Derivative sampling reconstruction of bandlimited stochastic processes in the almost sure sense
by
Tibor K. Pogány
Department of Maritime Studies, University of Rijeka, Croatia

The talk is concerning to the generalization of the Gaposhkin condition upon the a.s. sense uniformly spaced sampling reconstruction of the bandlimited weakly stationary stochastic processes. Namely, we generalize Gaposhkin's condition to the derivative sampling approach for stationary processes. So, let { \xi(t); t in R} be a band-limited (BL) weakly stationary stochastic process (WSSP). The sampling reconstruction of a BL WSSP by the Kotel'nikov - Shannon formula which uses samples of its r-1 derivatives, r in N, as well as those \xi(t) itself is called r^th-order derivative sampling restoration. Now, the derivative variant of the Gaposhkin's condition reads as follows.

Let { \xi(t); t in R} be a BL to r\pi WSSP. Then it holds

\xi(t) = sinr(\pit) å n in Z  (-1)nr \lceil\fracr-12\rceil å j=0  r-2j-1 å k=0  \frac\pij-r\Gammar, 2j(2j)!(r-2j-k-1)!\frac\xi(r-2j-k-1)(n)(t-n)k+1

in the a.s. sense iff

P ì í î lim m --> \infty  æ è ó õ [-r\pi, -r\pi+2-m]  Pr-1+(\lambda)\zeta(d\lambda) - ó õ [r\pi-2-m, r\pi]  Pr-1-(\lambda)\zeta(d\lambda) ö ø = 0 ü ý þ = 1 ,

where \Gamma_{r, j} :=lim_{z --> 0}\fracd^jdz^j( \fraczsinz )^r and

Pr-1 +/- (\lambda) = \lceil\fracr-12 \rceil å j=0  r-2j-1 å k=0  \frac4\lceilk/2 \rceil\pi2(j+\lceilk/2 \rceil)\Gammar, 2j(-1)jir-1(2j)!(r-2j-k-1)!\lambdar-2j-k-1 × [ 2\pi(1-(-1)k) \beta2\lceilk/2 \rceil+2(\lambda +/- r\pi)+i(1+(-1)k)\beta2\lceilk/2 \rceil+1(\lambda +/- r\pi) ].

Here \beta_s(·) denotes the periodic continuation of the Bernoulli polynomials of the degree s from its basic interval to [-r\pi, r\pi] and \zeta is the orthogonally scattered random spectral measure of \xi.

Taking r=1 in the previous forulæ we get the Gaposhkin result.

http://www.pfri.hr/~poganj

Date received: March 25, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caex-47.