Computationally-Efficient Blind Sub-Nyquist Sampling for Sparse Spectra

The sampling of spectrally sparse high-bandwidth signals below the Nyquist rate is of importance in many applications. In order to keep the front-end analog-to-digital converter sampling rate low, there has been a lot of recent interest in the literature to develop efficient and practical architectures for low-rate sampling and recovery of spectrally sparse signals. Our main contribution is a simple, implementation-friendly front-end sampling-architecture and a computationally efficient back-end algorithm for blind sub-Nyquist sampling of a class of band-limited sparse signals. In particular, we show that signals band-limited to W Hz consisting of k tones can be reliably recovered from sets of uniform samples obtained at a rate 4k Hz for asymptotic values of k and W, when k is sub-linear in W for noiseless observations of the signal. Further, the proposed reconstruction algorithm is computationally efficient, requiring only O(k log k) computations to recover the sinusoidal components. Our algorithm succeeds with high probability, with the probability of failure vanishing to zero asymptotically in the number of samples acquired. We provide simulation results that corroborate our theoretical findings, and also analyze the noise-robustness of the system.