## 4 Simulations, Results and Discussion  
### Simulations and Result Discussion on Complex Rao and Wald Tests for Bandedness Testing  
Consider an illustrative example, where we have the \(N=4\) observed data set \(\mathbf{X}=[\mathbf{x}_{0}^{T}\ \mathbf{x}_{1}^{T}\ \mathbf{x}_{2}^{T}\ \mathbf{x}_{3}^{T}]^{T}\), each \(\mathbf{x}_{n}\)'s is a \(4\times 1\) complex-valued IID Gaussian vector, \(\mathbf{x}_{n}\sim\mathcal{CN}(\mathbf{0},\mathbf{C})\). Also, \(\mathbf{C}^{-1}=\mathbf{D}^{H}\mathbf{D}\), and \(\mathbf{D}=\mathbf{D}_{B}+b_{1}\mathbf{\Phi}_{1}\) with \(\mathbf{\Phi}_{1}=\mathbf{e}_{4}\mathbf{e}_{1}^{T}\) and  
\[\mathbf{D}_{B}=\left[ \begin{array}{cccc}0.53&0&0&0\\ -0.26+0.25j&0.53&0&0\\ -0.12+0.1j&-0.33+0.28j&0.51&0\\ 0&-0.17-0.10j&0.2-0.27j&0.5\end{array} \right] \tag{48}\]  
We are testing whether the Cholesky factor matrix \(\mathbf{D}\) is banded and equal to the known \(\mathbf{D}_{B}\). It is equivalent to testing between \(b_{1}=0\) versus \(b_{1}\
eq 0\). The complex Rao test for this example can be shown to be (49)  
\[T_{R}(\mathbf{X}) = \frac{\text{Re}\left\{\left(4\text{tr}(\mathbf{D}_{B}^{-1} \mathbf{\Phi}_{1})-\sum_{n=0}^{3}\text{tr}(\mathbf{x}_{n}\mathbf{x}_{n}^{H  
The ROC curves of the derived complex Rao and Wald test is given in Figure 2.  
The results demonstrate that both the complex Rao and Wald tests exhibit commendable performance when confronted with limited data records. Notably, the complex Wald test outperforms the complex Rao test, aligning with expectations given that the latter generally demonstrates suboptimal performance. This performance disparity is expected considering that the complex Rao test, while less computationally intensive, as it does not necessitate the MLEs of the unknown parameters under \(\mathcal{H}_{1}\), inherently delivers slightly inferior performance. A second illustrative example is presented, where \(b_{1}=0.3+0.3j\), representing a more challenging scenario compared to the earlier instance. Figure 3 showcases the performance of both proposed detectors in this setting. It is evident that in comparison to the prior example, the detectors' performance declines due to the smaller magnitude of \(b_{1}\). Notably, the complex Wald test exhibits a slight edge over the complex Rao test in this more challenging scenario. Figure 1: Illustration of Monte Carlo Simulation Steps for each simulation setup. Figure 2: ROC curve of the complex Rao and Wald test detector with \(b_{1}=0.5+0.5j\). Next, we increase the number of IID samples from \(N=4\) to \(N=10\) and set the \(b_{1}=0.3+0.3j\).The comparative performance of the two detectors is depicted in Figure 4. A comparative analysis with Figure 3 reveals a conspicuous enhancement in the performance of both detectors, attributable to the increased availability of data samples. Notably, it is discernible that in this scenario, the performance of the two detectors converges significantly, indicating that the Complex Rao and Wald tests exhibit asymptotic equivalence as the dataset size grows. Both the Complex Rao test statistic and Complex Wald test under the null hypothesis \(\mathcal{H}_{0}\) are chi-squared distributed with one degree of freedom, \(T_{R}(\mathbf{X})\sim\chi_{1}^{2}\)[28]. The performance of the Rao test and Wald test can be found asymptotically or as \(N\to\infty\)  
Figure 4: ROC curve of the complex Rao and Wald test detector with \(b_{1}=0.3+0.3j\) and \(N=10\). Figure 3: ROC curve of the complex Rao and Wald test detector with \(b_{1}=0.3+0.3j\). Estimated probability density function (PDFs), shown as bar plots, for both the Rao test and the Wald test, and the theoretical asymptotic \(\chi_{1}^{2}\) PDF are shown in Figure 5. Notably, even with a limited data record of \(N=4\), the estimated PDFs remarkably align with the theoretical distribution. The detectors' performance is also dependent on the base matrix \(\mathbf{D}_{B}\). Next, we double the magnitude of each element of \(\mathbf{D}_{B}\); that is,  
\[\mathbf{D}_{B}=\left[