# A Channel Compensation Technique Based on Frequency-Hopping Binary Offset Carrier Modulated Signal
## 3 Performance Analysis
### Tracking Loop  
The tracking loop is aimed at synchronizing the local replica of the spreading code and carrier with the current received signal for demodulating the next incoming signal as accurately as possible. At the very beginning, the output of NCO is determined by the acquisition results of Doppler shift \(\hat{f}_{d}\) and the output of the PRN code generator by the code delay \(\hat{\tau}_{acq}\), as shown in Figure 7.  
Figure 5: ACF comparison between FH-BOC signals and BOC signals. For the start in each accumulation time period \(T_{a}\), the output of NCO is required to duplicate itself and adjust its phase to quadrature; then, the input signal needs to be multiplied by local carrier as demodulation. The following LPF is for filtering a high-frequency component and persevering the baseband signal only. Since the output signal from LPF itself is still spread with the PRN code, and code-phase has been affected by the transmission distance and Doppler shift, it is necessary to set the mismatched PRN code  
Figure 6: Power spectral density comparison between FH-BOC signals. Figure 7: Block diagram of tracking loop. replica as early and late, suggested as \(E\) and \(L\) for abbreviations. Either code alters its code delay half chip from the prompt code, in order to find out which code-phase is the most correlated with the incoming signal. Through all kinds of combinations, the accumulation of the output from different correlators provides certain control words, e.g., shown as \(w_{IE}\), as inputs of subsequent discriminator. The discriminator processes these control words to offer measurable quantities, and loop filters are responsible for reducing the noise in the discriminator's outputs. Finally, the renewed parameters on the carrier and PRN code of the received signal are sent back to the NCO and code generator, waiting for an incoming signal in next time period. Discriminators and loop filters are designed distinctly for different lock loops. For phase-lock loop (PLL), its discriminator can be designed as the most common one, given by  
\[e_{p}=a\tan(w_{QP}/w_{IP}), \tag{13}\]  
which is optimal for phase estimating, compared to the classic Costas loop discriminator, while requiring a relatively high computational complexity. Then, the implementation of loop filter must take the order of filter into consideration at first, since it can directly determine the loop's dynamic response towards the input signal. A higher order generally leads to more compatibility, with complex relative movement situations, although, in the meantime, it also produces a process delay. Additionally, a second-order loop filter is mostly introduced for being sensitive to acceleration stress. Additionally, the noise bandwidth is significant because it directly determines the difficulty tracking the instant baseband signal and the volume of incoming noise. It is assumed that the total loop gain is \(K=K_{d}K_{o}\), which consists of \(K_{d}\) from the discriminator and \(K_{o}\) from the NCO. The transfer function of our designed loop filter can be presented as  
\[F(s) =\frac{1}{K}(2\omega_{np}+2\omega_{np}^{2}\cdot\frac{1}{s}+\omega_{ np}^{3}\cdot\frac{1}{s^{2}}), \tag{14}\]  
where \(\omega_{np}\) represents the undamped natural frequency. By mapping from continuous a S-Domain to discrete Z-Domain through the equation  
\[\frac{1}{s}=\frac{T_{i}}{1-z^{-1}}, \tag{15}\]  
that \(T_{i}\) represents tracking loop's renew interval. Thus, the discrete transfer function can be derived and presented as  
\[F(z)=\frac{(2\omega_{np}+2\omega_{np}^{2}T_{i}+\omega_{np}^{3}T_{i}^{2})-(4 \omega_{np}+2\omega_{np}^{2}T_{i})z^{-1}+2\omega_{np}z^{-2}}{K(1-2z^{-1}+z^{-2 })}. \tag{16}\]  
The structure of loop filter in PLL is shown in Figure 8, and input \(e_{p}\) is the output of phase discriminator. Figure 8: Structure diagram of digital loop filter of PLL.