# Structural-Missing Tensor Completion for Robust DOA Estimation with Sensor Failure +
## 2 Tensor Signal Model with Array Sensor Failure  
As shown in Figure 1, we consider an \(M\times N\) URA  
\[\mathbb{U}{=}\{(x_{\mathbb{U}},y_{\mathbb{U}})|x_{\mathbb{U}}=[0,M-1]d,y_{ \mathbb{U}}=[0,N-1]d\}, \tag{1}\]  
where the unit inter-element spacing \(d\) equals to half of the signal wavelength. Assume that \(K\) narrowband far-field uncorrelated signals impinge on the URA \(\mathbb{U}\) from directions\((\theta_{k},\phi_{k})\) for \(k=1,2,\cdots,K\), where \(\theta_{k}\) and \(\phi_{k}\), respectively, denote the azimuth and elevation of the \(k\)-th source. For the conventional matrix-based signal processing methods, the array received signals at the \(t\)-th snapshot are modeled as a vector \(\mathbf{x}(t)\in\mathbb{C}^{MN}\), while the total \(T\) snapshots will then be stacked into a matrix  
\[\mathbf{X}=[\mathbf{x}(1),\mathbf{x}(2),\cdots,\mathbf{x}(T)]\in\mathbb{C}^{MN\times T}. \tag{2}\]  
Here, \(\mathbb{C}\) represents the set of complex numbers. Nevertheless, such operation fails to reflect the multi-spatial property of the URA signals, resulting in potential information loss. To preserve the multi-spatial signal characteristics, we model the \(t\)-th snapshot as a matrix \(\mathbf{X}(t)\in\mathbb{C}^{M\times N}\), and then concatenate \(T\) snapshots to a 3D tensor, defined as  
\[\mathbf{\mathcal{X}}=\sum_{k=1}^{K}\mathbf{a}(\mu_{k})\circ\mathbf{a}(\
u_{k})\circ\mathbf{s}_ {k}+\mathbf{\mathcal{N}}\in\mathbb{C}^{M\times N\times T}. \tag{3}\]  
Here,  
\[ \begin{split}\mathbf{a}(\mu_{k})&=[1,e^{-\jmath\pi\mu_{ k}},\cdots,e^{-\jmath\pi(M-1)\mu_{k}}]^{\mathrm{T}},\\ \mathbf{a}(\
u_{k})&=[1,e^{-\jmath\pi\
u_{k}},\cdots,e^{ -\jmath\pi(N-1)\
u_{k}}]^{\mathrm{T}}\end{split}  \tag{4}\]  
are, respectively, the steering vectors of \(\mathbb{U}\) along the \(x\)-axis and the \(y\)-axis with the angular projection factors  
\[ \begin{split}\mu_{k}&=\sin\phi_{k}\cos\theta_{k},\\ \
u_{k}&=\sin\phi_{k}\sin\theta_{k},\end{split}  \tag{5}\]  
\(\mathbf{s}_{k}=[s_{k}(1),s_{k}(2),\cdots,s_{k}(T)]^{\mathrm{T}}\) is the signal waveform of the \(k\)-th source, \(\circ\) denotes the outer product, \(\jmath=\sqrt{-1}\), \((\cdot)^{\mathrm{T}}\) denotes the transpose operator, \(\mathbf{\mathcal{N}}\) is an additive Gaussian white noise tensor, i.e., its \(t\)-th slice \(\mathbf{N}(t)\sim\mathcal{CN}(0,\delta_{n}^{2}\mathbf{\mathcal{I}})\), \(\forall t=1,2,\cdots,T\), where \(\delta_{n}^{2}\) denotes the noise power, and \(\mathbf{\mathcal{I}}\) denotes the identity tensor. In this way, the tensor signal model perfectly preserves the multi-spatial characteristics, which facilitates subsequent tensor optimizations for enhanced DOA estimation performance. Note that, the signal part of \(\mathbf{\mathcal{X}}\) in (3) follows a CP model, which is characterized by the sum of a certain number of rank-1 outer products. The formulated tensor in (3) offers an ideal signal model for a completely functional URA. Unfortunately, when sensor failure occurs, the signals received at the impaired sensors will be missing. In this case, the corresponding URA signals can be characterized by an incomplete tensor  
\[\mathbf{\tilde{\mathcal{X}}}=\mathbf{\mathcal{M}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\  
of the unimpaired sensor, whereas \(\mathbf{\mathcal{M}}_{(m,n,\cdot)}=0\) if \((m,n)\) is the position of the impaired sensor with \(m=1,2,\cdots,M\) and \(n=1,2,\cdots,N\). As demonstrated in (6), each impaired sensor results in an entirely missing fiber in the corresponding position of the incomplete tensor \(\mathbf{\mathcal{X}}\). Moreover, as the number of impaired sensors increases, there might even exist whole missing slices in \(\mathbf{\mathcal{\hat{X}}}\). As such, it is obvious that the sensor failure results in a structural-missing pattern for the tensor signal instead of the common random-missing pattern.