## 5 Results and Discussion  
This section presents the computer based simulation results of the proposed techniques and analysis of the obtained results. The normalized channel mean square error (NCMSE) and bit error rate (BER) have been used as the performance metrics for this purpose. The NCMSE for the proposed estimation techniques is defined as  
\[\text{NCMSE}=\frac{\sum_{n_{r}=1}^{N_{r}}\sum_{n_{t}=1}^{N_{t}}\sum_{\ell=0}^ {L-1}\left|\hat{h}_{n_{r}n_{t}}^{\ell}-h_{n_{t}n_{t}}^{\ell}\right|^{2}}{\sum_ {n_{t}=1}^{N_{r}}\sum_{n_{t}=1}^{N_{t}}\sum_{\ell=0}^{L-1}\left|h_{n_{t}n_{t}} ^{\ell}\right|^{2}}. \tag{28}\]  
A massive-MIMO system consisting of 64-antenna BS serving four UEs i.e., \(N_{r}=64\) and \(N_{t}=4\), is considered for the purpose of simulations. The underlying massive-MIMO sparse uplink channels are assumed to be time-invariant and frequency-selective. The channel vectors \(\mathbf{h}_{n_{t}n_{t}}\), having a fixed sparsity level \(Q/L\), are independently generated for each Monte Carlo run. The non-zero channel coefficients of \(h_{n_{t}n_{t}}\) are drawn from a zero-mean Gaussian distribution with variance \(1/(N_{r}(L+1))\). Moreover, a fixed channel length of \(L=14\) is used for all of the channels. The positions of non-zero channel taps of all the channels from a certain UE to all of the receive antennas at the BS is taken as the same because of the small separation distance of the antennas as compared to the line-of-sight (LoS) path length. The periodic and low power training sequence for each UE is generated by using the \(m-\) sequence based approach, as presented in [26]. For this purpose, a base sequence \(\{-1,-1,-1,1,1,1,1,-1,1,-1,1,1,-1,-1,1\}\) with period \(\bar{P}=15\) is used for all of the simulation results. AWGN is independently generated at each receiver for a specific signal-to-noise (SNR) ratio. The SNR at \(n_{r}\)-th receiver is defined as the ratio of power of received signal \(\sigma_{y_{m}}^{2}\) to noise power \(\sigma_{n_{r}}^{2}\), i.e., \(\text{SNR}_{n_{r}}=\sigma_{y_{m}}^{2}/\sigma_{n_{r}}^{2}\). The zero mean binary phase-shift keying (BPSK) modulated information sequences (\(\mathbf{b}_{n_{t}}\in\{1,-1\}\)) are generated mutually independent for each transmitter. Performance comparison of the proposed SiT-StOMP and SiT-GP with that of the SiT-LS is presented in Figure 2 for NCMSE and BER against SNR, respectively. For this purpose, length of the information sequence is set as \(M=1500\) bits, sparsity level of the channel vectors as \(Q/L=3/14\), and training-to-information ratio (TIR) as \(\sigma_{c_{n_{t}}}^{2}/\sigma_{b_{n_{t}}}^{2}=0.2\), respectively. From Figure 2a, it can be seen that the proposed schemes perform much better as compared to SiT-LS. In terms of NCMSE at \(SNR=10\) dB, the proposed techniques SiT-StOMP and SiT-GP give a performance gain of 18 dB and 13 dB, respectively, over the SiT-LS. The BER based performance comparison is shown in Figure 2b. For a BER of \(10^{-2}\), a performance gain of about 3 dB and 2.5 dB in SNR is achieved by proposed SiT-StOMP and SiT-GP, respectively, over SiT-LS. In order to demonstrate the performance of proposed schemes for different variants of channel's sparsity parameter, i.e., \(Q/L\), the NCMSE and BER are plotted in Figures 3 and 4, respectively, for several values of \(Q/L\) (i.e., \(Q/L=3/14,5/14\) and \(7/14\)) by keeping the value of TIR equal to 0.2. It can be observed from Figures 3 and 4 that NCMSE and BER decrease with increasing channel sparsity and vice versa. However, SiT-StOMP performs better than SiT-GP, in terms of NCMSE, even if the channel is less sparse. Figure 2: Normalized channel mean square error (NCMSE) and bit error rate (BER) based comparison of proposed SiT-StOMP and SiT-GP techniques with SiT-LS for massive-MIMO uplink communications, \(M=1500\) bits, \(Q/L=3/14\), and \(\sigma_{c_{n_{t}}}^{2}/\sigma_{b_{n_{t}}}^{2}=0.2\), (**a**) MSE based performance comparison; (**b**) BER based performance comparison.