## 2 Materials and Methods
### Proposed Joint CE and Turbo MUD
#### 2.2.2 Cat Swarm Optimization (CSO) Based Channel Estimation (CE)  
In MIMO-OFDM system, the pilot pattern can be optimized by minimizing \(\mu\left\{h_{q_{x}}\right\}\) to enhance the execution of the system. Due to the existence of strong interference, conventional methods (minimum mean-square error (MMSE) and least-square (LS)) has poor system performance. The pilot sequence becomes contaminated because of MAI. The performance of the channel is also generally limited due to pilot tone selection. To improve the performance, an optimal pilot pattern selection using the cat swarm optimization (CSO) algorithm is applied and thus, it improves the accuracy of sparse CE. During CE, the pilots are optimally selected using the CSO algorithm with the fitness of delay and phase, which reduces the mutual coherence. The two modes collected in it are the tracking and seeking modes. This algorithm identifies the global solutions compared to another existing optimization algorithm. We assumed that all the model parameters of the channel are associated with both the transmitter and receiver. Therefore, no feedback channel is necessary to define its error variance and no feedback is required for the encoder. Algorithm 1 describes the CSO based optimal pilot pattern for reducing the largest element in mutual coherence. The basic idea behind the search algorithm is to select the suboptimal pilot pattern of minimum mutual coherence from the set of pilot indices \(\Omega=\{1,2,\ldots,N\}\). To reduce the effect of multi-access interference (MAI), the active users should be continuously detected. For this activity detection, the pilot and data symbols are considered. In the measurement matrix calculation of optimal pilot placement, the optimal pilot allocation method is suggested to select the minimum mutual coherence. Therefore, for MUD, the mutual coherence should be minimized. The major objective is to minimize the mutual coherence \(\mu\{h_{q_{x}}\}\) during pilot allocation in MIMO-OFDM systems. \[Minimal\sum_{q_{x}=1}^{N_{x}}\left\|\hat{h}_{q_{x}}-h_{q_{x}}\right\|_{2}^{2} \tag{14}\]  
In order to tackle the optimization problem, the CSO optimization algorithm is utilized, which optimally selects the minimal value and hence, the system attains CE. We designated the set of pilot locations \(\Psi_{x}=z_{1},z_{2},\cdots z_{p}\) and the set of pilot distances as \(\phi=\phi_{1},\phi_{2},\cdots\phi_{p-1}\) with \(\phi_{i}=z_{i+1}-z_{i}\). The first step produces the pilot patterns \(\Psi_{x}=k_{x,1},k_{x,2},\cdots k_{x,p}\) and the mutual coherence \(\mu_{1},\ \mu_{2},\cdots,\mu_{q_{x}}\), but all the elements in the \(\mu_{1},\ \mu_{2},\cdots,\mu_{q_{x}}\) cannot achieve all their minimum levels at the same time. We estimated the fitness value for each \(N\) cat and the best fitness value of cat acts as the global best (gbest). The fitness for the individual function can be defined as:  
\[f(\varphi)=\left\{ \begin{array}{ll}\frac{1}{\mu_{\varphi}},&\text{if }0.6N\leq z_{p}\\ \frac{1}{1000},&\text{else}\end{array} \right. \tag{15}\]  
If the condition of \(z_{p}\leq 0.6N\) is satisfied, this points to a higher mutual coherence, while \(z_{p}\) specifies the pilot symbols collected from all subcarriers. ```
Initialization. Let different parameters for the population size be \(Ps=100\), the length of all the individuals is Len = \(P-1\) and the maximum generation is Mg. Create the initial population randomly \(\Phi i\), where \(i=1\), \(2\), \(\ldots\)\(Ps\). After this, compute the fitness for the initial population individually. 1: Produce \(N\)-cats, which represents pilot symbols. 2: Cats have \(M\)-dimensional space and arbitrarily gives a range of values for the maximum velocity of each cat. 3: By applying the position of cats in the fitness function, evaluate the fitness value for each cat. It represents the finest position of the cat (xbest) by calculating the mutual coherence of each pilot sequence. 4: Move the cat according to their modes, apply the process of the seeking mode if the cat is in seeking mode.