## 3 Performance Evaluation
### Code-Tracking Performance  
In the case that the front-end filter is assumed to be ideal, the CRLB of the code-tracking error can be approximated [39]:  
\[\tau_{\text{CRLB}}=\frac{B_{L}(1-0.5B_{L}T_{coh})}{(2\pi)^{2}\frac{C}{N_{0}}\int \limits_{-B_{r}/2}^{B_{r}/2}f^{2}G_{n}(f)\text{d}f} \tag{25}\]  
where \(B_{L}\)is the equivalent noise bandwidth of the code-tracking loops, \(C/N_{0}\) is the carrier power-to-noise density ratio, \(B_{r}\) is the pre-filtering bandwidth, and \(G_{n}(f)\) is the normalized PSD of the signal.  
Let us further suppose that the parameters of the code-tracking loops are fixed. In such a scenario, the code-tracking performance for different modulations is dependent on the integral term in (25), which is defined as the Gabor bandwidth:  
\[\Delta f_{\text{Gabor}}=\sqrt{\int\limits_{-B_{r}/2}^{B_{r}/2}f^{2}G_{n}(f) \text{d}f} \tag{26}\]  
The greater the Gabor bandwidth, the better the code-tracking performance will be. Furthermore, the variance in the code-tracking error with the thermal noise for a widely used non-coherent early-late processing (NELP) delay-locked loop (DLL) is [40]  
\[\sigma_{\text{NELP}}^{2}=\frac{B_{L}(1-0.5B_{L}T_{coh})\int\limits_{-B_{r}/2}^ {B_{r}/2}G_{n}(f)\sin(\pi fd)^{2}\text{d}f}{(2\pi)^{2}\frac{C}{N_{0}}\left( \int\limits_{-B_{r}/2}^{B_{r}/2}fG_{n}(f)\sin(\pi fd)\text{d}f\right)^{2}} \cdot\left(1+\frac{\int\limits_{-B_{r}/2}^{B_{r}/2}G_{n}(f)\cos(\pi fd)^{2} \text{d}f}{T_{coh}\frac{C}{N_{0}}\left(\int\limits_{-B_{r}/2}^{B_{r}/2}G_{n}( f)\cos(\pi fd)\text{d}f\right)^{2}}\right) \tag{27}\]  
where \(d\) is the early-late spacing of the correlator.  
Figure 9 shows the Gabor bandwidth and the code-tracking accuracy of the NELP DLL for all of the signals we compared. The pre-filtering bandwidth is 20 MHz, \(d=1\) chip, and \(B_{L}=1\). To analyze the signal tracking performance under extreme conditions, Figure 9b includes the theoretical tracking accuracies for the coherent integration time \(T_{coh}\) of 1 ms and 10 ms, \(C/N_{0}\) of 25 dB-Hz to 45 dB-Hz.  
The results indicate that pulsed signals have the same code-tracking accuracy compared to continuous signals. The longer the coherent integration time, the higher the tracking accuracy at the same modulation and \(C/N_{0}\). The code-tracking accuracy of OTFS(10,1023) and Pulse-OTFS(10,1023) is higher than that of BPSK(5) and Pulse-BPSK(5). This means that to achieve the same code-tracking accuracy, the \(C/N_{0}\) of BPSK(5) and Pulse-BPSK(5) is about 8 dB higher than that of OTFS(10,1023) and Pulse-OTFS(10,1023). Obviously, the proposed signal has the advantage of code-tracking accuracy.  
In a code-tracking loop, the discriminator curve (S-curve) is used to reflect the discriminator gain and the zero-crossing point of the received signal. Figure 10 shows the simulated S-curve of Pulse-OTFS(10,1023), Pulse-BPSK(5), OTFS(10,1023), and BPSK(5), where \(T_{coh}=1\) ms, \(d=1\) chip. Since a larger pre-filtering bandwidth does not necessarily give better performance in BPSK, we set the pre-filter bandwidth to 20 MHz to ensure a fair comparison. In contrast with BOC modulation, both OTFS(10,1023) and Pulse-OTFS(10,1023) have only one zero-crossing point within 1 chip delay offset and can therefore be received unambiguously. The discriminator gain is also higher than that of Pulse-BPSK(5) and BPSK(5), so the code-tracking performance of the proposed signal is superior to the latter.  
Figure 10: Comparison of S-curves for different signals with a 20 MHz pre-filtering bandwidth.  
Figure 9: Comparison of code-tracking performance for different signals. (**a**) Gabor bandwidth and (**b**) NELP DLL code-tracking error with a 20 MHz pre-filtering bandwidth.