# An Effective Off-Grid DOA Estimation Algorithm Using Difference Coarrays with Limited Snapshots
## 4 Performance Analysis  
### The CRB Derivation  
Next, we derive the CRB of the deterministically orthogonal signal model, as stated in Lemma 1. **Lemma 1**. : _Consider \(K\) far-field sources with DOAs \(\mathbf{\theta}=[\theta_{1},\dots,\theta_{K}]\) arriving at an \(N\)-element linear array. For the case of the deterministically orthogonal signal, the CRB lower bound is given by the following expression,_  
\[CRB(\mathbf{\theta}) = [\mathbf{J}_{\mathbf{\theta}\mathbf{\theta}}-\mathbf{J}_{\mathbf{ \theta}\mathbf{\theta}}P_{\mathbf{\theta}\mathbf{\theta}}^{-1}\mathbf{J}_{ \mathbf{\theta}\mathbf{\theta}}]^{-1} \tag{17}\] \[= \frac{1}{T}\Re\left[\frac{\mathbf{A}^{H}\mathbf{B}^{-1}\mathbf{A} }{\mathbf{R}_{\mathbf{\theta}}\mathbf{R}_{s}}\right],\]  
_where \(\mathbf{H}=\mathbf{\tilde{A}}^{H}\mathbf{B}^{-1}\mathbf{\tilde{A}}\mathbf{D}^ {H}\mathbf{B}^{-1}\mathbf{D}-\mathbf{D}^{H}\mathbf{B}^{-1}\mathbf{\tilde{A}} \mathbf{\tilde{A}}^{H}\mathbf{B}^{-1}\mathbf{D}\). Here, \(\mathbf{B}=(\mathbf{A}\mathbf{R}_{s}\mathbf{A}^{H})^{T}\otimes(\sigma_{n}^{2} \mathbf{I})+(\sigma_{n}^{2}\mathbf{I})\otimes(\mathbf{A}\mathbf{R}_{s}\mathbf{ A}^{H})+(\sigma_{n}^{2}\mathbf{I})\otimes(\sigma_{n}^{2}\mathbf{I})\), \(\mathbf{D}=[\mathbf{\tilde{a}}(\theta_{1}),\cdots,\mathbf{\tilde{a}}(\theta_ {K})]\) and \(\mathbf{\tilde{a}}(\theta_{i})=\frac{\partial\mathbf{a}(\theta_{i})}{\partial \theta_{i}}\). \(\Re\) is the real part of the complex variable._  
Proof. : The expected value of the vector \(\mathbf{r}\) is given by,  
\[\mathbf{m}=\mathbf{F}\mathbf{vec}(\mathbf{R})=\mathbf{F}\mathbf{\tilde{A}}( \mathbf{\theta})\mathbf{p}. \tag{18}\]The covariance matrix of the vector \(\mathbf{r}\) is,  
\[\mathbf{\Sigma} = \frac{1}{T}\{\mathbf{F}\mathbf{J}\{\mathbf{R}^{T}\otimes\mathbf{R}-[( \mathbf{AR}_{\mathbf{s}}\mathbf{A}^{H})^{T}\] \[\otimes (\mathbf{AR}_{\mathbf{s}}\mathbf{A}^{H})]\}\mathbf{J}^{H}\mathbf{F}^ {H}\} \tag{19}\] \[= \frac{1}{T}\{\mathbf{F}\mathbf{J}\mathbf{B}^{H}\mathbf{F}^{H}\}.\]  
We define the parameter vector \(\mathbf{\xi}=[\theta,\mathbf{p}^{T}]^{T}\), which concatenates the DOAs and the signal powers of the \(K\) sources. Considering the concatenated parameter vector \(\mathbf{\xi}=[\theta,\mathbf{p}^{T}]^{T}\), the Fisher information matrix (FIM) admits the representation,  
\[\mathbf{J}(\mathbf{\xi})=\left[ \begin{array}{cc}\mathbf{J}_{\theta\theta}& \mathbf{J}_{\theta\mathbf{p}}\\ \mathbf{J}_{\theta\mathbf{p}}^{T}&\mathbf{J}_{\mathbf{p}\mathbf{p}}\end{array}  \right], \tag{20}\]  
where \(\mathbf{J}_{\theta\mathbf{p}}^{T}=\mathbf{J}_{\mathbf{p}\theta}\). Each component in FIM is determined by the following formula,  
\[[\mathbf{J}]_{ij}=\frac{1}{2}\left\{\mathbf{tr}\left[\mathbf{\Sigma}^{-1} \frac{\partial\mathbf{\Sigma}}{\partial\xi_{i}}\mathbf{\Sigma}^{-1}\frac{ \partial\mathbf{\Sigma}}{\partial\xi_{j}}\right]\right\}+\Re\left[\frac{ \partial\mathbf{m}^{H}}{\partial\xi_{i}}\mathbf{\Sigma}^{-1}\frac{\partial \mathbf{m}}{\partial\xi_{j}}\right], \tag{21}\]  
where \([\mathbf{J}]_{ij}\) refers to the entry located at the \(i\)-th row and \(j\)-th column of the matrix.