# New Approach of Blind Adaptive Equalizer Based on Genetic Algorithms
## 3 Methods
### Adaptive Blind Equalization  
#### 3.3.1 Channel Model  
A traditional QAM (Quadrature Amplitude Modulation) data transmission system consists of a transmitter, a channel, and a receiver. The channel represents all interconnections between the transmitter and the receiver. An ideal communication channel allows the passage of all the frequency components of the signal, making changes in phase and amplitude as needed. However, in practical situations, communication channels allow only the passage of finite frequency ranges, according to their own physical characteristics. This results in interference between the symbols transmitted by the source, called inter-symbol interference (ISI). In transmission systems whose channel is based on free propagation, the multi-path phenomena are mainly responsible for the appearance of ISI. Figure 2 illustrates in a simplified way the multi-path phenomena, in which each path may be characterized by  
\[\rho_{i}\delta(t-\tau_{i}), \tag{3}\]  
where \(\delta(\cdot)\) is the impulse function, \(\rho_{i}\) is the complex gain (attenuation) of the channel, and \(\tau_{i}\) is the delay of the \(i\)-th path. The existence of noise in the communication system can also cause the symbols emitted by the source to be distorted, so that they are not correctly decoded by the receiver. Although it is possible to treat it, it cannot be eliminated from electronic systems [3]. The most common form of noise is thermal noise. Within physical and realization viability, ISI and symbolic distortions caused by noise must be compensated by the receiver. Figure 3 presents a simplified structure of a discrete base-band digital communication system with ISI, \(h(k)\), thermal noise, \(r(k)\), and a source of information transmitting complex symbols, \(a(k)\), belonging to a \(M\)-QAM alphabet, \(A=\{a_{0},\cdots,a_{M-1}\}\), of \(M\) possible symbols, so that all finite sub-sequences of the \(M\)-QAM alphabet symbols occur with non-zero probability. \(k\) indicates the discrete time index. The complex symbols, \(a(k)\), are expressed by  
\[a(k)=a^{I}(k)+ja^{Q}(k), \tag{4}\]  
where \(a^{I}(k)\) and \(a^{Q}(k)\) are the uni-dimensional phase components and quadrature, respectively, that constitute the bi-dimensional transmitted signal. The symbols \(a(k)\) are transmitted with a sampling period of \(T_{s}\) seconds by means of a channel which is assumed to be linear, causal, BIBO-stable (Bounded-Input Bounded-Output), and in which the impulse response, \(h(k)\), is described as  
\[h(k)=\sum_{i=0}^{L-1}\rho_{i}\delta(k-\tau_{i}), \tag{5}\]  
Figure 2: Multi-path phenomena between transmitter and receiver. \(\rho_{i}\) represents the attenuation of the channel, \(\delta(\cdot)\) is the impulse function, and \(\tau_{i}\) represents the delay of the \(i\)-th path. where \(L\) is the number of channel paths, \(\rho_{i}\) is the complex gain of the \(i\)-th path, and \(\tau_{i}\) is an integer value representing the \(i\)-th path in the instant \(k\). The signal \(x(k)\) represents the signal \(a(k)\) after undergoing channel attenuation. \(r(k)\) represents AWGN so that  
\[r(k)=r^{l}(k)+jr^{Q}(k), \tag{6}\]  
where \(r^{l}(k)\) and \(jr^{Q}(k)\), respectively, correspond to the phase and quadrature components of AWGN and they are random variables with Gaussian distribution of mean zero and variance \(\sigma_{r}^{2}\)[4]. Then, the equalizer processes the signal \(u(k)\), composed by the channel output, \(x(k)\), and the noise, \(r(k)\), and thus,  
\[u(k)=x(k)+r(k), \tag{7}\]  
\[x(k)=\rho_{0}(k)a(k-\tau_{0}(n))+\sum_{i=1}^{L-1}\rho_{i}(k)a(k-\tau_{i}(k)). \tag{8}\]  
The second term in Equation (8) represents the ISI. The structure of a blind adaptive equalizer is illustrated in Figure 4, where the output signal is expressed by  
\[\bar{a}(k-d_{eq}) = \sum_{l=0}^{N-1}w_{l}(k)u(k-l) \tag{9}\] \[= \sum_{l=0}^{N-1}w_{l}(k)x(k-l)+\sum_{l=0}^{N-1}w_{l}(k)r(k-l)\]  
where \(w_{l}\) is the \(l\)-th complex gain of the equalizer, \(N\) is the equalizer length, and \(d_{eq}\) is the equalization delay.