# An Improved Orthogonal Matching Pursuit Algorithm for CS-Based Channel Estimation
## 2 Materials and Methods
### Improvement of the OMP Algorithm
#### 2.2.2 Improvement of the OMP Algorithm
The least square solution is found: \[s_{t}=\arg\min_{s}\lVert y-A_{t}s_{t}\rVert=\left(A_{t}^{H}A_{t}\right)^{-1}A_ {t}^{H}y\]
5. New approximations and residuals are calculated : \(r_{t}=y-A_{t}\hat{s}_{t}\). 6. If \(t<K\), the process returns to step 2; otherwise, iteration is stopped and returns to step 7. 7. The non \(-\) zero element position set of \(\hat{s}\) obtained from the reconstruction is \(\Lambda_{K}=\{\lambda_{0},\lambda_{1},\cdots\lambda_{K-1}\}\), denotes \(\hat{L}=\max(\Lambda_{K})\), and the observation matrix \(\Phi=\Phi^{\prime}_{i}\) and observation vector \(y=R^{\prime}_{i}\) are updated. 8. The set of column vectors of \(\Phi\) is denoted as \(\Phi|_{\Lambda_{K}}\); with columns indexed using elements in \(\Lambda_{K}\), the least square solution is found: \[\hat{s}=\left(\Phi|_{\Lambda_{K}}^{H}\Phi|_{\Lambda_{K}}\right)^{-1}\Phi|_{ \Lambda_{K}}^{H}y\]

**Algorithm 1**Proposed improved OMP algorithm. It should be noted that, when applying CS-based channel estimation to OFDM systems, a mathematical model is established, as stated in Equation (10). When applying a greedy pursuit algorithm to reconstruct the time domain CIR, the observation vector length is equal to the number of frequency pilots, so all reference signals are utilized, and there is no possibility of further increasing the observation number. Therefore, the improved method is not applicable to OFDM systems using frequency domain pilots as reference signals. In particular, since the TDS-OFDM system uses the time domain TS as reference signals for synchronization and channel estimation, it is similar in structure to the SC-FDE system, and thus the improved scheme is also applicable to the TDS-OFDM system based on CS channel estimation. ## 3 Results and Discussion

This research sets up an SC-FDE system simulation platform to simulate the proposed improved algorithm. Multipath channel models are selected from tapped delay models, and the channel parameters of four paths and six paths are set according to the ionospheric channel characteristics of shortwave communication and the typical urban area channel model of wireless communication COST207 [28], as summarized in Table 1, where channel sparsity is the number of multipaths. Assuming that the system is ideally synchronized, the main parameters of the transmitter are shown in Table 2. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Channel Sparsity (K)** & **4** & **6** \\ \hline Channel length (\(L\)) & 40 & 50 \\ \hline Path delay (\(\tau_{k}\)) & Random & Random \\ \hline Path gain (\(\alpha_{k}\)) & \([0,-3,-6,-9]\) & \([-3,0,-2,-6,-8,-10]\) \\ \hline Doppler shift (\(f_{d}\)) & \(1\) Hz & \(100\) Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Channel model parameters. Channel length and sparsity are selected according to the following considerations. Based on the properties of ionospheric channels [29], the maximum multipath delay spread is \(\tau_{\max}=5\) ms. Considering that the system sampling rate is \(f_{\rm s}=8\) KSps, the tap number of the tapped delay channel model, and also the initialized value of the channel length, is \(L=f_{\rm s}\times\tau_{\max}=40\). ITU-R suggests test channels for the simulation of the performance of the shortwave digital communication system, and ITU-R F.1487 defines ten test channels at different latitudes and under different environmental conditions. These test channels are all Watterson models with a two-path tapped delay [29]. Six channel models are suggested according to the DRM digital broadcasting standard [30], in which the typical shortwave channel model is a four-path channel, and sparsity of \(K=4\) is chosen to better verify the reconstruction performance of the CS algorithm. Similarly, according to the typical urban area of the COST207 model, the maximum multipath delay spread is \(\tau_{\max}=5\) us, and the system sampling rate is taken as \(f_{\rm s}=10\) MSps, from which the initialized value of channel length is \(L=f_{\rm s}\times\tau_{\max}=50\).