# Tensor Rank Regularization with Bias Compensation for Millimeter Wave Channel Estimation
## 6 Numerical Experiments  
We present simulation results to demonstrate the performance of the proposed method. We consider a scenario where the BS employs a uniform linear array with \(N_{bs}=64\) antennas and RF chains \(M_{bs}=6\), the MS employs a uniform linear array with \(N_{ms}=32\) antennas and RF chains \(M_{ms}=6\). The separation between neighboring antenna elements is assumed to be half the signal wavelength. In our simulations, the mmWave MIMO channel is generated according to the wideband geometric channel model, in which the DoAs and DoDs are randomly distributed in \([0,2\pi]\), the delay spread \(\tau_{r}\) for each path is uniformly distributed between 0 and 100 nanoseconds, and the complex gain \(\alpha_{r}\) is a random variable following a circularly symmetric Gaussian distribution \((0,1/R)\). We set \(f_{c}=28\) GHz, the total number of subcarriers \(K_{0}=128\), the number of subcarriers selected for training \(K=6\), and the sampling rate \(f_{s}=0.32\) GHz. The beamforming matrix \(\mathbf{W}_{A}\) and the combining matrix \(\mathbf{F}_{A}\) are randomly generated with their entries uniformly chosen from a unit circle. Because the sparse scattering nature of mmWave channels, the number of paths \(R\) is usually small [15,16]. Thus, we select the number of paths equal to \(R=4\), \(5\), and \(6\) to form a low rank channel scenario in our simulations. We assume the initial rank \(\widetilde{R}\) to be an overestimation of the true rank. To compare the rank estimation performance, the signal-to-noise ratio (SNR) is defined as the ratio of the signal component to the noise component, i.e.,  
\[\text{SNR}\triangleq\frac{\left\|\mathcal{Y}-\mathcal{N}\right\|_{F}^{2}}{ \left\|\mathcal{N}\right\|_{F}^{2}}\]  
Figure 2 shows the estimated ranks versus regularization tuning parameter \(\lambda\) in the interval \(10^{-5}\) to \(1\) based on the baseline \(I1\) regularization LS method (11) when \(R=4\) with the selected SNR values. It is observed that tensor rank can be approximately recovered when \(\lambda=10^{-4}\). To evaluate the rank estimation performance of the proposed method as a function of weighting parameter \(\gamma\), Figure 3 depicts the performance comparison of rank estimation versus SNR with \(\lambda=10^{-4}\) and real rank \(R=4\). It also indicates the proposed method leadsto smaller rank estimation error when \(\gamma\) is no larger than 0.5 with SNRs between 20 dB and 30 dB than that of the \(l1\) regularization LS method. The comparison between the proposed method and the \(l1\) regularization LS in terms of robustness is shown in Figure 4. The simulation is based on the true rank \(R=4\), \(\lambda=10^{-4}\), \(\gamma=0.3\), SNR = 20 dB. We select the initial rank \(\widetilde{R}\) as some integers between 4 and 32 and calculate the estimated rank with the same stop criterion. The selection of the upper  
Figure 3: Comparison of the rank estimation performance of the proposed method as a function of \(\gamma\) and the \(l1\) regularization LS versus SNR with \(\lambda=10^{-4}\) and real rank \(R=4\). Figure 2: Rank estimation performance of the \(l1\) Regularization LS method as a function of \(\lambda\) with real rank \(R=4\). bound of the initial rank is bounded by the characteristics of \(\mathbf{C}^{(n)}\odot\mathbf{B}^{(n)}\), \(\mathbf{C}^{(n)}\odot\mathbf{A}^{(n+1)}\) and \(\mathbf{B}^{(n+1)}\odot\mathbf{A}^{(n+1)}\). We make these three Khatri-Rao products as overdetermined matrices to guarantee the ALS-based method has a unique solution. So, \(\bar{R}<M_{bs}K=36\). It can be observed from Figure 4 that the proposed method (the top red line) converges to the real rank \(R=4\) under the different initializations, while \(l1\) regularization LS method only converges at a large initial \(\bar{R}\). Thus, the rank estimation performance of the proposed method is much more robust than that of the \(l1\) regularization LS in terms of initialization.