$\m{Y}_{\m{\Omega}}^o$ means that we actually use a sparse linear array (SLA) that is obtained by retaining the antennas of the ULA indexed by $\m{\Omega}$. Therefore, the index set $\m{\Omega}$ refers to geometry of the SLA and a smaller $M$ means use of fewer antennas (note that SLAs are common in practice for obtaining a large aperture from a limited number of antennas, see, e.g., [@linebarger1993difference] and the references therein). Each frequency component corresponds to one source. The value of $f_k$ uniquely determines the direction of source $k$, and vice versa. Consequently, the problem of direction of arrival (DOA) estimation using a SLA $\m{\Omega}$ is exactly the frequency estimation problem in CCS given the measurement matrix $\m{Y}_{\m{\Omega}}^o$.

Due to its connections to line spectral estimation and DOA estimation, studies of the CCS problem have a long history while frequency estimation has been mainly focused on. Well known conventional methods include periodogram (or beamforming), Capon’s beamforming and subspace methods like MUSIC (see the review in [@stoica2005spectral]). Periodogram suffers from the so-called leakage problem and the Fourier resolution limit of $\frac{1}{N}$ even in the full data case when $M=N$ [@stoica2005spectral]. It therefore has difficulties in resolving two closely spaced frequencies. The situation becomes 