# Cognitive Frequency-Hopping Waveform Design for Dual-Function MIMO Radar-Communications System
## 4 Waveform Optimization  
In this section, we aim to enhance the target detection and feature estimation performance by maximizing the MI between the target response and the target returns in the first step, and then minimizing the MI between successive target scattering signals in the second step. These two stages correspond to the design of the ensemble of excitations and the selection of a suitable signal out of the ensemble, respectively. The two-step cognitive waveform design strategy is based upon continuous learning from the radar scene. The dynamic information about the target feature is utilized to design PM-based FH codes. In this way the transmitter adjusts its probing signals to suit the dynamically changing environment. Step 1: this step involves the design of PM-based FH waveforms for the dual-function transmit array. The main idea of waveform design is to maximize the MI between the target scattering signal and the target response, subject to the transmit power constraint. Step 2: once an ensemble of optimal transmit waveforms has been designed, then we select the most reasonable waveform for emission from the ensemble. The key concept of waveform selection is to minimize the MI between the target scattering signals at present and the next target-scattering signals. The step ensures that we continually obtain target scattering signals that are independent of each other in time, in order to achieve more feature information of the target at each time instant of reception. Figure 2 describes the architecture of an adaptive dual-function MIMO radar communication system. Step 1: during the i-th radar scan, the set of PM-based FH waveforms \(x_{m}(t,i),m=1,\ldots,M\) can be expressed as a matrix \(\mathbf{X}_{i}\in\mathbb{C}^{M\times K}\) after discrete sampling, where \(\mathbb{C}\) describes the number domain and \(K\) indicates the sample number. \(\mathbf{N}\in\mathbb{C}^{N\times K}\) is zero-mean white noise matrix. Therefore, the \(N\times K\) matric of the target scattering signals is written as:  
\[\mathbf{Y}_{i}=\mathbf{H}_{i}^{T}\mathbf{X}_{i}+\mathbf{N} \tag{23}\]  
Here \(\mathbf{H}_{i}\sim\mathbb{C}\Big{(}0\),\(\mathbf{R}_{\mathbf{H}_{i}}\Big{)}\) and \(\mathbf{N}\sim\mathbb{C}(0\),\(\mathbf{R}_{\mathbf{N}})\), and \(\mathbf{R}_{\mathbf{H}_{i}}=E\Big{[}\mathbf{H}_{i}\mathbf{H}_{i}^{H}\Big{]}\) and \(\mathbf{R}_{\mathbf{N}}=E\Big{[}\mathbf{N}^{H}\mathbf{N}\Big{]}\) indicate the covariance matrices of the channel response \(\mathbf{H}_{i}\) and the zero-mean white noise \(\mathbf{N}\), respectively. We intend to maximize the MI between the target-scattering signal and the target response given the transmit waveforms. This involves that the target-scattering signals would be more dependent upon the actual target feature information. According to the definition of MI, we have:  
\[I\big{(}\mathbf{Y}_{i};\mathbf{H}_{i}|\mathbf{X}_{i}\big{)}=H\big{(}\mathbf{Y }_{i}|\mathbf{X}_{i}\big{)}-H\big{(}\mathbf{N}\big{)} \tag{24}\]  
In (24), \(I\big{(}\mathbf{Y}_{i};\mathbf{H}_{i}|\mathbf{X}_{i}\big{)}\) represents the MI between two random variates \(\mathbf{Y}_{i}\) and \(\mathbf{H}_{i}\) given the transmit matrix \(\mathbf{X}_{i}\), and \(H\big{(}\mathbf{Y}_{i}|\mathbf{X}_{i}\big{)}\) indicates the conditional entropy that \(\mathbf{X}_{i}\) conveys about \(\mathbf{Y}_{i}\). The main objective of this step is to maximize \(I\big{(}\mathbf{Y}_{i};\mathbf{H}_{i}|\mathbf{X}_{i}\big{)}\) between \(\mathbf{Y}_{i}\) and \(\mathbf{H}_{i}\) given \(\mathbf{X}_{i}\). According to the definition of entropy, we have:  
\[H\big{(}\mathbf{Y}_{i}|\mathbf{X}_{i}\big{)}=\int-p\big{(}\mathbf{Y}_{i}| \mathbf{X}_{i}\big{)}ln\big{[}p\big{(}\mathbf{Y}_{i}|\mathbf{X}_{i}\big{)} \big{]}d\mathbf{Y}_{i} \tag{25}\]  
Here \(p\big{(}\mathbf{Y}_{i}|\mathbf{X}_{i}\big{)}\) indicates the conditional probability density function (PDF) of the received matrix \(\mathbf{Y}_{i}\) given transmit matrix \(\mathbf{X}_{i}\). The conditional PDF \(p\big{(}\mathbf{Y}_{i}|\mathbf{X}_{i}\big{)}\) can be expressed as follows:  
\[