## 3 Optimized Joint Signal Processing Schemes
### Sparsity Reconstruction-Based Improved Parameter Estimation
#### 3.1.3 Improved Parameter Estimation  
Generally speaking, the traditional method based on the modulation symbol domain can be used to realize target parameter estimation; that is, we can perform DFT on the signal in (17) along the pulse symbol dimension and perform an IDFT along the subcarrier dimension. Based on this, (17) can be updated to the following formula:  
\[\mathbf{\hat{Y}}=\mathbf{F}_{N_{c}}^{-1}\mathbf{Y}\mathbf{F}_{P}=N_{r}\sum_{q=1}^{Q}\beta_{q} \mathbf{F}_{N_{c}}^{-1}\mathbf{\tau}_{q}\otimes\mathbf{D}_{q}^{\text{T}}\mathbf{F}_{P}+\mathbf{ \hat{W}} \tag{18}\]  
where \(\mathbf{F}_{P}\) is a discrete Fourier transform (DFT) matrix and its size is \(P\times P\). However, due to the use of IM, there may be zero elements in the transmission symbols, which will cause some elements in the matrix \(\mathbf{Y}\) to converge to infinity. Therefore, it is very difficult to obtain a good parameter estimation performance by directly adopting the method in (18). We need to extract valid data through data filtering and use new methods to solve it. According to the relationship between the vectorization of a matrix product and Kronecker product [45], (17) can be transformed into the following formula:  
\[\text{vec}(\mathbf{Y})=\text{vec}\Bigg{(}N_{r}\sum_{q=1}^{Q}\beta_{q}\mathbf{\tau}_{q} \otimes\mathbf{D}_{q}^{\text{T}}\Bigg{)}+\text{vec}(\tilde{\mathbf{W}}) \tag{19}\]  
where \(\text{vec}(\cdot)\) indicates that the matrix is stacked in a vector form by columns. Suppose that \(\mathbf{R}_{q}=\mathbf{F}_{N_{c}}^{-1}\mathbf{\tau}_{q}\) and \(\mathbf{V}_{q}=\mathbf{D}_{q}^{\text{T}}\mathbf{F}_{P}\). According to the properties of the Kronecker product, (19) can be rewritten as  
\[\text{vec}(\mathbf{Y})= \text{vec}\Bigg{(}N_{r}\sum_{q=1}^{Q}\beta_{q}\mathbf{F}_{N_{c}}\mathbf{F }_{N_{c}}^{-1}\mathbf{\tau}_{q}\otimes\mathbf{D}_{q}^{\text{T}}\mathbf{F}_{P}\mathbf{F}_{P}^{- 1}\Bigg{)  
where  
\[\mathbf{V}_{q,h}=[e^{|2\pi f_{d}^{d}T_{p}},e^{|2\pi f_{d}^{d}2T_{p}},\cdots,e^{2\pi f_{ d}^{d}PT_{p}}]\mathbf{F}_{\bar{p}} \tag{28}\]  
According to the sparsity of \(\mathbf{\chi}_{h,R,v}\), (26) can be further equivalently transformed into an optimization problem, which can be described as  
\[f^{opt}=\arg\min_{\mathbf{\chi}_{h}}\left(\frac{1}{2}\|\mathbf{\dot{Y}}_{h}-\bar{\mathbf{ \dot{C}}}_{h}\mathbf{\chi}_{h,R,v}\|_{2}^{2}+\mu\|\mathbf{\chi}_{h,R,v}\|_{1}\right) \tag{29}\]  
A large amount of literature points out that there are many methods for solving the unconstrained optimization problem in (29). In this paper, an improved method with low complexity proposed in [46] is used to solve this problem. After solving the above problem, the matrix \(\mathbf{\chi}_{h,R,v}\) corresponding to the \(h\)th pulse group can be obtained, and then the range dimension information of the pulse group can be obtained by performing an IDFT on this matrix along the pulse group dimension, which can be represented as  
\[\mathbf{\psi}_{h,R}=\mathbf{\chi}_{h,R,v}F_{\bar{p}}^{-1} \tag{30}\]  
The \(H\) pulse groups are processed according to the above method, and all the results are then recombined. Then, DFT is performed on the reconstructed data, and finally the two-dimensional image information of multiple targets is obtained, which can be denoted as  
\[\mathbf{\hat{\chi}}_{R,v}= \mathbf{\psi}_{1,R},\mathbf{\psi}_{2,R},\cdots,\mathbf{\psi}_{H,R}|\mathbf{F}_{P}\] \[= \left[\mathbf{\chi}_{1,R,v}F_{\bar{p}}^{-1},\mathbf{\chi}_{2,R,v}F_{\bar {p}}^{-1}\cdots,\mathbf{\chi}_{H,R,v}F_{\bar{p}}^{-1}\right]\mathbf{F}_{P} \tag{31}\]  
To measure the estimation accuracy of the proposed method based on MIMO-OFDM-IM, we use the Cramer-Rao bound (CRB) as a standard to evaluate all unbiased estimators.