## 2 Chirp Transform Spectrometer
### Pulse Compression in CTS System
In the classical CTS system, the SAW filter is used for the final pulse compression. The SAW filter is also called the compressor, and its impulse response is also a chirp signal similar to the IF chirp signal, which can be expressed as:  
\[c(t)=a_{c}(t)\cdot\cos\left(2\pi\left(f_{bpfstop}t+\frac{1}{2}k_{c}t^{2} \right)+\varphi_{c0}\right)\ \ \ t\in\left(0,T_{c}\right) \tag{11}\]  
where \(a_{c}(t)\) is the amplitude response of the SAW filter, which can be seen as a constant; here, it is set to unity; \(\varphi_{c0}\) and \(T_{c}\) are the initial phase and the duration time of the compressor and \(k_{c}\) is the chirp rate. Considering a perfect match of the chirp rate between the expander and compressor (\(k_{c}=-k\)), and ignoring the phase constant, the output of the SAW filter can be expressed as:  
\[ \begin{array}{ll}\psi_{ss}&=\int_{-T_{c}/2}^{T_{c}/2}s_{ifc}(t)c(t-\tau)d \tau\\ &=\frac{1}{2}\sum\limits_{i=1}^{m}a_{i}\cdot\sqrt{T_{c}B_{c}}\frac{\sin\left[ \pi T_{c}B_{c}\left(1-\frac{|t-t_{i}|}{T_{c}}\right)\right]}{\pi T_{c}B_{c}} \cos\left(2\pi f_{bpfstop}(t-t_{i})\right)\ \ \ t\in\left(t_{i}t_{i}+T_{c}\right)\end{array}  \tag{12}\]  
where \(B_{c}\) and \(T_{c}\) are the bandwidth and time duration of the compressor, individually. \(\sqrt{T_{c}B_{c}}\) is the compression gain, and the output pulse has an envelope shape of sinc function. The bandwidth of the output pulse is approximately \(1/B_{c}\), and the frequency resolution is \(\Delta f\approx|\mu|/B_{c}\approx 1/T_{c}\). It can be seen that the character of the high frequency resolution in CTS depends on the property of the large compression time-bandwidth product \((T_{c}B_{c})\) of the SAW filter. Limited by the manufacturing difficulties of SAW filters with large time-bandwidth products, it is usually appropriate to utilize the minimum possible value of the compressor time-bandwidth product \((T_{c}B_{c})\)[22; 23]. This results in the expander bandwidth being twice the bandwidth of the compressor. Due to the bandwidth mismatch between of the expander and the compressor, the two-channel arrangement is used to make up the mismatch and guarantee the maximum sensitivity that can be obtained in the CTS system. However, in the classical M(l)-C(s) transform arrangement, the use of the SAW filter will give rise to some technological difficulties, and the implementation of the system also presents challenges [15]. Some challenges and disadvantages of the utilization of the SAW filter behaving as the compressor/convolver are listed below. * Influenced by the large insertion loss of the SAW convolution filters, which is usually larger than 40 dB, the operation bandwidth and dynamic range are both limited and cannot be very large. Therefore, the corresponding system performance, i.e., frequency resolution and sensitivity, are degraded. * Imperfect manufacturing of the SAW convolution filter will cause deviations from the ideal dispersion characteristics, which will degrade the system performance, e.g., frequency resolution and power spectral density accuracy. * The dispersive characteristic match between the generated expander chirp signal and the SAW filter behaving as the compressor/convolver need to be carefully considered. It is very hard to fabricate two SAW filters with perfectly matched dispersion characteristics.