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.
The second part of this paper adapts our analysis of the compressive matched filter to the problem of estimating the frequency of a pure sinusoidal tone from a small number of random time-domain samples. The recovery of signals that are sparse in the frequency domain based on compressive measurements is a problem that has been well-studied in the CS literature, although most work in this area has been concerned with signals that can be written as trigonometric polynomials @cite_22 @cite_19 @cite_24 @cite_8 . Some techniques for recovering off-grid frequency-sparse signals have been proposed that involve windowing @cite_8 or other classical techniques from the field of spectral estimation @cite_7 , and other work has considered the more general problem of recovering continuous-time signals based on a union of subspaces model @cite_5 , but the analysis that we present is more sharply focused on the statistics of the simpler pure tone estimation problem.
@cite_22 @cite_7 @cite_8 @cite_24 @cite_19 @cite_5
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This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal f spl isin C sup N and a randomly chosen set of frequencies spl Omega . Is it possible to reconstruct f from the partial knowledge of its Fourier coefficients on the set spl Omega ? A typical result of this paper is as follows. Suppose that f is a superposition of |T| spikes f(t)= spl sigma sub spl tau spl isin T f( spl tau ) spl delta (t- spl tau ) obeying |T| spl les C sub M spl middot (log N) sup -1 spl middot | spl Omega | for some constant C sub M >0. We do not know the locations of the spikes nor their amplitudes. Then with probability at least 1-O(N sup -M ), f can be reconstructed exactly as the solution to the spl lscr sub 1 minimization problem. In short, exact recovery may be obtained by solving a convex optimization problem. We give numerical values for C sub M which depend on the desired probability of success. Our result may be interpreted as a novel kind of nonlinear sampling theorem. In effect, it says that any signal made out of |T| spikes may be recovered by convex programming from almost every set of frequencies of size O(|T| spl middot logN). Moreover, this is nearly optimal in the sense that any method succeeding with probability 1-O(N sup -M ) would in general require a number of frequency samples at least proportional to |T| spl middot logN. The methodology extends to a variety of other situations and higher dimensions. For example, we show how one can reconstruct a piecewise constant (one- or two-dimensional) object from incomplete frequency samples - provided that the number of jumps (discontinuities) obeys the condition above - by minimizing other convex functionals such as the total variation of f. Compressive sensing (CS) is a new approach to simultaneous sensing and compression of sparse and compressible signals based on randomized dimensionality reduction. To recover a signal from its compressive measurements, standard CS algorithms seek the sparsest signal in some discrete basis or frame that agrees with the measurements. A great many applications feature smooth or modulated signals that are frequency-sparse and can be modeled as a superposition of a small number of sinusoids; for such signals, the discrete Fourier transform (DFT) basis is a natural choice for CS recovery. Unfortunately, such signals are only sparse in the DFT domain when the sinusoid frequencies live precisely at the centers of the DFT bins; when this is not the case, CS recovery performance degrades signicantly. In this paper, we introduce the spectral CS (SCS) recovery framework for arbitrary frequencysparse signals. The key ingredients are an over-sampled DFT frame and a restricted unionof-subspaces signal model that inhibits closely spaced sinusoids. We demonstrate that SCS signicantly outperforms current state-of-the-art CS algorithms based on the DFT while providing provable bounds on the number of measurements required for stable recovery. We also leverage line spectral estimation methods (specically Thomson’s multitaper method Wideband analog signals push contemporary analog-to-digital conversion (ADC) systems to their performance limits. In many applications, however, sampling at the Nyquist rate is inefficient because the signals of interest contain only a small number of significant frequencies relative to the band limit, although the locations of the frequencies may not be known a priori. For this type of sparse signal, other sampling strategies are possible. This paper describes a new type of data acquisition system, called a random demodulator, that is constructed from robust, readily available components. Let K denote the total number of frequencies in the signal, and let W denote its band limit in hertz. Simulations suggest that the random demodulator requires just O(K log(W K)) samples per second to stably reconstruct the signal. This sampling rate is exponentially lower than the Nyquist rate of W hertz. In contrast to Nyquist sampling, one must use nonlinear methods, such as convex programming, to recover the signal from the samples taken by the random demodulator. This paper provides a detailed theoretical analysis of the system's performance that supports the empirical observations. This article describes a computational method, called the Fourier sampling algorithm. The algorithm takes a small number of (correlated) random samples from a signal and processes them efficiently to produce an approximation of the DFT of the signal. The algorithm offers provable guarantees on the number of samples, the running time, and the amount of storage. As we will see, these requirements are exponentially better than the FFT for some cases of interest. Abstract We study the problem of reconstructing a multivariate trigonometric polynomial having only few non-zero coefficients from few random samples. Inspired by recent work of Candes, Romberg and Tao we propose to recover the polynomial by Basis Pursuit, i.e., by l 1 -minimization. In contrast to their work, where the sampling points are restricted to a grid, we model the random sampling points by a continuous uniform distribution on the cube, i.e., we allow them to have arbitrary position. Numerical experiments show that with high probability the trigonometric polynomial can be recovered exactly provided the number N of samples is high enough compared to the “sparsity”—the number of non-vanishing coefficients. However, N can be chosen small compared to the assumed maximal degree of the trigonometric polynomial. We present two theorems that explain this observation. One of them provides the analogue of the result of Candes, Romberg and Tao. The other one is a result toward an average case analysis and, unexpectedly connects to an interesting combinatorial problem concerning set partitions, which seemingly has not yet been considered before. Although our proofs follow ideas of they are simpler. A traditional assumption underlying most data converters is that the signal should be sampled at a rate exceeding twice the highest frequency. This statement is based on a worst-case scenario in which the signal occupies the entire available bandwidth. In practice, many signals are sparse so that only part of the bandwidth is used. In this paper, we develop methods for low-rate sampling of continuous-time sparse signals in shift-invariant (SI) spaces, generated by m kernels with period T . We model sparsity by treating the case in which only k out of the m generators are active, however, we do not know which k are chosen. We show how to sample such signals at a rate much lower than m T, which is the minimal sampling rate without exploiting sparsity. Our approach combines ideas from analog sampling in a subspace with a recently developed block diagram that converts an infinite set of sparse equations to a finite counterpart. Using these two components we formulate our problem within the framework of finite compressed sensing (CS) and then rely on algorithms developed in that context. The distinguishing feature of our results is that in contrast to standard CS, which treats finite-length vectors, we consider sampling of analog signals for which no underlying finite-dimensional model exists. The proposed framework allows to extend much of the recent literature on CS to the analog domain.