# Chaotic Orthogonal Composite Sequence for 5G NR Time Service Signal Capture Algorithm
## 3 5G NR Time Service Signal Acquisition Algorithm
### Signal Acquisition Algorithm
\begin{array}{ll}Z(k)&=\mathcal{F}\biggl{[}\sum\limits_{m=0}^{N-1}x(m)*y(n+ m)\biggr{]}\\ &=\sum\limits_{n=0}^{N-1}\sum\limits_{m=0}^{N-1}x(m)y(n+m)e^{-j2\pi kn/N}\\ &=\sum\limits_{m=0}^{N-1}x(m)e^{-j2\pi kn/N}\sum\limits_{n=0}^{N-1}y(n+m)e^{- j2\pi k(n+m)/N}\\ &=X^{*}(k)Y(k)\end{array}  \tag{20}\]In the equation above, \(X^{*}(k)\) refers to the complex conjugate of \(X(k)\). Therefore, by taking the complex conjugate of the Fourier transform of sequence x and multiplying it with the Fourier transform of sequence y in the frequency domain, and then performing the inverse Fourier transform, the result of the \"correlation\" operation between the two sequences can be obtained. This helps to identify the peak value of the \"correlation\" result, completing the signal acquisition process. Compared to traditional linear search and capture algorithms, the parallel code phase search algorithm based on FFT reduces the number of searches, thereby shortening the capture time. The number of searches required for the linear search and capture algorithm is:  
\[N_{cell}=\frac{(2f_{unc})(2t_{unc})}{f_{step}t_{step}}=\frac{f_{\Delta}}{f_{ step}}2M \tag{21}\]  
where \(N_{cell}\) represents the number of search units within the linear search range, \(f_{unc}\) and \(t_{unc}\) are the uncertainties in signal frequency and code phase, where \(2f_{unc}\) is equivalent to the frequency search range \(f_{\Delta}\), and \(2t_{unc}\) is equivalent to the code length M. \(f_{step}\) and \(t_{step}\) are the search steps for frequency and code phase, where \(f_{step}\) is set to 1 kHz for frequency search in this paper, and \(t_{step}\) is typically half a chip. The capture time required for the linear search and capture algorithm is:  
\[T_{tot}=N_{cell}T_{dwell} \tag{22}\]  
where \(T_{tot}\) denotes the time required for the linear search and capture algorithm to search the entire uncertain interval once, and \(T_{dwell}\) is the dwell time, i.e., the time taken to search for a signal in each unit. If the receiver determines that a signal has been detected during the sequential search process, it can stop the search. Assuming that the signal exists and its parameter values are randomly distributed within the two-dimensional search range, the average number of search units the receiver has searched before declaring signal acquisition is half of the total number of search units. Therefore, the estimated formula for the average acquisition time \(T_{acq}\) is [39]:  
\[T_{acq}=\frac{1}{2}T_{tot} \tag{23}\]  
The parallel code phase search algorithm based on FFT completes \(2M\) linear code phase searches in a single Fourier transform operation. The number of searches required for the parallel code phase search algorithm based on FFT is:  
\[N_{cellF}=\frac{f_{\Delta}}{f_{step}} \tag{24}\]  
where \(N_{cellF}\) represents the number of searches required for the parallel code phase search algorithm based on FFT. The capture time required for the parallel code phase search algorithm based on FFT is:  
\[T_{totF}=N_{cellF}T_{dwell} \tag{25}\]  
where \(T_{totF}\) represents the time required for the parallel code phase search algorithm based on FFT to search the entire uncertain interval once. The estimated formula for the average acquisition time \(T_{acq}\) for the parallel code phase search algorithm based on FFT is:  
\[T_{acqF}=\frac{1}{2}T_{totF} \tag{26}\]  
where \(T_{acqF}\) represents the average acquisition time for the parallel code phase search algorithm based on FFT. From the above formulas, it can be concluded that compared to the traditional linear search and capture algorithm, under the same conditions, the parallel code phase search algorithm based on FFT reduces both the capture time and the number of searches by a factor of \(2M\).