  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.
