Quantized Spectral Compressed Sensing: Cramer-Rao Bounds and Recovery Algorithms

TL;DR

This paper investigates the limits and develops algorithms for recovering spectrally-sparse signals from heavily quantized compressed measurements, balancing theoretical bounds with practical reconstruction methods.

Contribution

It introduces a new atomic norm-based recovery algorithm for quantized spectral compressed sensing and characterizes the Cramer-Rao bounds under Gaussian noise.

Findings

Spectral signals can be accurately reconstructed with high probability when measurements exceed K log n.

The proposed algorithm effectively handles both single and multiple measurement vectors.

Theoretical bounds reveal the trade-off between sample complexity and bit depth.

Abstract

Efficient estimation of wideband spectrum is of great importance for applications such as cognitive radio. Recently, sub-Nyquist sampling schemes based on compressed sensing have been proposed to greatly reduce the sampling rate. However, the important issue of quantization has not been fully addressed, particularly for high-resolution spectrum and parameter estimation. In this paper, we aim to recover spectrally-sparse signals and the corresponding parameters, such as frequency and amplitudes, from heavy quantizations of their noisy complex-valued random linear measurements, e.g. only the quadrant information. 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 soft thresholding for signal recovery, which is equivalent to proximal mapping of properly designed surrogate signals with respect to the atomic norm that motivates spectral sparsity. The proposed algorithm can be applied to both the single measurement vector case, as well as the multiple measurement vector case. It is shown that under the Gaussian measurement model, the spectral signals 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 level of spectral sparsity and $n$ is the signal dimension. Finally, numerical simulations are provided to validate the proposed approaches.