Accurate Reconstruction of Frequency-Sparse Signals from Non-Uniform Samples

摘要

With the advent of a new sampling theory in recent years, compressed sensing (CS), it is possible to reconstruct signals from measurements far below the Nyquist rate. The CS theory assumes that signals are sparse and that measurement matrices satisfy certain conditions. Even though there have been many promising results, unfortunately there still exists a gap between the theory and actual real world applications. This is because of the fundamental problem that the CS formulation is discrete. We propose a sampling and reconstructing method for frequency-sparse signals that addresses this issue. The signals in our scenario are supported in a continuous sparsifying domain rather than discrete. This work focuses on a typical case in which the unknowns are frequencies and amplitudes. However, directly looking for the unknowns that best fit the measurements in the least-squares sense is a non-convex optimization problem, because sinusoids are oscillatory. Our approach extends the utility of CS to simplify this problem to a locally convex problem, hence making the solutions tractable. Direct measurements are taken from non-uniform time-samples, which is an extension of the CS problem with a subsampled Fourier matrix. The proposed reconstruction algorithm iteratively approximates the solutions using CS and then accurately solves for the frequencies with Newton's method and for the amplitudes with linear least squares solutions. Our simulations show success in accurate reconstruction of signals with arbitrary frequencies and significantly outperform current spectral compressed sensing methods in terms of reconstruction fidelity for both noise-free and noisy cases.