# An Improved Orthogonal Matching Pursuit Algorithm for CS-Based Channel Estimation
## 2 Materials and Methods
### Improvement of the OMP Algorithm  
#### 2.2.1 CRLB for Parameterized Channel Estimation  
According to signal detection and estimation theory, for parameter estimation problems, the CRLB establishes a lower bound for the variance of any unbiased estimator; i.e., it is impossible to obtain an unbiased estimator with an estimated mean square error (MSE) smaller than the CRLB. Therefore, the difference between the MSE of the estimate and the CRLB is usually taken as an important indicator for evaluating the performance of an estimation method in both frequency offset and channel estimation [27].  
When the set \(\Omega\) of the channel path delay and sparsity \(K\) of the channel are obtained, the signal transmission model is as follows:  
\[R=\Phi_{\Omega}H_{K}+V \tag{17}\]where \(R\) is the received TS of the IBI-free region with length \(M\), and \(\Phi_{\Omega}\) is the observation matrix of \(M\times K\), which is a sub-matrix of \(\Phi\) whose column indices are elements in \(\Omega\). Also, Gaussian white noise \(V=\left[v_{1},v_{2},\cdots v_{M}\right]\) obeys the distribution of \(N\left(0,\sigma^{2}I_{M}\right)\), and \(H_{K}\) is the column vector of \(K\times 1\), giving only the channel sparse location coefficients.  
Assuming \(H_{K}\), the conditional probability density function (PDF) of \(R\) is denoted as follows [20]:  
\[p_{R|H_{K}}(R;H_{K})=\frac{1}{\left(2\pi\sigma^{2}\right)^{M/2}}\exp\biggl{\{}- \frac{1}{2\sigma^{2}}\|R-\Phi_{\Omega}H_{K}\|_{2}\biggr{\}} \tag{18}\]  
Using vector estimation theory [27], the CRLB of unbiased estimation \(\hat{H}_{K}=\Phi_{\Omega}^{\dagger}R\) can be introduced as expressed in Equation (19):  
\[CRLB=E\Bigl{\{}\|\hat{H}_{K}-H_{K}\|_{2}^{2}\Bigr{\}} \tag{19}\]  
According to Equation (17), the estimated value of \(H_{K}\) is obtained to satisfy  
\[\begin{array}{l}\hat{H}_{K}=\Phi_{\Omega}{}^{\dagger}R=\Phi_{\Omega}{}^{ \dagger}(\Phi_{\Omega}H_{K}+V)\\ =\Phi_{\Omega}{}^{\dagger}\Phi_{\Omega}H_{K}+\Phi_{\Omega}{}^{\dagger}V=H_{K} +\[PERSON]_{\Omega}{}^{\dagger}V\end{array} \tag{20}\]  
that is,  
\[\hat{H}_{K}-H_{K}=\[PERSON]_{\Omega}{}^{\dagger}V \tag{21}\]  
thus,  
\[CRLB=\Bigl{\{}\|\hat{H}_{K}-H_{K}\|_{2}^{2}\Bigr{\}}=E\Bigl{\{}\|\Phi_{\Omega} {}^{\dagger}V\|_{2}^{2}\Bigr{\}}=E\Biggl{\{}\|\left(\Phi_{\Omega}{}^{H}\[PERSON]_ {\Omega}\right)^{-1}\!\Phi_{\Omega}{}^{H}V\|_{2}^{2}\Biggr{\}} \tag{22}\]  
Considering \(G=\Phi_{\Omega}{}^{H}\Phi_{\Omega}\), when all elements in \(\Phi_{\Omega}\) are identically distributed random numbers with modulus 1, the diagonal elements of Gram matrix \(G\) are the constant \(M\), and the remaining elements are normally distributed random numbers with a mean of 0 and variance of \(1/M\). Expanding the matrix operations further yields  
\[CRLB=E\Bigl{\{}\|G^{-1}\Phi_{\Omega}{}^{H}V\|_{2}^{2}\Bigr{\}}=K\biggl{(}\frac {1}{M}\biggr{)}^{2}\sigma^{2}M=\frac{K}{M}\sigma^{2} \tag{23}\]  
From Equation (23), it can be seen that an increasing \(M\) decreases the CRLB; i.e., increasing the number of observations improves the probability that the OMP algorithm correctly recovers the original sparse signal. The relationship between \(M\) and the CRLB verified that increasing the number of observations effectively improves the performance of channel estimation. When channel multipath delay is calculated using the CS-based reconstruction algorithms, an accurate channel length can be obtained, which is usually shorter than the default maximum length; therefore, a larger number \(M\) of observations can be obtained, and a new observation matrix can be generated.