## 2 Chirp Transform Spectrometer
### Classical Two-Channel Structure of CTS  
Generally, there are two similar structures that can be applied to the CTS system. One is the \(M\left(s\right)-C\left(l\right)\) arrangement, in which the time period of the modulation chirp signal is shorter than that of the SAW matching filter. Another is the \(M\left(l\right)-C\left(s\right)\) arrangement, in which time duration of the modulation chirp signal is long [12; 13; 20]. The utilized symbol M represents signal multiplication for bandwidth expansion; C denotes signal convolution for matched filtering; and \(\left(l\right)\) and \(\left(s\right)\) mean long and short duration time, respectively. In this paper, we present the classical \(M\left(l\right)-C\left(s\right)\) arrangement shown in Figure 2.  
Figure 3 is the schematic diagram of signal expansion and compression procedure in the CTS. The time duration between the maximum point and first zero-crossing point of amplitude envelope of the output pulses is approximately equal to \(1/B_{c}\), and hence, the frequency resolution is \(\Delta f=\left|\mu\right|/B_{c}=1/T_{c}\), where \(B_{c}\) is the operation bandwidth of the SAW filter and \(T_{c}\) is the time length of the SAW filter's impulse response. When \(T_{c}\) is greater than 10 us, the frequency resolution \(\Delta f\) is less than 100 KHz. This means that the character of the high-frequency-resolution in CTS depends on the property of large time-bandwidth product \((T_{c}B_{c})\) of the compressor. For processing technology reasons, it is usually advisable to utilize the minimum possible value of the expander time-bandwidth product \((T_{c}B_{c})\)[12; 21]. This results that the expander bandwidth is twice the bandwidth of the compressor. As the bandwidth is different between the expander and the compressor, it is appropriate to adopt a push-pull arrangement with two uniform channels to make up the bandwidth mismatch between the expander and compressor, which can ultimately improve the system's sensitivity. Meanwhile, the push-pull structure can avoid spectrum leakage when dealing with nonstationary signals. Figure 4 shows the time-frequency schematic diagram of the classical two-channel structure. All the working parameters are the same as that in [22]. The frequency of the input signal ranges from 1.9 GHz to 2.3 GHz and the frequency of the LO input chirp signal ranges from 2.7 GHz to 3.5 GHz. The time period of the chirp signal is 20 us, the time duration of the impulse response of the SAW filter is 10 us, and its working bandwidth is 0.8-1.2 GHz. Here, we take three input signals with fixed frequencies (e.g., 1.9 GHz, 2.1 GHz, and 2.3 GHz) as an example to clarify the working principle of the classical two-channel structure of CTS. The three input signals will first be transformed to three chirp signals with 20-us time duration and 800-MHz bandwidth. Then, a common part with 400 MHz bandwidth (0.8-1.2 GHz) is filtered out from the three chirp signals as the input of the SAW filter for pulse compression. As the input times of the three chirp signals are different, the time of the output pulses will be different, which indicates the different frequencies of the input signals. The two-channel structure helps to make up the mismatch of time period between the expander and the SAW compressor. Obviously, the push-pull two-channel structure can avoid spectrum leakage and make sure that the output pulses can be obtained over the entire measured time. Figure 2: Classical \(M\left(l\right)-C\left(s\right)\) arrangement of the chirp transform spectrometer. For the classical push-pull arrangement of the CTS system, the two channels are separated; thus, there is little interference between the two channels. Reasonable structure design of the classical CTS can obtain very high frequency resolution and has real-time processing capability and very high stability. These high system performances require a perfect match between the two channels. Usually, some cost and challenges need to be considered when implementing the classical CTS shown as follows.