# A Low-Complexity Pilot-Based Frequency-Domain Channel Estimation for ICI Mitigation in OFDM Systems
## 4 LDL Based MMSE Equalization  
MMSE equalization is known to outperform LS criteria since it reduces the noise enhancement related to the latter scheme. With MMSE criteria, the detected symbol vector in the frequency domain can be expressed as follows:  
\[\underline{\mathbf{X}}=\underline{\mathbf{B}}^{H}\cdot\left(\underline{ \mathbf{B}}\cdot\underline{\mathbf{\hat{b}}}^{H}+\gamma^{-1}I_{N_{A}}\right) ^{-1}\cdot\underline{\mathbf{Y}} \tag{22}\]  
where \(\gamma\) is the signal-to-noise ratio (SNR), and \(\underline{\mathbf{\hat{b}}}\) the estimated coupling matrix. Although this scheme mitigates the ICI, the matrix inversion of (22) is complex, requiring \(O(N_{A}^{3})\) flops [31], where a flop represents a floating point operation (e.g., sum, product, division) of  
Figure 6: Sketch of proof for Remark 2: number of samples of the coupling matrix.  
two numbers. To alleviate this burden, a low-complexity \(LDL^{H}\) based MMSE equalization is used in this work. \(LDL^{H}\) factorization is a variant of classical Cholesky decomposition for Hermitian positive-definite matrices that avoids the computation of square roots related to Cholesky factorization [31]. With \(LDL^{H}\) factorization, an Hermitian positive-definite matrix, \(\underline{\mathbf{M}}\), is expressed as \(LDL^{H}\), being \(L\in\mathbb{C}^{N_{A}\times N_{A}}\) a lower unit triangular matrix, whose main diagonal elements are all 1 and \(D\in\mathbb{R}^{N_{A}\times N_{A}}\) is a diagonal matrix. As shown in [7], \(LDL^{H}\) factorization can be used to compute the inversion of the matrix \(\underline{\mathbf{M}}=\underline{\mathbf{\hat{B}}}\cdot\underline{\mathbf{ \hat{B}}}^{H}+\gamma^{-1}I_{N_{A}}\) that appears in the right hand side of (22). The involved steps are summarized as follows, [7]:  
1. Obtain the band matrix \(\underline{\mathbf{M}}=\underline{\mathbf{\hat{B}}}\cdot\underline{\mathbf{ \hat{B}}}^{H}+\gamma^{-1}\mathbf{I}_{N_{A}}\), where \(\underline{\mathbf{\hat{B}}}\) is the estimated coupling matrix as per (16).
2. Factorize \(\underline{\mathbf{M}}\) according to a \(LDL^{H}\) factorization, i.e., \(\underline{\mathbf{M}}=\underline{\mathbf{L}}\cdot\underline{\mathbf{D}} \cdot\underline{\mathbf{L}}^{H}\), being \(\underline{\mathbf{D}}\) a diagonal matrix and \(\underline{\mathbf{L}}\) a lower triangular matrix whose bandwidth is \(2Q\).
3. Invert the matrix \(\underline{\mathbf{M}}\), first, solving the system \(\underline{\mathbf{L}}\cdot\underline{\mathbf{f}}=\underline{\mathbf{Y}}\), which leads to \(\underline{\mathbf{f}}\); secondly, using \(\underline{\mathbf{f}}\) computed above and solving the system \(\underline{\mathbf{D}}\cdot\underline{\mathbf{g}}=\underline{\mathbf{f}}\), which leads to \(\underline{\mathbf{g}}\); and finally, using \(\underline{\mathbf{g}}\) computed above and solving the system \(\underline{\mathbf{L}}^{H}\underline{\mathbf{d}}=\underline{\mathbf{g}}\).
4. Estimate the transmitted vector, \(\underline{\mathbf{\Sigma}}\), as \(\underline{\mathbf{\hat{X}}}=\underline{\mathbf{\hat{B}}}^{H}\mathbf{d}\).  
As proven in [7], the \(LDL^{H}\) based MMSE decomposition reduces the complexity down to \(O(Q^{2}N_{A})\) flops, which is a considerable reduction compared to the direct MMSE criteria of (22).