# Comparison of CS-Based Channel Estimation for Millimeter Wave Massive MIMO Systems
## 5 Simulation Results
### Comprehensive Comparison of Estimation Quality  
In Figure 2, we compare the NMSE performance of different methods with respect to received SNR. We can observe that the performance of greedy iterative algorithms are best at the low SNR range. This is because the noise will be treated as the non-zero elements, which badly destroy the sparsity of the channel and reduce the amount of available pilot symbols. However, the gradient descent and iterative reweighting method are best at the high SNR range. It can also be observed that the performance of greedy iterative algorithms will remain unchanged with the SNR improving at the high SNR range because the estimated error caused by noise becomes smaller and smaller, but the quantization error decided by the resolution of angle quantization remains unchanged. Meanwhile, we can see the NMSE of Bayesian inference algorithm linearly decreases with the SNR improving, approximatively. In Figure 3, we illustrate the NMSE performance versus the sparseness \(L\). We can see that the NMSE of greedy iteration algorithms remains unchanged with the sparseness increasing, approximately, since the most relevant columns are selected in each iteration. The performance of IR algorithm also remains unchanged. Moreover, it is better than others at the low sparseness range. In addition, the performance of BCS and CVX algorithms improve as the sparseness increases. This is because the assumptions of prior distribution and the condition that let \(l_{1}\) norm replace \(l_{0}\) norm are fitted better when the sparseness increases. Furthermore, we can observe that the performance of BCS algorithm is better than greedy iterative algorithms when the sparseness is larger than 3 and the gap even reaches 8 dB when the sparseness is 7. Figure 4 depicts pilot symbol frames versus the NMSE performance where each frame is assumed to be length of 16 pilot symbols. It is obvious that the NMSE becomes smaller with the increasing of pilot symbols for the three categories of algorithms. However, for the OMP and BCS algorithms, the NMSE curves become flat as the number of pilot symbol frames becomes large enough. The NMSE of channel estimated by convex relaxation algorithm such as basis pursuit (BP) algorithm improves with the pilot symbol frames increasing. The figure shows that the performance of BP and BCS algorithms even make the channel estimation results unusable at the low SNR range, but they are obviously better than OMP algorithm at the high SNR range. The figure also shows that the OMP algorithm is optimal in the range of low SNR with a few pilot symbols. After all, the estimation quality of OMP algorithm is least affected by the pilot symbols, while the BP algorithm is most affected. Figure 2: NMSE of the estimated CSI versus received SNR with sparseness \(L=5\), pilot symbol frames \(M=16\), angle quantization \(G=32\) and without hardware impairment. In Figure 5, we illustrate the NMSE performance versus the angle quantization \(G\). We can see that the performance of BP algorithm is hardly affected by angle quantization, which means it can avoid the quantization error as the AoAs and AoDs are accurately estimated by gradient descent method in each iteration. However, the estimation quality of BP algorithm is worse than OMP and BCS algorithms when the angle quantization is larger than 20. For the OMP algorithm, the MNSE decreases as angle quantization increases, which means the angle quantization is a key factor that affects the estimation performance of OMP algorithm. As to the BCS algorithm, the NMSE curve goes down rapidly and then stays flat, which indicates there is an optimal angle quantization \(G\) for it. In Figure 6, we show the effect of transmitter's hardware impairments on channel estimation quality, where the noise contained in the SNR is the AWGN without considering the angle quantization error and the hardware impairments. We can see that the estimation quality of all algorithms is affected by the transmitter's hardware impairments, especially at the high SNR range.