This paper presents a spatially sparse precoding approach for millimeter wave (mmWave) multiple-input multiple-output (MIMO) systems. The goal is to design hybrid RF/baseband precoders that maximize spectral efficiency while adhering to hardware constraints. mmWave signals experience significantly higher pathloss than microwave signals, necessitating the use of large antenna arrays and beamforming techniques to overcome this. Traditional MIMO systems use digital processing at baseband, but mmWave systems face challenges due to the high cost and power consumption of mixed-signal devices, making analog RF processing more attractive. The paper proposes a method that leverages the sparse scattering nature of mmWave channels to formulate the precoding problem as a sparse reconstruction problem. By using the principle of basis pursuit, the authors develop algorithms that approximate optimal precoders and combiners that can be implemented in low-cost RF hardware. The approach involves designing hybrid precoders that combine RF and baseband processing, with the RF precoder using analog phase shifters and the baseband precoder using digital processing. The paper also considers receiver-side processing, showing that designing hybrid minimum mean-square error (MMSE) combiners can be cast as a simultaneously sparse approximation problem and solved via basis pursuit. The proposed framework is particularly suitable for limited feedback operations and can be efficiently compressed using simple scalar quantizers and low-dimensional Grassmannian subspace quantizers. The authors present a system model for mmWave communication, including the channel model and the hardware architecture. They show that the sparse scattering nature of mmWave channels allows for the design of hybrid precoders that can closely approximate the optimal unitary precoder. The paper introduces an algorithm based on orthogonal matching pursuit to find near-optimal precoders by selecting the best beamforming directions and forming appropriate linear combinations of the selected response vectors. The proposed algorithm is shown to generate beam patterns that closely resemble those generated by the optimal precoder, leading to favorable spectral efficiency performance. The paper concludes with design remarks, emphasizing that mmWave terminals do not need to know the exact angles of the channel matrix, and that the proposed approach can be implemented with practical hardware constraints.This paper presents a spatially sparse precoding approach for millimeter wave (mmWave) multiple-input multiple-output (MIMO) systems. The goal is to design hybrid RF/baseband precoders that maximize spectral efficiency while adhering to hardware constraints. mmWave signals experience significantly higher pathloss than microwave signals, necessitating the use of large antenna arrays and beamforming techniques to overcome this. Traditional MIMO systems use digital processing at baseband, but mmWave systems face challenges due to the high cost and power consumption of mixed-signal devices, making analog RF processing more attractive. The paper proposes a method that leverages the sparse scattering nature of mmWave channels to formulate the precoding problem as a sparse reconstruction problem. By using the principle of basis pursuit, the authors develop algorithms that approximate optimal precoders and combiners that can be implemented in low-cost RF hardware. The approach involves designing hybrid precoders that combine RF and baseband processing, with the RF precoder using analog phase shifters and the baseband precoder using digital processing. The paper also considers receiver-side processing, showing that designing hybrid minimum mean-square error (MMSE) combiners can be cast as a simultaneously sparse approximation problem and solved via basis pursuit. The proposed framework is particularly suitable for limited feedback operations and can be efficiently compressed using simple scalar quantizers and low-dimensional Grassmannian subspace quantizers. The authors present a system model for mmWave communication, including the channel model and the hardware architecture. They show that the sparse scattering nature of mmWave channels allows for the design of hybrid precoders that can closely approximate the optimal unitary precoder. The paper introduces an algorithm based on orthogonal matching pursuit to find near-optimal precoders by selecting the best beamforming directions and forming appropriate linear combinations of the selected response vectors. The proposed algorithm is shown to generate beam patterns that closely resemble those generated by the optimal precoder, leading to favorable spectral efficiency performance. The paper concludes with design remarks, emphasizing that mmWave terminals do not need to know the exact angles of the channel matrix, and that the proposed approach can be implemented with practical hardware constraints.