, 7-10 December 2009, Melbourne, Australia
1/27
Motivation
The scenario of distributed adaptive transmit beamforming
1
1
R. Mudumbai, R.D. Brown, U. Madhow, and H.V. Poor: Distributed Transmit Beamforming: Challenges
and Recent progress, IEEE Communications Magazine, 102-110, February 2009
Stephan Sigg
ISSNIP 2009, 7-10 December 2009, Melbourne, Australia
2/27
Motivation
The scenario of distributed adaptive transmit beamforming
2
2
R. Mudumbai, R.D. Brown, U. Madhow, and H.V. Poor: Distributed Transmit Beamforming: Challenges
and Recent progress, IEEE Communications Magazine, 102-110, February 2009
Stephan Sigg
ISSNIP 2009, 7-10 December 2009, Melbourne, Australia
3/27
Motivation
The scenario of distributed adaptive transmit beamforming
3
3
R. Mudumbai, R.D. Brown, U. Madhow, and H.V. Poor: Distributed Transmit Beamforming: Challenges
and Recent progress, IEEE Communications Magazine, 102-110, February 2009
Stephan Sigg
ISSNIP 2009, 7-10 December 2009, Melbourne, Australia
4/27
Motivation
The scenario of distributed adaptive transmit beamforming
4
4
R. Mudumbai, R.D. Brown, U. Madhow, and H.V. Poor: Distributed Transmit Beamforming: Challenges
and Recent progress, IEEE Communications Magazine, 102-110, February 2009
Stephan Sigg
ISSNIP 2009, 7-10 December 2009, Melbourne, Australia
5/27
Motivation
The scenario of distributed adaptive transmit beamforming
5
5
R. Mudumbai, R.D. Brown, U. Madhow, and H.V. Poor: Distributed Transmit Beamforming: Challenges
and Recent progress, IEEE Communications Magazine, 102-110, February 2009
Stephan Sigg
ISSNIP 2009, 7-10 December 2009, Melbourne, Australia
6/27
Outline
Algorithmic approaches to distributed adaptive beamforming
1
Motivation
2
A local random search approach
Scenario analysis
Simulations
3
An asymptotically optimal algorithm
Multivariable equations
4
Conclusion
Stephan Sigg
ISSNIP 2009, 7-10 December 2009, Melbourne, Australia
7/27
Scenario analysis and algorithmic improvement
Local random search
Global random search:
Synchronisation performance might deteriorate when the
optimum is near
With small local search space:
Majority of worse points excluded
Stephan Sigg
ISSNIP 2009, 7-10 December 2009, Melbourne, Australia
8/27
Local random search
An upper bound on the synchronisation performance
Assumptions :
Mutation probability: n−1
Uniform phase alteration
Initial distance to the optimum :
≥ n·log(k)
(Chernoff)
2
Technical assumption :
Fitness space divided in k
slices of identical width
Stephan Sigg
ISSNIP 2009, 7-10 December 2009, Melbourne, Australia
9/27
Local random search
An upper bound on the synchronisation performance
Analysis in two phases for the synchronisation process
Phase 1: Optimum outside search neighbourhood for at least
one node
Phase 2: Optimum within search neighbourhood for all nodes
Stephan Sigg
ISSNIP 2009, 7-10 December 2009, Melbourne, Australia
10/27
Local random search
An upper bound on the synchronisation performance
Phase 1: Optimum is outside the neighbourhood
Reach search point with improved fitness: ≥
Stephan Sigg
ISSNIP 2009, 7-10 December 2009, Melbourne, Australia
1
2
11/27
Local random search
An upper bound on the synchronisation performance
When i signals synchronised:
Improve n − i non-optimal signals
i already optimal ones unchanged:
1
2
1 i
n
<e < 1−
1 n−1
n
1
n
(n − i) ·
=
since 1 −
n−i
2n
1 n
n
· 1−
i
· 1 − n1
si ≥
·
n−i
2en
Expected number of mutations to
increase fitness bounded by si−1 .
Stephan Sigg
ISSNIP 2009, 7-10 December 2009, Melbourne, Australia
12/27
Local random search
An upper bound on the synchronisation performance
Time until optimum is within the neighbourhood?
Constant time to leave slice
k distinct slices
E [TP ] ≤
c·
Pk
2en
i=0 n−i
= 2cen ·
k+1
X
i −1
i=1
< 2cen · ln(k + 1) = O (n · log(k))
Stephan Sigg
ISSNIP 2009, 7-10 December 2009, Melbourne, Australia
13/27
Local random search
An upper bound on the synchronisation performance
Phase 2: Optimum within search neighbourhood
Worst case: Increase fitness with probability
Similar to consideration above:
1
N
O(N · n · log(k))
Overall synchronisation time: Weak estimation N = O(k) leads to
O(k · n · log(k)).
Stephan Sigg
ISSNIP 2009, 7-10 December 2009, Melbourne, Australia
14/27
Mathematical simulation environment
Impact of the node choice
Fitness measure:
v
u τ Pn
uX ( i=1 si + snoise (i) − s ∗ )2
RMSE = t
.
n
t=0
Stephan Sigg
ISSNIP 2009, 7-10 December 2009, Melbourne, Australia
15/27
Scenario analysis and algorithmic improvement
Local random search
Stephan Sigg
ISSNIP 2009, 7-10 December 2009, Melbourne, Australia
16/27
Outline
Algorithmic approaches to distributed adaptive beamforming
1
Motivation
2
A local random search approach
Scenario analysis
Simulations
3
An asymptotically optimal algorithm
Multivariable equations
4
Conclusion
Stephan Sigg
ISSNIP 2009, 7-10 December 2009, Melbourne, Australia
17/27
Multivariable equations
Received sum signal
Reduce the amount of randomness in the optimisation
Improve the synchronisation performance
Improve the synchronisation quality
Stephan Sigg
ISSNIP 2009, 7-10 December 2009, Melbourne, Australia
18/27
Multivariable equations
Received sum signal
Fitness function observed
by single node
Constant carrier phase
offset for n − 1 nodes
Fitness function:
F(Φi ) = A sin(Φi + φ) + c
Stephan Sigg
ISSNIP 2009, 7-10 December 2009, Melbourne, Australia
19/27
Multivariable equations
Received sum signal
Approach:
Measure feedback at 3
points
Solve multivariable
equations
Apply optimum phase
offset calculated
F(Φi ) = A sin(Φi + φ) + c
Stephan Sigg
ISSNIP 2009, 7-10 December 2009, Melbourne, Australia
20/27
Multivariable equations
Received sum signal
Problem:
Calculation not accurate when two or more nodes alter the
phase of their transmit signals
Stephan Sigg
ISSNIP 2009, 7-10 December 2009, Melbourne, Australia
21/27
Multivariable equations
Solution
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%
Stephan Sigg
ISSNIP 2009, 7-10 December 2009, Melbourne, Australia
22/27
Multivariable equations
Synchronisation process
1
Transmit with phase offsets γ1 6= γ2 6= γ3 ; measure feedback
2
Estimate feedback function and calculate γi∗
3
Transmit with γ4 = γi∗
4
If deviation smaller 1% finished, otherwise discard γi∗
Stephan Sigg
ISSNIP 2009, 7-10 December 2009, Melbourne, Australia
23/27
Multivariable equations
Received sum signal
Asymptotic synchronisation time:
O(n)
Classic approach:6
Θ(n · k · log(n))
6
Sigg, El Masri and Beigl, A sharp asymptotic bound for feedback based closed-loop distributed adaptive
beamforming in wireless sensor networks (submitted to Transactions on Mobile Computing)
Stephan Sigg
ISSNIP 2009, 7-10 December 2009, Melbourne, Australia
24/27
Multivariable equations
Performance estimation
Stephan Sigg
ISSNIP 2009, 7-10 December 2009, Melbourne, Australia
25/27
Conclusion
Algorithms for distributed adaptive beamforming in WSNs
Performance improvements possible by local random search
Upper bound on the synchronisation time:
O(k · n · log(k))
Asymptotically optimal optimisation approach
Synchronisation time O(n)
Synchronisation quality improved
Stephan Sigg
ISSNIP 2009, 7-10 December 2009, Melbourne, Australia
26/27
Questions?
Thank You for your attention.
sigg@ibr.cs.tu-bs.de
Stephan Sigg
ISSNIP 2009, 7-10 December 2009, Melbourne, Australia
27/27