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.