# A Low-Complexity Channel Estimation Based on a Least-Squares Algorithm in OFDM Systems
## 3 The Proposed Method Based on the LS Method  
The number of maximum delay points of a CIR was assumed to be \(N_{p}/2\). We divided the pilots into two groups according to their locations as even and odd multiples of \(\Delta p\). From (2), two groups of CFR with adjacent pilots spaced by \(2\Delta p\) were first observed, i.e.,  
\[H[2k\Delta p]=\sum_{n=0}^{N_{p}/2-1}h[n]e^{\frac{-2\pi m(2k\Delta p)}{N}},\ k=0,1 \cdots,N/(2\Delta p)-1, \tag{5}\]  
\[H[(2k+1)\Delta p]=\sum_{n=0}^{N_{p}/2-1}h[n]e^{\frac{-2\pi m(2k+1)\Delta p}{N}}, \ k=0,1\cdots,N/(2\Delta p)-1. \tag{6}\]  
The \(N_{p}/2\)-points of IDFTs of \(H[2k\Delta p]\) and \(H[(2k+1)\Delta p]\) are shown as (7) and (8), respectively:  
\[h[n]=\frac{2}{N_{p}}\sum_{k=0}^{(N_{p}/2)-1}H[2k\Delta p]e^{\frac{2\pi mk}{N_{ p}/2}},\ n=0,1,\ldots,N_{P}/2-1, \tag{7}\]  
\[h[n]=\frac{2}{N_{P}}e^{\frac{|2\pi m}{N_{p}}\sum_{k=0}^{(N_{p}/2)-1}H[(2k+1) \Delta p]e^{\frac{2\pi mk}{N_{p}/2}}},\ n=0,1,\ldots,N_{P}/2-1. \tag{8}\]  
(7) and (8) are represented by the following matrices:  
\[\mathbf{h}=\mathbf{G}_{p}\mathbf{H}_{p,even}, \tag{9}\]  
\[\mathbf{h}=\mathbf{DG}_{p}\mathbf{H}_{p,odd}, \tag{10}\]  
where  
\[\mathbf{h}=[h[0]\ h[1]\ \ldots\ h[(N_{p}/2)-1]]^{T},\]  
\[\mathbf{H}_{p,even}=[H[0]\ H[2\Delta p]\ \ldots\ H[(N_{p}-2)\Delta p]]^{T},\]  
\[\mathbf{H}_{p,odd}=[H[\Delta p]\ H[3\Delta p]\ \ldots\ H[(N_{p}-1)\Delta p]]^{T},\]  
\[\mathbf{D}=\text{diag}(1,e^{j2\pi/Np},\ldots,e^{j2\pi(Np/2-1)/Np}),\]  
\(\mathbf{G}_{p}\) is the inverse Fourier transform matrix of \(0.5N_{p}\times 0.5N_{p}\) with \((G_{p})_{i,k}=2e^{j2\pi ik/(Np/2)}/N_{p}\) and \(i,\)\(k=0,1,2,\)\(\ldots,\)\(N_{p}/2-1\). The CIR can be written as (9) and (10), so the two equations are equal as follows:  
\[H[(2k+1)\Delta p]=\sum_{n=0}^{N_{p}/2-1}h[n]e^{\frac{-2\pi m(2k+1)\Delta p}{N}}, \ k=0,1\cdots,N/(2\Delta p)-1. \tag{6}\]0 \begin{table}
\begin{tabular}{c l} \hline \hline
**Variable** & **Meaning** \\ \hline \(h[n]\) & The CIR in an OFDM symbol, \(0\leq n\leq N-1\). \\ \hline \(H[k]\) & The CFR in an OFDM symbol, \(0\leq k\leq N-1\). \\ \hline \(x[n]\) & The time-domain data transferred in an OFDM symbol, \(0\leq n\leq N-1\). \\ \hline \(X[n]\) & The frequency-domain data transferred in an OFDM symbol, \(0\leq k\leq N-1\). \\ \hline \(y[n]\) & The time-domain received data in an OFDM symbol, \(0\leq n\leq N-1\). \\ \hline \(Y[k]\) & The frequency-domain received data transferred in an OFDM symbol, \(0\leq k\leq N-1\). \\ \hline \(w[n]\) & The time-domain noise in an OFDM symbol at receiver, \(0\leq n\leq N-1\). \\ \hline \(W[k]\) & The frequency-domain noise in an OFDM symbol at receiver, \(0\leq k\leq N-1\). \\ \hline \(H[k_{p}]\) & The estimated CFRs at pilot subcarriers, \(kp\in\) pilot \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Variable definitions. The CFRs at the virtual even pilot subcarriers were selected as the target of estimation. Equation (11) was changed as follows:  
\[\mathbf{H}_{p,old}=0.5N_{p}(\mathbf{G}_{p})^{H}\mathbf{D}^{H}\mathbf{G}_{p} \mathbf{H}_{p,even}=\mathbf{A}\mathbf{H}_{p,even}, \tag{12}\]  
where (\(\cdot\))\({}^{H}\) represents the conjugate transposed operation of a matrix; \(0.5N_{p}\mathbf{G}_{p}^{H}\)\(=\)\(\mathbf{G}_{p}^{-1}\), \(\mathbf{D}^{H}=\mathbf{D}^{-1}\); and \(\mathbf{A}\) = \(0.5N_{p}(\mathbf{G}_{p})^{H}\mathbf{D}^{H}\mathbf{G}_{p}\) is a \(0.5N_{p}\times 0.5N_{p}\) matrix. For convenient representation, \(\mathbf{A}\) is expressed as follows:  
\[\mathbf{A}=\left[ \begin{array}{cc}\mathbf{A}_{1(0.25N_{p}^{out}\times 0.25N_{p} ^{out})}&\mathbf{A}_{2(0.25N_{p}^{out}\times 0.5N_{p}^{l})}&\mathbf{A}_{3(0.25N_{p}^{ out}\times 0.25N_{p}^{out})}\\ \mathbf{A}_{4(0.5N_{p}^{l}\times 0.25N_{p}^{out})}&\mathbf{A}_{5(0.5N_{p}^{l} \times 0.5N_{p}^{l})}&\mathbf{A}_{6(0.5N_{p}^{l}\times 0.25N_{p}^{out})}\\ \mathbf{A}_{7(0.25N_{p}^{out}\times 0.25N_{p}^{out})}&\mathbf{A}_{8(0.25N_{p}^{ out}\times 0.5N_{p}^{l})}&\mathbf{A}_{9(0.25N_{p}^{out}\times 0.25N_{p}^{out})} \end{array}