C Heisenberg group
C.1 Heisenberg group action on the contact space
We consider the unitary shift operators \(T_\tau,M_\omega\in\mathcal{U}(L^2(\mathbb{R}))\). The commutation relation of these unitary operators is given by \[\begin{equation} T_\tau M_\omega T_\tau^{-1} M_\omega^{-1} = e^{-2\pi i\omega\tau} \mathrm{Id} \end{equation}\]
Indeed, \(\forall s\in L^2(\mathbb{R})\) \[\begin{align*} T_\tau M_\omega T_\tau^{-1} M_\omega^{-1} s(t) &= T_\tau M_\omega T_\tau^{-1} e^{-2\pi i\omega t} s(t)\\ &= T_\tau M_\omega e^{-2\pi i\omega(t+\tau)} s(t+\tau)\\ &= T_\tau e^{-2\pi i\omega\tau} s(t+\tau)\\ &= e^{-2\pi i\omega\tau} s(t) \end{align*}\]
We also consider the representation \(U\) from the Heisenberg group \(\mathbb{H}^1\) defined \(U:\mathbb{H}^1\rightarrow\mathcal{U}(L^2(\mathbb{R}^2))\) by \[\begin{equation} U(\tau,\omega,\lambda) = e^{-2\pi i\lambda}T_\tau M_\omega \end{equation}\]
Therefore, the operator algebra generated by the unitary time and frequency shift operators coincides with the Heisenberg group \(\mathbb{H}^1\) [7].
C.2 Introducing the chirpiness to the Heisenberg group
An automorphism of a group \(G\) is a group isomorphism from \(G\) onto \(G\).
An isomorphism from \((G_1,*)\) to \((G_2,\cdot)\) is a bijective function \(f:G_1\rightarrow G_2\) such as \(\forall x,y\in G_1\)
\[\begin{equation} f(x*y) = f(x)\cdot f(y) \end{equation}\]
It follows that the automorphism of a group \((G,\cdot)\) is a bijective function \(g:G\rightarrow G\) such as \(\forall x,y\in G\)
\[\begin{equation}\label{eq:aut_g} g(x\cdot y) = g(x)\cdot g(y) \end{equation}\]
Let’s consider the affine group group \(H=\mathbb{R}_+\ltimes\mathbb{R}\) equipped with the square matrix multiplication.
\[\begin{equation} H = \left\{\left.\begin{pmatrix}a & t\\0 & 1\end{pmatrix}\right\vert a\in\mathbb{R}_+, t\in\mathbb{R}\right\} \end{equation}\]
We denote \((a,t)\equiv \begin{pmatrix}a & t\\0 & 1\end{pmatrix}\in H\)
We hence have \((a,t)(b,s)\equiv\begin{pmatrix}a & t\\0 & 1\end{pmatrix}\begin{pmatrix}b & s\\0 & 1\end{pmatrix}=\begin{pmatrix}ab & as+t\\0 & 1\end{pmatrix}\equiv(ab,as+t)\)
In order to find \(\mathrm{Aut}(H)\) we need to define the group action \((H,\psi)\) that verifies () \[\begin{equation}\label{eq:aut_h} \psi((a,t)(b,s)) = \psi(a,t)\psi(b,s) \end{equation}\] We notice that the right conjugation by \(h\) defined as \(\psi_h(x)=h^{-1}xh\) is a group action verifying (). For all \(x,y\in H\), \[\begin{equation} \psi_h(xy)=h^{-1}xy h=(h^{-1}xh)(h^{-1}yh)=\psi_h(x)\psi_h(y) \end{equation}\] \(\psi_h\) is therefore an inner automorphism of \(H\).
Let \(h=(b,s)\in H\) therefore \(\psi_h\) is defined \(\forall(a,t)\in H\) \[\begin{equation} \psi_h(a,t)=(b,s)^{-1}(a,t)(b,s)=\left(a, \frac{t}{b}+\frac{(a-1)s}{b}\right) \end{equation}\] Let \(h=(b,0)\in H\), the inner automorphism is defined as \(\psi_h(a,t)=(a,t/b)\).
This corresponds to multiplying the chirpiness by \(b\in\mathbb{R}_+\). Hence allowing to redefine the augmented space of \((\nu,a,t)\) as the semidirect product \(G=\mathbb{R}\ltimes H\) with respect to \(\psi\in\mathrm{Aut}(H)\). Nevertheless, further work is needed in order to construct such space from the Wavelet time-frequency representation.