# Structural-Missing Tensor Completion for Robust DOA Estimation with Sensor Failure +
## 4 Robust DOA Estimation with Array Sensor Failure
### Inverse Hankelization for Completed Covariance Tensor Derivation  
Based on the completed 6D dimensional-augmented Hankel tensor \(\mathbf{\mathcal{\hat{H}}}\), we propose the inverse Hankelization to recover a completed 3D tensor signal \(\mathbf{\mathcal{Y}}\) which serves as the equivalent signals received at an unimpaired URA \(\mathbb{U}\). The reduced dimensionality not only contributes to higher efficiency, but also facilitates the calculation of tensor statistics. Based on that, the completed covariance tensor of \(\mathbf{\mathcal{Y}}\) can be derived for DOA estimation.  
As shown in Figure 2, according to the spatio-temporal dimension augmentation of the incomplete tensor signal which involves dimensional duplication and folding, its inverse procedure can be represented as  
\[\mathbf{\mathcal{Y}}=unfold_{(\tau_{l})}(\mathbf{\mathcal{\hat{H}}})\times_{1}\mathbf{S}_ {1}^{\dagger}\times_{2}\mathbf{S}_{2}^{\dagger}\times_{3}\mathbf{S}_{3}^{\dagger}\in \mathbb{C}^{M\times N\times T}, \tag{18}\]  
where \((\cdot)^{\dagger}\) represents the Moore-Penrose pseudo-inverse operator, and \[unfold_{(\tau_{\rm{i}})}(\cdot)=fold_{(\tau_{\rm{i}})}^{-1}(\cdot) \tag{19}\]  
represents the inverse operator of \(fold_{(\tau_{\rm{i}})}(\cdot)\). Note that, the completed tensor signal \(\mathbf{\mathcal{Y}}\) is an approximate of the ideal tensor signal \(\mathbf{\mathcal{X}}\) in (3). Thus, \(\mathbf{\mathcal{Y}}\) should also conform to the CP representation as \(\mathbf{\mathcal{X}}\).  
Furthermore, the second-order tensor statistics of the URA \(\mathbb{U}\) can be derived from the completed tensor signal \(\mathbf{\mathcal{Y}}\) for DOA estimation. Specifically, we calculate the four-dimensional (4D) covariance tensor of \(\mathbf{\mathcal{Y}}\) with a CP representation as  
\[\mathbf{\mathcal{R}} =\mathbb{E}\bigg{\{}\frac{1}{T}\langle\mathbf{\mathcal{Y}},\mathbf{ \mathcal{Y}}^{*}\rangle_{3}\bigg{\}}\in\mathbb{C}^{M\times N\times M\times N} \tag{20}\] \[=\sum_{k=1}^{K}\sigma_{\tau_{\rm{k}}}^{2}\mathbf{a}(\mu_{k})\circ\mathbf{ a}(\
u_{k})\circ\mathbf{a}^{*}(\mu_{k})\circ\mathbf{a}^{*}(\
u_{k})+\sigma_{n}^{2}\mathbf{ \mathcal{I}},\]  
where \(\sigma_{\tau_{\rm{k}}}^{2}=\mathbb{E}\{s_{k}(t)s_{k}^{*}(t)\}\) denotes the power of \(k\)-th source signal, \(\sigma_{n}^{2}\) is the noise power, \(\mathbf{\mathcal{I}}\) denotes the 4D identity tensor, \((\cdot)^{*}\) denotes the conjugation operator, \(\mathbb{E}\{\cdot\}\) denotes the statistical expectation, and \(\langle\cdot,\cdot\rangle_{i}\) denotes the tensor contraction along the \(i\)-th dimension. Now, \(\mathbf{\mathcal{R}}\) can be regarded as the completed covariance tensor of the unimpaired URA \(\mathbb{U}\). Thus, we can decompose it to retrieve the angular parameters embedded in the CP factors of \(\mathbf{\mathcal{R}}\), leading to effective DOA estimation under sensor failure. The implementation of decomposition \(\mathbf{\mathcal{R}}\) is elaborated in the next subsection. In practice, the covariance tensor can be approximated by its sample version  
\[\hat{\mathbf{\mathcal{R}}}=\frac{1}{T}\langle\mathbf{\mathcal{Y}},\mathbf{ \mathcal{Y}}^{*}\rangle_{3} \tag{21}\]  
due to the finite number of snapshots.