# A Low-Complexity Channel Estimation Based on a Least-Squares Algorithm in OFDM Systems
## 3 The Proposed Method Based on the LS Method
\right] \tag{13}\]  
In this case, we assume that \(N_{p}^{l}\) and \(N_{p}^{l}\) are both even and equal and \(N_{p}^{out}\) is a multiple of 4. So \(N_{p}^{l}\) is also even. Otherwise, the matrix dimensions of \(\mathbf{A}_{1}\) to \(\mathbf{A}_{9}\) will vary according to different conditions. As the target of estimation is the CFRs at the virtual even pilot subcarriers, the unknown CFRs at the virtual subcarriers part in the left term \(\mathbf{H}_{p,old}\) of (12) were removed, and the equation at the righthand side of (12) was divided into two parts: CFRs at virtual and non-virtual pilot subcarriers. Equation (12) can be changed to the following:  
\[\mathbf{H}_{p,old}^{out}-\mathbf{A}_{C,R}^{out,out}\mathbf{H}_{p,even}^{out}= \mathbf{A}_{C,R}^{out,in}\mathbf{H}_{p,even}^{in} \tag{14}\]  
where  
\[\mathbf{H}_{p,old}^{out}=\left[\begin{array}{cc}H[\Delta p]&H[3\Delta p]& \cdots&H[(N_{p}^{l}-1)\Delta p]&H[(N_  
the regularization parameter before the calculation of the inverse matrix. Thus, Equation (15) was changed to:  
\[\widetilde{\mathbf{H}}_{p,even}^{in}=[(\mathbf{A}_{\mathrm{C},R}^{out,in})^{H} \mathbf{A}_{\mathrm{C},R}^{out,in}+\mathbf{\alpha}_{e}\mathbf{I}]^{-1}(\mathbf{ A}_{\mathrm{C},R}^{out,in})^{H}(\mathbf{\hat{H}}_{p,odd}^{out}-\mathbf{A}_{ \mathrm{C},R}^{out,out}\mathbf{\hat{H}}_{p,even}^{out}) \tag{16}\]  
where \(\mathbf{\alpha}_{e}\) is the regularization parameter, which is a constant; \(\mathbf{I}\) is a \(0.5N_{p}^{I}\times 0.5N_{p}^{I}\) identity matrix; and \(\widetilde{\mathbf{H}}_{p,even}^{in}\) is the final result estimated with the regularization parameter. However, (16) is not directly used for operation in actual hardware realization. The right side of Equation (16) was expanded as follows:  
\[\widetilde{\mathbf{H}}_{p,even}^{in}=[(\mathbf{A}_{\mathrm{C},R}^{out,in})^{H} \mathbf{A}_{\mathrm{C},R}^{out,in}+\mathbf{\alpha}_{e}\mathbf{I}]^{-1}( \mathbf{A}_{\mathrm{C},R}^{out,in})^{H}\mathbf{\hat{H}}_{p,odd}^{out}-[( \mathbf{A}_{\mathrm{C},R}^{out,in})^{H}\mathbf{A}_{\mathrm{C},R}^{out,in}+ \mathbf{\alpha}_{e}\mathbf{I}]^{-1}(\mathbf{A}_{\mathrm{C},R}^{out,in})^{H} \mathbf{A}_{\mathrm{C},R}^{out,out}\mathbf{\hat{H}}_{p,even}^{out} \tag{17}\]  
\[[(\mathbf{A}_{\mathrm{C},R}^{out,in})^{H}\mathbf{A}_{\mathrm{C},R}^{out,in}+ \mathbf{\alpha}_{e}\mathbf{I}]^{-1}(\mathbf{A}_{\mathrm{C},R}^{out,in})^{H} \mathbf{A}_{\mathrm{C},R}^{out,in}+\mathbf{\alpha}_{e}\mathbf{I}]^{-1}( \mathbf{A}_{\mathrm{C},R}^{out,in})^{H}\mathbf{A}_{\mathrm{C},R}^{out,out}\]  
worked out in advance and stored in the memory. Both of them are \(0.5N_{p}^{I}\times 0.5N_{p}^{out}\) matrices, and each has \(0.25N_{p}^{I}N_{p}^{out}\) complex numbers to be stored in the memory. The number of multiplication operations required by (17) is much smaller than that required by (16). Afterward, the CFRs at the virtual odd pilot subcarriers were estimated. Equation (11) was changed as in Equation (12) to obtain the following equation:  
\[\mathbf{H}_{p,even}=0.5N_{p}(\mathbf{G}_{p})^{H}\mathbf{D}\mathbf{G}_{p} \mathbf{H}_{p,odd}=\mathbf{B}\mathbf{H}_{p,odd}, \tag{18}\]  
where \(\mathbf{B}\) is a \(0.5N_{p}\times 0.5N_{p}\) matrix and \(\mathbf{B}=0.5N_{p}(\mathbf{G}_{p})^{H}\mathbf{D}\mathbf{G}_{p}\). For convenient representation, \(\mathbf{B}\) is expressed as follows:  
\[\mathbf{B}=\left[ \begin{array}{cc}\mathbf{B}_{1(0.25N_{p}^{out}\times 0.25N_{p}^{ out})}&\mathbf{B}_{2(0.25N_{p}^{out}\times 0.5N_{p}^{I})}&\mathbf{B}_{3(0.25N_{p}^{ out}\times 0.25N_{p}^{out})}\\ \mathbf{B}_{4(0.5N_{p}^{I}\times 0.25N_{p}^{out})}&\mathbf{B}_{5(0.5N_{p}^{I} \times 0.5N_{p}^{I})}&\mathbf{B}_{6(0.5N_{p}^{I}\times 0.25N_{p}^{out})}\\ \mathbf{B}_{7(0.25N_{p}^{out}\times 0.25N_{p}^{out})}&\mathbf{B}_{8(0.25N_{p}^{ out}\times 0.5N_{p}^{I})}&\mathbf{B}_{9(0.25N_{p}^{out}\times 0.25N_{p}^{out})}\end{array}