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Concept
Linear canonical transformation
Résumé
In Hamiltonian mechanics, the linear canonical transformation (LCT) is a family of integral transforms that generalizes many classical transforms. It has 4 parameters and 1 constraint, so it is a 3-dimensional family, and can be visualized as the action of the special linear group SL2(R) on the time–frequency plane (domain). As this defines the original function up to a sign, this translates into an action of its double cover on the original function space.
The LCT generalizes the Fourier, fractional Fourier, Laplace, Gauss–Weierstrass, Bargmann and the Fresnel transforms as particular cases. The name "linear canonical transformation" is from canonical transformation, a map that preserves the symplectic structure, as SL2(R) can also be interpreted as the symplectic group Sp2, and thus LCTs are the linear maps of the time–frequency domain which preserve the symplectic form, and their action on the Hilbert space is given by the Metaplectic group.
The basic properties of the transformations mentioned above, such as scaling, shift, coordinate multiplication are considered. Any linear canonical transformation is related to affine transformations in phase space, defined by time-frequency or position-momentum coordinates.
The LCT can be represented in several ways; most easily, it can be parameterized by a 2×2 matrix with determinant 1, i.e., an element of the special linear group SL2(C). Then for any such matrix with ad − bc = 1, the corresponding integral transform from a function to is defined as
Many classical transforms are special cases of the linear canonical transform:
Scaling, , corresponds to scaling the time and frequency dimensions inversely (as time goes faster, frequencies are higher and the time dimension shrinks):
The Fourier transform corresponds to a clockwise rotation by 90° in the time–frequency plane, represented by the matrix
The fractional Fourier transform corresponds to rotation by an arbitrary angle; they are the elliptic elements of SL2(R), represented by the matrices
The Fourier transform is the fractional Fourier transform when The inverse Fourier transform corresponds to
The Fresnel transform corresponds to shearing, and are a family of parabolic elements, represented by the matrices
where z is distance, and λ is wave length.
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