## 3 Receiver Design Based on Fractional Fourier Transform  
Figure 2 shows a block diagram of the proposed method. Firstly, a hopping pattern is generated and multiplied by the modulated transmission signal to perform frequency hopping, and the receiver utilizes a bandpass filter with a known hopping pattern. Then, the compression or expansion effect of the signal due to the Doppler shift can be corrected by introducing the matched filter results between the symbol estimation through the FrFT spectrum and the received signal. In the example shown in Figure 1, there are four symbols in a single hopping frequency interval. Each symbol is divided into a low-frequency band and a high-frequency band. The symbol with an up-slope in the low-frequency band is called \"Symbol Mapping #1\", and the symbol with a down-slope is called \"Symbol Mapping #2\". The symbol with up-slope in the high-frequency band is called \"Symbol Mapping #3\", and the symbol with down-slope is called \"Symbol Mapping #4\". Figure 2(a) shows the matched filter result, which is from the auto-correlation function of the \"Symbol Mapping #1\" in any hopping pattern. It can be seen that the output of the matched filter has a maximum value at the ideal point. Figure 2(b) shows the matched filter result, which is the cross-correlation function between the \"Symbol Mapping #1\" and \"Symbol Mapping #2\" in any hopping pattern. Since \"Symbol Mapping #1\" and \"Symbol Mapping #2\" have different slopes, the maximum point of the matched filter result cannot be correctly represented, and the magnitude is less than the ideal value [29]. Figure 3: _Cont._  
Figure 2: Block diagram of the proposed method. Figure 4a shows the matched filter result from the cross-correlation function between \"Symbol Mapping #1\" and \"Symbol Mapping #3\" in any hopping pattern. The mapping types have the same slopes but show a lower magnitude because the frequency bands are different. Figure 4b shows the matched filter result as a cross-correlation function between \"Symbol Mapping #1\" and \"Symbol Mapping #4\" in any hopping pattern. The mapping types have the same frequency, but show lower values due to their different slopes. Figure 4: Matched filter output in different frequency bands. (**a**) The same slope symbol; (**b**) different slope symbols. Figure 3: Matched filter output in the same frequency band. (**a**) The same slope symbol; (**b**) different slope symbols. In this paper, we propose a method to estimate the symbol in the received signal by applying the FrFT scheme. The FrFT is a generalization of the conventional Fourier transform concept, which makes it possible to convert the time domain signal into a time-frequency intermediate domain. That is, when the time domain is represented by the \(x\)-axis and the frequency domain by the \(y\)-axis, the FrFT is the result obtained by rotating an arbitrary angle \(\mathcal{O}\) in the counterclockwise direction. The FrFT can obtain the spectra with various characteristics depending on the transform orders defined as \(\alpha=2\mathcal{O}/\pi\). If the different signals have different frequency slopes, the signals will have peaks at different positions with different transform orders on the FrFT spectrum. In this way, the correct transform order with peaks in each FrFT spectrum is called the optimal transform order, which is denoted by \(\alpha_{opt}\)[27; 30]. The FrFT equation is  
\[F_{a}f(u)=\int K_{\alpha}(u,x)f(x)dx, \tag{7}\]  
where the term \(K_{\alpha}(u,x)\) is expressed as  
\[K_{\alpha}(u,x)=\left\{ \begin{array}{c}A_{\alpha}\exp\bigl{\{}i\pi\bigl{[} \cot(\alpha)\bigl{(}x^{2}+u^{2}\bigr{)}-2\csc(\alpha)ux\bigr{]}\bigr{\}},\ (\alpha\
eq n\pi)\\ \delta(u-x)\ (\alpha=2n\pi)\\ \delta(u+x)\ (\alpha=(2n\pm 1)\pi)\end{array} \right., \tag{8}\]  
where \(A_{\alpha}\) is the amplitude factor defined as \(\sqrt{1-icot(\alpha)}\). If \(\varnothing=\pi/2\), the FrFT is equivalent to the Fourier transform.