# Estimating the Performance of MIMO SC-FDE Systems Using SISO Measurements
## 4 Simulation Results  
To obtain theoretical values of BER performance as a function of \(E_{b}/N_{0}\) achieved by SISO and MIMO systems, simulation was used. The simulation parameters considered were 100 blocks with 256 transmitted data symbols (\(N=256\)), and a duration of 4 microseconds. As usual, \(E_{b}/N_{0}\) is the relation between the average bit energy for the receiving antenna, \(E_{b}\), and the unilateral power spectral density of the Additive White Gaussian Noise (AWGN) channel noise, \(N_{0}\). In both SISO and MIMO scenarios, the number of active users was always 1. The number of receiving antennas in that user varied from the SISO scenario to the MIMO scenario. In terms of simulation parameters, it is considered a MIMO scenario where the different pairs of antennas are uncorrelated and there is Rayleigh fading in the different multipath components. It is also important to point out that the average power is uncorrelated. We consider linear amplification and a perfect channel synchronisation and estimation. The theoretical values for the BER performance achieved by this procedure clearly show us the advantage of using previous DTs in a SISO system to infer the performance of a MIMO system. The obtained results (depicted in Figure 3, and only the first iteration of IB-DFE was considered) show that the greater the ratio between transmitting antennas (P) and receiving antennas (R) the higher the BER performance achieved, as it is supposed in a MIMO system. By using the iterative process of the receiver, the performance improves significantly, approaching the Matched Filter Bound (MFB), which represents the best performance that the receiver can ever achieve [24]. In this study, we have verified that the performance for the third iteration of IB-DFE already presents results very satisfactory and close to the MFB. It must be emphasised that this performance close to the MFB is only possible due to the use of IB-DFE. If we were only using linear FDE, as it is the case for the first iteration shown in Figure 3, the MIMO coverage would be worse and that is the main difference when we compare the results of Figures 3 and 4. Accordingly, it is also possible to verify that, for a given BER value, the \(E_{b}/N_{0}\) values of the different \(R/P\) ratios are closer, when compared with the first iteration. For each \(R/P\) ratio studied, the BER values considered for obtaining the performance were \(10^{-2}\), \(10^{-3}\) and \(10^{-4}\), for which the corresponding \(E_{b}/N_{0}\) values were obtained. By obtaining these \(E_{b}/N_{0}\) values, it was then possible to calculate the difference between the performance of SISO and MIMO systems, which is represented by \(G\Delta\). This difference, for the first iteration of IB-DFE, is shown in Figure 5, where the fitted lines for each case are also depicted. Figure 3: Theoretical BER performance for the first iteration when using IB-DFE for different \(R/P\) values. As depicted in Figure 6, the difference between the SISO and MIMO systems decreases for the third iteration of the IB-DFE. Figure 4: Theoretical BER performance for the third iteration when using IB-DFE for different \(R/P\) values. Figure 5: Difference between theoretical BER performance for SISO and MIMO systems for the first iteration when using IB-DFE. Through the fitted curves, shown in Figure 6, it is possible to predict the \(G\Delta\) (the difference between SISO and MIMO systems for a given BER performance), considering the variation in the \(R/P\) ratio. For the first iteration and considering a BER performance of \(10^{-2}\) the fitted curve is given by Equation (27). Equation (28) corresponds to a BER of \(10^{-3}\) and Equation (29) denotes the fitted curve for a BER of \(10^{-4}\).