## 3 LR-Aided MIMO Detector Design
### Performance Analysis  
In this section, we present the performance analysis of the ELR-aided detectors, where we compare the BER as well as complexity of the ELR- and LLL-aided detectors. Then, we analyse the performance difference between the ELR-aided K-Best detector and the K-Best detector, which forms the basis for our hardware implementation proposed architecture in Section 4. First, we compare the BER performance of the ELR-aided detectors with the benchmark techniques described in Section 3.1 using the LLL-aided detection, when employed to MIMO systems. This comparison is included in order to show the benefits of the proposed ELR-aided K-Best decoder design compared to its benchmarkers in large-scale MIMO systems. We performed Matlab simulation for a large-scale MIMO system employing \(N=200\) transmit and \(M=200\) receive antennas, while employing 64-QAM. In our simulations we considered Rayleigh fading channels, where the channels between the different transmit and receive antennas are independent. The simulation parameters are included in Table 2, where we have opted to use \(N=M=200\) to compare with the results reported in [29]. Additionally, in the simulation set up, we have assumed the channel state information is perfectly estimated at the receiver and we also consider perfect synchronisation of the transmit antennas. \begin{table}
\begin{tabular}{l l} \hline \hline
**Input:H** & **Output:T** \\ \hline (1) \(\widetilde{\mathcal{C}}=(H^{H}H)^{-1};\mathcal{T}^{\prime}=I_{m}\) \\ (2) Do \\ (3) & \(\lambda_{i,k}\leftarrow\lceil\frac{\widetilde{\mathcal{C}}_{i,k}}{\widetilde{ \mathcal{C}}_{i,i}}\rfloor,\forall i\
eq K\) \\ (4) & if all \(\lambda_{i,k}=0,\forall i\
eq K\) \\ (5) & break \\ (6) & end if \\ (7) & Find the largest reducible \(\widetilde{\mathcal{C}}_{k,k}\) \\ (8) & Choose i = arg \(max_{i}R_{i,k}\) \\ (9) & \(t_{k}^{\prime}\gets t_{k}^{\prime}+\lambda_{i,k}t_{i}^{\prime}\) \\ (10) & \(\widetilde{\mathcal{C}}_{k}\leftarrow\widetilde{\mathcal{C}}_{k}+\lambda_{i,k} \widetilde{\mathcal{C}}_{i}^{i}\) \\ (11) & \(\widetilde{\mathcal{C}}^{(k)}\leftarrow\widetilde{\mathcal{C}}^{(k)}+\lambda_{i,k }^{*}\widetilde{\mathcal{C}}^{(i)}\) \\ (12) & While (True) \\ (13) & \(T=(T^{\prime-1})^{H}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: ELR algorithm [28]. Figure 2 shows the BER performance comparison of the various decoding techniques (Note that in the figure we do not show the performance of the ML detector due to its extremely high complexity for simulation with our configuration). As shown in Figure 2 the BER performance of the ZF detector forms as an upper bound on the BER performance of the other detectors, while the ZF detector has the lowest computational complexity. When LR is combined with the ZF detection, Figure 2 shows that significant performance improvement can be attained compared to the ZF detector dispensing with LR. Additionally, it can be seen from Figure 2 that the ELR-aided ZF detector has a better performance that its LLL-aided counterpart. The aim of the LR-aided detectors is to attain a sub-optimal performance close to that of the ML, while requiring significantly lower complexity than the ML detector. The simulation results in Figure 2 for the LLL-aided detectors show BER performance improvements for the simulated large-scale MIMO compared to the ZF performance. Additionally, the proposed ELR-aided detectors show performance improvement compared to their LLL-aided counterparts. For example, the proposed ELR-aided K-Best detector is capable of attaining a 2 dB performance gain compared to its LLL-aided counterpart at BER of \(10^{-5}\). Additionally, the ELR-aided MMSE K-Best detector outperforms its LLL-aided counterpart by about 3 dBs at BER of \(10^{-5}\). Explicitly, \begin{table}
\begin{tabular}{l l} \hline Modulation & 64-QAM \\ \hline No. of Transmitter Antennas & 200 \\ \hline No. of Receiver Antennas & 200 \\ \hline Channel & Correlated Rayleigh Fading \\ \hline Normalised Doppler frequency & 0.01 \\ \hline \end{tabular}
\end{table} 
Table 2: System parameters.