## 3 Inter Symbol Interference  
Inter symbol interference (ISI) generally results from limited bandwidth being allocated to a channel, the presence of nonlinearities due to power amplifiers, and distortions due to multipath propagation. Indeed when the signal enters the channel, it suffers disturbances such as the addition of noise or its reflection by obstacles. This is equivalent to transmitting the signal through several separate channels, each having a different attenuation and delay [37]. Practically, the recipient receives the direct signal and also the contribution of the attenuated and delayed signals, each having followed a different path; this refers to inter symbol interference. The modeling of the digital transmission chain is represented in Figure 1:  
In Figure 1 the useful signal x[n], is transmitted over a channel with ISI, H(n), to which an additive white noise w(n) has been added. On the receiver side, the received signal is sent over an equalizer C(n) to minimize the effect of ISI. The signal y(n), corresponding to the channel output, can be represented as a discrete-time equation, such as [38]  
\[\mathrm{y(n)}=\underbrace{\sum_{k=0}^{\infty}h(k)x(n-k)}_{\mathrm{ISI}}+w(n), \tag{1}\]  
Regarding Equation (1), the convolution is represented by the weight function h(k), which moves over the input signal x(n), resulting in a sum of the input samples weighted by the filter coefficients, and w(n) represents the additive noise. Figure 1: The digital transmission chain [10]. If the h(0) term of the sum is written individually, the equivalent form of (1) can be written as  
\[\mathrm{y(n)=h(0)x(n)+\underbrace{\sum_{ \begin{subarray}{c}\mathrm{k=1}\\ \mathrm{k\
eq n}\end{subarray} }^{\infty}\mathrm{h(k)x(n-k)+w(n)}}_{\mathrm{ISI}}}+ \mathrm{w(n)}, \tag{2}\]  
In Equation (2), the first term x(n) represents the useful signal at time n, the second term is the inter-symbol interference (ISI) representing the contribution of neighboring symbols, and the last term denotes the additive noise. Consequently, it can be concluded that if a response h(k) is the following:  
\[\mathrm{h(k)= \begin{cases}1;&\mathrm{k=0}\\ 0;&\mathrm{k\
eq 0}\end{cases} } \tag{3}\]  
then the received signal y(n) at each sampling instant is influenced absolutely by the transmitted symbol, therefore eliminating all ISI. Thus, this is the time domain requirement for a channel without ISI. The signal transmitted through the channel suffers various distortions (such as inter symbol interference). To resolve this issue, an equalizer is employed, for instance, a filter positioned at the output of the transmission channel, to fix/recover the transmitted signal [39]. Different circumstances and constraints in the environment of communications affect the signal transmitted and these also cause the phenomenon of ISI, which consists of a partial overlapping of specific adjacent symbols, thus causing damaging effects to the signal. To limit the effect of the ISI, an equalizer can be situated downstream of the transmission chain, right before the decision block. Practically, the general objective is to apply an equalizer filter C(n) to the samples y(n), to restore the equivalent channel H(n). Here, \(\hat{\mathrm{x}}(n)\) represents the output corresponding to the equalizer, and \(\widetilde{\mathrm{x}}(n)\) represents the output of the decision block corresponding to the estimate of the input signal x(n). Figure 2 illustrates this construction. Subsequently, the equivalent discrete channel model will be considered, of which the configuration is presented in Figure 3.  
The equivalent discrete channel H(n) is defined by the L nonzero coefficients h(k). The linear time-invariant system is characterized by the associated weight function h(n), which represents the impulse response of the system, such as the output corresponding to a Dirac impulse applied to the input [40]. Figure 3: Discrete channel model [10]. Figure 2: The decision block is preceded by an equalizer. The operation performed by a filter block is effectively a convolution operation.