# Hybrid Precoder Using Stiefel Manifold Optimization for Mm-Wave Massive MIMO System
## 4 Simulation Results  
The performance of the proposed algorithm and its comparison with those of existing algorithms is compared in this section [14]. The system model is designed as an environment having 8 clusters and 10 rays per cluster. The system environment is considered to have a 10-degree angular speed that is uniformly distributed from 0 to 360 degrees [22]. The experiment considers an OFDM system with 128 multicarrier paths. The experimental setup considers an equal number of RF chains at the BS as well as the MS. This section also shows the comparison among the proposed algorithm, the OMP algorithm [11], the PE-AltMin algorithm [14], and the optimal digital precoder for a certain level.  
Figure 3 firstly presents that the spectral efficiency of the proposed algorithm is superior to that of the fast precoding algorithm for a range of SNRs [\(-\)15:10], with \(Ns\) = 4 and \(N_{RFT}\) = 6. The size of the antenna array taken for this simulation is 144 \(\times\) 36. This shows that the proposed algorithm gives a better result than the fast precoding algorithm [2].  
The spectral efficiency of the proposed algorithm is also compared with those of the optimal digital precoder for reference, PE-AltMin [14], and OMP [11] for different values of the number of RF chains that are equal at the transmitting and receiving ends. This comparison is shown in Figure 4. From this figure, it can be seen that the proposed algorithm produces steadily higher results than the PE-AltMin and OMP algorithms and is nearly close to the optimal precoder. The value of the number of symbols (Ns) is kept lower than twice the number of RF chains. This comparison is made with the SNR value of 0 dB. The performance of the proposed algorithm increases in proportion to the number of RF chains and is nearly close to that of the optimal precoder [24]. The performance of OMP also improves as compared with PE-AltMin upon increasing the number of RF chains. Figure 5 shows the spectral efficiency for a range of SNRs having different values of the number of RF chains.  
Figure 4: Spectral efficiency at 0 dB.  
Figure 3: Spectral efficiency of proposed of algorithm.  
_Comparison of Proposed Model with Existing Techniques_  
Table 1 shows the comparison of the proposed system with existing system models on the basis of performance parameters such as spectral efficiency (measured using \(-10\) dB, \(0\) dB, and \(10\) dB SNRs) and computational complexity. As the optimized digital precoder and the weight matrix have closed-form expressions and have much lower dimensions than the analog precoder, we ignore their computational complexity. Additionally, as both the analog precoder and combiner can be solved in the same procedure, we focus on the complexity analysis of the analog precoder. The proposed algorithm needs to calculate the Euclidean gradient defined in step 7 of the proposed algorithm, which is the dominant calculation cost of the proposed algorithm. Hence, the computational complexity for the gradient calculation is \(\mathcal{O}(N_{FFT}N_{T}Ns)\). Meanwhile, the calculation cost of the MO-AltMin algorithm is \(\mathcal{O}(N_{FFT}N_{T}^{3}N_{s}^{2})\)[14]. Furthermore, since PE-AltMin requires min{\(\mathcal{O}(N_{FFT}N_{s}^{2},\)\((N_{s}(N_{FFT})2)\)} for the SVD calculation per iteration, the proposed algorithm competes with the low-complexity PE-AltMin algorithm. Consequently, as summarized in Table 1, the proposed algorithm is comparable to or much faster at processing than state-of-the-art algorithms, and it produces near-optimal spectral efficiency.