## 4 Simulation and Analysis
### Simulation Results and Analysis
Figure 3: The output pulse of the single input frequency of 6.4 GHz based on CTS system. Frequency resolution of the output pulses using the classical time-domain pulse compression method (**a**) and the LPSA algorithm (**b**). Figure 4: Frequency resolution of the output pulses using the classical time-domain pulse compression method (**a**) and the LPSA algorithm (**b**). Figure 5: Output pulses calculated from five input frequencies. (**a**) is the result calculated by the classical time-domain pulse compression method. (**b**) is the result obtained by the novel LPSA method. Next, a more random signal with 11 input frequencies of 5.6 GHz, 5.7 GHz, 5.9 GHz, 5.99 GHz, 5.999 GHz, 6.0 GHz, 6.01 GHz, 6.03 GHz, 6.06 GHz, 6.1 GHz, 6.3 GHz and 6.4 GHz was measured using the novel LPSA algorithm with two-stage frequency resolution (3.1 MHz-100 kHz). In the calculation of the second-stage frequency resolution, the linear sampling rate ranged from 250 MHz to 500 MHz. The output pulses are shown in Figure 6. The average noise levels were very close in the two methods. There was also a little difference between the amplitudes obtained by the two methods. This was caused by the phase approximation and the number of the extracted sampling points. Higher phase approximation and more extracted sampling points yielded higher amplitude accuracy at the expense of relatively more computation. As mentioned above, the sampling rate will influence the phase accuracy of the extracted orthogonal sampling points. A higher sampling rate guarantees a smaller approximate factor p, leading to higher amplitude accuracy. Meanwhile, the increased number of the orthogonal sampling points will also reduce the impact of the phase approximation on amplitude accuracy. Figure 7 shows the amplitude accuracy versus different sampling rates and different numbers of the orthogonal sampling points. It is obvious that the amplitude accuracy improved with the increasing number of the orthogonal sampling points. The increase of the sampling rate helped to reduce the phase error, which also improved the amplitude accuracy significantly. The simulation indicated that 800 orthogonal sampling points under an 8 GHz sampling rate were enough in the second-stage frequency resolution for the fast pulse compression with an amplitude accuracy above 98%. To obtain an amplitude  
Figure 6: Output results of 11 input frequencies calculated by the classical time-domain pulse compression method (**a**) and the novel LPSA algorithm (**b**). accuracy above 99%, more than 1600 orthogonal sampling points in the second-stage frequency resolution under 8 GHz sampling rate would be needed. Considering a tradeoff between the sampling rate and the number of the orthogonal sampling points, a sampling rate of 8 GHz is a reasonable choice for the novel LPSA algorithm. To obtain the average amplitude accuracy and frequency resolution, the system's response to a sinusoidal input was measured at different input frequencies. For the simulation with 11 input frequencies, the measured amplitude accuracy, average frequency resolution and the amount of computations for the classical time-domain pulse compression method and the novel LPSA algorithm are shown in Table 1. The measured frequency resolutions of the two methods were very close. The classical time-domain pulse compression method required more computations, i.e., additions and multiplications. The computation of the novel LPSA algorithm was mainly focused on additions, which require relatively less machine instructions compared to the multiplications in the classical time-domain pulse compression method. In addition, to obtain the same amplitude accuracy as the classical time-domain pulse compression method, it needed 1600 orthogonal sampling points under the second-stage frequency resolution. Under this scenario, the total number of the addition increased to 1.0566 \(\times\) 10\({}^{6}\), and the multiplication remained the same.