## 4 Simulation Results  
### Parameter Setting  
The performance of the proposed HGA-based approach was evaluated by simulations in an MIMO system with \(N_{t}=4\) transmit antennas, \(N_{r}=4\) receive antennas and 64-QAM, 256-QAM, and 1024-QAM modulation. The channel entries are assumed to be i.i.d. complex Gaussian random variables and we assume perfect channel knowledge at the receiver.  
An important issue for GA-based detection is the choice of various parameters for the genetic operations. The parameters to be determined for the basic GA are the maximum number of generations \(G\), the population size \(P\), and the probability that each genetic operation is selected; and the parameters to be determined for the proposed HGA are the maximum number of generations \(G\), the number of nearest neighbors \(N\), \(C_{r}\), and \(M_{u}\). Unlike the basic GA, the population size of the proposed HGA is changed according to the result of the _selection_ operation for each generation, and is determined as follows  
\[\mathbf{P_{g}}=\begin{cases}1+C_{r}+(2N_{t}\cdot N\cdot M_{u}),&\text{for } \mathbf{s}_{g}=\mathbf{s}_{g-1}\\ 1+C_{r}+(2N_{t}\cdot N\cdot(M_{u}+1)),&\text{for }\mathbf{s}_{g}\
eq\mathbf{s}_{g-1} \end{cases} \tag{13}\]  
where \(\mathbf{s}_{g}\) is the individual selected by the _selection_ operation in the \(g^{\text{th}}\) generation. If the population size is too small, there is insufficient evolution, and there is a risk of the whole population becoming extinct. This will eventually drive the system into a local optimum (not the global optimum). Conversely, if the population size is too large, more evaluations of fitness function are needed, which will require extensive computing time. Therefore, we should determine the appropriate GA parameters to increase BER performance and reduce computational complexity.  
Unfortunately, there is no known method for determining GA parameters, so we need to make a heuristic decision. Figure 6 shows BER performance and the number of vectors to be searched in the HGA process for various parameters in a \(4\times 4\) MIMO with 64-QAM at a signal-to-noise ratio (SNR) of 30 dB. In this paper, we use the number of vectors that should be searched by the HGA as a measure of complexity. From Figure 6a, BER performance is improved as the \(G\) and \(C_{r}\) increase, but \(M_{u}\) can not significantly affect the BER performance. However, in Figure 6b, the complexity of the HGA process increases sharply as \(M_{u}\) increases, whereas the change in \(G\) and \(C_{r}\) slowly increase the complexity of the HGA. From these simulation results, we can select the HGA parameters with lower complexity and better BER performance. The selected parameters according to the modulation order for a \(4\times 4\) MIMO system are summarized in Table 1. In addition, for fair performance comparisons, the parameters for the basic GA have been chosen to have a similar population structure to that of the HGA, and are listed in Table 2.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **64-QAM** & **256-QAM** & **1024-QAM** \\ \hline \(G\) & 100 & 100 & 200 \\ \hline \(N\) & 2 & 2 & 2 \\ \hline \(C_{r}\) & 50 & 100 & 200 \\ \hline \(M_{u}\) & 20 & 50 & 100 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the proposed HGA.