## 3 HD-VLL for Hybrid DS/FH Spread Spectrum Signals
### Hd-Vll-Nco
Therefore, in high dynamic cases, it is necessary to comprehensively consider the precision and bit width when selecting the parameters for the multi-carrier NCO so as to ensure that the compensation of jerk and acceleration can be reflected on the frequency word while minimizing the waste of resources. Figure 6: Structure of multi-carrier NCO. Figure 7: Structure of HD-VLL-NCO. It can be seen in Figure 7 that there are three accumulators corresponding to three frequency words in the high dynamic multi-carrier NCO, with the name of jerk frequency word, acceleration frequency word, and frequency word, respectively. Three accumulators simultaneously accumulate these three frequency words, and the high \(N_{0}\) bits of the velocity accumulator are selected to perform a phase query. Finally, the multi-carrier NCO outputs the amplitude values of the sine and cosine wave signals. By appropriately setting the bit widths of the jerk frequency word, the acceleration frequency word, the frequency word, the accumulators, as well as the truncation, it is possible to achieve the accuracy and dynamic requirements while minimizing the waste of resources. The formulas and bit widths for the frequency words of each order are given below. The first-stage jerk accumulator sum2 accumulates the jerk frequency word \(f_{2w}\) and truncates \(c_{2}\) bits of the result and then adds it to the acceleration frequency word \(f_{1w}\), and the output \(\ddot{\theta}\) is  
\[\ddot{\theta}(n)=f_{1w}+\frac{1}{2^{c_{2}}}\sum_{m=1}^{n}f_{2w}=f_{1w}+\frac{ nf_{2w}}{2^{c_{2}}}. \tag{16}\]  
The second-stage acceleration accumulator sum1 accumulates \(\ddot{\theta}(n-1)\) and truncates \(c_{1}\) bits of the result and then adds it to the frequency word \(f_{0w}\), and the output \(\dot{\theta}\) is  
\[\dot{\theta}(n)=f_{0w}+\frac{1}{2^{c_{1}}}\sum_{m=1}^{n}\theta_{2}(m-1)=f_{0w }+\frac{nf_{1w}}{2^{c_{1}}}+\frac{f_{2w}(n^{2}-n)}{2\times 2^{c_{1}+c_{2}}}. \tag{17}\]  
The third-stage velocity accumulator sum0 accumulates \(\dot{\theta}(n-1)\) and then truncates \(c_{0}\) bits of the result, and the output \(\theta_{0}\) is  
\[\theta_{0}(n)=\frac{1}{2^{c_{0}}}\sum_{m=1}^{n}\theta_{1}(m-1)=\frac{nf_{0w}}{ 2^{c_{0}}}+\frac{f_{1w}(n^{2}-n)}{2\times 2^{c_{0}+c_{1}}}+\frac{f_{2w}(n^{3}- 3n^{2}+2n)}{6\times 2^{c_{0}+c_{1}+c_{2}}}. \tag{18}\]  
The phase variation caused by the filtered jerk \(\dot{\hat{a}}\), the filtered acceleration \(\hat{a}\), and the estimated velocity \(\hat{\sigma}_{d}\) can be expressed as  
\[\theta(t)=\frac{2\pi f_{cj}}{c}(\hat{\sigma}_{d}t+\frac{\hat{a}}{2}t^{2}+ \frac{\hat{\hat{a}}}{6}t^{3}). \tag{19}\]  
After discretizing the analog signal, that is, \(t=nT_{s}=n/f_{s}\), the equivalent phase sequence is defined as  
\[\theta(n)=\frac{2\pi f_{cj}}{c}(\frac{\hat{\sigma}_{d}}{f_{s}}n+\frac{\hat{a} }{2f_{s}^{2}}n^{2}+\frac{\hat{\hat{a}}}{6f_{s}^{3}}n^{3}). \tag{20}\]  
According to the principle of direct digital synthesis (DDS), the output of the accumulator has the following relationship with the phase of simulated signal:  
\[\frac{N_{sum}}{2^{N_{0}}}=\frac{\theta(n)}{2\pi}, \tag{21}\]  
where \(N_{sum}\) denotes the input of the lookup table and \(N_{0}\) denotes the bit width of the lookup table. Hence, \(N_{sum}\) can, in turn, be derived as follows:  
Figure 7.0  
By making the output \(\theta_{0}(n)\) of the HD-VLL based on the multi-carrier NCO equal to \(N_{sum}\), the formulas for the frequency word of each order can be obtained as follows:\[\left\{ \begin{array}{l}f_{0w}=\frac{f_{cl}}{c}\left(\frac{\dot{a}}{6f_{s}^{3}}+ \frac{\dot{a}}{2f_{s}^{2}}+\frac{\dot{a}}{f_{s}}\right)2^{N_{0}+c_{0}}\\ f_{1w}=\frac{f_{cl}}{c}\left(\frac{\dot{a}}{f_{s}^{3}}+\frac{\dot{a}}{f_{s}^{2}} \right)2^{N_{0}+c_{0}+c_{1}}\\ f_{2w}=\frac{f_{cl}}{c}\frac{\dot{a}}{f_{s}^{3}}2^{N_{0}+c_{0}+c_{1}+c_{2}} \end{array} \right.. \tag{23}\]  
The simplified formulas can be derived as follows:  
\[\left\{