## 4 Semi-Blind Receiver
Thus, the efficient transmission rates for the proposed coding scheme and the KRST coding scheme are \(\frac{RN-1}{PN}\) and \(\frac{M_{S}(N-1)}{PN}\), respectively. To ensure a fair comparison, the proposed coding scheme and the KRST coding

\[\mathbf{g}=\partial\phi\left(\mathbf{u}\right)/\partial\mathbf{u}=\mathbf{J} ^{H}\left(\mathbf{u}\right)\mathbf{z}\left(\mathbf{u}\right). \tag{33}\]6 Cost function versus the number of iterations. scheme should keep the same efficient transmission rate, i.e., \(N=\frac{M_{S}-1}{M_{S}-R}\). Thus, the system design parameters in this example are set equal to \(M_{S}=4\), \(P=7\), and \(R=N=3\). For TB-ST coding scheme, we divide \(P=P_{tr}+P_{d}\), where blocks \(P_{tr}=2\) and \(P_{tr}=5\) are used for channel training and data transmitting. Therefore, the length of the channel training sequence in the TB-ST receiver is \(N_{tr}=P_{tr}N=6\). The BER performance of different receivers versus SNR is shown in \[\mathbf{g}=\partial\phi\left(\mathbf{u}\right)/\partial\mathbf{u}=\mathbf{J} ^{H}\left(\mathbf{u}\right)\mathbf{z}\left(\mathbf{u}\right). \tag{33}\]7 It can be seen that the proposed semi-blind receiver outperforms the P-KRST and TB-ST receiver. The NMSE performance of the different receivers is demonstrated in \[\mathbf{g}=\partial\phi\left(\mathbf{u}\right)/\partial\mathbf{u}=\mathbf{J} ^{H}\left(\mathbf{u}\right)\mathbf{z}\left(\mathbf{u}\right). \tag{33}\]8 It can be seen from Figure 4 that the P-KRST receiver has the best performance of channel estimation, and the proposed semi-blind receiver yields a smaller NMSE compared with the TB-ST receiver. From [8], the per-iteration complexity in the PARAFAC based receiver is \(\mathcal{O}\left(M_{S}M_{D}PN\right)\). The complexity of the TB-ST scheme can be estimated as \(\mathcal{O}\left(N_{r}M_{S}\left(M_{D}+N_{r}\right)+RPM_{D}\left(N+R\right)\right)\). The per-iteration complexity of the proposed O-LM algorithm is given at the end of Section 4. The TB-ST scheme has the least computational complexity due to the use of the channel training sequence. Due to the adoption of the simple KRST coding scheme, the PARAFAC based receiver has lower complexity than that of the proposed receiver. However, the TB-ST receiver requires a long channel training sequence, the PARAFAC-based receiver needs to know the first column or row of the signal matrix to eliminate the scaling ambiguity, but the proposed receiver only needs to know one symbol of the signal matrix. In the third example, we evaluated and compared the performance of the traditional ALS (T-ALS) and optimized LM (O-LM) algorithms. We assume the system design parameters \(N=P=L\) and \(R=5\). Correlated MIMO channel is considered in this example, and the channel matrix \(\mathbf{H}\) is modeled as in [35], where \(\rho\) denotes the normalized correlation coefficient with magnitude \(|\rho|\leq 1\). We consider \(\rho=0\) (non-correlation) and \(\rho=0.8\) (strong correlation), respectively. For each Monte Carlo run, the T-ALS algorithm is initialized with ten different random matrices as in [20; 36]. The estimation performance is evaluated after selecting the best initialization, which is the one that results in the minimum value of \(\delta^{(l)}\). We observe from Figure 5 that the T-ALS and O-LM algorithms give a similar BER and NMSE performance, which means that these two algorithms converge to the same point. For the right subfigure of Figure 5, the NMSE of the T-ALS and O-LM algorithms is also shown in Table 1 for the sake of comparison. We can also observe from Figure 5 that for these two algorithms, BER and NMSE performance degrade when the channel becomes strongly correlated. The overall complexities of

\[\mathbf{g}=\partial\phi\left(\mathbf{u}\right)/\partial\mathbf{u}=\mathbf{J} ^{H}\left(\mathbf{u}\right)\mathbf{z}\left(\mathbf{u}\right). \tag{33}\]9 Bit error rate (BER) performance of different receivers versus signal-to-noise ratio (SNR). the O-LM algorithm and the ALS algorithm depend on the per-iteration complexity and the number of iterations.