## 2 Problem Description  
Consider an MISO-FIR system with time delays:  
\[y(t)=\sum_{i=1}^{r}z^{-d_{i}}B_{i}(z)u_{i}(t)+v(t) \tag{1}\]  
where \(u_{i}(t)\) is the \(i\)-th input, \(y(t)\) the output, \(d_{i}\) the time delay of the \(i\)-th input channel and \(v(t)\) a white noise with zero mean and variance \(\mathfrak{o}^{2}\), \(B_{i}(z)\), \(i=1,2,\cdots,r\) are the polynomials in the unit backward shift operator \(z^{-1}\) (_i.e._, \(z^{-1}y(t)=y(t-1)\)) and:  
\[B_{i}(z):=b_{i1}z^{-1}+b_{i2}z^{-2}+\cdots+b_{in_{bi}}z^{-n_{bi}}\]  
Assume that \(y(t)=0,u(t)=0\) and \(v(t)=0\) for \(t\leqslant 0\) and the orders \(n_{bi}\), the time delays \(d_{i}\), as well as the parameters \(b_{ij}\) are unknown. Thus, the identification goal is to estimate the unknown parameters \(b_{ij}\), orders \(n_{bi}\) and the time delays \(d_{i}\) from the observations.  
Define the information vector:  
\[\boldsymbol{\varphi}(t)\!:= \left[u_{1}(t-1),\cdots,u_{1}(t-d_{1}),u_{1}(t-d_{1}-1),\cdots,u_ {1}(t-d_{1}-n_{b_{1}}),\cdots,u_{1}(t-l),\cdots,\right.\] \[\left.u_{r}(t-1),\cdots,u_{r}(t-d_{r}),u_{r}(t-d_{r}-1),\cdots,u_ {r}(t-d_{r}-n_{b_{r}}),\cdots,u_{r}(t-l)\right]^{\intercal}\in\mathbb{R}^{N}\]  
where \(l\) is the input maximum regression length. \(l\) must be chosen to capture all of the time delays and the degrees of the polynomials \(B_{i}(z)\). Thus, \(l\geqslant\max(d_{i}+n_{bi})\). The dimension of \(\boldsymbol{\varphi}(t)\) is \(N=lr\). The corresponding parameter vector \(\boldsymbol{\theta}\) is:  
\[\boldsymbol{\theta}\!:=\![0,\cdots,0,b_{11},\cdots,b_{1n_{bi_{1}}},0,\cdots,0, b_{i1},\cdots,b_{in_{bi_{i}}},0,\cdots,0,b_{r1},\cdots,b_{rn_{b_{r}}},0, \cdots,0]^{\intercal}\in\mathbb{R}^{N}\]  
Then, Equation (1) can be written as:  
\[y(t)=\boldsymbol{\varphi}^{\intercal}(t)\boldsymbol{\theta}+v(t) \tag{2}\]  
Because of the unknown orders and time delays, the parameter vector \(\boldsymbol{\theta}\) contains many zeros, but only a few non-zeros. According to the CS theory, the parameter vector \(\boldsymbol{\theta}\) can be viewed as a sparse signal, and the number of non-zero entries \(K=\sum_{i=1}^{r}n_{bi}\) is the sparsity level of \(\boldsymbol{\theta}\)[8]. Let \(\|\boldsymbol{\theta}\|_{0}\) denote the number of non-zero entries in \(\boldsymbol{\theta}\). Then, the identification problem can be described as:  
\[\hat{\boldsymbol{\theta}}=\arg\min\|\boldsymbol{\theta}\|_{0},\quad\mathrm{s.t.}\quad\|\boldsymbol{y}-\boldsymbol{\varphi}^{\intercal}(t)\boldsymbol{\theta} \|_{2}<\varepsilon, \tag{3}\]  
where \(\hat{\boldsymbol{\theta}}\) is the estimate of \(\boldsymbol{\theta}\) and \(\varepsilon>0\) is the error tolerance.  
For \(m\) observations, define the stacked matrix and vectors as:  
\[\boldsymbol{\Phi}_{m}:=\begin{bmatrix}\boldsymbol{\varphi}^{\text{\tiny T}}(1)\\ \boldsymbol{\varphi}^{\text{\tiny T}}(2)\\ \vdots\\ \boldsymbol{\varphi}^{\text{\tiny T}}(m)\end{bmatrix}\in\mathbb{R}^{m\times N}, \ \boldsymbol{y}_{m}:=\begin{bmatrix}y(1)\\ y(2)\\ \vdots\\ y(m)\end{bmatrix}\in\mathbb{R}^{m},\ \boldsymbol{v}_{m}:=\begin{bmatrix}v(1)\\ v(2)\\ \vdots\\ v(m)\end{bmatrix}\in\mathbb{R}^{m}\]  
Then, we have:  
\[\boldsymbol{y}_{m}=\boldsymbol{\Phi}_{m}\boldsymbol{\theta}+\boldsymbol{v}_{m} \tag{4}\]  
If there are enough measurements, _i.e._, \(m\gg N\), according to the least squares principle, we can get the least squares estimate of \(\boldsymbol{\theta}\),  
\[\hat{\boldsymbol{\theta}}_{\text{LS}}=(\boldsymbol{\Phi}_{m}^{\text{\tiny T}} \boldsymbol{\Phi}_{m})^{-1}\boldsymbol{\Phi}_{m}^{\text{\tiny T}}\boldsymbol {y}_{m}\]  
However, from Equation (2), it is obvious that \(N\) is a large number, especially for systems with a large number of inputs. Therefore, it will take a lot of time and effort to get enough measurements. In order to improve the identification efficiency, in this paper, we aim to identify the parameter vector \(\boldsymbol{\theta}\) using less observations (\(K<m<N\)) based on the CS recovery theory.