Pseudo-differential representation of the metaplectic transform and its application to fast algorithms

The metaplectic transform (MT), also known as the linear canonical transform, is a unitary integral mapping which is widely used in signal processing and can be viewed as a generalization of the Fourier transform. For a given function $\psi$ on an $N$-dimensional continuous space $\textbf{q}$, the MT of $\psi$ is parameterized by a rotation (or more generally, a linear symplectic transformation) of the $2N$-dimensional phase space $(\textbf{q},\textbf{p})$, where $\textbf{p}$ is the wavevector space dual to $\textbf{q}$. Here, we derive a pseudo-differential form of the MT. For small-angle rotations, or near-identity transformations of the phase space, it readily yields asymptotic \textit{differential} representations of the MT, which are easy to compute numerically. Rotations by larger angles are implemented as successive applications of $K \gg 1$ small-angle MTs. The algorithm complexity scales as $O(K N^3 N_p)$, where $N_p$ is the number of grid points. We present a numerical implementation of this algorithm and discuss how to mitigate the associated numerical instabilities.