This paper investigates the problem of estimating the frequencies of a mixture of complex sinusoids from a random subset of regularly spaced samples. Unlike previous work in compressed sensing, the frequencies are not assumed to lie on a grid but can take any value in the normalized frequency domain [0, 1]. An atomic norm minimization approach is proposed to exactly recover the unobserved samples and identify the unknown frequencies, which is then reformulated as an exact semidefinite program. The paper demonstrates that $O(s \log s \log n)$ random samples are sufficient to guarantee exact frequency localization with high probability, provided the frequencies are well-separated. Extensive numerical experiments are performed to validate the effectiveness of the proposed method. The main contributions include a new semidefinite characterization of the atomic norm, a proof of exact recovery under mild resolution assumptions, and a detailed analysis of the dual solution for frequency localization.This paper investigates the problem of estimating the frequencies of a mixture of complex sinusoids from a random subset of regularly spaced samples. Unlike previous work in compressed sensing, the frequencies are not assumed to lie on a grid but can take any value in the normalized frequency domain [0, 1]. An atomic norm minimization approach is proposed to exactly recover the unobserved samples and identify the unknown frequencies, which is then reformulated as an exact semidefinite program. The paper demonstrates that $O(s \log s \log n)$ random samples are sufficient to guarantee exact frequency localization with high probability, provided the frequencies are well-separated. Extensive numerical experiments are performed to validate the effectiveness of the proposed method. The main contributions include a new semidefinite characterization of the atomic norm, a proof of exact recovery under mild resolution assumptions, and a detailed analysis of the dual solution for frequency localization.