## 2 Chirp Transform Spectrometer
### Pulse Compression in CTS System  
In the classical CTS system, two similar arrangements exist for the signal expansion and compression framework. One is the M(s)-C(l) transform arrangement, where the time duration of the expanding chirp signal is shorter compared to that of the matched SAW filter. Another similar arrangement is the M(l)-C(s) transform arrangement, where the time duration of the expanding chirp signal is greater than that of the matched SAW filter [20; 21]. Symbol M denotes the signal multiplication for bandwidth expansion, C means signal convolution for matched filtering, (l) and (s) denote long and short duration time, individually. The classical two-channel M(l)-C(s) transform arrangements for the pull-push CTS system is presented in Figure 1, assuming that there are m input cosine signals, which can be expressed as:  
\[s(t)=\sum_{i=1}^{m}a_{i}\cos(2\pi f_{i}t+\varphi_{i}) \tag{5}\]  
where \(a_{i}\), \(f_{i}\) and \(\varphi_{i}\) correspond to the amplitude, frequency and initial phase of the \(i\)-th input signal. The input signal \(s(t)\) is firstly mixed with a chirp signal \(s_{cc}(t)\) (usually called the expander chirp) to obtain the modulated chirp signal \(s_{mc}(t)\). The expander chirp signal can be written as:  
\[s_{cc}(t)=a_{cc}\cos(2\pi(f_{cc0}t+\frac{1}{2}kt^{2})+\varphi_{cc0})\ \ \ t\in(0,T_{0}) \tag{6}\]  
where subscript \"cc\" is short for expander chirp, \(a_{cc}\), \(f_{cc0}\) and \(\varphi_{cc0}\) are the amplitude, start frequency and initial phase of the expander chirp, k is the chirp rate, the time duration is \(T_{0}\) and the corresponding bandwidth is \(B=kT_{0}\). The modulated chirp signal can be expressed as:  
\[s_{mc}(t)=\frac{1}{2}\sum\limits_{i=1}^{m}\left(a_{i}\cos(2\pi((f_{cc0}+f_{i}) t+\frac{1}{2}kt^{2})+\varphi_{cc0}+\varphi_{i})+a_{i}\cos(2\pi((f_{i}-f_{cc0}) t-\frac{1}{2}kt^{2})-\varphi_{cc0}+\varphi_{i})\right)\ t\in(0,T_{0}) \tag{7}\]  
where subscript \"mc\" is short for modulated chirp. As the input signals are sent to the RF port and the expander chirp signal is sent to the LO port, the amplitudes of the modulated chirp signals are proportional to the amplitudes of the input signals. Only considering the difference frequency signal (\(s_{mcdf}\)) in (7), the modulated chirp signals can be rewritten as:  
\[s_{mcdf}(t)=\frac{1}{2}\sum\limits_{i=1}^{m}a_{i}\cos(2\pi((f_{i}-f_{cc0})t- \frac{1}{2}kt^{2})-\varphi_{cc0}+\varphi_{i})\ \ \ t\in(0,T_{0}) \tag{8}\]  
The subscript \"mcdf\" is short for modulated chirp difference frequency. Subsequently, the modulated chirp signal is sent to a band-pass filter (BPF). The output IF chirp signal from the BPF can be obtained:  
\[s_{ifc}(t)=\frac{1}{2}\sum\limits_{i=1}^{m}a_{i}\cos(2\pi((f_{i}-f_{cc0})t- \frac{1}{2}kt^{2})-\varphi_{cc0}+\varphi_{i})\ \ t\in\left(\frac{f_{i}-f_{bpfstop}-f_{cav}}{k},\frac{f_{i}-f_{bpfstart}-f_{ cav}}{k}\right) \tag{9}\]  
where subscript \"ifc\" is short for the IF chirp, \(f_{bpfstop}\) and \(f_{bpfstart}\) denote the stop and start frequency of the BPF, individually. The insertion loss of the BPF is not taken into consideration in (9). It can be seen that the IF chirp contains m components of chirp signals, which have the same start and stop frequencies, while the start times of each component are different; this is related to the frequencies of the input signals. Let \(t_{i}=\frac{f_{i}-f_{bpfstop}-f_{cav}}{k}\)  
Figure 1: Schematic of classical two-channel M(l)-C(s) transform arrangements for the expander-compressor. BPF represents band-pass filter, SAW represents surface acoustic. represent the start time of the i-th IF chirp signal and \(B_{bpf}=f_{bpfstop}-f_{bpfstart}\) represent the bandwidth of the BPF. Equation (9) can be rewritten as:  
\[s_{ifc}(t)=\frac{1}{2}\sum\limits_{i=1}^{m}a_{i}\cos(2\pi(f_{bpfstop}(t-t_{i})- \frac{1}{2}k(t-t_{i})^{2})-\varphi_{c0}+\varphi_{i})\ \ \ t\in\left(t_{i},t_{i}+\frac{B_{bpf}}{k}\right) \tag{10}\]  
The IF chirp signals shown in (10) contain the information of the input signals and will be ultimately compressed into pulses.