# Hybrid Precoding Design for Wireless Fading Channels
## 4 Simulation Results  
Computer simulations were performed to compare the performance of the proposed hybrid precoding scheme to existing ones [21; 22]. Assume that the number of transmitting antennas at the base station is M = 14, 28, and the number of receiving antennas for each user is one. Assume that the number of users belonging to group 1 applying average-based linear precoding is K, and the number of users belonging to group 2 applying symbol-based precoding is J. If the total transmission power for all users is \(P_{T}\) and the transmission power for each group is allocated in proportion to the number of users belonging to each group, the transmission power for each group can be expressed as \(P_{1}=\frac{K}{K+J}P_{T}\), \(P_{2}=\frac{J}{K+J}P_{T}\). It was assumed that quadrature phase-shift keying (QPSK) was applied as the modulation method for each symbol. The Rayleigh flat-fading channel model was used for the channel between the base station and each user, and it was assumed that the channel did not change during the number of symbol intervals, L. And since the design of the equalizer matrices \(\mathbf{A}_{1}\) and \(\mathbf{A}_{2}\) is beyond the scope of this paper, the special cases of \(\mathbf{A}_{1}=\mathbf{I}_{K}\) and \(\mathbf{A}_{2}=\mathbf{I}_{J}\) were considered. Figure 2 compares the SER performance of the proposed scheme, a linear MMSE precoding scheme, and two SLP schemes [21; 22] for the case where the number of base station antennas is M = 14 and (K + J) = 12. In the figure, the black, green, and red  
lines represent the conventional MMSE precoding scheme and the two conventional SLP schemes (SLP-1 and SLP-2) [21; 22], respectively. Also, the blue lines represent the proposed hybrid schemes. The performance of the proposed scheme was compared for K = 2, 6, and 10. In this figure, the horizontal axis represents the signal-to-noise ratio (SNR), while the vertical axis represents the symbol error rate (SER). To calculate the SER, 1000 independent channels were generated, and for each channel, 1000 independent received signals were generated to calculate the average error probability. From these results, it can be seen that the proposed scheme has slightly worse SER performance than the SLP-2 scheme [22] when K is 2, but it has better performance than the linear MMSE precoding and SLP-1 schemes [21]. And it can be seen that, as K decreases, the SER performance of the proposed technique approaches that of the SLP-2 scheme. Figure 3 compares the performance of the proposed scheme and the conventional ones for the case where the number of base station antennas is M = 28 and (K + J) = 26. The performance of the proposed scheme was compared for K = 4, 10, 16, and 22. As in Figure 2, it can be seen that, as K increases, the performance of the proposed scheme approaches that of the linear MMSE precoding scheme, and as K decreases, it approaches the performance of the SLP-2 scheme. Therefore, by adjusting the value of K, we can achieve the desired SER performance. Tables 1 and 2 show the results from comparing the computational complexity performance of the proposed scheme and conventional ones for various K values, and the units of the numbers are seconds. Here, the time required to calculate the precoding matrix and vector was defined as the computational complexity performance. The execution time taken to calculate precoding for each scheme was compared using a computer with an Intel Core (TM) i7-8700 3.19 GHz processor with 16 GB of memory. The period during which the channel does not change and maintains a constant value is expressed as the block length, L, and the performance was compared for the case where L is 1000. Figure 2: SER comparison of the proposed scheme and conventional ones when M = 14 and K + J = 12. (SLP-1 and SLP-2 refer to the schemes in [21] and [22], respectively). Table 1 shows the results from comparing the computational complexity for the case of M = 14 and K + J = 12.