In order to significantly reduce the time and space needed, compressive sensing builds upon the fundamental assumption of sparsity under a suitable discrete dictionary. However, in many signal processing applications there exists mismatch between the assumed and the true sparsity bases, so that the actual representative coefficients do not lie on the finite grid discretized by the assumed dictionary. Unlike previous work this paper introduces the unified compressive measurement operator into atomic norm denoising and investigates the problems of recovering the frequency support of a combination of multiple sinusoids from sub-Nyquist samples. We provide some useful properties to ensure the optimality of the unified framework via semidefinite programming (SDP). We also provide a sufficient condition to guarantee the uniqueness of the optimizer with high probability. Theoretical results demonstrate the proposed method can locate the nonzero coefficients on an infinitely dense grid over a wide range of SNR case.

Compressive sensing (CS) exploits the sparsity or compressibility of signals to recover themselves from a small set of nonadaptive, linear measurements. The number of measurements is much smaller than Nyquist-rate, thus signal recovery is achieved at relatively expense. Thus, many signal processing problems which do not require exact signal recovery have attracted considerable attention recently. In this paper, we establish a framework for parameter estimation of a signal corrupted by additive colored Gaussian noise (ACGN) based on compressive measurements. We also derive the Cramer-Rao lower bound (CRB) for the frequency estimation problems in compressive domain and prove some useful properties of the CRB under different compressive measurements. Finally, we show that the theoretical conclusions are along with experimental results.