# A Low-Complexity Channel Estimation Based on a Least-Squares Algorithm in OFDM Systems
## 2 Pilot Arrangement and CFR Estimation at Pilot Subcarrier  
Figure 1 shows the locations of the pilot of the OFDM system with a comb-type pilot arrangement. Related parameters are described in Table 1. When the time-varying channel is approximately invariant in an OFDM symbol, the digital CIR corresponding to some OFDM symbols will be set to \(h[n]\), where \(n\) is the index for time. If the number of maximum channel delay points is \(N_{p}/2\), \(h[n]=0\) when \(n>N_{p}/2-1\) or \(n<0\). Let the frequency domain data transferred in an OFDM symbol be \(X[k]\), where \(k\) is the index for frequency. The time-domain signal \(x[n]\) is derived from the \(N\)-points IDFT of \(X[k]\) (i.e., the content free of CP (cyclic prefix) in this OFDM symbol). Let the OFDM symbol content received by the receiver after the CP is removed be \(y[n]\). When the CP is long enough that two adjacent OFDM symbols will not disturb each other because of the channel, \(y[n]\) can be expressed as follows:  
\[y[n]=x[n]\otimes h[n]+w[n], \tag{1}\]  
where \(w[n]\) is white Gaussian noise and \(\otimes\) is the circular convolution operation. The CFR \(H[k]\) of this OFDM symbol can be derived from the \(N\)-points DFT of \(h[n]\); the mathematical expression is given as follows:  
\[H[k]=\sum_{n=0}^{N-1}h[n]e^{-\
icefrac{{-2\pi kn}}{{N}}}=\sum_{n=0}^{N_{p}/2-1 }h[n]e^{-\
icefrac{{-2\pi kn}}{{N}}},\ 0\ \leq\ k\ \leq\ N\ -\ 1. \tag{2}\]The \(N\)-points DFTs of \(x[n]\), \(y[n]\), and \(w[n]\) are \(X[k]\), \(Y[k]\), and \(W[k]\), respectively. In addition:  
\[\begin{array}{l}X[k]=\sum\limits_{n=0}^{N-1}x[n]e^{-j2\pi kn/N},\ 0\ \leq\ k\ \leq\ N\ -\ 1,\\ Y[k]=\sum\limits_{n=0}^{N-1}y[n]e^{-j2\pi kn/N},\ 0\ \leq\ k\ \leq\ N\ -\ 1,\\ W[k]=\sum\limits_{n=0}^{N-1}w[n]e^{-j2\pi kn/N},\ 0\ \leq\ k\ \leq\ N\ -\ 1, \end{array}\]  
where \(W[k]\) is also white Gaussian noise. The DFT of a circular convolution of two \(N\)-points sequences is the product of their respective DFTs. In an OFDM symbol, \(Y[k]\) can be expressed as follows:  
\[Y[k]=X[k]H[k]+W[k],\ 0\leq k\leq N-1. \tag{3}\]  
Afterward, CFRs at the pilot subcarriers are estimated by the LS method. The CFRs estimated at the pilot subcarriers can be written as follows:  
\[\hat{H}[k_{p}]=Y[k_{p}]/X[k_{p}]=H[k_{p}]+W[k_{p}]/X[k_{p}],\ k_{p}\ \in\ \text{ pilot}. \tag{4}\]  
All the above variables are described in Table 2.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Meaning** \\ \hline \(N\) & The number of FFT points. \\ \hline \(\Delta p\) & The spacing between pilot subcarriers. \\ \hline \(N_{P}^{l}\) & The number of pilot subcarriers on the left of the virtual subcarrier area. \\ \hline \(N_{P}^{l}\) & The number of virtual pilot subcarriers in the virtual subcarrier area. \\ \hline \(N_{P}^{R}\) & The number of pilot subcarriers on the right of the virtual subcarrier area. \\ \hline \(N_{p}\) & The total number of pilot subcarriers, including virtual pilot subcarriers (i.e., \(N_{p}=N/\Delta p=N_{P}^{l}+N_{P}^{l}+N_{P}^{R}\)). \\ \hline \(N_{P}^{out}\) & The number of pilot subcarriers out of the virtual subcarrier area and \(N_{P}^{out}=N_{P}^{l}+N_{P}^{R}\). \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameter definitions.  
Figure 1: Schematic diagram of the arrangement of OFDM system pilot signals.