## 3 Receiver Design Based on Fractional Fourier Transform
Next, the optimal transform order in a discrete region can be expressed as  
\[\alpha_{opt}=\frac{2}{\pi}\varnothing=\frac{2}{\pi}cot^{-1}\biggl{(}\frac{Mk }{f_{s}{}^{2}}\biggr{)}, \tag{9}\]  
where \(f_{s}\) is the sampling frequency, \(k\) is the chirp rate, and \(M\) represents the sampling point. If the sampling points and sampling frequency are fixed, the optimal transform order is determined from the value \(k\). Figure 5a shows a conceptual diagram of the FrFT for a chirp signal. In this case, we can assume that the chirp signal \(x_{u}(t)\) with the up-slope characteristic exists for the analysis period \(T\), and the chirp signal \(x_{u}(t)\) has the initial frequency \(f_{0}\) at the start time \((k\tau-\Delta T/2)\) and the frequency \(f_{1}\) at the end time \((k\tau+\Delta T/2)\) (generally, the frequency \(f_{1}\) is higher than the frequency \(f_{0}\)). Therefore, the point \(o_{u}\) which is the middle point between \(\mathrm{p}_{u}=\bigl{[}(k\tau-\Delta T/2)/\mathrm{s},\ f_{0}\mathrm{s}\bigr{]}\) and point \(\mathrm{q}_{u}=\bigl{[}(k\tau+\Delta T/2)/\mathrm{s},\ f_{1}\mathrm{s}\bigr{]}\) can be expressed as \([k\tau/\mathrm{s},\ f_{m}\mathrm{s}]\), where \(f_{m}\) is the center frequency and \(\mathrm{s}\) is the rescaling factor, and is defined as \(\mathrm{s}=T/f_{s}\). The \(k\) is the symbol number. The vector \(u_{0}\), which is the point \(o_{u}\) that corresponds to the u-axis, is  
\[u_{0}=|o_{u}|\sin\Bigl{(}\alpha_{opt}\frac{\pi}{2}\Bigr{)}=\biggl{[}f_{m} \mathrm{s}-\frac{k\tau}{\mathrm{s}}\biggr{]}\sin\Bigl{(}\alpha_{opt}\frac{ \pi}{2}\Bigr{)}. \tag{10}\]  
When the starting point for the signal \(x_{u}(t)\) is \(\tau=0\), the vector \(u_{0}\) can be written as:  
\[u_{0}=(f_{m}s)\sin\big{(}\alpha_{opt}\frac{\pi}{2}\bigr{)}. \tag{11}\]  
When considered in the frequency domain, the signal also has a negative frequency that is inversely proportional to the positive frequency. The negative frequency exists symmetrically with the positive frequency and forms a relatively wide and low spectrum on the FrFT as [\(p^{\prime}_{u}\)_to_\(q^{\prime}_{u}\)] in Figure 5a. On the other hand, in Figure 5b, the chirp signal \(x_{d}(t)\) with the down-slope characteristic will appear symmetrically with the signal \(x_{u}(t)\). Therefore, the vector \(u_{1}\) can be written as  
\[u_{1}=\bigl{(}-f^{\prime}_{m}s\bigr{)}\sin\Bigl{(}\alpha_{opt}\frac{\pi}{2} \Bigr{)}, \tag{12}\]where \(f_{m}^{\prime}\) is the center frequency of the signal \(x_{d}(t)\). If the signal \(x_{u}(t)\) and \(x_{d}(t)\) are in the same frequency band and have the same slope, the vector \(u_{1}\) can be rewritten as  
\[u_{1}=(-f_{m}s)\sin\Bigl{(}a_{opt}\frac{\pi}{2}\Bigr{)}=-u_{0}. \tag{13}\]  
The opposite characteristics of FrFT can be seen from Equation (13). If the center frequency and the frequency variation are same, the positions of the vectors \(u_{0}\) and \(u_{1}\) are the opposite and the same value. Figure 5(a) shows the symmetrical characteristics in the fractional domain to distinguish between the up-slope and the down-slope. This is an example of the FrFT spectrum for Figure 4(a). In the proposed method, with respect to the middle line of the fractional spectrum, the left side is the positive frequency domain, and the right side is the negative frequency domain (theoretically existing). Therefore, in the up-slope state (like Figure 4(a)), the energies are concentrated on the left side, and in the down-slope state (like Figure 4(b)), the energies are concentrated on the right side. On the other  
Figure 5: Relation between the chirp signal and FrFT coefficient \(\alpha\). (**a**) The up-slope chirp signal; (**b**) the down-slope chirp signal. hand, Figure 6b shows the FrFT spectrum of the conventional method that coexists with the up-slope and the down-slope in the symbol duration. Unlike the proposed method, the conventional method is processed only in the positive frequency domain and distinguishes between the up-slope and the down-slope by using the two transform orders which are dependent on each symbol.