## 4 Simulations, Results and Discussion
### Equivalence Among Complex Wald, Rao Test and GIRT for Linear Model
#### 4.2.5 The Root Cause of the Equivalence  
First,  
\[\frac{\partial\ln p_{\hat{\kappa}}(\hat{\kappa};\bar{\theta})}{\partial\bar{ \theta}^{*}}=\bar{\mathbf{I}}(\bar{\theta})(\hat{\bar{\theta}}-\bar{\theta}) \tag{56}\]  
where \(\bar{\theta}\) is the true value. Because the complex FIM is not dependent on the true value of the unknown parameters, \(\bar{\mathbf{I}}(\bar{\theta})=\bar{\mathbf{I}}(\hat{\bar{\theta}}_{1})\), we have  
\[\frac{\partial\ln p_{\hat{\kappa}}(\hat{\kappa};\bar{\theta})}{\partial\bar{ \theta}^{*}}=\bar{\mathbf{I}}(\hat{\bar{\theta}}_{1})(\hat{\bar{\theta}}- \bar{\theta}) \tag{57}\]  
So, by integrating with respect to \(\bar{\theta}^{*}\), it produces  
\[\ln p_{\hat{\kappa}}(\hat{\kappa};\bar{\theta})=-(\hat{\bar{\theta}}-\bar{ \theta})^{H}\bar{\mathbf{I}}(\hat{\bar{\theta}})(\hat{\bar{\theta}}-\bar{ \theta})+c(\hat{\bar{\theta}}) \tag{58}\]  
since the constant of integration must be \(c(\hat{\bar{\theta}})=\ln p_{\hat{\kappa}}(\hat{\kappa};\hat{\bar{\theta}})\).  
The GLRT becomes  
\[L_{G}(\hat{\kappa})=\frac{p(\hat{\kappa};\hat{\bar{\theta}}_{1})}{p(\hat{ \kappa};\bar{\theta}_{0})}=\frac{p(\hat{\kappa};\hat{\bar{\theta}}_{1})}{p( \hat{\kappa};\hat{\bar{\theta}}_{1})\exp\Bigl{[}-(\hat{\bar{\theta}}_{1}-\bar {\theta}_{0})^{H}\bar{\mathbf{I}}(\hat{\bar{\theta}}_{1})(\hat{\bar{\theta}}_ {1}-\bar{\theta}_{0})\Bigr{]}} \tag{59}\]  
Thus,  
\[2\ln L_{G}(\hat{\kappa})=2(\hat{\bar{\theta}}_{1}-\bar{\theta}_{0})^{H} \mathbf{I}(\hat{\bar{\theta}}_{1})(\hat{\bar{\theta}}_{1}-\bar{\theta}_{0}) \tag{60}\]  
This explains why the GLRT and the complex Wald test coincide for the complex linear model. Furthermore, to establish the equivalence between the complex Rao test and the Generalized Likelihood Ratio Test (GLRT), we use Equation (56)  
\[(\hat{\bar{\theta}}-\bar{\theta})=\bar{\mathbf{I}}^{-1}(\bar{\theta})\frac{ \partial\ln p_{\hat{\kappa}}(\hat{\kappa};\bar{\theta})}{\partial\bar{\theta} ^{*}} \tag{61}\]  
Specially,  
\[(\hat{\bar{\theta}}_{1}-\bar{\theta}_{0})=\bar{\mathbf{I}}^{-1}(\bar{\theta} _{0})\frac{\partial\ln p_{\hat{\kappa}}(\hat{\kappa};\bar{\theta})}{\partial \bar{\theta}^{*}}\bigg{|}_{\bar{\theta}=\bar{\theta}_{0}} \tag{62}\]  
By substituting (62) to (60), we have  
\[2\ln L_{G}(\hat{\kappa}) =2\Biggl{(}\bar{\mathbf{I}}^{-1}(\bar{\theta}_{0})\frac{\partial \ln p_{\hat{\kappa}}(\hat{\kappa};\bar{\theta})}{\partial\bar{\theta}^{*}} \bigg{|}_{\bar{\theta}=\bar{\theta}_{0}}\Biggr{)}^{H}\bar{\mathbf{I}}(\bar{ \theta}_{1})\Biggl{(}\bar{\mathbf{I}}^{-1}(\bar{\theta}_{0})\frac{\partial\ln p _{\hat{\kappa}}(\hat{\kappa};\bar{\theta})}{\partial\bar{\theta}^{*}}\bigg{|} _{\bar{\theta}=\bar{\theta}_{0}}\Biggr{)} \tag{63}\] \[=2\frac{\partial\ln p_{\hat{\kappa}}(\hat{\kappa};\bar{\theta})}{ \partial\bar{\theta}^{*}}\bigg{|}_{\bar{\theta}=\bar{\theta}_{0}}\bar{\mathbf{I }}^{-1}(\bar{\theta}_{0})\frac{\partial\ln p_{\hat{\kappa}}(\hat{\kappa};\bar{ \theta})}{\partial\bar{\theta}^{*}}\bigg{|}_{\bar{\theta}=\bar{\theta}_{0}}\]  
where we have used the property of \(\bar{\mathbf{I}}(\bar{\theta})=\bar{\mathbf{I}}(\bar{\theta}_{0})=\bar{\mathbf{I }}(\hat{\bar{\theta}}_{1})\). This completes the proof that the complex Rao test and the GLRT are equivalent for the complex linear model. Consequently,it also proves the equivalence among the aforementioned three detectors for the complex linear model. In summary, the equivalence stems from the fact that the complex FIM of the linear model \(\tilde{\mathbf{I}}(\tilde{\mathbf{\theta}})=\tilde{\mathbf{H}}^{H}\tilde{\mathbf{C}}^ {-1}\tilde{\mathbf{H}}\) does not depend on the true value of \(\tilde{\mathbf{\theta}}\). In particular, we have \(\tilde{\mathbf{I}}(\tilde{\mathbf{\theta}})=\tilde{\mathbf{I}}(\tilde{\mathbf{\theta}} _{0})=\tilde{\mathbf{I}}(\tilde{\mathbf{\theta}}_{1})\), and hence the complex Rao test and Wald test are both equivalent to the GLRT without extra constraints.