# New Approach of Blind Adaptive Equalizer Based on Genetic Algorithms
## 1 Introduction  
In digital communication systems, temporal dispersion caused by non-ideal characteristics of the channel frequency response or multi-path transmission may cause inter-symbol interference (ISI). The ISI is a critical manifestation of distortion, whereby symbols transmitted before and after a particular symbol corrupt the detection of this symbol. All physical channels, at high enough data rates, tend to exhibit ISI [1]. Therefore, the ISI has become a limiting factor in many communication systems. To minimize the ISI, bringing the bit error rate (BER) to acceptable values, one can make use of a digital channel equalizer. In general, the communication channel is time-varying; therefore, the equalizer needs to employ efficient adaptive algorithms, which conveniently adjust the coefficients of the equalizers (the parameters to be optimized), in order to attenuate the ISI [2, 3, 4]. The ISI cancellation using an adaptive equalizer has been studied for many years by the signal processing community. The conventional (supervised) adaptive channel equalization methods require the transmission of a reference signal to the receiver in order to calculate the equalizer coefficients, which considerably sacrifices the channel capacity. In contrast to these methods, the adaptive blind equalization methods operate without a training sequence [4], retrieving the information regarding the transmitted signal or the channel by analyzing the characteristics of its output as well as some information about the system or the transmitted sequence but not the sequence itself [5]. Over the past few years, many algorithms for adaptive blind equalization have been developed for the recovery of digitally modulated signals. The blind equalization techniques can be mainly classified as based on High-Order Statistics (HOS) or based on Second-Order Statistics. The Constant Modulus Algorithm (CMA), introduced in [6], is considered the simplest and most successful among HOS-based blind equalization algorithms. It is an optimization algorithm based on the descendant gradient technique. The CMA cost function is based only on the amplitude of the received signals. Nonetheless, the optimization function of the CMA does not provide full convergence, and presents undesirable local minima that can lead to the inefficient reduction in ISI [6; 7]. To avoid local minima issues, ref. [8] proposed a globally convergent convex cost function for blind equalization. The works presented in [9; 10] improved this technique by using linear programming (LP) algorithms to perform equalization. However, in a real-life situation, channel noise and inter-symbol interference act together, affecting the behavior of a data transmission system in a combined manner [2]. Despite the results obtained, these works do not include any analysis on noisy channels. So, the above studies are only valid for a deterministic-demand assumption, which is impractical to many possible scenarios. In this sense, the work [11] extends the function proposed by [8; 9; 10] in order to also consider the effects of channel noise, by proposing a new constraint function and applying other LP optimization algorithms in a blind adaptive equalizer. This new extended cost function has been proved to equalize signals with ISI and channel noise. Other work related to the LP methodology can be found in [12; 13]. Noise is a process of random nature, and adding it to the transmitted signal causes the resulting signal that is to be handled by the equalizer to also have a random characteristic. When one or more of the data elements in a linear program is represented by a random variable, this results in a stochastic linear program (SLP) [14]. Taking into account that traditional methods for solving linear problems, as presented in [11], are not the most suitable for approaches with stochastic variables, a more robust technique for this type of signal must be investigated.