# Research on Tracking Technique Based on BPSK-CSK Signals
## 2 Materials and Methods
### A BPSK-CSK Signal Tracking Method Based on Time-Division Multiplexing
#### 2.3.2 Code-Tracking Loop  
The basic code-tracking loop for CSK is the Delay-Locked Loop (DLL). The spreading codes of CSK signals have good autocorrelation and inter-correlation properties. The correlation can be maximised when codes of the same sequence are aligned. To improve accuracy, DLL will generate Early (\(E\)), Prompt (\(P\)), and Late (\(L\)) local codes.  
The mixing results of the I and Q branches of the received signal after carrier separation are correlated with the \(E\), \(P\), and \(L\) branch replicated spreading codes, respectively, and here the \(E\) code is taken as an example, and the expression of the correlation operation result is:  
\[I_{E}(N_{1})=\sqrt{C}\mathrm{d}(N_{1})\mathrm{R}(\tau_{\mathrm{E}})\mathrm{ sin}\,\mathrm{c}(\mathrm{f}_{\mathrm{e}}\mathrm{T}_{\mathrm{coh}})\cos\phi_{ \mathrm{e}}+N_{I} \tag{6}\]  
\[Q_{E}(N_{1})=\sqrt{C}\mathrm{d}(N_{1})\mathrm{R}(\tau_{\mathrm{E}})\mathrm{ sin}\,\mathrm{c}(\mathrm{f}_{\mathrm{e}}\mathrm{T}_{\mathrm{coh}})\cos\phi_{ \mathrm{e}}+N_{Q} \tag{7}\]  
where \(\tau_{\mathrm{E}}\) is the phase difference between the over-replicated code and the received code; \(R(\cdot)\) is the code autocorrelation function; \(\phi_{\mathrm{e}}\) is the carrier phase difference. After the coherent integration time \(T_{\mathrm{coh}}\), the noise component \(N_{I},\ N_{Q}\sim(0,1/T_{coh})\), and after the ith update cycle, the overdrive autocorrelation amplitude is:  
\[E(N_{1})=\sqrt{C}\mathrm{R}(\tau_{\mathrm{E}})\mathrm{|sin}\,\mathrm{c}(\mathrm{ f}_{\mathrm{e}}\mathrm{T}_{\mathrm{coh}})|+N_{E} \tag{8}\]  
Figure 5: Carrier-tracking loop.  
The design of the CSK code ring is shown in Figure 6. In this paper, the Noncoherent Early Minus Late Power (NELP) phase discriminator is used. Its calculation formula is shown in Equation (9):  
\[\delta_{cp}=\frac{1}{2}\frac{E^{2}-L^{2}}{E^{2}+L^{2}} \tag{9}\]  
where \(\delta_{cp}\) is the code phase difference after the BPSK cycle.  
In Figure 6, the code-tracking loop in this paper adopts DLL, which is mainly composed of the code NCO, code phase discriminator, and code loop filter, and generates three local codes with different phases through the code NCO, which are E code, P code, and L code, respectively, and respectively correlates these three local codes with the received signal at the same time, compares them to derive the position of the main peak, and determines the phase difference between the local P code and the received signal number, so as to make the local code and the received code momentarily aligned.  
When the correlation value of L code is maximum, the adjustment code generator decreases the code phase; when the correlation value of P code is maximum, it represents that the code-tracking loop tracks the signal correctly; and when the correlation value of E code is maximum, the adjustment code generator increases the code phase. After that, the pseudo-code phase difference is obtained by the code ring phase discriminator, and the code generator completes the dynamic adjustment of the whole code-tracking loop according to the code phase error to achieve the locking of the pseudo-code-tracking loop.