# Channel Estimation Algorithm Based on Parrot Optimizer in 5G Communication Systems
## 2 Channel Estimation Model
### MMSE-Based Channel Estimation Model  
Compared to LS, a more advanced channel estimation algorithm is MMSE, which utilizes channel statistical knowledge. In this algorithm, the LS estimate \(h_{LS}\) is multiplied by a weight matrix \(W\), providing the MMSE estimate \(h_{MMSE}\) as follows:  
\[h_{MMSE}=Wh_{LS} \tag{15}\]  
Considering the relationship between the LS estimate \(h_{LS}\) and the actual channel \(h\), it can be expressed as  
\[h_{LS}=h+e \tag{16}\]  
where \(e\) is the estimation error.  
The goal of MMSE estimation is to minimize the mean square error (MSE), which is  
\[\text{MSE}=E\{\|h_{MMSE}-h\|^{2}\} \tag{17}\]  
We aim to find a weight matrix \(W\) such that the weighted LS estimate \(Wh_{LS}\) is as close as possible to the true channel \(h\). The formula expands to  
MSE \[=E\{(Wh_{LS}-h)^{H}(Wh_{LS}-h)\}\] (18) \[=E\{(Wh_{LS})^{H}(Wh_{LS})-(Wh_{LS})^{H}h-h^{H}(Wh_{LS})+h^{H}h\}\]Using the linearity property of expectation, we can obtain  
\[\text{MSE}=E\{\left(Wh_{LS}\right)^{H}\left(Wh_{LS}\right)\}-E\{\left(Wh_{LS} \right)^{H}h\}-E\{h^{H}\left(Wh_{LS}\right)\}+E\{h^{H}h\} \tag{19}\]  
Simplifying the expectation terms. The first term is  
\[E\{\left(Wh_{LS}\right)^{H}\left(Wh_{LS}\right)\}=E\{h_{LS}^{H}W^{H}Wh_{LS}\}\] \[=\text{tr}(W^{H}WE\{h_{LS}h_{LS}^{H}\}) \tag{20}\] \[=\text{tr}(W^{H}WR_{LS})\]  
where \(R_{LS}=E\{h_{LS}h_{LS}^{H}\}\).  
For the second term, we obtain  
\[E\{\left(Wh_{LS}\right)^{H}h\}=E\{h_{LS}^{H}W^{H}h\}=\text{tr}(W^{H}E\{h_{LS} h^{H}\})=\text{tr}(W^{H}R_{hp}) \tag{21}\]  
The third term is  
\[E\{h^{H}\left(Wh_{LS}\right)\} =E\{h^{H}Wh_{LS}\}\] \[=\text{tr}(WE\{h_{LS}h^{H}\}) \tag{22}\] \[=\text{tr}(WR_{hp}^{H})\]  
The fourth term is  
\[E\{h^{H}h\}=\text{tr}(E\{hh^{H}\})=\text{tr}(R_{pp}) \tag{23}\]  
Combining all terms into the MSE expression, we obtain  
\[\text{MSE}=\text{tr}(W^{H}WR_{LS})-\text{tr}(W^{H}R_{hp})-\text{tr}(WR_{hp}^{ H})+\text{tr}(R_{pp}) \tag{24}\]  
Taking the derivative with respect to \(W\) and setting it to zero is expressed as  
\[\frac{\partial\text{MSE}}{\partial W}=0 \tag{25}\]  
We obtain  
\[\frac{\partial}{\partial W}\text{tr}(W^{H}WR_{LS})=2 WR_{LS}\] \[\frac{\partial}{\partial W}\text{tr}(W^{H}R_{hp})=R_{hp} \tag{26}\] \[\frac{\partial}{\partial W}\text{tr}(WR_{hp}^{H})=R_{hp}^{H}\]  
Solving for \(W\), we obtain  
\[2 WR_{LS}-R_{hp}-R_{hp}^{H}=0 \tag{27}\]  
Since \(R_{hp}\) is a covariance matrix, we can denote \(R_{hp}^{H}=R_{hp}\). Therefore, the above equation becomes  
\[2 WR_{LS}-2R_{hp}=0 \tag{28}\]  
Solving this equation, we obtain  
\[WR_{LS}=R_{hp} \tag{29}\]  
\[W=R_{hp}R_{LS}^{-1} \tag{30}\]  
Assuming \(h_{LS}=h+e\),  
\[R_{LS}=E\{h_{LS}h_{LS}^{H}\}=E\{\left(h+e\right)\left(h+e\right)^{H}\}=R_{pp}+ R_{ee}, \tag{31}\]assuming \(R_{ex}\) is the covariance matrix of the error term and \(e\) is white noise with a covariance matrix of \(R_{ex}=\dfrac{\beta}{\text{SNR}}I\). Specifically, \(\beta\) represents the Signal-to-Noise Ratio (SNR) scaling factor, which is used to adjust the covariance matrix \(R_{ex}\) of the error term in the MMSE-based channel estimation model. This factor is crucial for correctly modeling the impact of noise on the channel estimation process:  
\[R_{LS}=R_{pp}+\dfrac{\beta}{\text{SNR}}I \tag{32}\]  
Substituting \(R_{LS}\) into the expression for the weight matrix \(W\), we obtain  
\[W=R_{hp}(R_{pp}+\dfrac{\beta}{\text{SNR}}I)^{-1} \tag{33}\]