Signal Sampling and Recovery Under Dependent Errors

Abstract

The paper examines the impact of the additive correlated noise on the accuracy of the signal reconstruction algorithm originating from the Whittaker-Shannon (WS) sampling interpolation formula. A class of band-limited signals as well as signals which are non-band-limited are taken into consideration. The proposed reconstruction method is a smooth post-filtering correction of the classical WS interpolation series. We assess both the point-wise and global accuracy of the proposed reconstruction algorithm for a broad class of dependent noise processes. This includes short and long-memory stationary errors being independent of the sampling rate. We also examine a class of noise processes for which the correlation function depends on the sampling rate. Whereas the short-memory errors have relatively small influence on the reconstruction accuracy, the long-memory errors can greatly slow down the convergence rate. In the case of the noise model depending on the sampling rate further degradation of the algorithm accuracy is observed. We give quantitative explanations of these phenomena by deriving rates at which the reconstruction error tends to zero. We argue that the obtained rates are close to be optimal. In fact, in a number of special cases they agree with known optimal min-max rates. The problem of the limit distribution of the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> distance of the proposed reconstruction algorithm is also addressed. This result allows us to tackle an important problem of designing non- parametric lack-of-fit tests. The theory of the asymptotic behavior of quadratic forms of stationary sequences is utilized in this case.