## 4 Semi-Blind Receiver
Second, the low-rank property of the mmWave massive MIMO channel should be exploited. Due to very limited scattering of the mmWave channel and larger quantities of transmitting and receiving antennas, \(L_{m}\) is usually less than \(M_{S}\) and \(M_{D}\). Different from the conventional MIMO channel matrix that usually has full column or row rank, the rank of the mmWave massive MIMO channel matrix is much smaller than its dimension. This is called 'low-rank property' of the mmWave massive MIMO channel matrix. Therefore, the final part of the proposed **Algorithm 1** takes advantage of this low-rank constraint \(rank\left(\mathbf{H}^{(m)}\right)\leqslant L_{m}\) to further improve the estimation accuracy of the channel. ## 6 Simulation Results and Discussion

We studied the performance of the proposed semi-blind receiver through numerical simulations. The channel matrix \(\mathbf{H}\) has independent and identically distributed (i.i.d.) complex Gaussian entries with zero-mean and unit variance. The default values of the system parameters are set to \(M_{S}=M_{D}=4\), and the antenna-to-slot allocation matrix is all-ones matrix. Throughout the simulation, the coding tensor \(\underline{\mathbf{C}}\) is known at the receiver. Quadrature phase-shift keying (QPSK) constellations are used to modulate the transmitted symbols. All results are averaged over 10,000 independent Monte Carlo simulations. As in [8; 9], the signal-to-noise ratio (SNR) at the receiver is defined as:

\[\mathbf{g}=\partial\phi\left(\mathbf{u}\right)/\partial\mathbf{u}=\mathbf{J} ^{H}\left(\mathbf{u}\right)\mathbf{z}\left(\mathbf{u}\right). \tag{33}\]5

where \(\underline{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{  }}}}}}}}}}}}}}}\) denotes the noise-free signal tensor (the tensor-of-interest) containing both symbol and channel parameters. For each channel realization, the normalized mean square error (NMSE) for different receivers is computed as \(\left\|\mathbf{H}-\mathbf{\hat{H}}\right\|_{F}^{2}\middle/\left\|\mathbf{H} \right\|_{F}^{2}\), where \(\mathbf{\hat{H}}\) is the estimation of \(\mathbf{H}\) at convergence. In the first example, we evaluate the convergence performance of the optimized LM algorithm, which is used in our semi-blind receiver. We assume the system design parameters \(N=P=5\) and \(R=3\). In Figure 2, the average value of the cost function is plotted versus the number of iterations, for three SNR values. We observe from Figure 2 that for each SNR value, the cost function decreases as the number of iterations increases until the algorithm converges. We can also see that for the same number of iterations, the cost function decreases as SNR increases. The proposed algorithm needs few iterations to converge. For instance, the optimized LM algorithm achieves convergence in about 10 iterations at the SNR of 20 dB. In the second example, we evaluated the estimation performance of the proposed semi-blind receiver in terms of bit error rate (BER) and the NMSE of channel estimation. In particular, we compared the PARAFAC-based receiver with KRST (P-KRST) coding scheme in [8] and the training-based receiver with the space-time (TB-ST) coding scheme. For the TB-ST scheme, the symbol matrix is composed of two parts as in [9], i.e., the training symbol matrix and the unknown data symbol matrix. \(N_{tr}\) denotes the length of the channel training sequence in the TB-ST receiver. The transmission rates for the proposed coding scheme and the KRST coding scheme are \(\frac{RN}{PN}=\frac{R}{P}\) and \(\frac{M_{S}N}{PN}=\frac{M_{S}}{P}\) (data symbols per symbol period), respectively. However, the KRST coding scheme needs to know the first column of signal matrix \(\mathbf{S}\) to eliminate the scaling ambiguity, while the proposed coding scheme only needs to know \(s_{1,1}\) to eliminate the scaling ambiguity.