Compressive spectrum estimation using quantized measurements

摘要

In this paper, we aim to recover a spectrally-sparse signal from the quadrants of complex random linear measurements. We first characterize the Cramer-Rao bound under Gaussian noise, which highlights the trade-off between sample complexity and bit depth under different signal-to-noise ratios for a fixed budget of bits. Next, we propose a new algorithm based on atomic norm regularization, which is equivalent to proximal mapping of a properly designed surrogate signal with respect to the atomic norm that promotes the spectral sparsity. Moreover, the frequencies can be localized without knowing the model order a priori via a dual polynomial approach. It is shown that under the Gaussian measurement model, the signal can be reconstructed accurately with high probability, as soon as the number of quantized measurements exceeds the order of K log n, where K is the number of frequencies and n is the signal dimension. Our results can be extended to more general nonlinear measurements using generalized linear models.