## 3 Structure Design for FH Signal Tracking and Sorting
### Tracking of FH Signals  
Tracking of signals is a fundamental requirement for a real-time system. The purpose of FH signal tracking is to obtain the time and frequency of new hops as soon as possible, which is an important basis for separation. In order to apply CS to tracking, the system should be able to cope with changes in sparsity and update the processing results rapidly. Although the classic MWC structure works in the Fourier domain and is fit to conduct wideband spectrum sensing, the under-sampling board is designed by a given sparsity [33]. To make the system more suitable and feasible for FH signals, a novel dynamic programming MWC structure is proposed, which is depicted in Figure 2.  
Figure 1: Sketch of tracking and separation of frequency hopping (FH) signal. As shown in Figure 2, the dynamic programming MWC structure is a single antenna system consisting of \(m\) parallel sub-channels. In each sub-channel, the multiband signal \(x(t)\) is multiplied by a different mixing function \(p_{i}(t)\) and multi-frequency function \(q_{i}(t)\); \(h(t)\) is an ideal low-pass filter and \(s_{m}[n]\) is a smoothing filter that performs real-time sliding accumulation of energy. Once the mixing function of the sub-channel is determined, the response of the channel is relatively determined. Because of the pseudo-random property and expectation restricted isometry property (ExRIP) of the mixing function, \(p_{i}(t)\) is restricted and has certain random features [37]. The precise description of \(p_{i}(t)\) is provided in Appendix A and a brief description of ExRIP is provided in Appendix B. It is difficult to distinguish the signal information only by the mixing function. To solve these problems, a multi-frequency function \(q_{i}(t)\) is designed and implemented that depends on the mixing function and communication environment and is dynamically programmed by the estimation results of \(\sigma_{i}[n]\). Unlike the randomness of the mixing function, the multi-frequency function is determined for judgment in each sub-channel and strengthens the sensing ability for certain sub-bands in different sub-channels. As FH signals have wide frequency gaps, the frequency hopping can be sensed by tracking the power change of the sub-band. Thus, with a reasonable distribution of the sub-bands to the sub-channel, the frequency hopping can also be tracked by the power estimation of sub-channels. The discrete time Fourier transform (DTFT) of \(y_{i}[n]\) can be expressed as  
\[Y_{i}\Big{(}e^{j2\pi ft_{s}}\Big{)}=\sum\limits_{l=-L_{0}}^{+L_{0}}c_{il}X(f- lfp)+\sum\limits_{l=-L_{0}}^{+L_{0}}d_{il}X(f-lf_{p})\quad f\in\mathcal{F}_{s} \tag{3}\]  
where \(c_{il}\) and \(d_{il}\) are the Fourier series coefficients of \(p_{i}(t)\) and \(q_{i}(t)\), respectively, and \(d_{il}\) can be expressed as  
\[d_{il}=\frac{1}{T_{p}}\int_{0}^{T_{p}}q_{i}(t)e^{-j2\pi lfp}dt \tag{4}\]  
It is clear that \(q_{i}(t)\) is a \(T_{\rm p}\)-periodic function. In practical application, \(q_{i}(t)\) can be simply set as  
\[q_{i}(t)=\sum\limits_{l}\lambda_{il}e^{j2\pi lfp}\quad\ k\frac{T_{p}}{M}\leq t \leq(k+1)\frac{T_{p}}{M} \tag{5}\]  
Figure 2: Dynamic programming modulated wideband converter (MWC) structure. where \(M\) is the number of equal time intervals and \(\lambda_{il}\) is the amplitude of \(q_{i}(t)\). For each sub-channel, \(\lambda_{il}\) is dynamically programmed for estimation of the support set, which will be discussed in Section 3.2. The algorithm of signal tracking in the time domain is described as follows:  
Suppose there is no frequency hopping during the observation. Thus, the mixing signal \(x(t)\) is a wide-sense stationary process (WSS), the Fourier transform of which is \(X(f)\). \(x_{i}(t)\) is a part of \(x(t)\) and \(X_{i}(f)\) is the Fourier transform of \(x_{i}(t)\).