## 3 Performance Evaluation
### Acquisition Performance
#### 3.2.2 Acquisition Complexity  
The signal-to-noise ratio (\(SNR\)) of the Pulse-OTFS receiver output is  
\[SNR=\frac{I^{2}+Q^{2}}{N_{0}/T_{coh}}=\frac{PT_{coh}}{N_{0}}\Bigg{(}\frac{1}{M} \sum\limits_{k=0}^{M-1}e^{j2\pi kA/\tau}\Bigg{)}^{2}\bigg{(}\frac{\sin\pi f_{d} (T_{coh}-|\tau|)}{\pi f_{d}(T_{coh}-|\tau|)}\bigg{)}^{2}\bigg{(}1-\frac{|\tau|} {T}\bigg{)}^{2} \tag{20}\]  
In the case of a desired signal being in existence, the correlation value will satisfy the Rice distribution. Conversely, in the case of no desired signal being in existence, the correlation value will satisfy the Rayleigh distribution. The acquisition threshold and acquisition probability are as follows  
\[\left\{ \begin{array}{l}V_{t}=\sigma_{n}\sqrt{-2\ln P_{fa}}\\ P_{d}=\int\limits_{V_{t}}^{\infty}f(v)\mathrm{d}v\end{array} \right. \tag{21}\]  
where \(\sigma_{n}^{2}=N_{0}/T_{coh}\) is the noise power, \(P_{fa}\) is the false alarm probability, and \(f(v)\) is the Rice distribution probability density function (PDF). Doppler tolerance is referred to as the offset of the Doppler frequency when the signal processing gain is at its minimum allowable level. This directly affects the division of frequency grids in two-dimensional search, as well as the average acquisition time. The Doppler tolerance is defined by the following equation  
\[SNR_{Th}=func(\tau,f_{d,\text{max}}) \tag{22}\]  
where \(SNR_{Th}\) is the output \(SNR\) threshold during the acquisition process and \(f_{d,\text{max}}\) is the maximum Doppler offset that satisfies the minimum output \(SNR\) threshold. If the maximum Doppler offset \(f_{d,\text{max}}\) satisfies the false alarm probability \(P_{fa}\) and acquisition threshold \(V_{t}\), generally, \(2f_{d,\text{max}}\) is selected as the Doppler search interval to calculate the number of Doppler frequency searches. Substituting the \(SNR\) threshold \(SNR_{Th}=-2\ln P_{fa}\) and \(f_{d,\text{max}}\) into (20) yields  
\[\left|\frac{1}{M}\sum\limits_{k=0}^{M-1}e^{j2\pi kA/\tau}\bigg{(}\frac{\sin\pi f _{d}(T_{coh}-|\tau|)}{\pi f_{d}(T_{coh}-|\tau|)}\bigg{)}\bigg{(}1-\frac{|\tau| }{T}\bigg{)}\right|=\sqrt{\frac{N_{0}}{PT_{coh}}SNR_{Th}} \tag{23}\]  
Figure 6: Ambiguity function envelope of Pulse-OTFS(10,1023). The left part of (23) is the exact ambiguity function envelope; then, we have the peak envelope of the ambiguity function for different Doppler offset:  
\[ \begin{split} A(f_{d})&=\max_{\tau}\biggl{(}\bigg{|} \frac{1}{M}\sum_{k=0}^{M-1}e^{i2\pi k\Delta f\tau}\biggl{(}\frac{\sin\pi f_{d}( T_{coh}-|\tau|)}{\pi f_{d}(T_{coh}-|\tau|)}\biggr{)}\biggl{(}1-\frac{|\tau|}{T} \biggr{)}\biggr{|}\biggr{)}\\ &=\bigg{|}\frac{1}{M}\sum_{k=0}^{M-1}\mathrm{sinc}(f_{d}T_{coh}) \bigg{|}\end{split}  \tag{24}\]  
The Doppler search range of the LEO satellite at 1500 km altitude is (\(-40\) kHz, \(+40\) kHz) with a 2 GHz carrier frequency [38]. Using a parallel code phase search (PCS) scheme and signal period of 1 ms, the threshold on the right side of (23) is 0.64, then the corresponding search interval of both BPSK and OTFS is 500 Hz, and the number of serial searches for Doppler frequency is 161. However, the Doppler search interval of Pulse-OTFS and Pulse-BPSK are both 500 \(N\) Hz and the number of serial searches for Doppler frequency is \(160/N+1\), where the duty cycles are both \(1/N\) for Pulse-OTFS and Pulse-BPSK. Assuming that the same sampling rate and bit resolution are used for all signals to simplify the analysis of acquisition complexity, the same number of multiplications and additions are required to perform a PCS, denoted \(\Delta_{m}\) and \(\Delta_{a}\), respectively. Thus, the overall acquisition complexity depends on the number of complex multiplications and additions. The acquisition complexity of different signals is listed in Table 2. Figure 7 shows the multiplication and addition complexity ratio of Pulse-OTFS/Pulse-BPSK to OTFS/BPSK as a function of Doppler bins \(N\).