# Cognitive Frequency-Hopping Waveform Design for Dual-Function MIMO Radar-Communications System
## 2 System Configuration and Signal Model  
### Phase Modulation (PM)-Based Frequency Hopping (FH) Waveforms  
A PM-based method for embedding information into the radar emission was recently proposed in [31]. To deliver a finite number of binary bits per radar pulse, the PM-based method maps the binary data into a phase symbol that belongs to a phase dictionary of an appropriate size. During each radar pulse, the PM-based method embeds one phase symbol into the radar emission toward the intended communication direction. At the communication receiver, a phase detector is used to detect the embedded symbol and, subsequently, deciphers the corresponding binary sequence. Unlike the amplitude modulation (AM) and amplitude shift keying (ASK) methods [30], the PM-based method offers the ability to embed information toward communication receivers, regardless of whether they are located within the sidelobe or the main lobe. Since target detection is the main task of the dual-function radar-communications system, the transmit waveform should be considered primarily based on the requirements of the radar function. One fundamental requirement of the radar function is the high efficiency of transmitted power. So constant-modulus waveforms are selected. FH waveform enjoys the constant-modulus feature and is easy to generate. Furthermore, FH pulse waveform is immune to multipath channel fading and clutter interference under the circumstances. The configuration for a joint radar-communications system and the PM-based FH signal model was developed in [44]. In this section, we follow the methods of [44] and further develop a PM-based dual-function MIMO radar-communications system. We express the MIMO FH waveforms as  
\[\phi_{m}(\mathbf{\textit{t}})=\sum_{q=1}^{Q}e^{j2\pi c_{mq}\Delta ft}\textit{u} (\mathbf{\textit{t}}-\Delta\mathbf{\textit{t}}),m=1,\ldots,M \tag{1}\]where \(c_{m,q},m=1,\ldots,M;q=1,\ldots,Q\) describes the FH code, \(M\) and \(Q\) denote the number of transmit antennas and the length of FH code, respectively. \(\Delta f\) and \(\Delta t\) respectively denote the frequency step and the hopping interval duration, and  
\[u(t)= \begin{cases}1,&0<t<\Delta t\\ 0,&\text{otherwise}\end{cases}  \tag{2}\]  
The duration of a pulse is expressed as \(T_{0}=Q\Delta t\). We assume that the code is \(c_{m,q}\in\{1,\ldots,J\}\), where \(J\) denotes a predefined value. As a result, the bandwidth of the radar pulse can be approximately denoted by \(J\Delta f\). To realize transmit waveform orthogonality, FH code should be designed to meet the following requirement \(c_{m,q}=c_{m^{\prime},q},\forall q,m=m^{\prime}\). Several papers proposed FH waveform optimization for MIMO systems (see [19]; and references therein). Then, we propose that a dual-function MIMO radar-communications system with phase symbols yields the extended virtual data model at the radar receiver. We also present the information-embedding scheme and the associated transmission rate. Let \(\{\Omega_{m,q}\in[0,2\pi]\},m=1,\ldots,M,q=1,\ldots,Q\) be a set of \(M\)Q PM-based symbols. Hence, the set of FH waveforms is expressed as:  
\[x_{m}(t)=\sum_{q=1}^{Q}e^{i\Omega_{m,q}}e^{j2\pi c_{m,q}\Delta f \text{t}}u(t-\Delta t),m=1,\ldots,M \tag{3}\]  
In (3), \(\Delta t\) should be designed to meet the requirement:  
\[\int_{0}^{\Delta t}e^{j2\pi c_{m,q}\Delta f\text{t}}e^{-j2\pi c_{m^{\prime},q }\Delta f\text{t}}dt=0,m=m^{\prime},q=q^{\prime} \tag{4}\]  
By using (1) and the orthogonality between the FH waveforms stated in (4), it is easy to prove that PM-based FH waveforms \(x_{m}(t),m=1,\ldots,M\) are also orthogonal. The set of transmitted waveforms is presented in Figure 1.  
The \(N\times 1\) vector of the target scattering signals at the output of the radar receive antenna array can be expressed as:  
\[\mathbf{y}\mathbf{t},\mathbf{\hat{\jmath}}=\mathbf{H}^{T}(\mathbf{\hat{\jmath }}\mathbf{x}(t,\mathbf{\hat{\jmath}})+\mathbf{n}(t,\mathbf{\hat{\jmath}}) \tag{5}\]  
where \(i\) describes the index.