## 3 Digital Pulse Compression
### Linear Phase Sampling and Accumulating Method for Fast Pulse Compression  
In CTS structures, to solve the matching problem and to avoid large attenuation of the SAW filter, the digital pulse compression method seems to be a good solution. However, digital convolution in the time domain requires significant computation when dealing with large bandwidth chirp signals. The double time-frequency transform in the frequency domain still requires double FFT calculations, and is unable to determine the time difference of the chirp signals. Moreover, in many real applications, the measured signals are sparse in the frequency domain, which is not considered in classical digital pulse compression methods. Considering the quadratic phase characteristic of the IF chirp signal, a new LPSA method was developed for fast pulse compression in CTS systems. Figure 2 is a brief schematic diagram of a CTS system with the LPSA method for pulse compression. Only one channel is exhibited for signal expansion and compression in the system. The input signals are first mixed with the chirp signal to obtain the modulated chirp signals carrying the information of the input signal. After the amplifier and the band-pass filter, the output IF chirp signals are sampled and compressed into pulses by the digital algorithm. In CTS systems, the output IF chirp signals from the BPF have a fixed dispersive constant, start frequency and stop frequency. However, these IF chirp components have different time distributions, which correspond to the different input frequencies. So, if the time distributions of the IF chirp signals can be detected, the input frequencies can be measured. Considering the sparsity of the input signals, an LPSA algorithm with two-stage frequency resolution subdivision was developed in this paper for fast pulse compression. First, there are two sets of time sequences, \(t_{n}^{1}\) and \(t_{n}^{2}\), which satisfy the following mutually orthogonal equations:  
\[2\pi\bigg{(}f_{bpfstop}t_{n}^{1}-\frac{1}{2}k\Big{(}t_{n}^{1}\Big{)}^{2}\bigg{)} =\varphi_{r}+2n\pi\ \ \ n=1,2,\ldots,N \tag{16}\]  
\[2\pi\bigg{(}f_{bpfstop}t_{n}^{2}-\frac{1}{2}k\Big{(}t_{n}^{2}\Big{)}^{2}\bigg{)} =\varphi_{r}+\frac{\pi}{2}+2n\pi\ \ n=1,2,\ldots,N \tag{17}\]  
where \(\varphi_{r}\) is an arbitrary phase which is set to zero in the following derivations. The range of \(t_{n}^{1}\) and \(t_{n}^{2}\) is \(0\sim\frac{B_{bpf}}{k}\), and \(N=f_{bpfstop}\frac{B_{bpf}}{k}-\frac{1}{2}k\Big{(}\frac{B_{bpf}}{k}\Big{)}^{2}\) is the total number of \(t_{n}^{1}\) and \(t_{n}^{2}\), which is decided by the IF chirp signal and the chirp rate. \(n\) is a positive integer ranging from 1 to N. From (16) and (17), the time sequence can be derived as:  
\[t_{n}^{1}=\frac{f_{bpfstop}-\sqrt{f_{bpfstop}^{2}-2 nk}}{k} \tag{18}\]  
\[t_{n}^{2}=\frac{f_{bpfstop}-\sqrt{f_{bpfstop}^{2}-2(n+\frac{1}{4})k}}{k} \tag{19}\]  
It can be seen that the time intervals of \(t_{n}^{1}\) and \(t_{n}^{2}\) are not fixed, but change linearly, which is equivalent to linear phase sampling. By sampling the IF chirp signals using (10) via the time sequence of \(t_{n}^{1}\) and \(t_{n}^{2}\) from time \(t_{j}\), and then accumulating the sampling points, the following equation can be obtained:  
\[A_{1}=\frac{1}{2}\sum_{n=1}^{N}\sum_{i=1}^{m}a_{i}\cos(2\pi(f_{bpfstop}(t_{j}+ t_{n}^{1}-t_{i})-\frac{1}{2}k\Big{(}t_{j}+t_{n}^{1}-t_{i}\Big{)}^{2})-\varphi_{c0}+ \varphi_{i}) \tag{20}\]  
Figure 2: Simplified structure of the CTS system with the LPSA algorithm for fast pulse compression.