# Complexity Reduction of MLSE and MAP Equalizers Using Modified Prolate Basis Expansion
## 2 Materials and Methods
### System Model  
Consider the discrete-time baseband-equivalent model of a digital single carrier (SC) communication system, as shown in Figure 1.  
The data sequence \(\mathbf{b}=\left[b_{0},b_{1},\ldots,b_{L-1}\right]^{T}\) is considered a random sequence of \(L\) symbols \(b_{l}=\mathbf{b}[l]\) with zero mean and variance of \(\sigma_{b}^{2}=E(|b_{l}|^{2})\), whose elements come from an equally probable finite alphabet \(\mathcal{X}\) (\(b_{l}\in\mathcal{X}\)), and \(E(\cdot)\) denotes the expected value. Perfect synchronization between the transmitter and receiver with \(T\) as the symbol period is assumed. The symbols are propagated through the communication channel \(\mathbf{h}_{l}\), which is the discrete representation of the continuous impulse response of the baseband channel model \(h_{c}(t)\) in a given moment, where \(t\) represents the time variable and \(l\) is its equivalent discrete index. The aforementioned continuous model is composed of the convolution of \(f(t)\), \(v(t)\), and \(w(t)\). The propagation medium \(f(t)\), hereinafter assumed to be wide-sense stationary uncorrelated scattering (WSSUS) with a maximum time span of \(\tau_{max}\) seconds. \(v(t)\) and \(w(t)\) are the causal impulse response of the transmitter and receiver filters with a span of \(\tau_{v}\) and \(\tau_{w}\), respectively, a bandwidth \(W\) and a roll-off factor \(\beta\). Considering \(h_{c}(t)\) to be almost invariant during the transmission of data symbols with period \(T\) and time duration of \(\tau_{v}+\tau_{w}+\tau_{max}\), it can be written as follows:  
\[h_{c}(t,\tau) = \int_{\tau_{1}=0}^{\tau_{\max}}f(t,\tau_{1})v_{w}(\tau-\tau_{1}) \mathrm{d}\tau_{1}, \tag{1}\]  
where \(v_{w}(\tau)=\int_{\tau_{1}=-\infty}^{\infty}w(\tau_{3})v(\tau-\tau_{3}) \mathrm{d}\tau_{3}\), the convolution of the impulse responses of the transmitter and receiver filters and \(h_{c}(t,\tau)\) results in a wide-sense stationary correlated scattering. During the transmission of the sequence \(\mathbf{b}\), the CIR \(\mathbf{h}_{l}\) could be visualized as a variant finite impulse response (FIR) filter, with \(P\) coefficients; i.e., \(\mathbf{h}_{l}=\left[h_{l,0},h_{l,1},\ldots,h_{l,P-1}\right]^{T}\), where \(P=\left\lceil\frac{\tau_{v}+\tau_{w}+\tau_{max}}{T}\right\rceil\) and \(\left\lceil\cdot\right\rceil\) denotes the ceiling operation. The propagated signal is contaminated with a zero mean additive white Gaussian noise (AWGN) \(n_{l}\) with a variance of \(\sigma_{n}^{2}\). Thus, the received signal is expressed as \(y_{l}=\hat{y}_{l}+n_{l}\), where  
\[\hat{y}_{l}=\sum_{p=0}^{P-1}h_{l,p}b_{l-p} \tag{2}\]  
is the noiseless output of the channel. Figure 1: Baseband model of a digital communication system. If the channel is approximated using BEM, then the number of required coefficients for representing the channel could be less than \(P\) and consequently easier to implement. However, this reduction in complexity is firmly attached to the selection of the base, the optimal base being the Karnuhen-Loeve expansion (KLE). Nevertheless, to compute the KLE, it is necessary to know the autocorrelation of the channel, which is difficult, as it changes from one scenario to another. For overcoming these impairments, it is proposed in [17; 18] to take the impulse response of the transmitter and the receiver filters into account together with the maximum path delay for obtaining the MPFs. The channel BEM can be represented as  
\[h_{l,p}=\sum_{k=0}^{K-1}\left[\left(\mathbf{h}^{(k)}\right)^{H}\mathbf{h}_{l}\right]h_ {p}^{(k)}+e_{l,p}=\sum_{k=0}^{K-1}w_{l,k}h_{p}^{(k)}+e_{l,p}, \tag{3}\]  
where \(K<P\) is the number of required functions, \(w_{l,k}~{}=~{}\left(\mathbf{h}^{(k)}\right)^{H}\mathbf{h}_{l}\) is the corresponding weight of each invariant orthogonal sequence \(\mathbf{h}^{(k)}=\left[h_{0}^{(k)},h_{1}^{(k)},\ldots,h_{P-1}^{(k)}\right]^{T}\) and \(e_{l,p}\) is the modeling error in the \(p\)-th tap index, in the \(l\)-th time index.