## 3 Theoretical Analysis
### Detection Probability  
Instead of using threshold of the peak, the DC-XFAST method uses the maximum and second maximum value of correlation results and position of correlation peaks to judge if the acquisition process is succeeded. Provided that the carrier residual frequency is properly compensated, which means \(f_{d}=0\), the correlation result is constituted by three items: coherent integration result, input-related noise and self-noise result from folding. Since real and imaginary parts of the mixing noise are uncorrelated Gaussian noise with zero mean and variance of \(\sigma_{H}^{2}=NF\left(A^{2}+\sigma^{2}\right)\), the absolute magnitude of the correlation result will be Rice distribution with probability density function of:  
\[\mathrm{P}_{\mathrm{Ricean}}(r)=\frac{r}{\mathrm{No}_{H}^{2}}\exp\left(-\frac{ r^{2}+s_{k}^{2}}{2\mathrm{No}_{H}^{2}}\right)I_{0}(\frac{rs_{k}}{\mathrm{No}_{H}^{ 2}}) \tag{7}\]  
With \(I_{0}\) the zero order modified Bessel function. \(s_{k}=N\) is the summation of the mean of the signals on the in phase and quadrature branches for bin k. Partial correlation also follows the Rice distribution with \(s_{k}=k\) for \(R_{1,m-1,\delta}\), \(s_{k}=N\) for \(R_{1,m,\delta}\) and \(s_{k}=N-k\) for \(R_{1,m+1,\delta}\). After stacking the correlation result of \(m-1\) and \(m+1\), \(s_{k}=N\), which means the magnitude of full peak which contained two partial peaks is not affected by k. The correlation result is shown in Figure 3.  
When there are only self-noise and input-related noise in the coherent integration results, the absolute magnitude of the correlation result will be Rayleigh distribution with probability density function of:  
\[\mathrm{P}_{\mathrm{Rayleigh}}(r)=\frac{r}{\mathrm{No}_{H}^{2}}\exp\left(- \frac{r^{2}}{2\mathrm{No}_{H}^{2}}\right),\ r\geq 0 \tag{8}\]  
And cumulative distribution function of:  
\[\mathrm{F}_{\mathrm{Rayleigh}}(r)=1-\exp\left(-\frac{r^{2}}{2\mathrm{No}_{H}^ {2}}\right),\ r\geq 0 \tag{9}\]  
alarm probability in each \(\mathrm{N_{th}}\) as \(\mathrm{N_{th}}=4\), so when four noise peaks with relative position of 0 or N are detected, then a false alarm arises. Since the probability that the noise value at any certain position is the largest value within a cell is 1/N, the probability that four noise values arise simultaneously is 1/N4. Considering the total number of possible positions is N, the false alarm probability is derived as:  
\[\mathrm{P_{fa}}|_{\mathrm{N_{th}}\to 4}=\mathrm{P_{fa}}\frac{\mathrm{N}}{ \mathrm{N^{4}}}=\mathrm{P_{fa}}\frac{1}{\mathrm{N^{3}}} \tag{16}\]  
\(\mathrm{N_{th}}=3\), the probability that three noise peaks with detected relative position is:  
\[\mathrm{P_{fa}}|_{\mathrm{N_{th}}=3}=\mathrm{P_{fa}}\frac{4\mathrm{N(N-1)+N}}{ \mathrm{N^{4}}} \tag{17}\]  
\(\mathrm{N_{th}}=2\), the probability that two noise peaks with relative detected position is:  
\[\mathrm{P_{fa}}|_{\mathrm{N_{th}}=2}=\mathrm{P_{fa}}\frac{4\mathrm{N(N-1)+6N(N -1)^{2}+N}}{\mathrm{N^{4}}} \tag{18}\]  
\(\mathrm{N_{th}}=1\), the probability that one noise peaks with detected relative position is:  
\[\mathrm{P_{fa}}|_{\mathrm{N_{th}}=1}=\mathrm{P_{fa}} \tag{19}\]