## 3 Design of Novel Single-Channel Structure for CTS  
To deal with the matching problem and simplify the structure, a single-channel arrangement is developed. The circuit frame diagram of the developed novel single-channel arrangement is demonstrated in Figure 5. In the new structure, one channel is replaced by only adding a small number of devices. The front part used for signal expansion is the same as the classical CTS structure, and the extension signal is then divided into two parts by the power divider. One is used for signal mixing, and the outputs have both the up-and-down conversion signals. This part is then combined with another part to obtain a complete signal distribution over the entire measuring time. At last, the complete modulated chirp signals resulting from the combiner are delivered to the SAW device for pulse compression. The signal send to the LO port of the mixer is a sine signal with a fixed frequency that is equal to the compression bandwidth. A matching network is added to adjust the little difference of the line delay and eliminate the influence of the added mixer. Usually, in real hardware implementation, the dispersion of mixer, power splitter, and combiner is very small and the introduced mismatch can be ignored. In the \(M\left(l\right)-C\left(s\right)\) arrangement, the real input signal is firstly premultiplied by the chirp signal \(C_{1}\left(t\right)\) for signal expansion, then passed though the SAW filter \(H_{0}\left(t\right)\). The output of the filter is given by a convolution integral  
\[S\left(t\right)=\int_{-\infty}^{+\infty}f\left(\tau\right)C_{1}\left(\tau \right)H_{0}\left(t-\tau\right)d\tau. \tag{5}\]  
Figure 5: Frame diagram of the novel single-channel arrangement. Considering that the impulse response of the premultiplying chirp \(C_{1}\left(t\right)\) and the SAW filter \(H_{0}\left(t\right)\) are both of the form  
\[C_{1}\left(t\right)=\prod\left(\frac{t-T_{1}}{T_{1}}\right)W_{1}\left(t\right) \cdot\cos\left(\omega_{1}t-\pi\mu t^{2}+\phi_{1}\right), \tag{6}\]  
where \(\prod\left(\frac{t-T_{1}}{T_{1}}\right)\) is a rectangular gating function with duration \(T_{1}\) centered on time \(T_{1}\). \(W_{1}\left(t\right)\) is an arbitrary weighting function and here it is set to be unity. Notation \(\mu\) is the chirp rate of the chirp signal and \(\phi_{1}\) is a phase term. Thus, Equation (5) can be rewritten as  
\[S\left(t\right)=\prod\left(\frac{t-t^{\prime}}{T^{\prime\prime}}\right)\int_{- \infty}^{+\infty}\hat{f}\left(\tau\right)\cdot\cos\left(\omega_{1}\tau-\pi\mu t ^{2}+\phi_{1}\right)\cdot\cos\left(\omega_{0}\left(t-\tau\right)+\pi\mu\left(t -\tau^{2}\right)+\phi_{0}\right)d\tau, \tag{7}\]  
where  
\[\hat{f}\left(\tau\right)=f\left(\tau\right)\prod\left(\frac{t-\tau-\frac{1}{2 }T_{0}}{T_{0}}\right), \tag{8}\]  
\[t^{\prime}=\frac{1}{2}\left(T_{0}+T_{1}\right)\quad T^{\prime\prime}=T_{1}-T_ {0}\quad\left(T_{1}>T_{0}\right). \tag{9}\]  
Multiplication of the cosine terms produces integrals that involve terms in \(\left(\omega_{1}+\omega_{0}\right)\) and \(\left(\omega_{1}-\omega_{0}\right)\). Ignoring those terms depends on \(\left(\omega_{1}+\omega_{0}\right)\); Equation (7) can be expanded in the form  
\[S\left(t\right)=\prod\left(\frac{t-t^{\prime}}{T^{\prime\prime}}\right)\left[ \int_{-\infty}^{+\infty}x\left(\tau\right)\cdot\exp\left(-j\Omega\tau\right)d \tau+\int_{-\infty}^{+\infty}x^{\ast}\left(\tau\right)\cdot\exp\left(j\Omega \tau\right)d\tau\right], \tag{10}\]  
where \(\Omega=\omega_{0}-\omega_{1}+2\pi\mu t\) and  
\[x\left(\tau\right)=\hat{f}\left(\tau\right)\cdot\exp\left[j\left(\omega_{0}t+ \pi\mu t^{2}+\phi_{0}+\phi_{1}\right)\right]. \tag{11}\]  
The notation \(\ast\) means complex conjugation.