## 3 Methods
### Complex Parameter Wald Test
#### 3.2.3 The Complex Wald Test for Testing Bandedness  
This section delves into deriving the complex Wald test statistic for the aforementioned problem of testing bandedness. Recall that  
\[\ln p(\mathbf{X};\underline{\mathbf{b}})=\ln(\frac{1}{\pi^{NL}})-\sum_{n=0}^{ N-1}\mathbf{x}_{n}^{H}\mathbf{D}^{H}\mathbf{D}\mathbf{x}_{n}+N\ln\det( \mathbf{D}^{H}\mathbf{D}), \tag{42}\]  
and  
\[\begin{split}\frac{\partial\ln p(\mathbf{X};\underline{\mathbf{b }})}{\partial b_{k}}&=N\frac{\partial\ln\det(\mathbf{D}^{H} \mathbf{D})}{\partial b_{k}}-\sum_{n=0}^{N-1}\frac{\partial\mathbf{x}_{n}^{H} \mathbf{D}^{H}\mathbf{D}\mathbf{x}_{n}}{\partial b_{k}}\\ &=N\frac{\partial\ln\det(\mathbf{D}^{H}\mathbf{D})}{\partial b_{k }}-\sum_{n=0}^{N-1}\frac{\partial\text{tr}(\mathbf{D}\mathbf{x}_{n}\mathbf{x}_ {n}^{H}\mathbf{D}^{H})}{\partial b_{k}},\end{split} \tag{43}\]  
for \(k=1,2,\ldots,M\), where  
\[\frac{\partial\ln\det(\mathbf{D}^{H}\mathbf{D})}{\partial b_{k}}=\text{tr}( \mathbf{D}^{-1}\mathbf{\Phi}_{k}), \tag{44}\]  
and  
\[\frac{\partial\text{tr}(\mathbf{D}\mathbf{x}_{n}\mathbf{x}_{n}^{H}\mathbf{D} ^{H})}{\partial b_{k}}=\text{tr}(\mathbf{x}_{n}\mathbf{x}_{n}^{H}\mathbf{D} ^{H}\mathbf{\Phi}_{k}). \tag{45}\]  
We aim to determine the Maximum Likelihood Estimates (MLEs) of \(b_{k}\). Considering that \(b_{k}\) are relatively small, we are interested in testing whether they are equal to zero, and we approximate \(D^{-1}\) as \(D_{B}^{-1}\).  
\[\begin{split}\frac{\partial\ln p(\mathbf{X};\underline{\mathbf{b }})}{\partial b_{k}}&=N\text{tr}(\mathbf{D}^{-1}\mathbf{\[PERSON]}_{k})- \sum_{n=0}^{N-1}\text{tr}(\mathbf{x}_{n}\mathbf{x}_{n}^{H}\mathbf{D}^{H}\mathbf{ \[PERSON]}_{k})\\ &=N\text{tr}(\mathbf{D}_{b}^{-1}\mathbf{\[PERSON]}_{k})-\sum_{n=0}^{N-1} \text{tr}(\mathbf{x}_{n}\mathbf{x}_{n}^{H}(\mathbf{D}_{b}+\sum_{i=1}^{M}b_{i} \mathbf{\[PERSON]}_{i})^{H}\mathbf{\[PERSON]}_{k})\\ &=N\text{tr}(\mathbf{D}_{b}^{-1}\mathbf{\[PERSON]}_{k})-\sum_{n=0}^{N-1} \text{tr}(\mathbf{x}_{n}\mathbf{x}_{n}^{H}\mathbf{D}_{b}^{H}\mathbf{\[PERSON]}_{k})-b_{k }^{*}\sum_{n=0}^{N-1}\text{tr}(\mathbf{x}_{n}\mathbf{x}_{n}^{H}\mathbf{\[PERSON]}_{k} ^{H}\mathbf{\[PERSON]}_{k})\end{split} \tag{46}\]  
therefore\[\dot{b}_{k}^{\star}=\frac{N\text{tr}(\mathbf{D}_{b}^{-1}\mathbf{\[PERSON]}_{k})-\sum_{n= 0}^{N-1}\text{tr}(\mathbf{x}_{n}\mathbf{x}_{n}^{H}\mathbf{D}_{b}^{H}\mathbf{ \[PERSON]}_{k})}{\sum_{n=0}^{N-1}\text{tr}(\mathbf{x}_{n}\mathbf{x}_{n}^{H}\mathbf{ \[PERSON]}_{k}^{H}\mathbf{\Phi}_{k})} \tag{47}\]  
Up to this point, we have obtained the MLEs of unknown parameters under the alternative hypothesis. Computation of these MLEs constitutes extra computational cost in the Wald test compared to the Rao test. To put it simply, for each additional unknown parameter, the Wald test requires roughly three times the workload of \(N\times N\) matrix multiplications in contrast to the Rao test.  
Upon substituting the MLEs into both the complex FIM equation and the complex Wald test equation in (31), we derive the Wald test statistic. It is noteworthy that given the presence of the unknown parameter within the covariance matrix, the conditions required for the application of the specialized complex Wald test theorem, as delineated in [19], are not satisfied in this case due to the absence of the requisite special form in the FIM.