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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

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Sensor Network Scenario

Sensors Distributed in an Information Field, transmitting data to a central location

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A Communication Abstraction

Assumptions

• Single hop transmission from sensor nodes to receiver
• No inter sensor communication

Problem Statement in Most General Form

• Given b ∼ N(0, B), cost on transmission C(xk), distortion metric d

“ b, e b ”

• Design Encoder/Decoder to minimize distortion E

h d “ b, e b ”i subject to the constraint: PM

k=1 E [C(xk)] = Cavg

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Our System Model

r = SP

1 2 b + n

e b = C⊤r C(xk) = Pk;

M

X

k=1

Pk = Ptot(Total Power) d “ b, e b ” = E »“ e b − b ”⊤ “ e b − b ”– (TMSE)

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TMSE and the Optimization Problem

TMSE = tr

• C⊤SP

1 2 BP 1 2 S⊤C + σ2C⊤C − 2C⊤SP 1 2 B + IM

• The optimization problem is:

min

S,P,C TMSE subject to tr (P) = Ptot

Optimization is in Two Stages

• 1. Rx Side Optimize C for a given S and P

1 2

• 2. Tx Side Optimize S and P

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Optimal Receiver Filter

The optimum receiver for given S and P

1 2 is the LMMSE filter:

C⋆ =

• SP

1 2 BP 1 2 S⊤ + σ2IL

−1 SP

1 2 B

• The corresponding TMSE expression is:

TMSE = σ2tr

• σ2B−1 + A⊤A

−1 Where A = SP

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Transmitter Optimization

The optimization problem can be rewritten as: min

A∈A tr

»“ σ2B−1 + A⊤A ”−1– where, A is the set of all L × M matrices such that tr(A⊤A) = tr(P) = Ptot Lemma 1 Sufficient to search over a restricted space A′ where A⊤A and B commute i.e. they have the same eigenvectors

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Graphical Illustration of Optimal Solution

Let {u1, . . . , uM} and {λ1 ≥ . . . ≥ λM } be the eigenvectors and eigenvalues of B {v1, . . . , vL} and {µ1 ≥ . . . ≥ µL} be the eigenvectors and eigenvalues of A⊤A

• Question: How to choose µis and vis optimally ?

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Graphical Illustration of Optimal Solution

Let {u1, . . . , uM} and {λ1 ≥ . . . ≥ λM } be the eigenvectors and eigenvalues of B {v1, . . . , vL} and {µ1 ≥ . . . ≥ µL} be the eigenvectors and eigenvalues of A⊤A

• Question: How to choose µis and vis optimally ?
• Answer: Align vis with uis and waterfill over K = min (L, rank(B))

eigenvalues of B−1

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Optimal Sequences: Examples

Example 1 Sensors take independent measurements: B = IM. L ≤ M. Optimal Solution - Only L out of M transmitters transmit using

• rthogonal codewords with equal powers.

Note: Same TMSE can be achieved if all M transmitters transmit using WBE sequences with equal powers

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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

• rthogonal/random sequences for an arbitrary B?

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Comparison of Optimal vs Random/Orthogonal Codewords

B =      1 ρ . . . ρ ρ 1 . . . ρ . . . . . . ... . . . ρ . . . ρ 1     

0.2 0.4 0.6 0.8 1 0.5 1 1.5 2

ρ

TMSE

Random Codewords Orthogonal Codewords Optimal Codewords

(M = L = 4 SNR = 10dB)

0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5 3 3.5

ρ

TMSE

Random Codewords Orthogonal Codewords Optimal Codewords

(M = L = 4 SNR = 0dB)

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Comparison of Optimal vs Random/Orthogonal Codewords

B =      1 ρ12 . . . ρ1M ρ21 1 . . . ρ2M . . . . . . ... . . . ρM1 ρM2 . . . 1     

20 40 60 80 100 0.5 1 1.5 2 Indices of chosen B matrices TMSE

Random Codewods Orthogonal codewords Optimal codewords

(M = L = 4 SNR = 10dB)

20 40 60 80 100 1 1.5 2 2.5 3 3.5 Indices of chosen B matrices TMSE

Random Codewords Orthogonal Codewords Optimal Codewords

(M = L = 4 SNR = 0dB) The B matrices are sorted in order of decreasing TMSE, when random codewords are chosen

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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

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Thank You

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