Channel estimation is a key task for beamforming in communication systems operating at millimeter-wave (mmWave) frequencies. This thesis focuses on improving baseband channel estimates by developing denoising techniques that rely on the sparsity of such channel vectors. Specifically, we present a novel and computationally-efficient channel-vector denoising algorithm for multi-antenna basestation designs. In addition, we adapt our algorithm for basestations that rely on 1-bit analog-to-digital converters to reduce system costs and power. Moreover, we develop a denoiser for cell-free communication systems with block-sparse channel matrices. Finally, we propose blind estimators that efficiently track key quantities for denoising, such as noise power and signal power, and present a nonparametric channel denoising algorithm, which can be utilized in a wide range of emerging wireless communication systems.