## 3 Prefiltered Frequency-Domain Equalization
### Symbol Detection  
For the modulation index \(h=1/2\), the equalized signal \(\hat{b}_{n}\) obtained by taking the inverse DFT on \(\hat{B}_{k}\) can be de-mapped by  
\[a_{n}=\hat{b}_{n}\exp(j\frac{\pi}{2}(n+1)) \tag{13}\]  
Because the real part \(\text{Re}(a_{n})\) and imaginary part \(\text{Im}(a_{n})\) of \(a_{n}\) contain the information regarding the current and previous moments, respectively, \(\text{Re}(a_{n})\) and \(\text{Im}(a_{n})\) are extracted, respectively, and \(\text{Im}(a_{n})\) is delayed one symbol. Denote \(y_{n}\in\){\(\text{Re}(a_{n})\), \(\text{Im}(a_{n})\)} and assume that \(\text{Re}(a_{n})\) and \(\text{Im}(a_{n})\) satisfy the Gaussian distribution after energy normalization. Then, \(y_{n}\) can be expressed as follows:  
\[y_{n}=d_{n}+\sigma^{2} \tag{14}\]  
where \(d_{n}\in\){\(\pm 1\)} and \(\sigma^{2}\) denotes the variance of the output noise. Hence, the conditional probability distribution function of \(y_{n}\) can be expressed as follows:  
\[\text{P}(y_{n}=d_{n}|\hat{x}_{n})=\frac{1}{\sqrt{2\pi}\sigma}\exp(-\frac{(y_{n }-d_{n})^{2}}{2\sigma^{2}}) \tag{15}\]  
Therefore, the log-likelihood estimate of the symbol \(u_{n}\), which is to be estimated, is given by\[ \begin{array}{ll}\mathrm{L}(u_{n})&=\mathrm{L}(\mathrm{Re}(a_{n})=d_{n}|\hat{x}_ {n})\ +\ \mathrm{L}(\mathrm{Im}(\mathit{a}_{n+1})\ =d_{n}|\hat{x}_{n})\\ &=\mathrm{ln}\frac{P(\mathrm{Re}(a_{n})\ =\ 1|\hat{x}_{n})P(\mathrm{Im}(a_{n+1}) \ =\ 1|\hat{x}_{n})}{P(\mathrm{Re}(a_{n})\ =\ -1|\hat{x}_{n})P(\mathrm{Im}(a_{n+1})\ =\ -1|\hat{x}_{n})}\end{array}  \tag{16}\]  
The transmission information \(\hat{x}_{n}\) can be obtained from a hard decision on \(u_{n}\). \[\hat{x}_{n}=\left\{ \begin{array}{ll}1,\ \ u_{n}u_{n-1}>0\\ 0,\ \ u_{n}u_{n-1}<0\end{array} \right. \tag{17}\]  
In the above calculation, \(\sigma\) can be calculated by the known CP information at the tail of the sequence \(\mathbf{r}\). The likelihood equation about \(\sigma\) is given by  
\[\frac{\partial}{\partial\sigma^{2}}\ln f(y_{n};b_{n},\sigma^{2})=-\frac{\mathrm{ n}}{2\sigma^{2}}-\frac{1}{2{(\sigma^{2})}^{2}}\sum_{i=N-M}^{n}{(y_{i}-b_{i})^{2}}=0 \tag{18}\]  
In addition, the maximum likelihood estimate of \(\sigma^{2}\) can be calculated by  
\[\sigma^{2}=\frac{1}{M}\sum_{i=N-M}^{N-1}{(y_{i}-b_{i})^{2}} \tag{19}\]  
For binary CPM signals whose modulation index is not \(1/2\), the symbol detection can be simplified [9]:  
\[\widetilde{y}_{n}=\frac{1}{\pi\hbar}\mathrm{arg}(\hat{b}_{n}\hat{b}_{n-1}^{*}) \tag{20}\]  
Based on the modulation index, the theoretical phase value of \(x_{n}\) can be defined as \(\widetilde{d}_{n}=\pm\pi\hbar t\), and the conditional probability distribution function of \(\widetilde{y}_{n}\) is obtained:  
\[\mathrm{P}(y_{n}=\widetilde{d}_{n}|\hat{x}_{n})=\frac{1}{\sqrt{2\pi\sigma}} \exp(-\frac{{(y_{n}-\widetilde{d}_{n})}^{2}}{2\sigma^{2}}) \tag{21}\]  
Moreover, the theoretical phase value of \(\hat{b}_{n}\) without forward information can be given by \(\mathrm{arg}(\widetilde{b}_{n})\in\{k\pi\hbar\}\), where \(k\) is an integer. Considering that the forward information exists in \(\hat{b}_{n}\) as phase, we can describe the phase interference caused by the forward information as follows:  
\[\gamma\triangleq\arctan(\frac{g(2)\sin(\pi\hbar)}{1+g(2)\cos(\pi\hbar)}) \tag{22}\]  
where \(g\) is the transfer function. The conditional probability distribution function of forward information can be expressed as follows:  
\[\mathrm{P}(\mathrm{arg}(b_{n+1})-\mathrm{arg}(\widetilde{b}_{n+1})=\gamma \widetilde{d}_{n}|\hat{x}_{n})=\frac{1}{\sqrt{2\pi\sigma_{b}}}\exp(-\frac{{( \mathrm{arg}(b_{n+1})-\mathrm{arg}(\widetilde{b}_{n+1})-\gamma\widetilde{d}_ {n})}^{2}}{2\sigma_{b}^{2}}) \tag{23}\]  
Then, the transmission information \(\hat{x}_{n}\) can be obtained from a hard decision on the log-likelihood estimate \(\mathrm{L}(u_{n})\) of the symbol.