## 3 Design of Novel Single-Channel Structure for CTS
Since \(f\left(t\right)\) is real, Equation (10) can be written as [20]  
\[S\left(t\right)=\prod\left(\frac{t-t^{\prime}}{T^{\prime\prime}}\right)\left( X\left(\Omega\right)+X^{\ast}\left(\Omega\right)\right), \tag{12}\]  
where \(X\left(\Omega\right)\) is the Fourier transform of \(x\left(t\right)\); and Equation (12) can be further written as  
\[S\left(t\right)=\int_{-\infty}^{+\infty}f\left(\tau\right)C_{1}\left(\tau \right)H_{0}\left(t-\tau\right)d\tau. \tag{5}\]0  
where Re means the real part and \(\hat{F}\left(\Omega\right)=\hat{F}\left(\omega_{0}-\omega_{1}+2\pi\mu t\right)\) is the Fourier transform of \(\hat{F}\left(t\right)\). Considering \(\hat{F}\left(\Omega\right)\) in form of the magnitude and phase angle  
\[S\left(t\right)=\int_{-\infty}^{+\infty}f\left(\tau\right)C_{1}\left(\tau \right)H_{0}\left(t-\tau\right)d\tau. \tag{5}\]1  
Equation (13) becomes  
\[S\left(t\right)=\int_{-\infty}^{+\infty}f\left(\tau\right)C_{1}\left(\tau \right)H_{0}\left(t-\tau\right)d\tau. \tag{5}\]2  
Equation (15) indicates that the envelope of the signal \(s\left(t\right)\) resulting from the SAW filter is the amplitude spectrum of the (truncated) real input signal \(f\left(t\right)\). It is important to stress that the gating function shown in Equation (15) is sliding across \(f\left(t\right)\), only embracing different parts of \(f\left(t\right)\) while at the same time performing Fourier transformation in a time-ordered manner [20]. For the classical CTS structure described in [22], the time duration of the premultiplying chirp signals (20 us) is twice that of the SAW filters (10 us), thus, only half of the input signal is measured according to Equation (15) for one \(M\left(l\right)-C\left(s\right)\) arrangement. So, the push-pull two-channel structure is adopted to obtain the full duty cycle of the expander-compressor arrangement. In the single-channel structure, an additional up-and-down conversion mixing process is adopted. So, the output of the SAW filter for the additional up-and-down conversion signals can be written as  
\[S\left(t\right)=\int_{-\infty}^{+\infty}f\left(\tau\right)C_{1}\left(\tau \right)H_{0}\left(t-\tau\right)d\tau. \tag{5}\]3  
where \(\cos\left(2\pi\mu T_{0}\tau\right)\) means the added mixer with a fixed LO input frequency of \(\mu T_{0}\). Equation (16) can be further written as  
\[S\left(t\right)=\int_{-\infty}^{+\infty}f\left(\tau\right)C_{1}\left(\tau \right)H_{0}\left(t-\tau\right)d\tau. \tag{5}\]4  
where  
\[S\left(t\right)=\int_{-\infty}^{+\infty}f\left(\tau\right)C_{1}\left(\tau \right)H_{0}\left(t-\tau\right)d\tau. \tag{5}\]5  
Let \(\Omega_{1}=\omega_{0}-\omega_{1}+2\pi\mu\left(t\pm T_{0}\right)\) and \(x_{1}\left(\tau\right)=\hat{f}\left(\tau\pm T_{0}\right)\cdot\exp\left[j\left( \omega_{0}t+\pi\mu t^{2}+\phi_{0}+\phi_{1}\right)\right]\); Equation (17) can be written as  
\[S\left(t\right)=\int_{-\infty}^{+\infty}f\left(\tau\right)C_{1}\left(\tau \right)H_{0}\left(t-\tau\right)d\tau. \tag{5}\]6  
where \(X_{1}\left(\Omega\right)\) is the Fourier transform of \(x_{1}\left(t\right)\). This indicates that \(M\left(l\right)-C\left(s\right)\) arrangement with the additional up-and-down conversion mixing process essentially represents the Fourier transform of \(\hat{f}\left(\tau\pm T_{0}\right)\)--meaning the previous and later parts of \(f\left(t\right)\) with time interval of \(T_{0}\). This actually has the same effect with the push-pull two-channel structure of obtaining the full duty cycle of the expander-compressor arrangement. Here, we assume that the working parameters are the same as that in [22]. The schematic diagram with corresponding temporal relations of the novel structure is shown in \[S\left(t\right)=\int_{-\infty}^{+\infty}f\left(\tau\right)C_{1}\left(\tau \right)H_{0}\left(t-\tau\right)d\tau. \tag{5}\]7  
In Figure 6, two input signals with frequencies of 2.3 GHz and 2.1 GHz are modulated by a chirp signal firstly and then are up-and-down converted. The obtained up-and-down conversion modulated chirp signals will be compressed into output pulses though the SAW filter.