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.
As mentioned above, similar probabilistic tools have been employed in CS, but for the analysis of the sparse signal recovery problem @cite_30 @cite_26 @cite_4 @cite_2 @cite_9 . While in principle one could view the matched filter problem as that of recovering a @math -sparse signal from a dictionary @math of possible candidates, such a dictionary would have infinite size and extremely high coherence, preventing the application of most standard recovery analysis techniques. One recent work @cite_1 has formalized the matched filter problem using signal recovery principles and a union of subspaces model. However, this work is quite different from ours in that it does not theoretically study noise sensitivity and relies on a non-random sampling architecture that is carefully designed to facilitate the solution of the recovery problem. Interestingly, outside the field of CS, very similar random processes to those that we study have also arisen in the analysis of the spectral norm of random Toeplitz matrices @cite_28 .
@cite_30 @cite_26 @cite_4 @cite_28 @cite_9 @cite_1 @cite_2
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This paper improves upon best-known guarantees for exact reconstruction of a sparse signal f from a small universal sample of Fourier measurements. The method for reconstruction that has recently gained momentum in the sparse approximation theory is to relax this highly nonconvex problem to a convex problem and then solve it as a linear program. We show that there exists a set of frequencies Ω such that one can exactly reconstruct every r-sparse signal f of length n from its frequencies in Ω, using the convex relaxation, and Ω has size k(r, n) = O(r log(n)·log 2 (r) log(r logn)) = O(r log 4 n ). A random set Ω satisfies this with high probability. This estimate is optimal within the log log n and log 3 r factors. We also give a relatively short argument for a similar problem with k(r, n) ≈ r[12 + 8 log(n r)] Gaussian measurements. We use methods of geometric functional analysis and probability theory in Banach spaces, which makes our arguments quite short. Compressed sensing seeks to recover a sparse vector from a small number of linear and non-adaptive measurements. While most work so far focuses on Gaussian or Bernoulli random measurements we investigate the use of partial random circulant and Toeplitz matrices in connection with recovery by 1-minization. In contrast to recent work in this direction we allow the use of an arbitrary subset of rows of a circulant and Toeplitz matrix. Our recovery result predicts that the necessary number of measurements to ensure sparse reconstruction by 1-minimization with random partial circulant or Toeplitz matrices scales linearly in the sparsity up to a log-factor in the ambient dimension. This represents a significant improvement over previous recovery results for such matrices. As a main tool for the proofs we use a new version of the non-commutative Khintchine inequality. This paper considers the problem of estimating a discrete signal from its convolution with a pulse consisting of a sequence of independent and identically distributed Gaussian random variables. We derive lower bounds on the length of a random pulse needed to stably reconstruct a signal supported on [1, n]. We will show that a general signal can be stably recovered from convolution with a pulse of length m ≳ n log5 n, and a sparse signal which can be closely approximated using s ≲ n log5 n terms can be stably recovered with a pulse of length n. Suppose that @math is a Toeplitz matrix whose entries come from a sequence of independent but not necessarily identically distributed random variables with mean zero. Under some additional tail conditions, we show that the spectral norm of @math is of the order @math . The same result holds for random Hankel matrices as well as other variants of random Toeplitz matrices which have been studied in the literature. In the theory of compressed sensing, restricted isometry analysis has become a standard tool for studying how efficiently a measurement matrix acquires information about sparse and compressible signals. Many recovery algorithms are known to succeed when the restricted isometry constants of the sampling matrix are small. Many potential applications of compressed sensing involve a data-acquisition process that proceeds by convolution with a random pulse followed by (nonrandom) subsampling. At present, the theoretical analysis of this measurement technique is lacking. This paper demonstrates that the sth-order restricted isometry constant is small when the number m of samples satisfies m ≳ (s logn)^(3 2), where n is the length of the pulse. This bound improves on previous estimates, which exhibit quadratic scaling. Time-delay estimation arises in many applications in which a multipath medium has to be identified from pulses transmitted through the channel. Previous methods for time delay recovery either operate on the analog received signal, or require sampling at the Nyquist rate of the transmitted pulse. In this paper, we develop a unified approach to time delay estimation from low-rate samples. This problem can be formulated in the broader context of sampling over an infinite union of subspaces. Although sampling over unions of subspaces has been receiving growing interest, previous results either focus on unions of finite-dimensional subspaces, or finite unions. The framework we develop here leads to perfect recovery of the multipath delays from samples of the channel output at the lowest possible rate, even in the presence of overlapping transmitted pulses, and allows for a variety of different sampling methods. The sampling rate depends only on the number of multipath components and the transmission rate, but not on the bandwidth of the probing signal. This result can be viewed as a sampling theorem over an infinite union of infinite dimensional subspaces. By properly manipulating the low-rate samples, we show that the time delays can be recovered using the well-known ESPRIT algorithm. Combining results from sampling theory with those obtained in the context of direction of arrival estimation, we develop sufficient conditions on the transmitted pulse and the sampling functions in order to ensure perfect recovery of the channel parameters at the minimal possible rate. This paper considers the problem of estimating the channel response (or the Green's function) between multiple source–receiver pairs. Typically, the channel responses are estimated one-at-a-time: a single source sends out a known probe signal, the receiver measures the probe signal convolved with the channel response and the responses are recovered using deconvolution. In this paper, we show that if the channel responses are sparse and the probe signals are random, then we can significantly reduce the total amount of time required to probe the channels by activating all of the sources simultaneously. With all sources activated simultaneously, the receiver measures a superposition of all the channel responses convolved with the respective probe signals. Separating this cumulative response into individual channel responses can be posed as a linear inverse problem. We show that channel response separation is possible (and stable) even when the probing signals are relatively short in spite of the corresponding linear system of equations becoming severely underdetermined. We derive a theoretical lower bound on the length of the source signals that guarantees that this separation is possible with high probability. The bound is derived by putting the problem in the context of finding a sparse solution to an underdetermined system of equations, and then using mathematical tools from the theory of compressive sensing. Finally, we discuss some practical applications of these results, which include forward modeling for seismic imaging, channel equalization in multiple-input multiple-output communication and increasing the field-of-view in an imaging system by using coded apertures.