# 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
As shown in Figures 6 and 8, the MSEs of the CS-based methods are lower than those of an MMSE method, except that the conventional OMP algorithm is not as good as MMSE at using a

Figure 4: MSE corresponding to different observation numbers (K = 4, SNR = 15 dB). Figure 5: MSE corresponding to different observation numbers (K = 6, SNR = 15 dB). dual UW at a low SNR. Compared with the conventional OMP and SOMP algorithms, the improved algorithms pull down the CRLB and, thus, achieve lower MSEs. For the four-path channel, as illustrated in Figure 6, at \(MSE=10^{-2}\), P-OMP achieves a SNR gain of about 1.0 dB compared to OMP and P-SOMP achieves a SNR gain of about 1.1 dB compared to SOMP. As shown in Figure 7, using the conventional OMP and SOMP algorithms, SNR gains of about 0.7 dB and 1.8 dB are achieved at \(BER=10^{-5}\) compared to the MMSE, respectively. When adopting the improved algorithms, an extra 0.3 dB and 0.2 dB in SNR gains can be obtained compared to the conventional algorithms, respectively. For the six-path channel, Figure 8 illustrates that, at \(MSE=10



In terms of algorithm complexity, MMSE estimation requires a matrix-inverse operation with a time complexity of O(\(P^{3}\)), and the complexity is higher for larger matrix orders. The application of iterative methods such as most rapid descent and conjugate gradient methods are able to avoid the matrix inverse, and channel estimation is realized by solving system equations. It was found that those methods converge faster, with O(\(P^{2}\)) computations per iteration. For CS-based schemes, the main part of both conventional and improved algorithms lies in the iteration of a greedy algorithm. The iteration numbers of both algorithms are the same; they are equal to the channel sparsity \(K\). The time complexities of the conventional OMP and SOMP algorithms are both O(\(K\)ML), and compared with conventional methods, the improved methods only add the step of solving for the exact path gain according to the maximum likelihood rule (MLR), as stated in Equation (28). The increased time complexity is O(\(M^{\prime}K^{2}+K^{3}\)) [19]; i.e., the computational time is exponentially increased with the increase in sparsity. For wireless channels with significant sparsity, we have \(K<<L\), and the computations achieved via the improved algorithms are acceptable. The observation vector length \(M^{\prime}\) for calculating the path gain has a smaller effect on the complexity. The time complexities of different estimation schemes are summarized in Table 3, where the complexity of MMSE is based on the computation of a standard matrix inversion method. Figure 8: MSE with different SNR values (K = 6, M = 24). Figure 9: BER with different SNR values (K = 6, M = 24). In summary, when using CS-based methods, compared with the dual UW-based MMSE method, the BER performance of the channel estimator is effectively improved with a substantial reduction of the reference signal; i.e., the effectiveness and reliability of the communication system are simultaneously enhanced. Compared with the conventional OMP and SOMP algorithms, the proposed scheme pulls down the CRLB and further improves the performance of channel estimation. Therefore, adopting CS-based estimation methods and applying an improved algorithm are of great practical significance for the reliable transmission of numerous wireless channels with sparse characteristics. ## 4 Conclusions

The OMP algorithm is an important greedy algorithm that is more extensively applied to CS-based channel estimation. This research model's wireless channel estimation problem, as a sparse signal reconstruction problem, analyzes the factors affecting OMP algorithm performance, proposes an improved scheme applicable to SC-FDE systems, which is applied to the OMP and SOMP algorithms, and finally develops system simulations and experiments for the two channel models.