# Cognitive Frequency-Hopping Waveform Design for Dual-Function MIMO Radar-Communications System
## 4 Waveform Optimization
 \tag{32}\]  
where \(d_{i,i+1}^{m}(d_{i,i+1}^{1}\geq d_{i,i+1}^{2}\geq\ldots\geq d_{i,i+1}^{M})\) is the diagonal element of the diagonal matrix \(\mathbf{D}_{i,i+1}\). \(\mathbf{D}_{i,i+1}\) is achieved by singular value decomposition (SVD) of the covariance matrix, which can be denoted as:  
\[\mathbf{R}_{\mathbf{Y}_{i}}\mathbf{Y}_{i+1}=E\Big{(}\mathbf{Y}_{i}^{H}\mathbf{ Y}_{i+1}\Big{)}=\mathbf{X}_{i}^{H}\mathbf{R}_{\mathbf{H}_{i}}\mathbf{X}_{i+1} \tag{33}\]  
In (33), \(\mathbf{R}_{\mathbf{Y}_{i},\mathbf{Y}_{i+1}}\) is the cross-covariance of the expressions for \(\mathbf{Y}_{i}\) and \(\mathbf{Y}_{i+1}\). Hence, the MI minimization criterion can be expressed as:\[ \begin{array}{l}\min\limits_{\mathbf{x}_{i+1}\in\mathbb{C}_{\hat{\mathbf{X}}_{i}} }\left\{-N\sum\limits_{m=1}^{M}In\left\{1-\left(d_{i,i+1}^{m}\right)^{2} \right\}\right\}\\ s.t.\ \text{tr}\left[\mathbf{X}_{i+1}^{H}\mathbf{X}_{i+1}\right]\leq P_{0}\\ c_{m,q}\
eq c_{m^{\prime},q},\ \ \text{for}\ m=m^{\prime},\forall q\ \ \Delta t\Delta f=1\end{array}  \tag{34}\]  
We can estimate the values for \(\mathbf{Y}_{i+1}\) over all possible values of \(\hat{\mathbf{X}}_{i+1}\in\mathbb{C}_{\hat{\mathbf{X}}_{i}}\) using (23). Thus, we can also form an estimate of all the values of the corresponding \(\mathbf{D}_{i,i+1}\ (d_{i,i+1}^{m}(d_{i,i+1}^{1}\geq d_{i,i+1}^{2}\geq\ldots\geq d_{i,i+1}^{M}))\) and choose the value for \(\hat{\mathbf{X}}_{i+1}\) that minimizes (34). The above waveform optimization problem (34) is convex. We can obtain the optimal solution directly by using a MATLAB optimization toolbox, such as CVX. Step 1 designs the optimal ensemble \(\mathbb{C}_{\hat{\mathbf{X}}_{i}}\) based on MI maximization criterion over the spatial domain, and step 2 selects the optimized PM-based FH waveform for each dual-functional transmit antenna from the ensemble \(\mathbb{C}_{\hat{\mathbf{X}}_{i}}\) based on MI minimization criterion over the temporal domain. The proposed adaptive waveform design and selection procedures can be summarized as Algorithm 1. ```
Initialize the transmit matrix \(\mathbf{X}_{0}\) and the covariance matrix of the noise \(\mathbf{R}_{\mathbf{N}}\). 1. At the initial time \(t=0\), measure the target scattering signal \(\mathbf{Y}_{0}\) and calculate the covariance of the target-scattering signal \(\mathbf{R}_{\mathbf{Y}_{0}}\). The covariance matrix of the channel response \(\mathbf{R}_{\mathbf{H}_{0}}\) can be obtained through successive measurements with uniform power allocation over the transmit antennas by solving (36). 2. The optimized waveform ensemble \(\mathbb{C}_{\hat{\mathbf{X}}_{0}}\) can be obtained through the waveform design process by solving (30). 3. At time \(t=1\), measure \(\mathbf{Y}_{1}\) and calculate \(\mathbf{R}_{\mathbf{Y}_{1}}\) and the cross-covariance matrix \(\mathbf{R}_{\mathbf{Y}_{0},\mathbf{Y}_{1}}\). The actual channel response \(\mathbf{H}_{1}\) is obtained by de-convolving the target-scattering signal with the transmit signal by using (23). 4. Calculate the corresponding singular values \(d_{0,1}^{m}\), \(m=1,\ldots,M\) of \(\mathbf{R}_{\mathbf{Y}_{0},\mathbf{Y}_{1}}\) and \(\hat{\mathbf{X}}_{1}\in\mathbb{C}_{\hat{\mathbf{X}}_{0}}\) can be acquired through the waveform selection process by solving (38). 5. At time \(t=1\), emission \(\hat{\mathbf{X}}_{1}\) and observe the corresponding received signal \(\hat{\mathbf{Y}}_{1}\) to achieve \(\mathbf{R}_{\mathbf{H}_{1}}\) by using (36). 6. If \(i=I_{\text{max}}\), the iterative procedure ends; or else, repeat steps 2-5 iteratively. ```  
**Algorithm 1** The adaptive waveform design and selection algorithm  
It is worth noting that adaptation is included in the waveform design and selection procedures through the feedback process and numerous interactions with the radar channel. In summary, the adaptive waveform optimization strategy for target detection is implemented according to the block diagram in Figure 3.