# Cognitive Frequency-Hopping Waveform Design for Dual-Function MIMO Radar-Communications System
## 3 Multiple-Input Multiple-Output (MIMO) Radar Ambiguity Function  
In this section, we consider a radar target at \(\chi(\tau,v,f)\) where \(\tau\) describes the delay of the target range, \(v\) denotes the Doppler frequency, \(f=2\pi\frac{d_{R}}{\lambda}\sin\theta\) indicates the spatial frequency. Here \(\theta\) and \(\lambda\) are the angle of the target and the wavelength, respectively. \(d_{R}\) and \(d_{T}\) are the distance between the transmit antennas and between the receiver antennas, respectively. To simplify the discussion, we assume \(d_{T}=d_{R}\) in this paper. \(x_{m}(t),m=1,\dots,M\) denotes the waveform radiated by the \(m\)-th transmit antenna. In [17], the MIMO radar ambiguity function can be defined as:  
\[\chi(\tau,v,f,f^{{}^{\prime}})=\sum_{m=1}^{M}\sum_{m^{\prime}=1}^{M}\chi_{m,m^ {\prime}}(\tau,v)e^{j2\pi(fm-f^{\prime}m^{\prime})} \tag{12}\]  
where  
\[\chi_{m,m^{\prime}}(\tau,v)=\int_{-\infty}^{\infty}x_{m}(t)x_{m^{\prime}}^{*} (t+\tau)e^{j2\pi vt}dt \tag{13}\]  
In (13), \(\chi_{m,m^{\prime}}(\tau,v)\) describes the cross ambiguity function, which implicates two radar waveforms \(x_{m}(t)\) and \(x_{m^{\prime}}(t)\). \(\chi_{m,m^{\prime}}(\tau,v)\) is analogous to the SIMO ambiguity function presented in [19]. We now discuss the MIMO radar ambiguity function for the case when \(x_{m}(t)\) is composed of the shifted forms of a rectangular pulse \(u_{m}(t)\). \[x_{m}(t)=\sum_{l=1}^{L}u_{m}(t-T_{l}) \tag{14}\]  
where \(l\) denotes the number of the rectangular pulse and \(T\) describes pulse repetition interval (PRI). The cross ambiguity function of the waveform \(x_{m}(t)\) is defined as:  
\[\chi_{m,m^{\prime}}(\tau,v)=\sum_{l^{\prime}=1}^{L}\sum_{l=1}^{L}\chi_{m,m^{ \prime}}^{u}(\tau+T_{l}-T_{l^{\prime}},v)e^{j2\pi vTl} \tag{15}\]  
where \(\chi_{m,m^{\prime}}^{u}(\tau,v)\) describes the cross ambiguity function of the rectangular pulses \(u_{m}(t)\) and \(u_{m^{\prime}}(t)\). We assume that the pulse duration \(T_{0}\) and the Doppler frequency \(v\) are small enough so that \(T_{0}v\)\(\approx\)0. At a result, the envelope of the Doppler frequency keeps unchanged within an entire pulse period. The cross ambiguity function \(\chi_{m,m^{\prime}}^{u}(\tau,v)\) reduces to the cross correlation function \(r_{m,m^{\prime}}^{u}(\tau)\), which is not a function of \(v\) any more, we obtain:  
\[\chi_{m,m^{\prime}}^{u}(\tau,v)\)\(\approx\)\(r_{m,m^{\prime}}^{u}(\tau) \tag{16}\]  
We further assume that little reflections take place at these second trip ranges. Therefore, the cross ambiguity function of \(x_{m}(t)\) can be rewritten as:  
\[\chi_{m,m^{\prime}}(\tau,v)\simeq r_{m,m^{\prime}}^{u}(\tau)\sum_{l=1}^{L}e^ {j2\pi vT_{l}} \tag{17}\]  
The MIMO ambiguity function of \(x_{m}(t)\) can be expressed as:  
\[\chi(\tau,v,f,f^{{}^{\prime}})=\sum_{m=1}^{M}\sum_{m^{\prime}=1}^{M}r_{m,m^{ \prime}}^{u}(\tau)e^{j2\pi(fm-f^{\prime}m^{\prime})}\sum_{l=1}^{L}e^{j2\pi vT _{l}} \tag{18}\]  
It is worth noting that the MIMO ambiguity function \(\chi(\tau,v,f,f^{{}^{\prime}})\) depends on the cross correlation functions \(r_{m,m^{\prime}}^{u}(\tau)\). Furthermore, the shifted forms of a rectangular pulse \(\{u_{m}(t)\}\) just have an impact on the range and spatial resolution. The pulses have no effect on the Doppler resolution. Consequently,to gain a sharp MIMO ambiguity function, these waveforms should be designed such that the function \(\Omega(\tau,f,f^{\prime})\) is denoted as:  
\[\Omega(\tau,f,f^{\prime})=\sum\limits_{m=1}^{M}\sum\limits_{m^{\prime}=1}^{M}r_{ m,m^{\prime}}^{u}(\tau)e^{|2\pi(fm-f^{\prime}m^{\prime})} \tag{19}\]  
Next, we discuss the MIMO radar ambiguity function of the PM-based information embedding FH waveforms. The proposed waveforms can be expressed as (3). We intend to obtain the expression for the function \(\Omega(\tau,f,f^{\prime})\) in terms of \(\{c_{m,q}\}\) and \(\{\Omega_{m,q}\}\). To derive \(\Omega(\tau,f,f^{\prime})\), we therefore begin with the cross correlation function of the PM-based FH waveform.