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 that is widely used in signal processing and can be viewed as a generalization of the Fourier transform. For a given function $ \psi $ψ on an $ {N} $N-dimensional continuous space $ {\textbf q} $q, the MT of $ \psi $ψ is parameterized by a rotation (or more generally, a linear symplectic transformation) of the $ 2{N} $2N-dimensional phase space $ ({\textbf q},{\textbf p}) $(q,p), where $ {\textbf p} $p is the wavevector space dual to $ {\textbf q} $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 differential representations of the MT, which are easy to compute numerically. Rotations by larger angles are implemented as successive applications of $ {K} \gg 1 $K≫1 small-angle MTs. The algorithm complexity scales as $ {O}({K}{{N}^3}{{N}_p}) $O(KN³N_p), where $ {{N}_p} $N_p is the number of grid points. Here, we present a numerical implementation of this algorithm and discuss how to mitigate the associated numerical instabilities.