# Multi-User Detection Based on Improved Cheetah Optimization Algorithm
## 2 Methods  
### Robust Multi-User Detection Model  
In complex channel environments, conventional multi-user detectors are not resistant to the adverse effects of disturbances such as impact noise on the detector and thus require a robust and improved design of the detector.  
Assuming the system has \(P\) users with a pseudo-code period of \(T\) for each user, the multi-user detection model for a conventional CDMA communication system is  
\[y(t)=\sum_{i=1}^{P}\sum_{i=-M}^{M}A_{i}B_{i}[i]s_{i}(t-iT-\tau_{i})+\sigma n(t) \tag{1}\]  
where \(A_{i}\) represents the amplitude, \(B_{i}[i]\) denotes a sequence of information consisting of 1 or \(-1\), \(n(t)\) represents the channel noise, \(\tau_{i}\in[0,T)\) is the size of the signal time delay for the \(l\) user, and \(s_{l}(t)\) denotes the pseudo-code sequence. Assuming the sampling period is equal to the width of a single code slice, the system output after sampling can be expressed as [13]  
\[y=\mathbf{SAB}+\mathbf{\delta n} \tag{2}\]  
where \(\mathbf{S}=\left[\mathbf{s}_{1}\,\mathbf{s}_{2}\,\ldots\,\mathbf{s}_{P}\right]\), \(\mathbf{A}=\mathit{diag}(A_{1},A_{2},\ldots,A_{P})\), \(\mathbf{B}=\left[B_{1},\ldots,B_{P}\right]^{T}\), \(\mathbf{n}\) is the impact noise vector [13].  
Non-Gaussian wireless transmission channels containing impact noise are commonly modeled with a steady distribution model, and the eigenfunction of the steady distribution model is denoted as  
\[\varphi(t)=\left\{\begin{matrix}\exp\{jut-y|t|^{\alpha}\big{[}1+j\beta\text{ sgn}(t)\tan(\frac{\alpha\pi}{2})\big{]}\},\alpha\
eq 1\\ \exp\{jut-y|t|^{\alpha}\big{[}1+j\beta\text{sgn}(t)\frac{\alpha}{2}\log[t]\big{]} \},\alpha=1\end{matrix}\right. \tag{3}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Abbreviation** & **Full From** \\ \hline HCO & Hybrid Cheetah Optimizer \\ CO & Cheetah Optimizer \\ BER & Bit Error Rate \\ NP & Non-deterministic Polynomial-time \\ DE & Differential Evolution \\ GA & Genetic Algorithm \\ GWO & Grey Wolf Optimization \\ GOA & Grasshopper Optimization Algorithm \\ WOA & Whale Optimization Algorithm \\ \hline \hline \end{tabular}
\end{table}
Table 1: A table of acronyms.  
where \(\alpha\in(0,2]\) is the eigenvector, the symmetry parameter is denoted by \(\beta\in[-1,1]\), the dispersion coefficient is denoted by \(\gamma>0\), and the location coefficient is denoted by \(-\infty<u<\infty\)[24].  
According to the literature [25], robust multi-user detection is obtained:  
\[\hat{b}=\underset{b}{\text{argmin}}\sum_{g=1}^{M}\rho\left(y_{g}-\sum_{l=1}^{ p}s_{gl}A_{l}B_{l}\right) \tag{4}\]  
The Huber M estimator's penalty function is formulated as  
\[\rho\big{(}x_{g}\big{)}=\begin{cases}\frac{x_{g}^{2}}{2z_{g}},\ \left(\left|x_{g} \right|\leq\epsilon_{g}z_{g}^{2},\epsilon_{g}=\frac{3}{2z_{g}}\right)\\ \epsilon_{g}\left|x_{g}^{2}\right|-\frac{\epsilon_{g}z_{g}^{2}}{2},\ \left|x_{g} \right|>\epsilon_{g}z_{g}^{2}\end{cases} \tag{5}\]  
where \(\gamma_{g}=\sqrt{2}z_{g}\), \(\gamma_{g}\) denotes the noise divergence. Equation (4) is the robust multi-user detection expression; the detector can detect the abnormal wild values of the signal and effectively limit the magnitude size of the wild values, which plays an important role in shock resistance without destroying the original data.