# Cognitive Frequency-Hopping Waveform Design for Dual-Function MIMO Radar-Communications System
## 4 Waveform Optimization
\begin{array}{rl}p(\mathbf{Y}_{i}|\mathbf{X}_{i})&=\prod\limits_{n=1}^{N}p \Big{(}\mathbf{y}_{n,i}|\mathbf{X}_{i}\Big{)}\\ &=\prod\limits_{n=1}^{N}\frac{1}{n^{k}\text{det}\big{(}\chi_{i}^{H}\mathbf{R} _{\mathbf{H}_{i}}\mathbf{X}_{i}+\mathbf{R}_{\mathbf{N}}\big{)}}e^{-\mathbf{y} _{n,i}^{H}\big{(}\chi_{i}^{H}\mathbf{R}_{\mathbf{H}_{i}}\mathbf{X}_{i}+ \mathbf{R}_{\mathbf{N}}\big{)}^{-1}\mathbf{y}_{n,i}}\\ &=\!\frac{1}{n^{N}\text{det}\big{(}\text{det}\big{(}\chi_{i}^{H}\mathbf{R}_{ \mathbf{H}_{i}}\mathbf{X}_{i}+\mathbf{R}_{\mathbf{N}}\big{)}^{N}\big{)}^{N}}e^ {-tr\big{(}[\chi_{i}^{H}\mathbf{R}_{\mathbf{H}_{i}}\mathbf{X}_{i}+\mathbf{R}_{ \mathbf{N}}\big{)}^{-1}\chi_{i}^{H}\mathbf{Y}_{i}]}\end{array}  \tag{26}\]  
Figure 2: The architecture of an adaptive dual-function MIMO radar communication system. Solving (25) and (26) gives rise to the following result for the conditional entropy [40]:  
\[H(\mathbf{Y}_{i}|\mathbf{X}_{i})=NKIn(\pi)+NK+NIn\Big{[}\text{det}\Big{(} \mathbf{X}_{i}^{H}\mathbf{R}_{\mathbf{H}_{i}}\mathbf{X}_{i}+\mathbf{R}_{ \mathbf{N}}\Big{)}\Big{]} \tag{27}\]  
Similarly, the result for the entropy of the noise can be derived as follows:  
\[H(\mathbf{N})=NKIn(\pi)+NK+NIn[\text{det}(\mathbf{R}_{\mathbf{N}})] \tag{28}\]  
Making use of Equations (24), (27) and (28), the MI between the target-scattering signal and the channel response given transmit waveforms can be rewritten as follows:  
\[I(\mathbf{Y}_{i};\mathbf{H}_{i}|\mathbf{X}_{i})=NIn\Big{[}\text{det}\Big{(} \mathbf{X}_{i}^{H}\mathbf{R}_{\mathbf{H}_{i}}\mathbf{X}_{i}+\mathbf{R}_{ \mathbf{N}}\Big{)}\Big{]}-NIn[\text{det}(\mathbf{R}_{\mathbf{N}})] \tag{29}\]  
Therefore, we can formulate the MI maximization criterion as follows:  
\[ \begin{array}{l}\underset{\mathbf{X}_{i}}{\max}\Big{\{}NIn\Big{[}\text{det} \Big{(}\mathbf{X}_{i}^{H}\mathbf{R}_{\mathbf{H}_{i}}\mathbf{X}_{i}+\mathbf{R}_ {\mathbf{N}}\Big{)}\Big{]}-NIn[\text{det}(\mathbf{R}_{\mathbf{N}})]\Big{\}} \frac{n!}{n!(n-r)!}\\ s.t.\text{tr}\Big{[}\mathbf{X}_{i}^{H}\mathbf{X}_{i}\Big{]}\leq P_{0}\\ c_{m,q}\
eq c_{m^{\prime},q},\text{ for }m=m^{\prime},\forall q\ \ \ \Delta t\Delta f=1\end{array}  \tag{30}\]  
where \(P_{0}\) indicates the transmit power. The work of [40] has given a rigorous solution of the above optimization problem (30). Then, we can obtain the ensemble \(\mathbb{C}_{\hat{\mathbf{X}}_{i}}\) out of the whole set of PM-based FH sequences, and the corresponding power allocation vector over diverse dual-function transmit antennas. We start the design procedure with the transmit waveforms from the PM-based FH matrix, and allocate the power on a pulse level as well as across the transmit antennas based on the MI maximization criterion. Step 2: we then proceed to the waveform selection procedure, in which the successive target scattering signals are different from each other. The step ensures that we achieve more information of the target feature at each time instant of reception. We denote the MI between the successive target scattering signals at time \(i\) and at time \(i+1\) as:  
\[I(\mathbf{Y}_{i+1},\mathbf{Y}_{i})=H\Big{(}\mathbf{Y}_{i+1}\big{|}\mathbf{X}_ {i+1}\Big{)}+H(\mathbf{Y}_{i}|\mathbf{X}_{i})-H\Big{(}\mathbf{Y}_{i+1}, \mathbf{Y}_{i}\big{|}\mathbf{X}_{i+1},\mathbf{X}_{i}\Big{)} \tag{31}\]  
In (31), the term \(H(\mathbf{Y}_{i}|\mathbf{X}_{i})\) (or \(H\Big{(}\mathbf{Y}_{i+1}\big{|}\mathbf{X}_{i+1}\Big{)}\)) denotes the entropy of \(\mathbf{Y}_{i}\) (or \(\mathbf{Y}_{i+1}\)) at time \(i\) (or \(i+1\)) given the knowledge of \(\mathbf{X}_{i}\) (or \(\mathbf{X}_{i+1}\)). The term \(H\Big{(}\mathbf{Y}_{i+1},\mathbf{Y}_{i}\big{|}\mathbf{X}_{i+1},\mathbf{X}_{i} \Big{)}\) is defined similarly. According to the literature [6; 34], the above Equation (31) can be rewritten as follows:  
\[ \begin{array}{l}I(\mathbf{Y}_{i},\mathbf{Y}_{i+1}) =-NIn\Big{(}\text{det}\Big{(}\mathbf{I}_{(M\times M)}-\mathbf{D}_ {i,i+1}^{2}\Big{)}\Big{)}\\ =-N\sum\limits_{m=1}^{M}In\Big{(}1-\Big{(}d_{i,i+1}^{m}\Big{)}^{2}\Big{)}\end{array}