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

Description

$ 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 & %

• point processes


• stationary point


• tnk ?bn


• additive displacement


• point process


• noise


• intensity enr


• ????? ≥

Sujets

• Stationary point
• Point process

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 9
• Langue: English

Extrait

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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1

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. ✪

2

AWGN DISPLACEMENT OF A POINT PROCESS

 n : (simple) stationary ergodic point process on IR n . λ n = e nR : intensity of  n . { T n k } : points of  n ( codewords ). IP n 0 : Palm probability of  n . { D k n } : i.i.d. sequence of displacements, independent of  n : D k n = ( D k n (1)      D k n ( n )) i.i.d. over the coordinates and N (0  σ 2 ) ( noise ). Z k n = T k n + D k n : 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 k n } : 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 T k k n ∈ 1 T B k n n ( ∈ 0  B A n ) (0 1  Z A k n )  ∈V k n heorem 1-wgn Poltyrev [94] 1. If R < − 2 1 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 > − 2 1 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. ✪

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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. ✪

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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) ∼ π π n 2 n  r 2 n n  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 > − 2 1 log(2 πeσ ) . Capacity and Error Exponents of Stationary Point Processes with Additive Displacement Noise V. A. & F. B.

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