Regularized signal reconstruction for level-crossing sampling using Slepian functions

Abstract

In this paper, we propose a method for efficient signal reconstruction from non-uniformly spaced samples collected using level-crossing sampling. Level-crossing (LC) sampling captures samples whenever the signal crosses predetermined quantization levels. Thus the LC sampling is a signal-dependent, non-uniform sampling method. Without restriction on the distribution of the sampling times, the signal reconstruction from non-uniform samples becomes ill-posed. Finite-support and nearly band-limited signals are well approximated in a low-dimensional subspace with prolate spheroidal wave functions (PSWF) also known as Slepian functions. These functions have finite support in time and maximum energy concentration within a given bandwidth and as such are very appropriate to obtain a projection of those signals. However, depending on the LC quantization levels, whenever the LC samples are highly non-uniformly spaced obtaining the projection coefficients requires a Tikhonov regularized Slepian reconstruction. The performance of the proposed method is illustrated using smooth, bursty and chirp signals. Our reconstruction results compare favorably with reconstruction from LC-sampled signals using compressive sampling techniques.

Introduction

A uniformly sampled signal can be reconstructed from its samples using sinc interpolation whenever the signal is band-limited. However, in many practical applications, such as medical imaging, geophysics, astronomy and communication systems, it is often not appropriate to assume uniformly sampled signals. Furthermore, non-uniform sampling occurs inevitably in many applications due to imperfect sensors, mismatched clocks or event-triggered phenomena [1]. One can find several methods in the literature for the signal reconstruction from non-uniform samples. In general, the choice of the reconstruction method depends on the spectral size, signal length and sampling density [2]. If one wants to obtain non-uniform samples which are dependent on the structure of the signal, it can be done using level-crossing (LC) sampling [3], [5]. In the LC sampling scheme, samples are taken whenever the signal crosses fixed quantization levels. The local properties of the signal affects the density of samples taken so that more samples are collected in the information-rich regions of the signal and fewer samples otherwise. Therefore, the LC sampling method is both non-uniform and signal dependent. Several reconstruction and processing methods have been developed for non-uniformly sampled signals the performance of which depends heavily on the distribution of the sampling times [7], [8]. As indicated in [9], [15], even in the band-limited case, the inversion of the sinc matrix in the Shannon interpolation becomes highly ill-conditioned for random sampling, so that even small perturbations on the coefficient estimates lead to large reconstruction errors. One difficulty comes from the fact that Poisson's formula is no longer applicable and that the use of the sinc basis is not appropriate for representing finite-support signals [14], [20], [21]. Moreover, recovery methods of the signal from the level-crossing information [10], [11] are not always robust to sampling noise and timing quantization. In this paper, it is shown that by using a more appropriate basis for the interpolation than the sinc functions, the prolate spheroidal wave functions (PSWF), and applying the Tikhonov regularization in the reconstruction, a robust method to recover the original signal from the LC-sampled signal is obtained.

Slepian [12] identified a set of functions, known as the PSWF or Slepian functions, that are time-limited and that have maximum energy concentration within a specified bandwidth. Although no function can be simultaneously band-limited and time-limited, the PSWF come the closest to fit this description. Using the Slepian functions as an orthogonal basis [14], [20] it is possible to reduce the reconstruction error. Recently we have shown that a non-uniformly sampled signal can be reconstructed by means of the Slepian projection [21] and also by means of iterative projection onto convex sets (POCS) method [22].

In many applications it is required to sample signals that deliver information in bursts rather than in constant streams. According to Nyquist sampling theory, such a signal requires a large sampling frequency to acquire the information in the bursty intervals, although the samples in the non-bursty intervals provide no significant information. As indicated in [6] the nature of LC sampling is similar to compressive sampling. The support of a sparse signal in some basis is smaller than in the base the signal is represented. By sampling where the signal information is available, LC exploits the sparse or bursty nature of the signal. Sparse signals can be reconstructed from a small number of random projections by means of convex optimization whenever the measurements are not noisy. In the LC sampling, the number of quantization levels also affect the reconstruction. In this paper, we provide a new method for reconstructing a signal from LC samples, and in general from random samples, under the assumption that the signal is time-limited and nearly band-limited. If such a signal is well approximated in a low-dimensional subspace, the reconstruction can be achieved using the Slepian projection. There are cases, however, where the number of levels of the LC sampler does not permit an acceptable reconstruction. For a more general solution, we formulate the estimation of the projection coefficients using the Tikhonov regularization [17]. The main objective of the regularization is to incorporate more information in order to get the desired solution with the Slepian projection being a special case. We demonstrate the effectiveness of the proposed method in terms of the reduction in the number of necessary samples for reconstruction, and in the case of noisy sampling by comparing the results of our method with those from the compressive sampling (CS) technique [23].

The rest of the paper is organized as follows. In Section 2, we discuss briefly level-crossing sampling and compressive sampling. In Section 3, we consider the Slepian projection and reconstruction. In Section 4, we present the Tikhonov regularized Slepian reconstruction for the LC sampled signals. Simulations compare the regularized Slepian reconstruction and compressive sampling methods for different types of signals. Conclusions follow.