Capacity and Error Exponents of Stationary Point Processes with Additive Displacement Noise

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$ Capacity and Error Exponents of Stationary Point Processes with Additive Displacement Noise F. Baccelli INRIA & ENS Joint work with V. Anantharam, UC Berkeley Lille, April 2011 & %

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Publié par

profil-urra-2012

Langue

English

Stationary point Point process

Capacity and Error Exponents of Stationary Point Processes with Additive Displacement Noise

F. Baccelli

INRIA & ENS

Joint work with V. Anantharam, UC Berkeley

Lille, April 2011

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Structure of the Lecture

Capacity and Error Exponents for – Additive White Gaussian Noise AWGN Displacement of a Point Process – Additive Stationary Ergodic Noise ASEN Displacement of a Point Process Capacity and Error Exponents for – AWGN Channel with constraints – ASEN Channel with constraints

Capacity and Error Exponents of Stationary Point Processes with Additive Displacement Noise

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V. A. & F. B. ✪

AWGN DISPLACEMENT OF A POINT PROCESS

 n : (simple) stationary ergodic point process on IR n . λ n = e nR : intensity of  n . { T nk } : points of  n ( codewords ). IP n 0 : Palm probability of  n . { D kn } : i.i.d. sequence of displacements, independent of  n : D kn = ( D kn (1)      D kn ( n )) i.i.d. over the coordinates and N (0  σ 2 ) ( noise ). Z kn = T kn + D kn : displacement of the p.p. ( received messages )

Capacity and Error Exponents of Stationary Point Processes with Additive Displacement Noise V. A. & F. B.

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AWGN UNDER MLE DECODING {V kn } : Voronoi cell of T n in  n . k Error probability under MLE decoding : T p e ( n ) = IP n 0 ( Z 0 n ∈  V 0 n ) = IP n 0 ( D 0 n ∈  V 0 n ) = A li → m ∞ P k 1 P Tk kn ∈ 1 TB knn ( ∈ 0 BA n )(0 1 ZA kn )  ∈V kn heorem 1-wgn Poltyrev [94] 1. If R < − 21 log(2 πeσ 2 ) , there exists a sequence of point pro-cesses  n (e.g. Poisson) with intensity e nR s.t. p e ( n ) → 0  n → ∞

2. If R > − 21 log(2 πeσ 2 ) , for all sequences of point processes  n with intensity e nR ,

p e ( n ) → 1  n → ∞

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Capacity and Error Exponents of Stationary Point Processes with Additive Displacement Noise V. A. & F. B. ✪

Proof of 2 [AB 08]

– V n ( r ) : volume of the n -ball or radius r . – By monotonicity arguments, if |V 0 n | = V n ( nL n ) , IP n 0 ( D n ∈  V 0 n ) ≥ IP n 0 ( D 0 n ∈  B n (0  nL n )) = IP n 0 1 n i = n X 1 D 0 n ( i ) 2 ≥ L 2 n 0

– By the SLLN, IP n 0  1 n i = n X 1 D 0 n ( i ) 2 − σ 2  ≥ ǫ ! = η ǫ ( n ) → n →∞ 0

– Hence IP n 0 ( D 0 n ∈  V 0 n ) ≥ IP n 0 ( σ 2 − ǫ ≥ L 2 n ) − η ǫ ( n ) = 1 − IP n 0 ( V n ( n ( σ 2 − ǫ )) < |V 0 n | ) − η ǫ ( n )

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Capacity and Error Exponents of Stationary Point Processes with Additive Displacement Noise V. A. & F. B. ✪

Proof of 2 [AB 08] ( continued )

– By Markov ineq. IP n 0 ( |V n | ) > V n ( n ( σ 2 − ǫ ))) ≤ V n ( IE n 0 n (( |V 0 n | ) 0 σ 2 − ǫ )) – By classical results on the Voronoi tessellation IE n 0 ( |V 0 n | ) = λ 1 n = e − nR

– By classical results n V n ( r ) = Γ( π 2 n 2 + r n 1) ∼ ππn 2 n  r 2 nn  n 2 e

– Hence IE n 0 ( |V 0 n | ) ∼ e − nR e − 2 n log(2 πe ( σ 2 − ǫ )) → n →∞ 0 V ( n ( σ 2 − ǫ )) 2 since R > − 21 log(2 πeσ ) . Capacity and Error Exponents of Stationary Point Processes with Additive Displacement Noise V. A. & F. B.

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