# Uni-Cycle Genetic Algorithm as an Adaptation Engine for Wireless Channel Equalizers
## 3 Uni-Cycle Genetic Algorithm  
The genetic Algorithm is an iterative technique that simulates the natural evolution process by applying the methods of selection, crossover, and mutation of the \"genetic material\" according to some probabilistic rules [13]. The \"genetic material\" (called chromosome) represents the details of the given optimization problem solution, called individual. Several individuals co-exist and constitute a population. The conventional genetic algorithm has been designed to solve optimization problems that do not evolve over time [14; 15]. By the application of various GA operators, the structure of the population continuously evolves (the individuals whose chromosomes exhibit the poorest fitness die, the newborns constitute a new generation; there are also some random mutations applied to the chromosomes to apply some minor random changes). By creating a new generation, the algorithm takes a single step on its path towards the optimal (fixed) solution. In the case of channel equalizer task, the role of the equalizer is to continuously track the current channel state, so there is no one fixed optimization goal. Conceptually, it is possible to consider several GA generations in order to pursue a solution (the channel taps or, equivalently, the decision on one transmitted symbol) that is optimal for a single signaling interval, and then start from scratch to consider the next signaling interval. Such  
Figure 1: Adaptive Equalizer block diagram. a solution would be similar to the one presented in [9], (The reference solution from [9] is even more difficult, as it incorporates the decoder, additionally.). According to the proposed UCGA strategy, only one generation per one signaling interval is assumed to limit the computational payload. The resultant population from one signaling interval is considered the initial population for the next UCGA instance, related to the next signaling interval, and so on, as shown in Figure 2.  
As the channel evolution is rather slow, it can be considered as if the global optimum were fixed over a few subsequent signaling intervals which allows for the UCGA convergence. Obviously, the algorithm cannot find the exact optimum, which is slinking, but a rough estimation of the channel coefficients is generally sufficient to compensate for the channel effect efficiently. The cycle of the UCGA operations gives the chance to adapt the current set of suggested solutions to evolving channel state. The implementation details of the UCGA are as follows. * Problem representation  
The chromosomes have the meaning of the equalizer coefficients to be optimized. They are represented by vectors containing real numbers. The data type used during the experiment is floating-point [14]. (It is noteworthy that in general one individual may comprise more than one chromosome; in the current system, the individual is determined by exactly one chromosome). In detail, the chromosome represents both real and imaginary parts of the candidate coefficients:  
\[\overset{\star}{\mathbf{c}}=\left[\hat{c}_{1}^{\Re}\cdots\hat{c}_{L}^{\Re} \,\hat{c}_{1}^{\Im}\cdots\hat{c}_{L}^{\Im}\right] \tag{3}\]  
where \(\hat{c}_{l}^{\Re},\hat{c}_{l}^{\Im}\in\mathbb{R}\) define the candidate coefficient \(\hat{c}_{l}\) of the \(l\)th equalizer tap (or more precisely, its real and imaginary part, respectively). * Initialization  
The UCGA initialization is executed in the 1 st signaling interval. Two strategies are considered. According to the first approach, all the chromosomes are generated randomly; the values of either real or imaginary parts are constrained to the range from \(-1\) to \(1\). The second strategy assumes random generation of all chromosomes (as above) with the exception for the first individual, for which \(\hat{c}_{\lfloor L/2\rfloor}^{\Re}=1\), \(\hat{c}_{l\
eq\lfloor L/2\rfloor}^{\Re}=0\) and \(\hat{c}_{l=1...L}^{\Im}=0\). Such settings would be ideal for a single-path transmission, but we expect it to boost the GA convergence even in the case of multipath Rayleigh fading channel.