## 3 Structure Design for FH Signal Tracking and Sorting
### Tracking of FH Signals
The power spectrum of \(x(t)\) can be expressed as:  
\[P_{x}(f)=E\Big{[}\big{|}X(f)\big{|}^{2}\Big{]}=E\Big{[}\big{|}X_{i}(f)\big{|}^{ 2}\Big{]} \tag{6}\]  
where \(E[\bullet]\) denotes expectation and \(P_{x}(f)\) is the Fourier transform of the autocorrelation function:  
\[P_{x}(f)=\int_{-\infty}^{+\infty}r_{x}(\tau)e^{-j2\pi/\tau}d\tau \tag{7}\]  
Based on [38], the power spectrum of the WSS signal is only related to itself and the result of cross correlation is equal to the power spectrum:  
\[E\Big{[}X_{i}(f_{1})X_{i}^{*}(f_{2})\Big{]}=P_{x}(f_{1})\delta(f_{1}-f_{2}) \tag{8}\]  
where \((\bullet)^{*}\) denotes conjugate and \(f_{1}\), \(f_{2}\) are arbitrary frequencies in the band of \(x(t)\). Thus, the power spectrum of \(y_{i}[n]\) can be expressed as:  
\[P_{y_{i}}(f)=E\Big{[}\big{|}Y_{i}(f)\big{|}^{2}\Big{]}=\sum_{l=-L_{0}}^{+L_{0} }|\epsilon_{il}+d_{il}|^{2}P_{x}(f-lf_{p})\hskip 14.226378 ptf\in\mathcal{F}_{s} \tag{9}\]  
Based on Parseval theorem,  
Figure 1:0  
As \(h(t)\) is an ideal low-pass filter, all the power of \(y_{i}[n]\) is limited in \(\mathcal{F}_{p}=\left[-\frac{f_{p}}{2},\frac{f_{p}}{2}\right]\). Thus:  
Figure 1:1 Figure 1:2 Figure 1:3  
Thus, the estimation of signal power in the time domain and of the MWC sub-channel in the frequency domain are connected. In the proposed structure, we obtain  
Figure 1:4  
where \(l_{s}\) is the length of the smoothing filter \(s_{m}[n]\) and \(a_{s}\) is the amplitude, and \(\otimes\) denotes convolution. Combining Equations (11) and (12), it can be concluded that \(\sigma_{i}[n]\) is a power estimation of \(y_{i}[n]\). For an appropriate filter length \(l_{s}\), \(\{\sigma_{i}[n]\}\)\(\quad i=1,2,\ldots,m\) can track the power change between sub-channels. Based on Equation (9), the power of certain sub-bands can be reflected in \(\{\sigma_{i}[n]\}\)\(\quad i=1,2,\ldots,m\). Therefore, as long as the signal-to-noise ratio (SNR) is high enough in its sub-band, the frequency hopping of the signal can be detected and tracked by estimating the energy of the sub-channels in the time domain. Suppose the threshold of energy detecting is \(\varepsilon\), which is based on the length of the smoothing filter and original signal power. The TOA estimation of a single hop is  
\[ \begin{array}{c}T_{H}[n]=\frac{\frac{(n-1)+\pi}{2}+\frac{(n+\tau-1)+\pi+\tau}{ 2}}{2}=\frac{2\pi+\tau-1}{2}\\ if\left\{\begin{array}{cc}\sigma_{i}[n-1]>\varepsilon,&\sigma_{i}[n]\leq \varepsilon\\ \sigma_{j}[n+\tau-1]\leq\varepsilon,&\sigma_{j}[n+\tau]>\varepsilon\end{array}\right. \end{array}  \tag{13}\]  
where \(i,j\) (\(i\
eq j\)) are the sub-channel numbers, \(\tau\) is a positive integer and less than the length of the smoothing filter. Suppose there is a single hop \(s_{i}(t)\) in sub-band \(l_{a}\) and there are \(\{l_{k}\}\) sub-bands including \(l_{a}\), amplified in sub-channel \(m_{c}\). Set \(\lambda_{m_{c}l_{i}}=0\quad l_{i}\
otin\{l_{k}\}\). Thus, the expectation of the output SNR \(\eta_{\infty}\) of \(\sigma_{m_{c}}[n]\) is  
\[\eta_{\infty}=\frac{\left(d_{m_{c}l_{a}}+c_{m_{c}l_{a}}\right)^{2}}{card(\{l_ {k}\})\bullet(d_{m_{c}}+c_{m_{c}})^{2}+(L-card(\{l_{k}\}))c_{i}^{2}}\eta_{i} \tag{14}\]  
where \(card(\bullet)\) denotes the cardinality of sets, \(d_{m_{c}}\) and \(c_{m_{c}}\) are the expectation of the Fourier series coefficients of \(p_{m_{c}}(t)\) and \(q_{m_{c}}(t)\), respectively, in sub-channel \(m_{c}\), and \(\eta_{i}\) is the expectation of input equivalent narrowband SNR of the signal. However, if we bypass the multi-frequency function, we obtain  
\[\eta_{o}=\frac{c_{i}^{2}}{Lc_{i}^{2}}\eta_{i}=\frac{1}{L} \tag{15}\]  
where \(c_{i}\) is the expectation of the Fourier coefficient of \(p_{i}(t)\). If \(d_{i}\gg c_{i}\), the output SNR of sub-channel \(m_{c}\) could be significantly increased. It is more efficient to process directly in the time domain than in the frequency domain, which is conducive for signal tracking. At the same time, FH signals generally have a high equivalent narrowband SNR, which meets the requirement of processing.