# 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 typical urban area channel model given via COST207 usually has six significant paths; therefore, the sparsity is chosen to be \(K=6\). The iterations of the OMP and SOMP algorithms are equal to the sparsity level, and our simulations assume that the channel sparsity is known. In order to illustrate the superiority of the proposed scheme, the improved OMP and SOMP algorithms (with jointness \(J=16\)) were simulated and compared with the MMSE method [24] and conventional OMP [9] and SOMP [16] algorithms in terms of both spectral efficiency and channel estimation performance, respectively. When the MMSE method is applied, TS is a dual UW, which is a Chu sequence with a constant envelope property [31]. The first is used to resist multipath interference due to unknown data, and the second is applied as the reference signal for channel estimation. When the CS-based methods are used, UW is a pseudo-random (PN) sequence. The UW length needs to be longer than the channel maximum delay spread, and it is set to \(P=64\) and \(P=74\) for two channel conditions, respectively. In our simulation, fifteen symbol blocks were used for each simulation run, and each symbol block consisted of a TS of \(P\) samples and a data block of \(N\) samples. The average values of the MSE and bit error rate (BER) were calculated using 5000 simulation runs. The curves of the MSE versus the number of observations for conventional and improved algorithms are illustrated in Figures 4 and 5 for four-path and six-path channel conditions, respectively, with a fixed signal-to-noise ratio (SNR). The SNR was chosen to be 15 dB. P-OMP and P-SOMP in the legend represent the improved algorithms proposed in this research. In the figure, the red and blue dashed lines are the CRLB using conventional and improved methods, respectively. Obviously, the channel estimator using improved algorithms reduces the CRLB and obtains a smaller MSE, which effectively improves the estimation performance. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Channel Sparsity (K)** & **4** & **6** \\ \hline Sample rate (\(f_{\rm s}\)) (Hz) & \(8\times 10^{3}\) & \(10\times 10^{6}\) \\ \hline Modulation & QPSK & QPSK \\ \hline Channel coding & Convolutional encoding & Convolutional encoding \\ \hline Code rate & 0.5 & 0.5 \\ \hline Training sequence & Chu/PN & Chu/PN \\ \hline IBI-free region length (\(M\)) & 24 & 24 \\ \hline Training sequence length & & \\ (\(P=M+L\)) & 64 & 74 \\ \hline Data block length (\(N\)) & 256 & 256 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Transmitter parameters. As shown in Figure 4, the OMP algorithm achieves a mean square error of \(MSE=10^{-2}\) when \(M=22\) for the four-path channel model. However, the P-OMP algorithm achieves a mean square error of \(MSE=10^{-2}\) when \(M=19\). Similarly, the numbers of observations required for the SOMP and P-SOMP algorithms to achieve a mean square error of \(MSE=10^{-2}\) are \(M=17\) and \(M=15\), respectively. In other words, the improved methods achieve the same performance as the conventional methods using fewer observations; however, at a constant observation number, the improved methods can significantly decrease the MSE and improve the performance of the channel estimator. It is also evident that the MSEs of the SOMP methods applied to recover the joint sparse signal are closer to the CRLB than the OMP methods; that is, SOMP is able to utilize the temporal correlation of the channel state information (CSI) in consecutive frames, and it can use a shorter observation vector to achieve the same reconstruction probability as the OMP algorithm. When the observation vectors are the same as those of the OMP algorithms, the SOMP methods effectively reduce the MSEs and obtain more superior reconstruction performance for a sparse signal. Figures 6-9 illustrate the curves of the MSE and BER of different estimation methods with respect to the SNR, according to the two channel conditions.