## 3 Performance Evaluation
### Quantization Loss  
Most RF transmitters and receivers in modern communication or navigation systems use digital-to-analog converters (DACs) and ADCs to implement the transmission of  
Figure 4: Comparison of ACFs for Pulse-OTFS signals with BPSK(5), and Pulse-BPSK(5).  
Figure 3: Comparison of PSDs for Pulse-OTFS signals with BPSK(5), and Pulse-BPSK(5).  
continuous-time signals and the reception of discrete-time signals. Due to limited resolution, ADCs tend to introduce quantization errors into the system, which causes a loss in system performance. Unlike BPSK, the amplitude of the OTFS time-domain signal does not have a constant envelope, and a higher bit resolution may be required for amplitude quantization; therefore, the quantization loss of the proposed signal is analyzed in this section. To simplify the problem, we assume that the signal is quantized before it is transmitted. Therefore, the quantization loss is for the entire system. The receiver does not suffer additional quantization loss.  
When the signal is \(Q\)-bit quantized (\(Q\geq 1\)), there are \(2^{Q}-1\) quantization levels and \(2^{Q}\) quantization results, and the quantization function can be expressed as:  
\[\mathbb{Q}[n]=\left\{\begin{array}{l}-2^{Q}+1,\;s[n]\leq(2^{Q-1}-1)L\\ 2i+1,\;iL<s[n]\leq(i+1)L\\ 2^{Q}-1,\;s[n]>\big{(}2^{Q-1}-1\big{)}L\end{array}\right. \tag{15}\]  
where \(i\in\{-2^{Q-1}+1,-2^{Q-1}+2,\ldots,2^{Q-1}-2,\}\). \(L\) is the quantization interval, and the value of \(L\) is related to the quantization threshold and the number of quantization levels. If the quantization threshold \(H\) is determined, we have  
\[L=\frac{H}{2^{Q-1}} \tag{16}\]  
In general, the quantization threshold is set to a constant value.  
According to the Wiener-Khintchine theorem, it is known that the ACF of a signal is the inverse Fourier transform of its corresponding PSD. Therefore, the PSD of the quantized signal also reflects the quantization loss of the ACF. The closer the power ratio of the main lobe of the quantized PSD is to that before quantization, the less the loss of the ACF. The simulation results of the PSD of OTFS(10,1023) with different quantization bits are given in Figure 5. The main lobe bandwidth of the signal is 10.23 MHz, the sampling rate is 20 MHz, and +\(\infty\) represents no quantization. We can observe from the figure that the sidelobe of the OTFS rises as the quantized bits of the signal decrease, which makes the power ratio of the main lobe decrease and then the loss of the ACF becomes greater.  
Figure 5: Simulation PSDs with different quantization bits for OTFS(10,1023).  
Table 1 gives the corresponding quantization losses for OTFS(10,1023) with different quantization bits \(Q\). We can also calculate the main lobe power ratio of BPSK(5) as 92.45% in all cases, which is because the amplitude of BPSK has a constant envelope. Comparing the results in Table 1, we can see that the main lobe power ratio of OTFS is lower than that of BPSK when the quantization bits \(Q\leq 2\). At this moment, the utilization of the transmitter power is not as efficient as that of BPSK; the quantization loss is around \(-0.75\) dB. The main lobe power ratio of OTFS is higher than that of BPSK and its quantization loss is significantly reduced if \(Q\geq 3\). Therefore, 3 bits can be selected to exploit the performance of OTFS when hardware resources are sufficient.