# An Improved Orthogonal Matching Pursuit Algorithm for CS-Based Channel Estimation
## 1 Introduction
Due to an increasing data rate and system bandwidth, the channel responses experienced with adjacent symbols were no longer independent from each other, but they presented a time-dependent joint sparse structure [14]. To describe this feature, a joint sparse model (JSM) [15] was proposed, and accordingly, the simultaneous orthogonal matching pursuit (SOMP) [16; 17], which was based on distributed compressed sensing (DCS), has been widely studied. Reference [18] improved the SOMP algorithm by first jointly estimating the same support set and the elements of the channel response for multiple consecutive symbols and then estimating different parts of the support set, symbol by symbol. The obtained results showed that the improved method was applicable to both the JSM-1 and JSM-2 models. The sparse reconstruction algorithms mentioned above were mostly applied to CS-based channel estimation for OFDM systems employing frequency domain pilots. References [19; 20; 21] discussed the application of CS-based sparse channel estimation in time-domain-synchronous OFDM (TDS-OFDM) systems, which utilized a known training sequence (TS) in the time domain as a guard interval for data blocks. The frame structure of the single-carrier frequency domain equalization (SC-FDE) system is very similar to that of TDS-OFDM. Reference [22] investigated the application of the OMP algorithm to sparse channel estimation in SC-FDE systems. The algorithm proposed in reference [20] required a small amount of frequency domain pilots, and reference [22] proposed inserting the cyclic prefix (CP) of the TS in the header of a frame. All these methods required additional pilots, which sacrificed spectral efficiency while obtaining superior performance. For SC-FDE systems, an improved scheme is proposed in this research, which was applied to both the OMP and SOMP algorithms, and the improved algorithms are abbreviated as P-OMP and P-SOMP. The contribution of this research is to focus on the improvement of the algorithmic structure to achieve superior channel estimation performance without resorting to additional pilots or TS. Compared with conventional algorithms, the improved algorithms calculate the path gain after obtaining the exact path delay and updating the observation matrix. With a longer observation vector for estimating the path gain, the improved algorithms reduce the Cramer-Rao lower bound (CRLB) and achieve better estimation performance; therefore, they effectively improve the reliability of the communication system. Due to the similarity of the transmission frame structure, the proposed method is also applicable to TDS-OFDM systems. In addition, the main processes of all the matching pursuit algorithms are the same, and the proposed method is also applicable to other matching pursuit algorithms. Therefore, the developed method is important for the reliable transmission of wireless communication systems based on the transmission block structure employing time-domain pilots. Notation: \(\left(\cdot\right)^{T},\left(\cdot\right)^{H},\left(\cdot\right)^{\dagger}, \left(\cdot\right)^{-1}\) and \(\|\cdot\|_{p}\) denote transpose, conjugate transpose, Moore-Penrose matrix inversion, matrix inversion, and \(l_{p}\) norm operations, respectively. \(I_{n}\) denotes the identity matrix of dimension \(n\).