Optimal Signature Sets for Transmission of Correlated Data over a Multiple Access Channel WINLAB, Fall 2004 IAB J. Acharya, R. Roy, J. Singh, C. Rose { joy,rito,jasingh,crose } @winlab.rutgers.edu. Wireless Information Network Laboratory (WINLAB) 73, Brett Road, Piscataway, NJ - 08854 Optimal Signature Sets for Transmission of Correlated Data over a Multiple Access Channel – p.1/14
Sensor Network Scenario Sensors Distributed in an Information Field, transmitting data to a central location Optimal Signature Sets for Transmission of Correlated Data over a Multiple Access Channel – p.2/14
A Communication Abstraction Assumptions • Single hop transmission from sensor nodes to receiver • No inter sensor communication Problem Statement in Most General Form “ ” • b , e Given b ∼ N ( 0 , B ) , cost on transmission C ( x k ) , distortion metric d b h “ ”i • b , e Design Encoder/Decoder to minimize distortion E subject to the d b constraint: P M k =1 E [ C ( x k )] = C avg Optimal Signature Sets for Transmission of Correlated Data over a Multiple Access Channel – p.3/14
Our System Model 1 2 b + n r = SP e b = C ⊤ r M X C ( x k ) = P k ; P k = P tot (Total Power) k =1 »“ ”– “ ” ” ⊤ “ b , e e e = E b − b b − b (TMSE) d b Optimal Signature Sets for Transmission of Correlated Data over a Multiple Access Channel – p.4/14
TMSE and the Optimization Problem � � 1 1 1 C ⊤ SP 2 BP 2 S ⊤ C + σ 2 C ⊤ C − 2 C ⊤ SP 2 B + I M TMSE = tr The optimization problem is: S , P , C TMSE subject to tr ( P ) = P tot min Optimization is in Two Stages 1 1. Rx Side Optimize C for a given S and P 2 1 2. Tx Side Optimize S and P 2 Optimal Signature Sets for Transmission of Correlated Data over a Multiple Access Channel – p.5/14
Optimal Receiver Filter 1 2 is the LMMSE filter: The optimum receiver for given S and P � − 1 � C ⋆ = � 2 S ⊤ + σ 2 I L � 1 1 1 2 BP 2 B SP SP The corresponding TMSE expression is: �� σ 2 B − 1 + A ⊤ A � − 1 � TMSE = σ 2 tr 1 Where A = SP 2 Optimal Signature Sets for Transmission of Correlated Data over a Multiple Access Channel – p.6/14
Transmitter Optimization The optimization problem can be rewritten as: »“ ” − 1 – σ 2 B − 1 + A ⊤ A A ∈ A tr min where, A is the set of all L × M matrices such that tr ( A ⊤ A ) = tr ( P ) = P tot Lemma 1 Sufficient to search over a restricted space A ′ where A ⊤ A and B commute i.e. they have the same eigenvectors Optimal Signature Sets for Transmission of Correlated Data over a Multiple Access Channel – p.7/14
Graphical Illustration of Optimal Solution Let { u 1 , . . . , u M } and { λ 1 ≥ . . . ≥ λ M } be the eigenvectors and eigenvalues of B { v 1 , . . . , v L } and { µ 1 ≥ . . . ≥ µ L } be the eigenvectors and eigenvalues of A ⊤ A • Question: How to choose µ i s and v i s optimally ? Optimal Signature Sets for Transmission of Correlated Data over a Multiple Access Channel – p.8/14
Graphical Illustration of Optimal Solution Let { u 1 , . . . , u M } and { λ 1 ≥ . . . ≥ λ M } be the eigenvectors and eigenvalues of B { v 1 , . . . , v L } and { µ 1 ≥ . . . ≥ µ L } be the eigenvectors and eigenvalues of A ⊤ A • Question: How to choose µ i s and v i s optimally ? • Answer: Align v i s with u i s and waterfill over K = min ( L, rank ( B )) eigenvalues of B − 1 Optimal Signature Sets for Transmission of Correlated Data over a Multiple Access Channel – p.8/14
Optimal Sequences: Examples Example 1 Sensors take independent measurements: B = I M . L ≤ M . Optimal Solution - Only L out of M transmitters transmit using orthogonal codewords with equal powers. Note: Same TMSE can be achieved if all M transmitters transmit using WBE sequences with equal powers Optimal Signature Sets for Transmission of Correlated Data over a Multiple Access Channel – p.9/14
Optimal Sequences: Examples Example 2 Sensors take identical measurements. Optimal Solution - All M transmitters use identical codewords with equal powers. Leads to lower TMSE than using orthogonal codewords with equal power or a single codeword with full power ( M -fold beamforming gain) How much do we gain by using optimal sequences over orthogonal/random sequences for an arbitrary B ? Optimal Signature Sets for Transmission of Correlated Data over a Multiple Access Channel – p.10/14
Comparison of Optimal vs Random/Orthogonal Codewords 1 ρ . . . ρ 1 ρ . . . ρ B = . . . ... . . . . . . 1 ρ . . . ρ 2 3.5 3 1.5 2.5 TMSE TMSE 1 2 1.5 Random Codewords 0.5 Random Codewords Orthogonal Codewords Orthogonal Codewords Optimal Codewords 1 Optimal Codewords 0 0.5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ρ ρ ( M = L = 4 SNR = 10 dB ) ( M = L = 4 SNR = 0 dB ) Optimal Signature Sets for Transmission of Correlated Data over a Multiple Access Channel – p.11/14
Comparison of Optimal vs Random/Orthogonal Codewords 1 ρ 12 . . . ρ 1 M 1 ρ 21 . . . ρ 2 M B = . . . ... . . . . . . 1 ρ M 1 ρ M 2 . . . 2 3.5 Random Codewods Orthogonal codewords Optimal codewords 3 1.5 TMSE 2.5 TMSE 1 2 0.5 Random Codewords 1.5 Orthogonal Codewords Optimal Codewords 0 1 0 20 40 60 80 100 0 20 40 60 80 100 Indices of chosen B matrices Indices of chosen B matrices ( M = L = 4 SNR = 10 dB ) ( M = L = 4 SNR = 0 dB ) The B matrices are sorted in order of decreasing TMSE, when random codewords are chosen Optimal Signature Sets for Transmission of Correlated Data over a Multiple Access Channel – p.12/14
Conclusion Summary • To minimize end to end distortion, correlation between sensor measurements can be exploited by mapping it to the transmitted waveforms • This work derives the optimal codewords and power allocations for minimizing TMSE between sensor measurements and their estimates at the receiver Future Work • Derive optimal codewords when each node has an individual power constraint • Design a distributed scheme for updating codewords at sensor nodes based on feedback from receiver Optimal Signature Sets for Transmission of Correlated Data over a Multiple Access Channel – p.13/14
Thank You Optimal Signature Sets for Transmission of Correlated Data over a Multiple Access Channel – p.14/14
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