Alias-free Discrete-time FIR System Realisation Using Hybrid Stratified Sampling

This paper proposes a method for system realisation, where the realised system is described by a continuous-time, finite-duration impulse response. The proposed discrete-time implementation deploys Digital Alias-free Signal Processing. It means that despite the use of digital signal processing, the produced results do not suffer from aliasing. However, owing to the use of random sampling, the approach relies on constructing a suitable estimator of the system output. This paper shows that the proposed estimator is unbiased. It is also consistent, i.e. its variance goes to zero when the density of signal samples increasing. It is proven that under moderately restrictive assumptions, the estimator goes to zero proportionally to the fifth power of the average distance between the samples

FIR filters for systems with input clock jitter

A method of designing fixed-coefficient FIR filters whose input signals are sampled irregularly due to clock jitter is presented. The approach does not require direct measuring of the jitter. Instead it is assumed that the jitter is a strictly stationary stochastic process for which some statistical information is available. Preliminary analysis of degradation of filter performance due to presence of jitter is also presented. Some numerical analyses illustrate the main assertions of the paper

Antithetical Stratified Sampling Estimator for Filtering Signals with Discontinuities

A novel approach to signal filtering using digital alias-free signal processing (DASP) is presented in this paper. We propose an unbiased, fast-converging estimator of the output of a finite impulse response (FIR) continuous-time filter. The estimator processes 2N signal samples collected with the use of random antithetical stratified (AnSt) sampling technique. To assess the estimator convergence rate as the function of N, we consider various forms of smoothness of the input signal, filter impulse response and windowing function. The cases are piecewise-continuous second-order derivative (SOD), piecewise-continuous first-order derivative (FOD) and piecewise-continuous zero-order derivative (ZOD). In each case we assume that the respective derivative has a finite number of bounded discontinuities. We prove that the proposed estimator converges to the true filter output at the rate of N^(-5) in the first case. But for the other two the rate drops to N^(-4) and N^(-2) respectively

Filtering Nonuniformly Sampled Grid-Based Signals

This paper presents an example application of digital alias-free signal processing, where a sequence of irregularly spaced, yet uniformly gridded, samples of a bandlimited discrete-time signal is filtered by using an oversampled finite impulse response filter. The mathematical model of the proposed filter is introduced, and a new interpolation formula for calculating the convolution operation of the filter, based on nonuniform sampling, is derived. In addition, uniform grid versions of Total Random, Stratified and Antithetical Stratified random sampling techniques are demonstrated. We carry out numerical comparison between these techniques and the proposed one in terms of Fourier transform estimates of the filtered output signal. The proposed interpolation technique shows enhancements over other sampling techniques after certain number of sampling points. Furthermore, it has a faster uniform convergence rate of the normalized root mean squared error than other techniques

Spectral analysis of randomly sampled signals: suppression of aliasing and sampler jitter

Nonuniform sampling can facilitate digital alias-free signal processing (DASP), i.e., digital signal processing that is not affected by aliasing. This paper presents two DASP approaches for spectrum estimation of continuous-time signals. The proposed algorithms, named the weighted sample (WS) and weighted probability (WP) density functions, respectively, utilize random sampling to suppress aliasing. Both methods produce unbiased estimators of the signal spectrum. To achieve this effect, the computational procedure for each method has been suitably matched with the probability density function characterising the pseudorandom generators of the sampling instants. Both proposed methods are analyzed, and the qualities of the estimators they produce have been compared with each other. Although none of the proposed spectrum estimators is universally better than the other one, it has been shown that in practical cases, the WP estimator produces generally smaller errors than those obtained from WS estimation. A practical limitation of the approaches caused by the sampling-instant jitter is also studied. It has been proven that in the presence of jitter, the theoretically infinite bandwidths of WS and WP signal analyses are limited. The maximum frequency up to which these analyses can be performed is inversely proportional to the size of the jitter

Estimation of Fourier Transform Using Alias-free Hybrid-Stratified Sampling

This paper proposes a novel method of estimating the Fourier Transform (FT) of deterministic, continuous-time signals, from a finite number \u1d441 of their samples taken from a fixed-length observation window. It uses alias-free hybrid-stratified sampling to probe the processed signal at a mixture of deterministic and random time instants. The FT estimator, specifically designed to work with this sampling scheme, is unbiased, consistent and fast converging. It is shown that if the processed signal has continuous third derivative, then the estimator's rate of uniform convergence in mean square is \u1d441^−5. Therefore, in terms of frequency-independent upper bounds on the FT estimation error, the proposed approach significantly outperforms existing estimators that utilize alias-free sampling, such as total random, stratified sampling, and antithetical stratified whose rate of uniform convergence is \u1d441^−1. It is proven here that \u1d441^−1 is a guaranteed minimum rate for all stratified-sampling-based estimators satisfying four weak conditions formulated in this paper. Owing to the alias-free nature of the sampling scheme, no constraints are imposed on the spectral support of the processed signal or the frequency ranges for which the Fourier Transform is estimated

Digital filtering of band-limited signals using Periodic Nonuniform Sampling

We examine the problem of digital filtering of band-limited signals by means of a linear digital filter with one or more stopbands. The main target of the study is to filter the signals using lower than Landau sampling rates, where the Landau rate is defined as the total bandwidth of the input signal. In order to reach such low rates Periodic Nonuniform Sampling is employed. We derive necessary and sufficient conditions for perfect filtering, and propose a practical algorithm for constructing PNS grids that allow for sub-Landau sampling and filtering. Finally, we present a reconstruction system and provide a numerical result illustrating the proposed method

Comparison Between Uniform and Nonuniform Interpolation Techniques for Digital Alias-free FIR Filtering

In this paper, we propose three grid-based nonuniform interpolation techniques to find the AUC of the convolution operation of a digital alias-free FIR filter. Up to the authors’ knowledge, these techniques were not addressed in literature before. We call them composite 3-nonuniform-sample (C3NS), composite 4-nonuniform-sample (C4NS) and composite 5-nonuniform-sample (C5NS) rules. They are named after the traditional composite Simpson’s 1/3 rule which is usually used in second-order polynomial interpolation of equally-spaced sampling points. The proposed new rules shows better estimated results than the uniform-based ones when the number of sampling points doesn’t match the required Nyquist rate. Moreover, we prove that composite Simpson’s 1/3 rule is more accurate than composite Simpson’s 3/8 rule mathematically and by simulation

Optimal periodic sampling sequences for nearly-alias-free digital signal processing

Alias-free DSP (DASP) is a methodology of processing signals digitally inside bandwidths that are wider than the famous Nyquist limit of half of the sampling requency. DASP is facilitated by suitable combination of nonuniform sampling and appropriate processing algorithms. In this paper we propose a new method of constructing sampling schemes for the needs of DASP. Unlike traditional approaches that rely on randomly selected sampling instants we use deterministic schemes. A method of optimizing such sequences aimed at minimization of aliasing is proposed. The approach is tested numerically in an experiment where an undersampled signal is processed using DASP; first to estimate the signal's spectrum support function and then the spectrum itself. We demonstrate advantages of the proposed approach over those that use random sampling

FIR Filtering of Discontinuous Signals: A Random-Stratified Sampling Approach

This paper presents a novel approach, based on random stratified sampling (StSa) technique, to estimate the output of a finite impulse response (FIR) filter when the input signal is either a piecewise-continuous function having first-derivative discontinuities (FDDs), or a piecewise-discontinuous function, i.e. having zero-derivative discontinuities (ZDDs). The proposed approach investigates the implications of such discontinuities on the output signal and its statistical properties. Mainly, we devise mathematical expressions for the variance of the StSa estimator in the two cases above, along with other minor special cases. It is found that the uniform convergence rate of the estimator, in the FDDs case, is N^{-3}, where N is the number of random samples. However, the variance in the ZDDs case is adversely affected by the existence of discontinuities. We prove that it converges more slowly with a uniform rate of N^{-2}