# 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{19}\]  
As the target of estimation is the CFRs at the virtual odd pilot subcarriers, the unknown CFRs at the virtual subcarriers part in the left term \(\mathbf{H}_{p,even}\) of (18) were removed, and the equation at the right-hand side of (18) was divided into two parts: CFRs at virtual and non-virtual pilot subcarriers. Equation (18) can be changed to the following:  
\[\mathbf{H}_{p,even}^{out}-\mathbf{B}_{\mathrm{C},R}^{out,out}\mathbf{H}_{p,odd }^{out}=\mathbf{B}_{\mathrm{C},R}^{out,in}\mathbf{H}_{p,odd}^{in}\]  
where  
\[\mathbf{H}_{p,odd}^{in}=\left[ \begin{array}{cc}H[(N_{p}^{L}+1)\Delta p]&H[( N_{p}^{L}+3)\Delta p]&\cdots&H[(N_{p}^{L}+N_{p}^{I}-1)\Delta p]\end{array}  \right]^{T}\]  
\(\mathbf{H}_{p,odd}^{in}\) is the CFR vector at the virtual odd pilot subcarriers that we aimed to estimate. \(\mathbf{B}_{\mathrm{C},R}^{out,out}\) is a \(0.5N_{p}^{out}\times 0.5N_{p}^{out}\) matrix and  
\[\mathbf{B}_{\mathrm{C},R}^{out,out}=\left[ \begin{array}{cc}\mathbf{B}_{1}& \mathbf{B}_{3}\\ \mathbf{B}_{7}&\mathbf{B}_{9}\end{array} \right]\text{. }\]  
\(\mathbf{B}_{\mathrm{C},R}^{out,in}\) is a \(0.5N_{p}^{out}\times 0.5N_{p}^{I}\) matrix and  
\[\mathbf{B}_{\mathrm{C},R}^{out,in}=\left[ \begin{array}{c}\mathbf{B}_{2}\\ \mathbf{B}_{8}\end{array} \right]\text{. }\]\(\mathbf{B}_{C,R}^{out,out}\) and \(\mathbf{B}_{C,R}^{out,in}\) are two known matrices. Afterward, the LS method was used to estimate the CFRs at virtual odd pilot subcarriers, expressed as follows:  
\[\widetilde{\mathbf{H}}_{p,odd}^{in}=[(\mathbf{B}_{C,R}^{out,in})^{H}\mathbf{B}_{ C,R}^{out,in}+\alpha_{o}\mathbf{I}]^{-1}(\mathbf{B}_{C,R}^{out,in})^{H}( \hat{\mathbf{H}}_{p,even}^{out}-\mathbf{B}_{C,R}^{out,out}\hat{\mathbf{H}}_{p,odd}^ {out}) \tag{20}\]  
where \(\hat{\mathbf{H}}_{p,even}^{out}\) and \(\hat{\mathbf{H}}_{p,odd}^{out}\) are the CFRs estimated by (4). The regularization parameter \(\alpha_{o}\) is used to overcome the problem that \((\mathbf{B}_{C,R}^{out,in})^{H}\mathbf{B}_{C,R}^{out,in}\) approaches a singular matrix. Equation (20) was not directly used in actual hardware implementation. The right side of Equation (20) was expanded as follows:  
\[\widetilde{\mathbf{H}}_{p,odd}^{in}=[(\mathbf{B}_{C,R}^{out,in})^{H}\mathbf{B}_ {C,R}^{out,in}+\alpha_{o}\mathbf{I}]^{-1}(\mathbf{B}_{C,R}^{out,in})^{H}\hat{ \mathbf{H}}_{p,even}^{out}-[(\mathbf{B}_{C,R}^{out,in})^{H}\mathbf{B}_{C,R}^{ out,in}+\alpha_{o}\mathbf{I}]^{-1}(\mathbf{B}_{C,R}^{out,in})^{H}\mathbf{B}_{C,R}^{ out,out}\hat{\mathbf{H}}_{p,odd}^{out} \tag{21}\]  
\[[(\mathbf{B}_{C,R}^{out,in})^{H}\mathbf{B}_{C,R}^{out,in}+\alpha_{o}\mathbf{I}] ^{-1}(\mathbf{B}_{C,R}^{out,in})^{H}\text{ and }[(\mathbf{B}_{C,R}^{out,in})^{H}\mathbf{B}_{C,R}^{ out,in}+\alpha_{o}\mathbf{I}]^{-1}(\mathbf{B}_{C,R}^{out,in})^{H}\mathbf{B}_{C,R}^{ out,out}\]  
calculated 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 (21) is much smaller than that required by (20). According to the results of (4), (17), and (21), the CFRs vector \(\hat{\mathbf{H}}_{p,PLS}\) at the pilot and virtual pilot subcarriers can be obtained. The CIR \(\hat{\mathbf{R}}_{PLS}\) of the \(N_{p}\) points can be calculated by the \(N_{p}\)-points IFFT of \(\hat{\mathbf{H}}_{p,PLS}\). According to the assumptions of this paper, the number of maximum delay points of a CIR is smaller than or equal to half the number of pilots (including virtual pilot) (i.e., \(0.5N_{p}\) points). Therefore, the values of \(0.5N_{p}\) points of the second half of \(\hat{\mathbf{R}}_{PLS}\) can be regarded as the estimation error and discarded; that is, let the values of \(0.5N_{p}\) points of the second half of \(\hat{\mathbf{R}}_{PLS}\) be zero to obtain \(\hat{\mathbf{h}}_{PLS}\). The complete CFR \(\hat{\mathbf{H}}_{PLS}\) can be obtained by \(N\)-points DFT of \(\hat{\mathbf{h}}_{PLS}\).