# An Improved Orthogonal Matching Pursuit Algorithm for CS-Based Channel Estimation
## 2 Materials and Methods
### Wireless Communication System Model Based on CS Channel Estimation
#### 2.1.2 SC-FDE System Based on CS Channel Estimation  
A training sequence (TS) can be applied as both a guard interval and a reference signal. Figure 1 shows the frame structure of the TS-based SC-FDE system. Generally, the training sequences of SC-FDE systems apply the same unique word (UW), and the most common form of the UW is a CAZAC sequence with a constant envelope characteristic. In order to avoid the interference of data blocks due to the multipath delay spread, reference [24] applied a dual UW (DUW) as a guard interval, which effectively improved the channel estimation performance, but the channel overhead was high because the length of each UW was equivalent to the maximum multipath delay. To  
Figure 1: SC-FDE frame structure based on the training sequence. decrease the channel overhead, reference [25] developed an iterative method whose total UW length was equivalent to the maximum multipath delay, but it was not suitable for fast time-varying channels due to their slow convergence, which made it impossible to acquire the channel state quickly. Reference [19] explored CS-based TDS-OFDM transmission methods. The frame structure of the TDS-OFDM system was very similar to that of SC-FDE, which utilized a known TS in the time domain as a guard interval for data blocks. The research results showed that, when the SC-FDE system applied the CS-based channel estimation method, it only needed to insert a single UW to obtain better performance than conventional methods, which improved the reliability and effectiveness of the system at the same time. Figure 2a shows the block transmission structure of the TS-based SC-FDE system, where the known sequence \(C_{i}=\left[c_{i,0},c_{i,1},\ldots,c_{i,P-1}\right]^{T}\) is a UW with length \(P\). Therefore, for any \(i\) and \(j\), \(c_{i}=c_{j}\). Also, \(X_{i}=\left[x_{i,0},x_{i,1},\ldots,x_{i,N-1}\right]^{T}\) is a payload data block with length \(N\). The discrete time channel impulse response with the maximum multipath spread length \(L\) is assumed to be \(H_{i}=\left[h_{i,0},h_{i,1},\ldots,h_{i,L}\right]^{T}\), which satisfies \(P\geq L\). \(M=P-L\) is denoted. Figure 2b illustrates the received signal after channel transmission and time-frequency synchronization. The received signal \(D_{i}=\left[d_{i,0},d_{i,1},\ldots,d_{i,P-1}\right]^{T}\) corresponding to a known sequence is stated as follows:  
\[D_{i}=\Psi_{i}H_{i}+N_{i} \tag{11}\]  
where \(N_{i}\) is additive Gaussian white noise, and \(\Psi_{i}\) is as follows:  
\[\Psi_{i}= \begin{bmatrix}c_{i,0}&x_{i,N-1}&x_{i,N-2}&\cdots&x_{i,N-L}\\ c_{i,1}&c_{i,0}&x_{i,N-1}&\cdots&x_{i,N-L+1}\\ c_{i,2}&c_{i,1}&c_{i,0}&\cdots&x_{i,N-L+2}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ c_{i,L}&c_{i,L-1}&c_{i,L-2}&\cdots&c_{i,0}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ c_{i,P-1}&c_{i,P-2}&c_{i,P-3}&\cdots&c_{i,P-L-1}\end{bmatrix} _{P\times(L+1)} \tag{12}\]  
It is evident in matrix \(\Psi_{i}\) that the last \(M\) samples of the received signal \(D_{i}\) are not interfered with unknown data \(X_{i}\), which is known as the inter-block-interference (IBI)-free region of TS and is recorded as \(R_{i}=\left[d_{i,L},d_{i,L+1},\ldots,d_{i,P-1}\right]^{T}\). Also, \(V_{i}\) is the vector that contains the last \(M\) samples of \(N_{i}\); then:  
\[R_{i}=\Phi_{i}H_{i}+V_{i} \tag{13}\]  
Figure 2: Block transmission and multipath interference. (**a**) Block transmission of transmitter signals; (**b**) multipath interference. where \(\Phi_{i}\) is a Toeplitz array, stated as follows:  
\[\Phi_{i}= \begin{bmatrix}c_{i,L}&c_{i,L-1}&c_{i,L-2}&\cdots&c_{i,0}\\ c_{i,L+1}&c_{i,L}&c_{i,L-1}&\cdots&c_{i,1}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ c_{i,P-1}&c_{i,P-2}&c_{i,P-3}&\cdots&c_{i,P-L-1}\end{bmatrix} _{M\times(L+1)} \tag{14}\]  
For the SC-FDE of DUW, \(M=L+1\), the matrix \(\Phi_{i}\) is the column full rank. Then, Equation (13) is applied for conventional methods, such as the least square (LS) and minimum mean square error (MMSE) algorithms, to estimate channel state \(H_{i}\).