## 4 Numerical Results and Discussion  
In order to analyze the performance of the proposed algorithms, the following simulations are conducted. Fifty random Rayleigh fading channels are generated, and the pathloss exponent is set to 2. The variances of noises are assumed as \(\sigma_{r}^{2}=\sigma_{S_{i}}^{2}=\sigma^{2}\), the transmit powers are set as \(p_{i}=18E_{s}\) and \(p_{r}=12E_{s}\), and the signal noise ratio (SNR) is calculated from \(SNR=10 log_{10}(E_{s}/\sigma^{2})\), where \(E_{s}\) is the power of signal. Meanwhile, the energy-harvesting requirement \(\bar{e}_{i}=0.1\). \(N=4\), \(M=2\) and \(N=2\), \(M=2\) are both considered, and the data stream \(S=2\). Moreover, the carrier frequency of the system is given by \(f_{c}=5\) GHz. Four schemes are simulated: 1. The unaided scheme, which means that the beamformers are set as initial matrices; 2. the proposed FPP-SCA scheme; 3. the proposed low-complexity scheme; and 4. the semidefinite relaxation (SDR) scheme [46] used in the previous literature. In order to show the impact of noise, the impact of different values of \(\beta\), and the number of antennas, we do the corresponding simulations. Figure 2 and Table 1 show the performance under different \(\beta\) for the proposed FPP-SCA scheme and the Low-Complexity scheme. From the simulation results, obviously, a larger \(\beta\) leads to a higher system performance for both schemes, since more signals can be used for decoding the information in the receiver shown in Equation (6). In order to make the comparision with the existing works, we choose to use \(\beta=0.5\). The convergence property of different schemes is evaluated in Figure 3, where the total-MSE is plotted versus the iterations ranging from 0 to 50 in \(SNR=5\) dB and \(SNR=20\) dB when \(N=M=2\). From Figure 3, as the increment in the number of iterations, the FPP-SCA scheme always converges slower and requires more iterations for a convergence as SNR increases than the proposed low-complexity one. Furthermore, the FPP-SCA scheme exhibits a better performance than the low-complexity one for different SNRs when the curve converges. Meanwhile, comparing the FPP-SCA and conventional SDR scheme, we can find that the SDR scheme always converges slower than the FPP-SCA one and that the MSE of it is always higher than that of the FPP-SCA one (e.g., 2.07 vs. 2.04 for \(SNR=5\) dB and 0.50 vs. 0.48 for \(SNR=20\) dB) under 50 iterations, which implies an advantage of the proposed FPP-SCA scheme. Figure 2: The bit error rate (BER) versus signal noise ratio (SNR) for the proposed schemes under different \(\beta\). It can be claimed that the proposed FPP-SCA scheme has a lower level of total-MSE while it has a higher iteration complexity than the low-complexity counterpart. The performance comparison of different schemes is indicated in Figure 4, Tables 2 and 3, where in Figure 4, the bit error rate (BER) is plotted against SNR ranging from 0 dB and 30 dB under conditions \(N=M=2\) and \(N=4,M=2\) with respect to 50 iterations. From the results illustrated in Figure 4, obviously as the SNR increases, the BER decreases for all schemes. Meanwhile, in both conditions of antenna, the FPP-SCA one is the best in terms of the performance of all schemes, which increases the performance 0.003 compared with the Unaided Scheme, \(0.4\times 10^{-4}\) compared with the Low-Complexity Scheme, and \(4.0\times 10^{-6}\) compared with the SDR Scheme for \(N=M=2\) under \(SNR=20\) dB and 0.38 compared with the Unaided Scheme, 0.33 compared with the Low-Complexity Scheme, and \(1.5\times 10^{-5}\) compared with the SDR Scheme for \(N=4,M=2\) under \(SNR=15\) dB, which are shown in Tables 2 and 3.