In this paper, we study a simple correlation-based strategy for estimating the unknown delay and amplitude of a signal based on a small number of noisy, randomly chosen frequency-domain samples. We model the output of this “compressive matched filter” as a random process whose mean equals the scaled, shifted autocorrelation function of the template signal. Using tools from the theory of empirical processes, we prove that the expected maximum deviation of this process from its mean decreases sharply as the number of measurements increases, and we also derive a probabilistic tail bound on the maximum deviation. Putting all of this together, we bound the minimum number of measurements required to guarantee that the empirical maximum of this random process occurs sufficiently close to the true peak of its mean function. We conclude that for broad classes of signals, this compressive matched filter will successfully estimate the unknown delay (with high probability and within a prescribed tolerance) using a number of random frequency-domain samples that scales inversely with the signal-to-noise ratio and only logarithmically in the observation bandwidth and the possible range of delays.
We say that a signal is randomly sampled when the samples are taken at random instants of time. The study of random sampling and randomly sampled signals is motivated both by practical and theoretical interests. The first one includes spectral analysis (estimation of spectra from a finite number of samples) and quality of service (signal reconstruction), and the second one includes statistical analysis of reconstruction methods. The present paper focuses on the computation of the (theoretical) spectrum of randomly sampled signals and on the computation of the reconstruction error. Using a point process approach, we obtain general formulas for spatial random sampling, providing powerful tools for the analysis and the processing of randomly sampled signals. A class of spectral estimates of continuous-time stationary stochastic processes X(t) from a finite number of observations X(t_ n ) ^ N _ n =l taken at Poisson sampling instants t_ n is considered. The asymptotic bias and covariance of the estimates are derived, and the influence of the spectral windows and the sampling rate on the performance of the estimates is discussed. The estimates are shown to be consistent under mild smoothness conditions on the spectral density. Comparison is made with a related class of spectral estimates suggested in [15] where the number of observations is random . It is shown that the periodograms of the two classes have distinct statistics. The notion of alias-free sampling is generalized to apply to random processes x(t) sampled at random times t_n ; sampling is said to be alias free relative to a family of spectra if any spectrum of the family can be recovered by a linear operation on the correlation sequence r(n) , where r(n) = E[x(l_ m+n ) x(t_m) ] . The actual sampling times t_n need not be known to effect recovery of the spectrum of x(t) . Various alternative criteria for verifying alias-free sampling are developed. It is then shown that any spectrum whatsoever can be recovered if t_n is a Poisson point process on the positive (or negative) half-axis. A second example of alias-free sampling is provided for spectra on a finite interval by periodic sampling (for t t_o or t t_o ) in which samples are randomly independently skipped (expunged), such that the average sampling rate is an arbitrarily small fraction of the Nyquist rate. A third example shows that randomly jittered sampling at the Nyquist rate is alias free. Certain related open questions are discussed. These concern the practical problems involved in estimating a spectrum from imperfectly known r(n) . Estimation of the parameters of a single-frequency complex tone from a finite number of noisy discrete-time observations is discussed. The appropriate Cramer-Rao bounds and maximum-likelihood (MI.) estimation algorithms are derived. Some properties of the ML estimators are proved. The relationship of ML estimation to the discrete Fourier transform is exploited to obtain practical algorithms. The threshold effect of one algorithm is analyzed and compared to simulation results. Other simulation results verify other aspects of the analysis. Abstract Let X = X(t), − ∞ φ X (λ) of φX(λ) based on the discrete-time observation X(τk), τk are considered. Asymptotic expressions for the bias and covariance of φ X (λ) are derived. A multivariate central limit theorem is established for the spectral estimators φ X (λ) . Under mild conditions, it is shown that the bias is independent of the statistics of the sampling point process τk and that there exist sampling point processes such that the asymptotic variance is uniformly smaller than that of a Poisson sampling scheme for all spectral densities φX(λ) and all frequencies λ.