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We consider the estimation of millimeter wave (mmWave) channel gains following the now well-established approach of sparse signal processing. Within that context, we offer two major contributions. The first is a complete frame-theoretical treatment of the sensing matrix design (beam management) problem, compactly described by a pair of Lemmas that together with efficient low-coherence frame construction algorithms offer a general solution for the optimal transmitter (Tx) and receiver (Rx) beamforming components of the sparse mmWave channel estimation problem. The second contribution is a pair of novel sparse recovery algorithms, which unlike the majority of sparse solvers found in the literature, is not based on the relaxation of <inline-formula> <tex-math notation="LaTeX">$\ell _{0}$ </tex-math></inline-formula>-norm into an <inline-formula> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula>-norm, but rather on a smooth approximation of the <inline-formula> <tex-math notation="LaTeX">$\ell _{0}$ </tex-math></inline-formula>-norm handled further via fractional programming (FPG). The two algorithms differ from each other in that in the first (slightly more accurate), the resulting convex problem is solved via interior point methods, while the second (stand-alone) makes use of the alternating direction method of multipliers (ADMM). As a bonus, an original ADMM variation of the well-known basis pursuit denoising (BPDN)-<inline-formula> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula>-reweighted sparse solver is also given. Simulation results confirm the channel estimation accuracy improvements obtained by both contributions.

Frame Theory and Fractional Programming for Sparse Recovery-Based mmWave Channel Estimation