Feedback based distributed adaptive transmit beamforming Algorithmic 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 beamforming Distributed nodes synchronise the carrier frequency and phase offset of transmit signals Low power and processing devices Non-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 synchronisation 1 Receiver Receiver Receiver Receiver Source Transmitter Transmitter Transmitter Transmitter Source Source Source Source Receive node broadcasts Receiver transmits the relative Receive nodes bounce the Synchronised nodes transmit common master beacon phase offset of each node on these beacon back on distinct as a distributed beamformer to all source nodes CDMA channels CDMA channels 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 synchronisation 2 Receiver Receiver Receiver Source Transmitter Transmitter Transmitter Source Source Master Source Transmit nodes synchronise their The receiver broadcasts a sinusoidal The synchronised nodes transmit frequency and local oscillators in signal for open−loop synchronisation as a distributed beamformer to the a closed−loop synchronisation to the transmit nodes receiver 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 3 Superimposed received sum signal Distributed synchronisation schemes Receiver feedback Cosed loop feedback based 2 carrier synchronisation a 4 f 1 t+ γ 1 2 π f 2 t+ γ 2 2 π a R. Mudumbai, J. Hespanha, U. Madhow, G. Barriac, Mutation Distributed transmit beamforming using feedback control , IEEE Transactions on Information Theory 56(1), volume 56, January 2 π f n i t+ γ n 2 π f n t+ γ n 2010 i i+1 i+1 1 Iteration i Iteration i+1 Frequency 1 Time Stephan Sigg | Feedback based distributed adaptive beamforming | 6
Introduction Upper bound Lower bound Optimum Algorithm Adaptive protocol Conclusion Introduction 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 Receiver feedback Optimisation time can be improved by factor 2 when erroneous decisions are not discarded but inverted b Phase and frequency synchronisation feasible c a R. Mudumbai, J. Hespanha, U. Madhow, G. Barriac, Distributed transmit beamforming using feedback control , IEEE Transactions on Information Theory 56(1), volume 56, January 2010 b J. Bucklew, W. Sethares, Convergence of a class of decentralised beamforming algorithms , IEEE Transactions on Signal Processing 56(6), volume 56, 2008 c M. 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
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 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 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 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 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 nodes Each node changes the phase of its carrier signal with probability 1 n Carrier phase altered uniformly at random from [0 , 2 π ] 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 2 π k Between any two carrier signals For arbitrary k Divide phase space into k intervals of width 2 π 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 � n − 1 � · 1 n · 1 � 1 − 1 k · 1 n � n − 1 � n − i � � 1 − 1 = · n · k n 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 � n � n − 1 � 1 − 1 < 1 � 1 − 1 e < n n Probability that L i is left for partition j , j > i : n − i P [ L 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 above by P [ L i ] − 1 : n − 1 e · n · k � E [ T P ] ≤ n − i i =0 n 1 � = e · n · k · i i =1 < 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 progress After initialisation, phases of carrier signals are identically and independently distributed. Each bit in the binary representation of search point s ζ has equal probability 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 ( s opt , s ζ ) ≤ l ; l ≪ n · log( k ) to global optima s opt at most l � � n · log( k ) k � P [ h ( s opt , s ζ ) ≤ l ] = · n · log( k ) − i 2 n · log( k ) − i i =0 ( n · log( k )) l +2 ≤ 2 n · 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 l � � k n · log( k ) � P [ h ( s opt , s ζ ) ≤ l ] = · n · log( k ) − i 2 n · log( k ) − i i =0 ( n · log( k )) l +2 ≤ 2 n · log( k ) − l This means that with high probability (w.h.p.) the hamming distance to the nearest global optimum is at least l . Stephan Sigg | Feedback based distributed adaptive beamforming | 19
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