# Low Complexity Equalization of Orthogonal Chirp Division Multiplexing in Doubly-Selective Channels
## 3 Low Complexity Equalizers for UP-OCDM under Doubly Selective Channels
### Band MMSE Block Equalizer
#### 3.2.1 Band MMSE Equalization Algorithm  
At the receiver, we perform an FFT algorithm on the received signal first, and we have  
\[\mathbf{r}_{F} =\mathbf{Fr} \tag{18}\] \[=\mathbf{FHs}+\mathbf{Fw}\] \[=\mathbf{FHF}^{H}\mathbf{F\Phi}\mathbf{F}^{H}\mathbf{Fd}+\mathbf{w}_{F}\] \[=\mathbf{H}_{E}\mathbf{d}_{F}+\mathbf{w}_{F},\]  
where \(\mathbf{H}_{E}=\mathbf{FH}\mathbf{F}^{H}\mathbf{F\Phi}\mathbf{F}^{H}\) is the equivalent channel matrix between \(\mathbf{r}_{F}\) and \(\mathbf{d}_{F}\), and \(\mathbf{d}_{F}=\mathbf{Fd}\) is the transmitted symbols in the frequency domain. Therefore, the MMSE estimate of \(\mathbf{d}_{F}\) is  
\[\mathbf{\hat{d}}_{F-MMSE}=\mathbf{H}_{E}^{H}(\mathbf{H}_{E}\mathbf{H}_{E}^{H}+\gamma^{-1}\mathbf{I} _{N})^{-1}\mathbf{r}_{F}, \tag{19}\]  
where \(\gamma={\sigma_{\mathbf{d}_{F}}}^{2}/{\sigma_{\mathbf{w}_{F}}}^{2}={\sigma_{\mathbf{d}}}^{ 2}/{\sigma_{\mathbf{w}}}^{2}\). It can be seen from (19) that, the computational complexity of obtaining \(\mathbf{\hat{d}}_{FMMSE}\) is also \(\mathcal{O}(N^{3})\), which is the same as its time-domain counterpart in Section 3.1. Note that \(\mathbf{H}_{E}\) is the product of \(\mathbf{FHF}^{H}\) and \(\mathbf{F\Phi}\mathbf{F}^{H}\), where \(\mathbf{FHF}^{H}\) is the channel frequency response (CFR) matrix and is denoted as \(\mathbf{H}_{F}=\mathbf{FHF}^{H}\) in this paper. We will show that by employing the approximately banded structure of the CFR matrix of doubly selective channels and the circulant structure of \(\mathbf{\Phi}\), \(\mathbf{H}_{E}\) can be approximated as a banded matrix, so that the complexity of MMSE equalization would be considerably reduced. Firstly, due to the presence of severe intercarrier interference (ICI) in doubly selective channels, \(\mathbf{H}_{F}\) is not diagonal as in the time-invariant scenarios; see Reference [22]. Whereas, with a relatively high Doppler spread factor, \(\mathbf{H}_{F}\) is shown nearly banded, and each diagonal is associated with a discrete Doppler frequency introducing ICI [23]. In this paper, we denote the banded approximation matrix of \(\mathbf{H}_{F}\) by \(\mathbf{H}_{B}\). We assume the number of subdiagonals and superdiagonals retained from \(\mathbf{H}_{F}\) is \(Q\), so that the bandwidth of \(\mathbf{H}_{B}\) is \(2Q+1\). It is easy to figure out that, the higher \(Q\) we choose, the smaller approximation error matrix \(\mathbf{H}_{B}\) has. With a chosen \(Q\), \(\mathbf{H}_{B}\) can be obtained by  
\[\mathbf{H}_{B}=\mathbf{H}_{F}\odot\mathbf{T}, \tag{20}\]  
where \(\odot\) is the element-wise product, and \(\mathbf{T}\) is an \(N\times N\) banded matrix, with its lower and upper bandwidth both equal to \(Q\), and all elements inside the band set as one. Secondly, with the circulant structure of \(\mathbf{\Phi}\), it can be easily proved that \(\mathbf{F\Phi}\mathbf{F}^{H}\) is a diagonal matrix with the \((k,k)\)th entry equaling to \(e^{j\pi(\frac{i^{2}}{N}-\frac{1}{4})}\). We define \(\mathbf{\Lambda}=\mathbf{F\Phi}\mathbf{F}^{H}\), therefore \(\mathbf{H}_{E}=\mathbf{H}_{F}\mathbf{\Lambda}\). Note that the modulus of the diagonal elements of \(\mathbf{\Lambda}\) equals to 1, therefore, the modulus of the \((i,j)\)th entry of \(\mathbf{H}_{E}\) equals to that of \(\mathbf{H}_{F}\), which means \(\mathbf{H}_{E}\) can also be approximated as a banded matrix with bandwidth \(2Q+1\). We denote the band approximation of \(\mathbf{H}_{E}\) as \(\mathbf{E}\), so that \(\mathbf{E}=\mathbf{H}_{E}\odot\mathbf{T}=\mathbf{H}_{B}\mathbf{\Lambda}\). Therefore, the matrix \((\mathbf{E}\mathbf{H}^{H}+\gamma^{-1}\mathbf{I}_{N})\) in (19) is also a banded matrix with bandwidth \(2Q+1\), and is denoted as \(\mathbf{R}\) in this paper. Thus, the inversion of \(\mathbf{R}\) can be computed by low complexity algorithms. Note that \(\mathbf{R}\) is symmetric, in this paper, we solve the inversion of \(\mathbf{R}\) via LDL\({}^{H}\) factorization, which is reported below.