## 3 HD-VLL for Hybrid DS/FH Spread Spectrum Signals  
### A Description of the Problem  
It can be seen in (3) that the changing carrier frequency of each hopping time slot leads to the variable Doppler frequency, and the Doppler frequency jump between two adjacent hopping time slots can be expressed as follows:  
\[f_{d(j+1)}-f_{dj}=\frac{v_{j+1}}{c}[f_{c}+h(j+1)\Delta f]-\frac{v_{j}}{c}[f_{c }+h(j)\Delta f]. \tag{4}\]  
In general, it can be assumed that the velocity is almost the same in two adjacent frequency hopping time slots due to the high frequency hopping rate, so (4) can be written as follows:  
\[f_{d(j+1)}-f_{dj}=\frac{v}{c}[h(j+1)-h(j)]\Delta f. \tag{5}\]  
Equation (5) indicates that the Doppler frequency jump between adjacent hopping time slots is small when the relative motion velocity between the transmitter and receiver is small. In this case, an FLL can be used for carrier tracking, and the Doppler frequency jump will be huge when the relative motion velocity becomes large. This Doppler frequency jump continuously introduces a frequency step excitation to the FLL, causing a continuous response of the frequency step excitation, which makes it difficult to maintain steady-state tracking. To address the above problem, considering that the relative motion velocity is almost constant while the Doppler frequency is severely hopping between the two adjacent hopping time slots, we replace the FLL with a VLL in hybrid DS/FH spread spectrum signal carrier tracking. A VLL is much the same as an FLL except for the velocity discriminator. After the synchronization of the FH code, both the transmitter and receiver know the prior information of the FH sequence; thus, the velocity value can be converted to the corresponding Doppler frequency value in the multi-carrier NCO module. The structure of the VLL is shown in Figure 1.  
The FLL and serially assisted PLL and the FLL with a parallelly assisted PLL are commonly used coupled structures of FLLs and PLLs for DS spread spectrum signal carrier tracking. Since a VLL is used instead of an FLL for hybrid DS/FH spread spectrum signal carrier tracking, there are corresponding structures of a VLL with a serially assisted PLL (VLL-SA-PLL) and a VLL with a parallelly assisted PLL (VLL-PA-PLL), as shown in Figures 2 and 3, respectively. In addition, note that both the VLL and HD-VLL are second-order and the PLL is third-order in the following discussion and simulation. Taking the second-order VLL loop filter as an example, the filter output is fixed during each loop update period (the loop update period is generally the same as the coherent integration period). Assuming that in the \(k\)th loop update period, the initial phase of the multi-carrier NCO accumulator is \(\phi_{0k}\), the filter output \(\vartheta_{dk}\) in the last period plus velocity by acquisition \(v_{acq}\) is taken as an estimated value of the velocity \(\hat{v}_{d}\), which is converted into Doppler frequency according to the carrier frequency of the current hopping time slot. Then, the local instantaneous phase generated by the multi-carrier NCO during the \(k\)th loop update period is given by (6) as follows:  
\[\phi_{ki}=\phi_{0k}+\frac{2\pi f_{cj}}{c}\big{(}\vartheta_{dk}+v_{acq}\big{)} iT_{s}=\phi_{0k}+\frac{2\pi f_{cj}}{c}\vartheta_{d}iT_{s},\;\;i=0,1,2,\ldots,M-1, \tag{6}\]  
where \(T_{s}\) denotes the sampling interval, and \(i\) and \(M\) denote the \(i\)th sampling point and the number of sampling points in a loop update period, respectively. Practically, the instantaneous phase of the received signal is not only affected by velocity, but also by acceleration and jerk.