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.
To the best of our knowledge, our framework for studying the compressive matched filter is novel. Prior statistical analysis for compressive inference problems has focused specifically on problems of signal detection or classification from a finite model set @cite_6 @cite_20 @cite_0 or employed a geometric point-of-view based on a stable embedding of signal family from an original finite-dimensional signal space into a lower-dimensional measurement space @cite_11 @cite_3 . Our work takes a substantially different approach, considering the inference of a continuous-valued shift parameter from a continuous-time received signal, and more thoroughly characterizing the statistics of the problem using the language and tools of empirical processes.
@cite_3 @cite_6 @cite_0 @cite_20 @cite_11
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The recently introduced theory of compressive sensing enables the recovery of sparse or compressible signals from a small set of nonadaptive, linear measurements. If properly chosen, the number of measurements can be much smaller than the number of Nyquist-rate samples. Interestingly, it has been shown that random projections are a near-optimal measurement scheme. This has inspired the design of hardware systems that directly implement random measurement protocols. However, despite the intense focus of the community on signal recovery, many (if not most) signal processing problems do not require full signal recovery. In this paper, we take some first steps in the direction of solving inference problems-such as detection, classification, or estimation-and filtering problems using only compressive measurements and without ever reconstructing the signals involved. We provide theoretical bounds along with experimental results. Compressive sampling (CS), also called compressed sensing, entails making observations of an unknown signal by projecting it onto random vectors. Recent theoretical results show that if the signal is sparse (or nearly sparse) in some basis, then with high probability such observations essentially encode the salient information in the signal. Further, the signal can be reconstructed from these "random projections," even when the number of observations is far less than the ambient signal dimension. The provable success of CS for signal reconstruction motivates the study of its potential in other applications. This paper investigates the utility of CS projection observations for signal classification (more specifically, m-ary hypothesis testing). Theoretical error bounds are derived and verified with several simulations. Compressive sampling (CS) refers to a generalized sampling paradigm in which observations are inner products between an unknown signal vector and user-specified test vectors. Among the attractive features of CS is the ability to reconstruct any sparse (or nearly sparse) signal from a relatively small number of samples, even when the observations are corrupted by additive noise. However, the potential of CS in other signal processing applications is still not fully known. This paper examines the performance of CS for the problem of signal detection. A generalized restricted isometry property (GRIP) is introduced, which guarantees that angles are preserved, in addition to the usual norm preservation, by CS. The GRIP is leveraged to derive error bounds for a CS matched filtering scheme, and to show that the scheme is robust to signal mismatch. The recently introduced theory of compressed sensing (CS) enables the reconstruction or approximation of sparse or compressible signals from a small set of incoherent projections; often the number of projections can be much smaller than the number of Nyquist rate samples. In this paper, we show that the CS framework is information scalable to a wide range of statistical inference tasks. In particular, we demonstrate how CS principles can solve signal detection problems given incoherent measurements without ever reconstructing the signals involved. We specifically study the case of signal detection in strong inference and noise and propose an incoherent detection and estimation algorithm (IDEA) based on matching pursuit. The number of measurements and computations necessary for successful detection using IDEA is significantly lower than that necessary for successful reconstruction. Simulations show that IDEA is very resilient to strong interference, additive noise, and measurement quantization. When combined with random measurements, IDEA is applicable to a wide range of different signal classes The theory of compressive sensing (CS) enables the reconstruction of a sparse or compressible image or signal from a small set of linear, non-adaptive (even random) projections. However, in many applications, including object and target recognition, we are ultimately interested in making a decision about an image rather than computing a reconstruction. We propose here a framework for compressive classification that operates directly on the compressive measurements without first reconstructing the image. We dub the resulting dimensionally reduced matched filter the smashed filter. The first part of the theory maps traditional maximum likelihood hypothesis testing into the compressive domain; we find that the number of measurements required for a given classification performance level does not depend on the sparsity or compressibility of the images but only on the noise level. The second part of the theory applies the generalized maximum likelihood method to deal with unknown transformations such as the translation, scale, or viewing angle of a target object. We exploit the fact the set of transformed images forms a low-dimensional, nonlinear manifold in the high-dimensional image space. We find that the number of measurements required for a given classification performance level grows linearly in the dimensionality of the manifold but only logarithmically in the number of pixels samples and image classes. Using both simulations and measurements from a new single-pixel compressive camera, we demonstrate the effectiveness of the smashed filter for target classification using very few measurements.