# Structural-Missing Tensor Completion for Robust DOA Estimation with Sensor Failure +
## 4 Robust DOA Estimation with Array Sensor Failure
### Covariance Tensor CPD for Robust DOA Estimation  
By decomposing the covariance tensor \(\mathbf{\mathcal{R}}\) to retrieve the estimated steering vectors \(\mathbf{a}(\hat{\mu}_{k})\) and \(\mathbf{a}(\hat{\
u}_{k})\), the estimated angular projection factors \(\hat{\mu}_{k}\) and \(\hat{\
u}_{k}\) can be obtained, leading to the closed-form solution of both azimuth and elevation. Considering that \(\mathbf{\mathcal{R}}\) is represented in a CP model, we apply the CPD to \(\mathbf{\mathcal{R}}\) by solving the following least squares problem  
\[\min_{\hat{A}(\mu),\hat{A}(\
u)}\big{\|}\mathbf{\mathcal{R}}-\mathbf{ \mathcal{J}}\times_{1}\hat{A}(\mu)\times_{2}\hat{A}(\
u)\times_{3}\hat{A}^{*} (\mu)\times_{4}\hat{A}^{*}(\
u)\big{\|}_{\rm{F}^{\prime}}^{2}, \tag{22}\]  
where  
\[\hat{A}(\mu) =\big{[}\mathbf{a}(\hat{\mu}_{1}),\mathbf{a}(\hat{\mu}_{2}),\cdots,\mathbf{a }(\hat{\mu}_{K})\big{]}\in\mathbb{C}^{M\times K}, \tag{23}\] \[\hat{A}(\
u) =\big{[}\mathbf{a}(\hat{\
u}_{1}),\mathbf{a}(\hat{\
u}_{2}),\cdots,\mathbf{a }(\hat{\
u}_{K})\big{]}\in\mathbb{C}^{N\times K},\]  
are the estimated steering matrices of \(\mathbb{U}\), \(\mathbf{\mathcal{J}}\in\mathbb{C}^{K\times K\times K}\) is the signal power tensor with \(\sigma_{\tau_{\rm{k}}}^{2}\) on its main diagonal.  
The above least squares problem (22) can be solved by iteratively updating \(\hat{A}(\mu)\) and \(\hat{A}(\
u)\) as  
\[\hat{A}(\
u) =\underset{\hat{A}(\
u)}{\text{argmin}}\Big{\|}[\mathbf{\mathcal{R}} ]_{1}-\hat{A}(\mu)\big{(}\hat{A}^{*}(\
u)\odot\hat{A}^{*}(\mu)\odot\hat{A}( \
u)\big{)}^{\mathsf{T}}\Big{\|}_{\rm{F}^{\prime}}^{2}, \tag{24}\] \[\hat{A}(\
u) =\underset{\hat{A}(\
u)}{\text{argmin}}\Big{\|}[\mathbf{\mathcal{R}} ]_{2}-\hat{A}(\
u)\big{(}\hat{A}^{*}(\
u)\odot\hat{A}^{*}(\mu)\odot\hat{A}( \mu)\big{)}^{\mathsf{T}}\Big{\|}_{\rm{F}^{\prime}}^{2},\]  
where \(\odot\) denotes the Khatri-Rao product. For each iteration, the closed-form solution of the estimated steering matrices can be computed as \[\hat{A}(\mu) =[\mathbf{\mathcal{R}}]_{1}\Big{[}\big{(}\hat{\mathbf{A}}^{*}(\
u)\odot\hat{ \mathbf{A}}^{*}(\mu)\odot\hat{\mathbf{A}}(\
u)\big{)}^{\rm T}\Big{]}^{\dagger}, \tag{25}\] \[\hat{A}(\
u) =[\mathbf{\mathcal{R}}]_{2}\Big{[}\big{(}\hat{\mathbf{A}}^{*}(\
u)\odot \hat{\mathbf{A}}^{*}(\mu)\odot\hat{\mathbf{A}}(\mu)\big{)}^{\rm T}\Big{]}^{\dagger}.\]  
The iterations repeat until the relative error of the decomposed covariance tensor between successive iterations is smaller than a convergence threshold, and the resulting estimation of steering matrices can be used for retrieving the angular parameters.  
As such, the estimation of the angular projection factors \(\hat{\mu}_{k}\) and \(\hat{\
u}_{k}\) can be obtained as  
\[\hat{\mu}_{k} =\frac{1}{M}\sum_{m=1}^{M}\angle(\mathbf{a}_{m+1}(\hat{\mu}_{k})/\mathbf{a }_{m}(\hat{\mu}_{k}))/\pi, \tag{26}\] \[\hat{\
u}_{k} =\frac{1}{N}\sum_{n=1}^{N}\angle(\mathbf{a}_{n+1}(\hat{\
u}_{k})/\mathbf{ a}_{n}(\hat{\
u}_{k}))/\pi,\]  
where \(\mathbf{a}_{m}(\hat{\mu}_{k})\), \(\mathbf{a}_{n}(\hat{\
u}_{k})\), respectively, represent the \(m\)-th element and \(n\)-th element of \(\hat{\mathbf{a}}(\mu_{k})\), \(\hat{\mathbf{a}}(\
u_{k})\), and \(\angle\) denotes the phase of a complex number. Finally, according to the relationship between angular projection factors \((\mu_{k},\
u_{k})\) and \((\theta_{k},\phi_{k})\) as defined in (5), the azimuth and elevation of the \(k\)-th source can be calculated in closed-forms as  
\[\hat{\theta}_{k} =\arctan(\hat{\
u}_{k}/\hat{\mu}_{k}), \tag{27}\] \[\hat{\phi}_{k} =\arcsin(\sqrt{\hat{\
u}_{k}^{2}+\hat{\mu}_{k}^{2}}).\]