## 3 Structure Design for FH Signal Tracking and Sorting
### Sorting of FH Signals  
Based on the tracking algorithm in Section 3.1, the estimated hopping time is obtained. However, the power estimation of \(\sigma_{i}[n]\) is not an estimation of source because the received signal \(x(t)\) is modulated by \(p_{i}(t)\) and \(q_{i}(t)\). Furthermore, the multiple signal is not separated at the output. To solve these problems, in this section, the dynamic programming method is taken for the support set estimation of each single hop. The signals are separated according to their own sources based on hopping time and source power. Equation (3) can be given by the following matrix form [31]  
\[\mathbf{Y}(f)=\mathbf{A}\mathbf{Z}(f)\quad\ f\in\mathcal{F}_{s} \tag{16}\]  
where \(\mathbf{Z}(f)=\left[X(f-L_{0}f_{p}),\cdots,X(f),\cdots,X(f+L_{0}f_{p})\right]^ {T}\) and \(\mathbf{A}\) is the sensing matrix. It is clear that \(X(f+lf_{p}),l=-L_{0},\ldots,0,\ldots,L_{0}\) are independent and arranged in order of sub-bands. The support set can be regarded as the active sub-bands that contain the signals. The algorithms for solving the support set are the core of separation. Suppose the support set is \(S\), Equation (16) can be transformed to [31]  
\[\mathbf{Y}(f)=\mathbf{A}_{S}\mathbf{Z}_{S}(f)\quad\ f\in\mathcal{F}_{s} \tag{17}\]As the source signal is sparse, the number of elements in \(S\) is smaller than the number of observed signals. Thus [31]  
\[ \begin{array}{l}\mathbf{z}_{s}[n]=\mathbf{A}_{S}^{\dagger}\mathbf{y}[n]\\ z_{l}[n]=0,1\
otin S\end{array}  \tag{18}\]  
where \(\mathbf{z}[n]=\left[z_{-L_{0}}[n],\ldots,z_{L_{0}}[n]\right]^{T}\), \(l\in[-L_{0},\ldots 0,\ldots L_{0}]\), and \(z_{l}[n]\) is the inverse discrete time Fourier transform (DTFT) of \(Z_{l}(f)\). \(\mathbf{A}_{S}^{\dagger}\) is the generalized inverse matrix of \(\mathbf{A}_{S}\). Thus, the corresponding relation between source signals and received signals is established. The separated signal can be recovered in an order manner from \(\mathbf{y}[n]\). In Section 3.1, the support set was roughly estimated by the tracking of sub-channels. The accurate estimation of the support set by dynamic programming is shown in Figure 3. The situation of sub-bands (-1\(\sim\)\(L\)) is symmetrical with that of sub-bands (\(L\)\(\sim\)1), not shown in Figure 3.  
Without loss of generality, three sub-bands correspond to one sub-channel in Figure 3. In the tracking state, the amplitudes of multi-frequency functions \(\lambda_{i\bar{i}}\) are equal for the sub-bands. Once the power information of the single hop is detected in the \(i\)th sub-channel by \(\sigma_{i}[n]\), the amplitudes of multi-frequency functions for the \(i\)th sub-channel are dynamically programmed and other sub-channels stay the same. Suppose that the corresponding sub-bands for the \(i\)th sub-channel are \(k\), \(k\)+1 and \(k\)+2. In the tracking state, the amplitudes of multi-frequency functions are  
\[\lambda_{i,k}=\lambda_{i,k+1}=\lambda_{i,k+2}=\lambda_{t} \tag{19}\]  
The power estimation of the \(i\)th sub-channel is \(\sigma_{t}\). If the observed single hop does not cross sub-channels, there are 5 kinds of relationships between the single hop and sub-bands:  
\(H0\): The hop is only in the \(k\)th sub-band (as the single hop C in Figure 3). \(H1\): The hop crosses the \(k\)th sub-band and (\(k\)+1)th sub-band. \(H2\): The hop is only in the (\(k\)+1)th sub-band. \(H3\): The hop is crosses the (\(k\)+1)th sub-band and (\(k\)+2)th sub-band (as the single hop A in Figure 3). \(H4\): The hop is only in the (\(k\)+2)th sub-band. In the proposed system, the relationships are identified by dynamic programming. Figure 3: Dynamic programming strategy for support set estimation. First, a gain factor \(\beta(\beta>1)\) is applied for the amplitudes of multi-frequency functions. Thus we obtain  
\[\left\{ \begin{array}{l}\lambda_{i,k}=\frac{1}{\sqrt{\beta}}\lambda_{t}\\ \lambda_{i,k+1}=\lambda_{t}\\ \lambda_{i,k+2}=\sqrt{\beta}\lambda_{t}\end{array} \right.