# Chaotic Orthogonal Composite Sequence for 5G NR Time Service Signal Capture Algorithm
## 3 5G NR Time Service Signal Acquisition Algorithm
### Signal Acquisition Algorithm  
The linear acquisition algorithm is a basic signal search and capture method that utilizes digital correlators to scan the signal in the time domain for both the Doppler frequency shift and the code phase [36]. The advantage of linear search is that it requires only a few digital correlators, simplifying the capture process and reducing hardware complexity. However, it sequentially searches all chips in the frequency band centered around the frequency band center, and if the desired signal is not found, the frequency value of the band center is changed, and the search is repeated for all chips. Because searches a single search unit at a time rather than a group of search units, this results in slower search speeds. This paper adopts a parallel code phase search algorithm [37; 38] that utilizes fast Fourier transform to achieve parallel search for frequency and code phase. It effectively replaces the correlation operation of digital correlators with the digital signal processing technique of Fourier transform, reducing the computational requirements for signal acquisition and speeding up the signal search process. The specific flowchart of the parallel code phase search algorithm is shown in the Figure 10 below. According to the flowchart in Figure 10, the specific process of the parallel code phase search algorithm based on FFT is as follows:  
1. Multiply the input 5G NR time service intermediate frequency signal by the in-phase and quadrature signals from the local carrier generator, and filter the baseband complex signal \(S(t)\) using a low-pass filter. 2. Perform FFT on the baseband complex signal \(S(t)\). Figure 10: Flowchart of Signal Acquisition Principle Based on FFT. Figure 9: Auto-correlation Plot of Chaotic Orthogonal Combination Sequence with \(a=0.5\). 3. Perform FFT on the pseudo-code signal generated by the local pseudo-code generator and take its conjugate. 4. Multiply the results from steps 2 and 3, then perform IFFT on the product. 5. Take the modulus of the result from step 4 and apply threshold decision. If strong peaks appear, signal acquisition is achieved. The peak corresponds to the pseudo-code phase, and the frequency value of the local carrier at that time is the carrier frequency of the signal. If no strong peaks appear, reset the frequency of the local carrier generator to the value of the next frequency well and repeat steps 1 to 4. The specific algorithm for the parallel code phase search and capture based on FFT is as follows:  
For a finite sequence \(X(n)\) of length \(N\), the Fourier transform can be represented as:  
\[X(k)=\sum\limits_{n=0}^{N-1}x(n)e^{-j2\pi kn/N} \tag{16}\]  
The cross-correlation of two finite sequences of length \(N\) can be represented as:  
\[z(n)=\sum\limits_{m=0}^{N-1}x(m)y(n+m) \tag{17}\]  
The convolution of two finite sequences \(x(n)\) and \(y(n)\) of length \(N\) can be represented as:  
\[z(n)=x(n)*y(n)=\sum\limits_{m=0}^{N-1}x(m)y(n-m) \tag{18}\]  
The symbol \(*\) represents the convolution operation. From the above equations, it can be seen that the cross-correlation and convolution differ only in the sign of \(y(n+m)\) and \(y(n-m)\). The Fourier transform of the convolution of sequences \(x\) and \(y\) can be represented as:  
\[ \begin{array}{ll}Z(k)&=\mathcal{F}[x(n)*y(n)]\\ &=\mathcal{F}\biggl{[}\sum\limits_{m=0}^{N-1}x(m)*y(n-m)\biggr{]}\\ &=\sum\limits_{n=0}^{N-1}\sum\limits_{m=0}^{N-1}x(m)y(n-m)e^{-j2\pi kn/N}\\ &=\sum\limits_{m=0}^{N-1}x(m)e^{-j2\pi kn/N}\sum\limits_{n=0}^{N-1}y(n-m)e^{- j2\pi k(n-m)/N}\\ &=\mathcal{F}[x(n)]\mathcal{F}[y(n)]\\ &=X(k)Y(k)\end{array}  \tag{19}\]  
Here, \(\mathcal{F}\) represents the Fourier transform. From the above equation, it can be observed that the Fourier transform of the time-domain convolution of sequences \(x\) and \(y\) is equivalent to the product of the individual Fourier transforms of sequences \(x\) and \(y\) (in the frequency domain). The Fourier transform of the \"correlation\" operation between sequences \(x\) and \(y\) is:  
\[