# Joint Design of Transmit Waveforms for Object Tracking in Coexisting Multimodal Sensing Systems
## 2 Orthogonal Waveforms with Nonlinear Phase Function  
A general form of a time-varying (TV) waveform can be given in terms of a TV amplitude function \(a(t)\) and a TV phase function \(\xi(t/t_{r})\) as:  
\[s(t)=a(t)\,e^{\beta\pi b\,\xi(t/t_{r})}\,,\quad t\in\Xi_{t}, \tag{1}\]  
where \(b\)\(\in\)\(\mathbb{R}\) is the frequency-modulation (FM) rate, \(\Xi_{t}\) is the range of the time values \(t\), and \(t_{r}>0\) is a normalization time constant. An orthogonal TV waveform can be obtained from Equation (1) under certain constraints [38]. The first constraint is that the TV amplitude function is given by:  
\[a(t)=\sqrt{|\
u(t)|}=\left|\frac{d}{dt}\xi(t/t_{r})\right|^{1/2}, \tag{2}\]  
where \(\
u(t)\) is the waveform's instantaneous frequency, obtained as the derivative of the phase function. The second constraint is that the TV phase function \(\xi(t/t_{r})\) must be a monotonic function whose range is \(\gamma=\xi(t/t_{r})\)\(\in\)\(\mathbb{R}\). Then, assuming that both constraints hold, we define the \(m\)th TV orthogonal waveform, \(m\)\(\in\)\(\mathbb{N}\), with FM rate \(b_{m}\), as \(s_{m}(t)=\sqrt{|\
u(t)|}\,e^{\beta\pi b_{m}\,\xi(t/t_{r})}\), where the inverse function of \(\xi(t/t_{r})\)exists with range \(\Xi_{t}\), \(\xi(t/t_{r}){\in}\mathbb{R}\), \(t_{r}{>}0\), and \(\
u(t)=d/dt(\xi(t/t_{r}))\) in Equation (2). It can then be shown that the correlation of the two orthogonal signals \(s_{m}(t;b_{m})\) and \(s_{l}(t;b_{l})\) is given by:  
\[\int_{\Xi_{t}}s_{m}(t;b_{m})\,s_{l}^{*}(t;b_{l})\,dt=\int_{\Xi_{t}}\left|\
u(t )\right|e^{j2\pi(b_{m}-b_{l})\,\xi(t/t_{r})}\,dt=\int_{\mathbb{R}}e^{j2\pi(b_{ m}-b_{l})\,\gamma}\,d\gamma=\delta(b_{m}-b_{l})\,, \tag{3}\]  
where \(\delta(\cdot)\) is the Dirac delta function and \(m,l{\in}\mathbb{N}\). An example of a TV waveform that satisfies Equation (3) is the orthogonal linear frequency-modulated (OLFM) waveform that is specifically defined with quadratic phase function \(\xi(t/t_{r})=\mathrm{sgn}(t)|t/t_{r}|^{2}\), \(t{\in}\mathbb{R}\), where \(\mathrm{sgn}(t)\) is \(\pm 1\) depending on the sign of \(t\), and TV amplitude amplitude function given by \(a(t)=\sqrt{2|t|/t_{r}^{2}}\) in Equation (2). When the waveforms are used in practice with finite duration, orthogonality no longer holds. However, approximate orthogonality conditions can be obtained by minimizing the correlation between two finite duration waveforms. Specifically, the \(m\)th OLFM waveform, \(m=1,\ldots,M\), with finite duration \(T\) and FM rate \(b_{m}\) is given by:  
\[s_{m}(t;b_{m})=\sqrt{2\,t}\,e^{j2\pi b_{m}\,t^{2}},\quad t\in(0,T)\,. \tag{4}\]  
Here, \(t_{r}=1\) without loss of generality. For approximate orthogonality, the absolute correlation between any two finite duration OLFM waveforms, \(s_{m}(t;b_{m})\) and \(s_{l}(t;b_{l})\), \(m,l=1,\ldots,M\), must satisfy:  
\[\Gamma_{m,l}=\left|\,\int_{0}^{T}2\,t\,e^{j2\pi(b_{m}-b_{l})t^{2}}\,dt\right| =\left|\,\int_{0}^{T^{2}}\,e^{j2\pi(b_{m}-b_{l})\tau}d\tau\right|=\left|\,T^{ 2}\,\mathrm{sinc}\Big{(}(b_{m}-b_{l})\,T^{2}\Big{)}\right|=\delta[m-l], \tag{5}\]  
where \(\mathrm{sinc}(x)\triangleq\sin(\pi x)/\pi x\) and \(\delta[m]\) is the Kronecker delta function. Thus, the condition in Equation (5) holds, provided the FM rates, \(b_{m}\) and \(b_{l}\), of the OLFM waveforms are selected to satisfy:  
\[b_{m}-b_{l}=\frac{\ell}{T^{2}},\quad\ell=0,\pm 1,\pm 2,\ldots \tag{6}\]  
If we only select positive FM rates, \(b_{m}{\in}\mathbb{R}^{+}\), then Equation (6) simplifies to \(b_{m}-b_{l}=\ell/T^{2}\), \(\ell\in\mathbb{N}^{+}\), for \(m>l\). Note that the values of the FM rates are constrained by the waveform bandwidth \(B\), as the instantaneous frequency of the \(m\)th OLFM waveform in Equation (4) is given by \(\
u(t)=2\,b_{m}\,t,{\in}(0,T)\). Thus, the largest possible FM value \(b_{M}\) of the OLFM waveform must satisfy \(B=2\,b_{M}\,T\). The maximum number \(M\) of approximate OLFM waveforms that can be assigned a unique positive FM rate that satisfy Equation (5) is thus given by \(M=\left\lfloor TB/2\right\rfloor\).