Multichannel Sparse Recovery for Constant Modulus Signals via $\ell _{{\infty}, 1}$ Minimization

Compressed sensing techniques have extensive applications in radar signal processing. Convex optimization approaches, such as $\ell _{2,1}$ minimization, are used for multichannel sparse signal recovery. However, when jointly sparse signals also exhibit the constant modulus (CM) property, $\ell _{2,1}$ minimization cannot utilize this prior information. In this article, we focus on utilizing $\ell _{\infty, 1}$ minimization to recover sparse signals with the CM property. We first establish a sufficient recovery condition for jointly sparse signals. Based on the duality theory, our main theorem sheds light on the superiority of $\ell _{\infty, 1}$ minimization over $\ell _{2, 1}$ minimization in the CM signal recovery. In addition, we provide an average-case analysis for $\ell _{\infty, 1}$ minimization. These results are applicable to the direction-of-arrival estimation with a nonuniform linear array and have practical relevance. A fast algorithm based on the alternating direction method of multipliers is proposed, and extensive numerical simulations are carried out to validate the results obtained.