# Channel Estimation Algorithm Based on Parrot Optimizer in 5G Communication Systems
## 2 Channel Estimation Model
### PO-Based Channel Estimation Model  
Optimization problems have always garnered significant attention due to their critical role in mathematics and various scientific and technical fields. Numerous algorithms, such as dynamic programming and branch and bound, can provide globally optimal solutions for all nonlinear problems. Due to their efficiency and global characteristics, meta-heuristic optimization algorithms inspired by biological or physical phenomena have gained popularity in technical applications. Population-based techniques that mimic the behavior of animal groups have shown remarkable performance in evolutionary algorithms and are well-suited for optimization needs in high-speed communication applications. As an effective contemporary research technique, stochastic optimization methods have shown outstanding performance in solving complex optimization challenges. Reference [38] introduced the Parrot Optimizer (PO), an optimization method inspired by the behavior of trained Pyrrhura Molinae parrots. The study presented qualitative analysis and comprehensive experiments to demonstrate PO's unique characteristics in handling various optimization problems. Performance evaluations included benchmarking on 35 functions, covering classical cases and issues from the IEEE CEC 2022 test set, and comparisons with eight popular algorithms, highlighting PO's competitive advantage in exploration and exploitation [38]. The mathematical model of the PO in this paper is discussed following the work of [PERSON] et al. in [38]. Considering the population size \(N\), the maximum number of iterations \(\text{Max}_{\text{iter}}\), and the upper and lower bounds of the search space, \(\text{ub}\) and \(\text{lb}\), respectively, the initialization formula for PO can be expressed as  
\[X_{i}^{0}=\text{lb}+\text{rand}(0,1)\cdot(\text{ub}-\text{lb}) \tag{34}\]  
where \(\text{rand}(0,1)\) represents a random number in the range \([0,1]\), and \(X_{i}^{0}\) represents the position of the \(i\)-th parrot at the initial stage. During the foraging behavior of PO, parrots primarily estimate the approximate location of food by observing the position of the food or considering the owner's position, then fly to the corresponding position. The movement of the position follows the equation  
\[X_{i}^{t+1}=(X_{i}^{t}-X_{\text{best}})\cdot\text{Levy}(\text{dim})+\text{rand }(0,1)\cdot\left(1-\dfrac{t}{\text{Max}_{\text{iter}}}\right)^{\frac{2t}{ \text{Max}_{\text{iter}}}}\cdot X_{\text{mean}}^{t} \tag{35}\]  
In Equation (35), \(X_{i}^{t}\) represents the current position, and \(X_{i}^{t+1}\) represents the next updated position. \(X_{\text{mean}}^{t}\) denotes the average position of the current population, and \(\text{Levy}(\text{dim})\) denotes the Levy distribution [39], which is used to describe the parrot's flight. \(X_{\text{best}}\) represents the best position found from initialization to the current search, and it also represents the current position of the owner. \(t\) represents the current iteration number. The term \((X_{i}^{t}-X_{\text{best}})\cdot\text{Levy}(\text{dim})\) indicates movement based on the individual's position relative to the owner's position. The term \(\text{rand}(0,1)\cdot\left(1-\dfrac{t}{\text{Max}_{\text{iter}}}\right)^{\frac{ 2t}{\text{Max}_{\text{iter}}}}\cdot X_{\text{mean}}^{t}\) represents determining the direction of the food by observing the position of the entire population. The average position of the current population \(X_{\text{mean}}^{t}\) is calculated using the formula  
\[X_{\text{mean}}^{t}=\dfrac{1}{N}\sum_{k=1}^{N}X_{k}^{t} \tag{36}\]  
The Levy distribution can be obtained from the formula, with \(\gamma\) typically valued at 1.5. \[\left\{ \begin{array}{ll}\text{Levy}(\text{dim})=\frac{\mu\cdot\sigma}{\left|v \right|^{1/\gamma}}\\ \mu\,\sim N(0,\text{dim})\\ v\,\sim N(0,\text{dim})\\ \sigma=\left(\dfrac{\text{r}(1+\gamma)\cdot\sin\left(\dfrac{\pi\gamma}{2} \right)}{\text{r}\left(\dfrac{1+\gamma}{2}\right)\cdot\text{$\gamma$}^{2(1+ \gamma)/2}}\right)^{\gamma+1}\\ \end{array} \right.