Fast Algorithms For The Digital Computation Of Linear Canonical Transforms PDF Download

Although it is straightforward to determine the relationship between the in-focus image and the object of a simple optical system such as a lens, it is far more challenging to compute the input/output relationships of general first-order and astigmatic optical systems. Such optical systems are known as quadratic-phase systems (QPS) and they include the Fresnel propagation in free space, propagation in graded-index media, passage through thin lenses, and arbitrary concatenations of any number of these, including anamorphic, astigmatic, nonorthogonal elements. Such computation is accomplished by representing the physical system with a general mathematical framework of integrations against kernels and then distilling the entire system into one input-output relationship that can be represented by a linear integral transform. The underlying mathematical integral transforms can be applied to a wider field of signal processing where they are known as the linear canonical transform (LCT) of a signal. Conventional numerical integration methods have a computational complexity of O(N^2) where N is the space-bandwidth product of the sampling scheme, e.g. the number of pixels in the field for an optical system. The algorithms described here yield a complexity of only O(Nlog N). The key is the use of different decompositions (or factorizations) of a given input/output relationship into simpler ones. Instead of following the general physical subparts in cascaded systems and computing input-output relations separately, these algorithms use the simplest possible decompositions to represent the entire system in terms of least possible number of steps. The algorithms are Fast Fourier Transform (FFT) based methods and the only essential deviation from exactness arises from approximating a continuous Fourier transform (FT) with the discrete Fourier transform (DFT). Thus the algorithms work with a performance similar to that of the fast Fourier transform algorithm in computing the Fourier transform, both in terms of speed and accuracy. Unlike conventional techniques these algorithms also track and control the space-bandwidth products, in order to achieve information that is theoretically sufficient but not wastefully redundant.