# Complexity Reduction of MLSE and MAP Equalizers Using Modified Prolate Basis Expansion
## 3 Results and Discussion  
In this section, the performance evaluation of the proposed equalizer for an SC communication system is presented. Transmission of data blocks composed of 800 symbols at a symbol period of \(T=10^{-7}s\) was assumed. These are considered QPSK and 16 quadrature amplitude modulation (QAM) modulations. The pulse shaping and matched filter were square-root raised cosine with a roll-off factor of 0.5 and a length of six symbol periods. The orthogonal equalizers were simulated for a communication system with a power delay profile defined by the following normalized path's power: \([0,-1.5,-1.4,-3.6,-0.6,-9.1,-7,-12,-16.9]\) dB, and corresponding path delay: \([0,3,15,31,37,71,109,173,251]\) ns, each path with Jakes Doppler profile and with a maximum normalized Doppler spread \(f_{max}T=0.001\). The latter corresponds to a CIR of size \(P=9\). Given the former system values, the number of MPFs needed to expand the channels results in \(K=5\). In Figures 2 and 3, there are depicted the bit error rate (BER) performance of the equalizers versus the signal to noise ratio (SNR) under the assumption of perfect knowledge of the CSI and over the pilot-assisted channel estimation as described in [23]. The simulation results confirm that the performance of the orthogonal equalizers is quite similar to that of the MLSE. Moreover, the prediction in the chosen number of eigenfunctions \(K\) is valid: The performance is not affected when the number of computations is reduced. The performance of the equalizer proposed in [15], based on the Fourier basis, is also presented in Figures 2 and 3. In that work, the channel BEM is developed in the time domain, in contrast to this work, which considers a BEM in the time-delay domain. The BEM in time, as developed in [15], \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithms** & **Classical** & **Orthogonal Fourier** & **Orthogonal** \\  & & [15] & **MPF** \\ \hline Multiply-add operations & \(LPMP^{P}\) & \(LK_{1}M^{P}\) & \(LK_{2}M^{P}\) \\ Memory locations & Not required & \(K_{1}M^{P}\) & \(K_{2}M^{P}\) \\ Memory update period & Not required & \(L\) & Not required \\  [samples] & Not required & \(K_{1}PM^{P}\) & \(K_{2}PM^{P}\) \\ Multiplications needed & Not required & \(K_{1}L_{1}P\) & \(K_{2}LP\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Complexity and memory-requirement comparison to obtain \(\{\tilde{y}_{l}|\mathbf{b}_{l}\}_{l=0}^{L-1}\). Modified versions of the prolate functions (MPFs), basis expansion modeling (BEM). causes the necessity of computing the \(K_{1}\) BEM coefficients and updating the memory locations every \(L\) samples. Furthermore, it does not provide an expression to define \(L\) and \(L_{1}\). For the simulations in this work, we chose the parameters \(K_{1}=5\), \(L=800\) and \(L_{1}=1600\), used in [15] to expand the channel for the given normalized Doppler spread. Due to the lack of accuracy in the representation through the Fourier basis, the simulation results for [15] in Figures 2 and 3 exhibit the presence of an error floor, which is worst when the modulation order is increased. A better resolution in Fourier basis can reduce the modeling error and leads to the absence of error floor, but this is achieved either by extending the observation window and introducing lags in the estimated data and extra memory or by an iterative process of joint channel estimation and equalization. Table 3 shows the number of multiply-add complex operations, memory locations, and their update period in the simulated scenario for the classical and orthogonal equalizers for QPSK and 16-QAM. It is possible to observe that Fourier and MPF algorithms have the same computational complexity, achieving both a reduction in the number of multiplications of 55.6% concerning the classical MLSE. The latter because we are considering the same number of functions (\(K_{1}=K_{2}\)). However, as it was shown, MPFs outperforms the Fourier implementation when the same number of functions is used (Figures 2 and 3).