Pseudo-differential representation of the metaplectic transform and its application to fast algorithms.
Pseudo-differential representation of the metaplectic transform and its application to fast algorithms.
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元变换变换的伪微分表示及其在快速算法中的应用。
DOI：
10.1364/josaa.36.001846
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
I. Dodin
中科院分区：
文献类型：
--
作者：
N. Lopez;I. Dodin
广义变换（MT），也称为线性正则变换，是一种广泛应用于信号处理的酉积分映射，可以看作是傅里叶变换的推广。对于$ {N} $N维连续空间$ {\textbf q} $q上的给定函数$ \psi $ψ, $ \psi $ψ的MT通过$2 {N} $ 2n维相空间$({\textbf q},{\textbf p}) $（q,p）的旋转（或更一般地说，线性辛变换）来参数化，其中$ {\textbf p} $p是$ {\textbf q} $q的波向量空间对偶。在这里，我们推导了MT的伪微分形式。对于小角度旋转或相空间的近恒等变换，它很容易产生MT的渐近微分表示，这很容易在数值上计算。更大角度的旋转是通过$ {K} \gg 1 $K²1个小角度mt的连续应用来实现的。算法复杂度尺度为$ {O}({K}{{N}^3}{{N}_p}) $O(KN3Np)，其中$ {{N}_p} $Np是网格点的个数。在这里，我们提出了该算法的数值实现，并讨论了如何减轻相关的数值不稳定性。
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(KN3Np), where $ {{N}_p} $Np is the number of grid points. Here, we present a numerical implementation of this algorithm and discuss how to mitigate the associated numerical instabilities.