# A New Design of Codebook for Hybrid Precoding in Millimeter-Wave Massive MIMO Systems
## 1 Introduction
However, the training cost is high for each update, that is to say, the computational complexity is high. The digital precoding codebook and analog precoding codebook are designed separately to reduce the complexity of the algorithm in [9; 10], but the relationship between them is not considered and exploited, resulting in the loss of system spectral efficiency. The hybrid precoding algorithm proposed in [11] reduces the complexity of the algorithm by avoiding singular value decomposition (SVD) and matrix inversion but at the cost of spectral efficiency. The hybrid precoding algorithm in [12; 13] is only applicable to the frequency division duplexing (FDD) system with known channel state information (CSI), but the analog codebook construction scheme is no longer applicable in the scenario where the codebook size and phase shifter resolution are determined so that application scenarios are not flexible enough. Inspired by the aforementioned algorithms, we propose an improved iterative algorithm of the hybrid precoding codebook to improve the spectral efficiency of a system. Firstly, we convert the design problem of hybrid precoding codebook into a minimization problem, which aims to minimize the Euclidean distance between the designed hybrid precoder and the optimal unconstrained precoder. Further, we apply the CoSaMP algorithm to reconstruct the digital precoding matrix. Then, with the aid of channel sparsity in mmWave Massive MIMO System, after taking the relationship between analog and digital precoders into consideration, we jointly generate the hybrid codebook by combining reconstruction algorithms with dictionary learning algorithms. Finally, the qualified size digital precoding codebook is generated using the clustering algorithm. Simulation results show that the hybrid precoding matrix generated by the proposed algorithm is obviously better than the OMP-based codebook [4] algorithm and the hybrid precoding with quantization [1] algorithm in low-SNR regimes and with different numbers of RF chains. The remainder of this paper is organized as follows. The extended Saleh-Valenzuela [14] channel model and the system model of hybrid precoding based on the mmWave massive MIMO system is introduced and the objective function is also derived in Section 2. The iterative hybrid precoding codebook algorithm is proposed to improve the spectral efficiency in Section 3. A simulation analysis of the algorithm and comparative discussion with the existing algorithm are presented in Section 4. Finally, we conclude this paper in Section 5. Notations: For a matrix \(\mathbf{A}\), the transpose and conjugate transpose of \(\mathbf{A}\) are represented by \(\mathbf{A}^{T}\) and \(\mathbf{A}^{H}\); \(|\mathbf{A}|\) denotes the determinant; \(\|\mathbf{A}\|_{F}\) and \(\|\mathbf{A}\|_{0}\) denote the Frobenius norm and zero norm of \(\mathbf{A}\); \(diag(\mathbf{A})\) is a vector formed by the diagonal elements of \(\mathbf{A}\); \(\mathbf{A}(\cdot,n)\) and \(\mathbf{A}(m,\cdot)\) denote the \(n\)th column and \(m\)th row of \(\mathbf{A}\), respectively; \(\mathbf{A}(m,n)\) denotes the \(m\)th row and \(n\)th column element of \(\mathbf{A}\).