## 3 Identification Algorithm  
The CS theory, which was introduced by [PERSON]\(\grave{e}\)s, [PERSON] [12,13] and [PERSON] [14], has been widely used in signal processing. It indicates that an unknown signal vector can be recovered from a small set of measurements under the assumptions that the signal vector is sparse and some conditions on the measurement matrix. The identification problem can be viewed as a CS recovery problem in that the parameter vector \(\boldsymbol{\theta}\) is sparse. From Equation (4), we can see that the output vector \(\boldsymbol{y}_{m}\) can be written as a linear combination of the columns of \(\boldsymbol{\Phi}_{m}\) plus the noise vector \(\boldsymbol{v}_{m}\), _i.e._,  
\[\boldsymbol{y}_{m}=\boldsymbol{\phi}_{1}\boldsymbol{\theta}_{1}+\boldsymbol{ \phi}_{2}\boldsymbol{\theta}_{2}+\cdots+\boldsymbol{\phi}_{i}\boldsymbol{ \theta}_{i}+\cdots+\boldsymbol{\phi}_{N}\boldsymbol{\theta}_{N}+\boldsymbol{v }_{m} \tag{5}\]  
where \(\boldsymbol{\theta}_{i}\) is the \(i\)-th element of \(\boldsymbol{\theta}\) and \(\boldsymbol{\phi}_{i}\) is the \(i\)-th column of \(\boldsymbol{\Phi}_{m}\). Because there are only \(K\) non-zero parameters in \(\boldsymbol{\theta}\), the main idea is to find the \(K\) non-zero items at the right-hand side of Equation (5). In this paper, we propose a TH-OMP algorithm to do the identification. First, we give some notations. Let \(k=1,2,\cdots\) be the iterative number and \(\lambda_{k}\) be the solution support of the \(k\)-th iterative number; \(\Lambda_{k}\) is a set composed of \(\lambda_{i}\), \(i=1,2,\cdots,k\); \(r_{k}\) denotes the residual of the \(k\)-th iterative number; \(\boldsymbol{\Phi}_{\Lambda_{k}}\) is the sub-information matrix composed by the \(k\) columns of \(\boldsymbol{\Phi}_{m}\) indexed by \(\Lambda_{k}\). \(\hat{\boldsymbol{\theta}}_{k}\) is the parameter estimation of the \(k\)-th iterative number. The TH-OMP algorithm is initialized as \(\boldsymbol{r}_{0}=\boldsymbol{y}_{m}\), \(\Lambda_{0}=\varnothing\) and \(\hat{\boldsymbol{\theta}}_{\Lambda_{0}}=\boldsymbol{0}\). At the \(k\)-th iteration, minimizing the criteria function:  
\[\varepsilon(i)=\min\lVert\boldsymbol{r}_{k-1}-\boldsymbol{\phi}_{i}\boldsymbol {\theta}_{i}\rVert^{2},\ i=1,2,\cdots,N\]  
with respect to \(\boldsymbol{\theta}_{i}\) yields:  
\[\boldsymbol{\theta}_{i}=\frac{\boldsymbol{\phi}_{i}^{\text{\tiny T}} \boldsymbol{r}_{k-1}}{\lVert\boldsymbol{\theta}_{i}\rVert^{2}}\]Plugging it back into \(\varepsilon(i)\), we have:  
\[\varepsilon(i) = \min_{\theta_{i}}\|\mathbf{\phi}_{i}\mathbf{\uptheta}_{i}-\mathbf{r}_{k-1}\|^{2} \tag{6}\] \[= \left\|\frac{\mathbf{\phi}_{i}^{\mathrm{ T}}\mathbf{r}_{k-1}}{\|\mathbf{\phi}_{i}\|^{2}}\mathbf{\phi}_{i}-\mathbf{r}_{k-1}\right\|^{2}\] \[= \|\mathbf{r}_{k-1}\|^{2}-2\frac{(\mathbf{\phi}_{i}^{\mathrm{ T}}\mathbf{r}_{k-1})^{2}}{\|\mathbf{\phi}_{i}\|^{2}}+\frac{(\mathbf{\phi}_{i}^{\mathrm{ T}}\mathbf{r}_{k-1})^{2}}{\|\mathbf{\phi}_{i}\|^{2}}\] \[= \|\mathbf{r}_{k-1}\|^{2}-\frac{(\mathbf{\phi}_{i}^{\mathrm{ T}}\mathbf{r}_{k-1})^{2}}{\|\mathbf{\phi}_{i}\|^{2}}\] \[= \|\mathbf{r}_{k-1}\|^{2}-(\frac{1}{\|\mathbf{\phi}_{i}\|}\mathbf{\phi}_{i}^{ \mathrm{ T}}\mathbf{r}_{k-1})^{2}\]  
From the above equation, we can see that the quest for the smallest error is equivalent to the quest for the largest inner product between the residual \(\mathbf{r}_{k-1}\) and the normalized column vectors of \(\mathbf{\Phi}_{m}\). Thus, the \(k\)-th solution support can be obtained by:  
\[\lambda_{k}=\arg\max_{i=1,2,\cdots,N}|\langle\mathbf{r}_{k-1},\frac{\mathbf{\phi}_{i} }{\|\mathbf{\phi}_{i}\|}\rangle| \tag{7}\]  
Update the support set \(\Lambda_{k}\) and the sub-information matrix \(\mathbf{\Phi}_{\Lambda_{k}}\) by:  
\[\Lambda_{k}:=\Lambda_{k-1}\cup\lambda_{k},\quad\mathbf{\Phi}_{\Lambda_{k}}:=\mathbf{ \Phi}_{\Lambda_{k-1}}\cup\mathbf{\phi}_{\lambda_{k}} \tag{8}\]  
The next step is to find the \(k\)-th provisional parameter estimate from the sub-information matrix \(\mathbf{\Phi}_{\Lambda_{k}}\) and measurement vector \(\mathbf{y}_{m}\).