# Tensor Rank Regularization with Bias Compensation for Millimeter Wave Channel Estimation
## 6 Numerical Experiments
The performance comparisons between the proposed method and the \(l1\) regularization LS in terms of robustness in success rate is shown in Figure 5. The simulation settings are the same as that of Figure 4. In Figure 5, the success rate is defined as the ratio of the number of the correctly estimated ranks over the total number of trails. This result shows the proposed method can converge to the true rank value under different initial rank selections with a success rate of over 90%. Figure 4: Robustness of the proposed method regarding different initial rank values with \(R=4\), SNR = 20 dB, \(\lambda=10^{-4}\) and \(\gamma=0.3\). Figure 5: Robustness of the proposed method in success rate regarding different values of initial rank with \(R=4\), SNR = 20 dB, \(\lambda=10^{-4}\) and \(\gamma=0.3\). The performance comparisons between the proposed method and the \(l1\) regularization LS (\(\gamma=0\)) in terms of success rate for different values of \(\gamma\) with same initial rank 24 are shown in Figure 6. It shows the proposed method with some values of \(\gamma\) can achieve much better rank estimation success rate than that of the baseline \(l1\) regularization method when SNR = 20 dB. For example, the rank estimation success rate is above 98% when \(\gamma=0.3\). The plots in Figure 7a-d show the relative error versus the number of iterations converging to the threshold when \(\gamma=0\), 0.3, 0.5, 0.7, respectively. For example, the proposed method takes about 300 iterations to reach the threshold in Figure 7b, compared to the baseline \(l1\) regularization with roughly 200 iterations in Figure 7a. Rank estimation is crucial for channel estimation when the number of multipath is unknown. It is worthy to get a significantly improved rank estimation success rate at the expense of slightly more iterations. We then turn to the overall channel estimation performance of our proposed scheme, which is measured by the normalized mean squared error. Channel NMSE = \(\sum_{k=1}^{K}\left\|\hat{\mathbf{H}}_{k}-\mathbf{H}_{k}\right\|_{F}^{2}/\sum_{ k=1}^{K}\left\|\mathbf{H}_{k}\right\|_{F}^{2}\). Figure 8 depicts the channel NMSE performance versus the system SNR derived by the proposed scheme, which is compared with that of \(l1\) regularization LS method (\(\gamma=0\)). It is observed that some values of \(\gamma\) (i.e., \(\gamma=0.3\) at SNR = 20 dB) give us smaller NMSE or better channel estimation performance. The robustness of the proposed method for different true rank \(R\) in terms of success rate is shown in Table 1. The simulation is based on the true rank \(R\) = 4, 5, and 6, \(\lambda=10^{-4}\), \(\gamma=0.3\), SNR = 20 dB. It can be observed that our proposed method can converge to the true ranks with high probability for different true rank \(R\). Figure 4.0 Success rate regrading different values of \(\gamma\) with \(R=4\), SNR = 20 dB, \(\lambda=10^{-4}\). Figure 4.1 NMSEs of channel estimation schemes versus the system SNR and values of \(\gamma\) with \(R=4\), \(\lambda=10^{-4}\). Figure 4.2 Relative error vs. Iteration using the proposed method for different values of \(\gamma\) with \(R=4\), SNR = 20 dB, \(\lambda=10^{-4}\). (**a**) \(\gamma=0\), (**b**) \(\gamma=0.3\), (**c**) \(\gamma=0.5\), (**d**) \(\gamma=0.7\). The rank estimation performance versus different \(\gamma\) and ranks is shown in Figure 4.3 In this simulation, we assume the true rank \(R=4\), \(5\), and \(6\), \(\lambda=10^{-4}\), SNR = 20 dB. When \(\gamma=0\) in (28) the proposed algorithm is equivalent to \(\Omega\) regularization LS (11). It can be observed that the average estimated ranks are very close to the true ranks with weighting parameter \(\gamma\) from 0.1 to 0.4. Otherwise, we get large rank estimation errors. Figure 10 depicts the number of iterations that the proposed method converges for each trail as a function of weighting parameter \(\gamma\) with real rank \(R=4\), \(5\), and \(6\), SNR = 20 dB, \(\lambda=10^{-4}\). Our simulation shows fast convergence for the weighting parameter \(\gamma<0.5\).