## 4 Simulation Results  
In the following, the simulation results are provided to verify the performance of the proposed scheme for sparse FIR MIMO DFE design in two different channels. As a comparison, the performance of the optimum MMSE DFE and the OMP method in [15] are presented respectively. Two different multipath Rayleigh fading channels, namely ITU (International Telecommunication Union) Vehicular A channel [23] and an exponential decay channel, are simulated. ITU Vehicular A channel has a sparse power delay profile (PDP) with 6-paths spanning about 13 symbol periods. The exponential decay channel has a non-sparse PDP as \(\rho(n)=10^{-n/10},0\leq n\leq 12\) with 13-paths. The simulation system uses binary phase shift keying (BPSK) modulation. The decision delay is set to be \(\Delta=N_{f}\)-1, which is optimum when \(N_{b}=\
u\) for DFEs [24]. The small increase \(\theta\) in the equalizer MSE of the sparse DFE compared with the MMSE DFE is determined by the maximum tolerable performance loss \(\eta_{\max}\) as \(\theta=(10^{\eta_{\max}/10}-1)\xi_{\min}\), where \(\xi_{\min}\) is the MSE for the MMSE DFE. We define the signal-to-noise (SNR) as \(\text{SNR}\triangleq 10\lg(\varepsilon_{x}/\sigma^{2})\), set the oversampling factor \(l\) = 2 and MIMO DFEs with \(n_{i}\) = 2 and \(n_{o}\) = 2. Figure 1 plots the nonzero taps percentage for the sparse SISO DFEs and MIMO DFEs versus the performance loss \(\eta_{\max}\) using the proposed scheme respectively. It is shown that the number of nonzero taps for DFE is reduced with the increase of \(\eta_{\max}\). When \(\eta_{\max}\) = 0.4 dB, the nonzero taps for both the sparse SISO DFEs and MIMO DFEs are reduced by more than 70% and 60% in the two different channels respectively. Therefore, comparing the ITU Vehicular A channel with sparse PDP, the FIR DFEs needs more nonzero taps to eliminate the ISI caused by the multipath effect in the exponential decay channel with more paths. Furthermore, in MIMO systems, the DFEs needs more nonzero taps to eliminate both ISI and inter-antenna interference, as shown in Figure 1, MIMO DFEs has a larger percentage of nonzero taps when compared with SISO DFEs. Figure 2 gives a single example of the SISO DFE by the three design methods in ITU Vehicular A channel. Note that only the nonzero taps are marked, and the tap indices below 25 are insignificant to show. It is shown that there are many taps whose values are close to zero in the non-sparse MMSE DFE. Compared with the MMSE DFE, most of the taps for sparse FIR DFEs designed by the OMP method and the proposed scheme are zero, and the tap positions selected by the two methods are not identical. To verify the performance of the proposed scheme, we compare the bit error rate (BER) and the normalized mean square error (NMSE) for the three design methods. Figures 3 and 4 compare the BER performance and the equalizer NMSE respectively in ITU Vehicular A channel. The sparse DFEs  
Figure 1: Nonzero taps percentage for the sparse SISO DFEs and MIMO DFEs versus \(\eta_{\max}\) with \(\text{SNR}=20\) dB and \(N_{f}=30\). designed by the proposed scheme and the OMP algorithm are constrained to have the same sparsity level in the simulation. The simulation results show that the maximum SNR loss for the SISO DFEs and MIMO DFEs designed by the proposed scheme are only 0.4 dB and 0.6 dB when the number of nonzero taps is reduced by more than 83% and 80%, respectively. That is, the BER performance and the NMSE for the sparse DFEs designed by the proposed scheme are similar to the MMSE DFE performance. As shown in Figure 3, the BER performance for the sparse DFEs designed by the proposed scheme outperform the OMP algorithm with the same sparsity level. Besides, by comparing the NMSE of the two methods, it is known that the design accuracy of the proposed scheme is superior to the OMP method. Considering that the MIMO system has a certain diversity gain, the simulation results in Figure 3 show that the BER performance of the MIMO DFEs have a significant improvement than the SISO DFEs.