# Chaotic Orthogonal Composite Sequence for 5G NR Time Service Signal Capture Algorithm
## 3 5G NR Time Service Signal Acquisition Algorithm  
### Chaotic Orthogonal Combination Sequence  
This paper proposes a chaotic orthogonal combination sequence, which is a combination of Walsh sequences and Skew Tent-Map chaotic sequences, and thus can improve the deficiencies of traditional spread spectrum codes.  
Walsh sequences are a class of orthogonal sequences widely used in communication fields such as error correction coding and secure coding. They can be derived from Hadamard matrices. A Hadamard matrix \(H\) is an orthogonal square matrix composed of the elements \"\(+1\)\" and \"\(-1\)\", where any two rows or columns are orthogonal to each other. This property can be mathematically expressed as:  
\[HH^{T}=NI \tag{11}\]  
Figure 5: Principle Diagram of the Integration of 5G NR Time Service Signal with 5G Signal.  
where \(H^{T}\) represents the transpose of \(H\), \(N\) is the order of \(H\), and \(I\) is the identity matrix. When we discuss Hadamard matrices of order \(N=2^{n}\) (where \(n\) is a positive integer), the Hadamard matrix \(H_{n}\) can be generated by the following recursive relation:  
\[H_{n}=\begin{bmatrix}H_{n-1}&H_{n-1}\\ H_{n-1}&-H_{n-1}\end{bmatrix},H_{0}=1 \tag{12}\]  
By considering each row of the Hadamard matrix as a binary sequence, an \(N=2^{n}\) order matrix \(H_{n}\) yields a total of \(N\) sequences, each of length \(N\). Any two sequences in this orthogonal set are mutually orthogonal, forming the Walsh sequences.  
Chaos is a deterministic, random-like process that occurs in nonlinear dynamic systems. This process is neither periodic nor convergent and exhibits a high sensitivity to initial conditions, presenting pseudo-random characteristics. Due to its sensitive dependence on initial conditions, it can provide a large number of uncorrelated, pseudo-random yet deterministic signals that are easy to generate and regenerate, making it suitable for use as pseudo-random codes.  
The Tent-Map is a classic chaotic mapping model. The Skew Tent-map used in this paper is a special form of the Tent map. It is a one-parameter, piecewise-linear, weakly-unimodal, map of the interval [0, 1]. Its iterative equation is simple, and the sequence generator is easy to implement. It can be applied to any scenario requiring the transmission of encrypted data through a common channel, offering good security and is suitable for use as a spreading code in spread spectrum systems [34; 35].  
Its mapping equation is:  
\[x_{k+1}=\begin{cases}\frac{x_{k}}{a},&0<x_{k}<a\\ \frac{1-x_{k}}{1-a},&a\leq x_{k}\leq 1\end{cases} \tag{13}\]  
The range of the parameter \(a\) in the equation is \(0<a<1\). The below figure (Figure 6) shows the chaotic mapping effect of the Skew Tent-Map.  
By multiplying corresponding bits of the same set of Walsh sequences and chaotic sequences generated by Skew Tent-Map with the same order, we can obtain the chaotic orthogonal combination sequence.  
If we denote the chaotic mapping sequence as \(x_{1}(t)\), the Walsh sequence as \(x_{2}(t)\), and the chaotic orthogonal combination sequence as \(x(t)\), then the expression for the chaotic orthogonal combination sequence will be:  
\[x(t)=x_{1}(t)x_{2}(t) \tag{14}\]  
Figure 6: Skew Tent-Map Chaotic Mapping Effect Diagram.  
We verify the cross-correlation of the chaotic orthogonal combination sequence \(x(t)\) by selecting sequences of the same order from different groups (denoted as group \(i\) and group \(j\)) and integrate over one period.  
\[\int\limits_{T}x_{i}(t)x_{j}(t)dt=\int\limits_{T}x_{1}^{2}(t)x_{2i}(t)x_{2j}(t) dt=\int\limits_{T}x_{2i}(t)x_{2j}(t)dt=0 \tag{15}\]  
As per the above equation, it is evident that the chaotic orthogonal combination sequence still retains the property of mutual orthogonality between any two sequences within the same family of Walsh sequences.