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.
Finally, we would like to point out some of the differences between the tone estimation problem considered in this paper and the classical problem of estimating the power spectrum of a random process from samples at random locations (see @cite_10 @cite_27 @cite_12 @cite_29 ). In Sections and , we will show how the output of the compressive matched filter is a random process whose mean is the template autocorrelation function. This random process is completely specified by the samples we have observed, and rather than merely estimating its second-order statistics, we will be interested in establishing a uniform bound on its deviation from the template; this will allow us to conclude that it peaks at or near the correct location. It is also worth mentioning that our work differs from Rife and Boorstyn's classical analysis of the single-tone parameter estimation problem @cite_25 . Specifically, our work permits sampling below the Nyquist rate, and with high probability we provide an absolute bound on the accuracy of the frequency estimate, rather than involving the Cram ' e r-Rao bound.
@cite_29 @cite_27 @cite_10 @cite_25 @cite_12
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A constructive solution of the irregular sampling problem for band- limited functions is given. We show how a band-limited function can be com- pletely reconstructed from any random sampling set whose density is higher than the Nyquist rate, and give precise estimates for the speed of convergence of this iteration method. Variations of this algorithm allow for irregular sampling with derivatives, reconstruction of band-limited functions from local averages, and irregular sampling of multivariate band-limited functions. In the irregular sampling problem one is asked whether and how a band- limited function can be completely reconstructed from its irregularly sam- pled values f(xi). This has many applications in signal and image processing, seismology, meteorology, medical imaging, etc. Finding constructive solutions of this problem has received considerable attention among mathematicians and engineers. The mathematical literature provides several uniqueness results (1, 2, 17, 18, 19). It is now part of the folklore that for stable sampling the sampling rate must be at least the Nyquist rate (18). These results, as deep as they are, have had little impact for the applied sciences, because they were not constructive. If the sampling set is just a perturbation of the regular oversampling, then a reconstruction method has been obtained in a seminal paper by Duffin and Schaeffer (6) (see also (29)): if for some L > 0, a > 0, and o > 0 the sampling points xk , k e Z , satisfy (a) - ok a, k ^ I, then the norm equivalence A iR (x) ) with w < n o. This norm equivalence implies that it is possible to reconstruct through an iterative procedure, the so-called frame method. Most of the later work on constructive methods consists of variations of this method (3, 21, 22, 26). The above conditions on the sampling set exclude random irregular sampling sets, e.g., sets with regions of higher sampling density. A partial, but undesirable remedy, to handle highly irregular sampling sets, would be to force the above conditions by throwing away information on part of the points and accept a very slow convergence of the iteration. We study the problem of sampling a random signal with sparse support in frequency domain. Shannon famously considered a scheme that instantaneously samples the signal at equispaced times. He proved that the signal can be reconstructed as long as the sampling rate exceeds twice the bandwidth (Nyquist rate). Candes, Romberg, Tao introduced a scheme that acquires instantaneous samples of the signal at random times. They proved that the signal can be uniquely and efficiently reconstructed, provided the sampling rate exceeds the frequency support of the signal, times logarithmic factors. In this paper we consider a probabilistic model for the signal, and a sampling scheme inspired by the idea of spatial coupling in coding theory. Namely, we propose to acquire non-instantaneous samples at random times. Mathematically, this is implemented by acquiring a small random subset of Gabor coefficients. We show empirically that this scheme achieves correct reconstruction as soon as the sampling rate exceeds the frequency support of the signal, thus reaching the information theoretic limit. The recovery of signal parameters from noisy sampled data is a fundamental problem in digital signal processing. In this paper, we consider the following spectral analysis problem: Let f be a real-valued sum of complex exponentials. Determine all parameters of f, i.e., all different frequencies, all coefficients, and the number of exponentials from finitely many equispaced sampled data of f. This is a nonlinear inverse problem. In this paper, we present new results on an approximate Prony method (APM) which is based on [1]. In contrast to [1], we apply matrix perturbation theory such that we can describe the properties and the numerical behavior of the APM in detail. The number of sampled data acts as regularization parameter. The first part of APM estimates the frequencies and the second part solves an overdetermined linear Vandermonde-type system in a stable way. We compare the first part of APM also with the known ESPRIT method. The second part is related to the nonequispaced fast Fourier transform (NFFT). Numerical experiments show the performance of our method. We consider inference in a general data-driven object-based model of multichannel audio data, assumed generated as a possibly underdetermined convolutive mixture of source signals. We work in the short-time Fourier transform (STFT) domain, where convolution is routinely approximated as linear instantaneous mixing in each frequency band. Each source STFT is given a model inspired from nonnegative matrix factorization (NMF) with the Itakura-Saito divergence, which underlies a statistical model of superimposed Gaussian components. We address estimation of the mixing and source parameters using two methods. The first one consists of maximizing the exact joint likelihood of the multichannel data using an expectation-maximization (EM) algorithm. The second method consists of maximizing the sum of individual likelihoods of all channels using a multiplicative update algorithm inspired from NMF methodology. Our decomposition algorithms are applied to stereo audio source separation in various settings, covering blind and supervised separation, music and speech sources, synthetic instantaneous and convolutive mixtures, as well as professionally produced music recordings. Our EM method produces competitive results with respect to state-of-the-art as illustrated on two tasks from the international Signal Separation Evaluation Campaign (SiSEC 2008).