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 $\mathbf{\text{q}}$, the MT of $\psi$ is parameterized by a rotation (or more generally, a linear symplectic transformation) of the $2N$-dimensional phase space $(\mathbf{\text{q}},\mathbf{\text{p}})$, where $\mathbf{\text{p}}$ is the wavevector space dual to $\mathbf{\text{q}}$. 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 $\mathit{\text{differential}}$ 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.