Linear canonical transformation

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).

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.

Definition

The LCT can be represented in several ways; most easily, it can be viewed as a 2x2 matrix with determinant 1, i.e., an element of the special linear group SL₂(R). Taking a matrix $\left({\begin{smallmatrix}a&b\\c&d\end{smallmatrix}}\right)$, with $ad-bc=1\,$, the corresponding integral transform is:

$X_{(a,b,c,d)}(u)={\sqrt {-i}}\cdot e^{-i\pi {\frac {d}{b}}u^{2}}\int _{-\infty }^{\infty }e^{-i2\pi {\frac {1}{b}}ut}e^{i\pi {\frac {a}{b}}t^{2}}x(t)\,dt\,,$	when $b\neq 0\,,$
$X_{(a,0,c,d)}(u)={\sqrt {d}}\cdot e^{-i\pi cdu^{2}}x(du)\,,$	when $b=0\,.$

Special cases of LCT

Many classical transforms are special cases of the Linear Canonical Transform:

• The Fourier transform corresponds to rotation by 90°, represented by the matrix:

${\begin{bmatrix}a&b\\c&d\end{bmatrix}}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}.$

• The fractional Fourier transform corresponds to rotation by an arbitrary angle; they are the elliptic elements of SL₂(R), represented by the matrices:

${\begin{bmatrix}a&b\\c&d\end{bmatrix}}={\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{bmatrix}}.$

• The Fresnel transform corresponds to shearing, and are a family of parabolic elements, represented by the matrices:

${\begin{bmatrix}a&b\\c&d\end{bmatrix}}={\begin{bmatrix}1&\lambda z\\0&1\end{bmatrix}}.$

where z is distance and $\lambda$ is wave length.

Composition

Composition of LCTs corresponds to multiplication of the corresponding matrices; this is also known as the "additivity property of the WDF".

In detail, if we denote the LCT by $O_{F}^{(a,b,c,d)}\,,$ i.e.

$X_{(a,b,c,d)}(u)=O_{F}^{(a,b,c,d)}[x(t)]\,$

then

$O_{F}^{(a2,b2,c2,d2)}\left\{O_{F}^{(a1,b1,c1,d1)}[x(t)]\right\}=O_{F}^{(a3,b3,c3,d3)}[x(t)]\,,$

where

${\begin{bmatrix}a3&b3\\c3&d3\end{bmatrix}}={\begin{bmatrix}a2&b2\\c2&d2\end{bmatrix}}{\begin{bmatrix}a1&b1\\c1&d1\end{bmatrix}}.$

History

Paraxial optical systems implemented entirely with thin lenses and propagation through free space and/or graded index (GRIN) media, are Quadratic Phase Systems (QPS). The effect of any arbitrary QPS on an input wavefield can be described using the linear canonical transform (LCT), a unitary, additive, four-parameter class of linear integral transform.

The former appeared a couple of times before Moshinsky and Quesne (1974) called attention to their significance in connection with canonical transformations in quantum mechanics. A particular case of the latter was developed by Segal (1963) and Bargmann(1961) in order to formalized Fok's boson calculus (1928).

Applications

Canonical transforms provide a fine tool for the analysis of a class of differential equations. These include the diffusion, the Schrödinger free-particle, the linear potential (free-fall), and the attractive and repulsive oscillator equations. It also includes a few others such as the Fokker-Planck equation. Although this class is far from universal, the ease with which solutions and properties are found makes canonical transforms an attractive tool for problems such as these.

Wave propagation travel through air, lens, and dishes are discussed in here. All of the computations can be reduced to 2x2 matrix algebra. This is the spirit of LCT.

Electromagnetic wave propagation

If we assume the system look like this, the wave travel from plane x_i, y_i to the plane of x and y. We can use the Fresnel transform to describe the electromagnetic wave propagation in the air:

$U_{0}(x,y)=-{\frac {i}{\lambda }}{\frac {e^{ikz}}{z}}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{j{\frac {k}{2z}}[(x-x_{i})^{2}-(y-y_{i})^{2}]}U_{i}(x_{i},y_{i})\,dx_{i}dy_{i},$

with

$k={\frac {2\pi }{\lambda }}\,$	: wave number;
$\lambda \,$	: wavelength;
$z\,$	: distance of propagation.

This is equivalent to LCT (shearing), when

${\begin{bmatrix}a&b\\c&d\end{bmatrix}}={\begin{bmatrix}1&\lambda z\\0&1\end{bmatrix}}.$

When the travel distance (z) is larger, the shearing effect is larger.

Spherical lens

With the above lens from the image, and refractive index = n, we get:

$U_{0}(x,y)=e^{ikn\Delta }e^{-j{\frac {x}{2f}}[x^{2}+y^{2}]}U_{i}(x,y)$

with $f\,$ the focal length and $\Delta \,$ the thickness of the lens.

The distortion passing through the lens is similar to LCT, when

${\begin{bmatrix}a&b\\c&d\end{bmatrix}}={\begin{bmatrix}1&0\\{\frac {-1}{\lambda f}}&1\end{bmatrix}}.$

This is also a shearing effect, when the focal length is smaller, the shearing effect is larger.

Satellite dish

Dish is equivalent to LCT, when

${\begin{bmatrix}a&b\\c&d\end{bmatrix}}={\begin{bmatrix}1&0\\{\frac {-1}{\lambda R}}&1\end{bmatrix}}.$

This is very similar to lens, except focal length is replaced by the radius of the dish. Therefore, if the radius is larger, the shearing effect is larger.

Example

If the system is considered like the following image. Two dishes, one is the emitter and another one is the receiver, and the signal travel through a distance of D. First, for dish A (emitter), the LCT matrix looks like this:

${\begin{bmatrix}1&0\\{\frac {-1}{\lambda R_{A}}}&1\end{bmatrix}}.$

Then, for dish B (receiver), the LCT matrix looks like this:

${\begin{bmatrix}1&0\\{\frac {-1}{\lambda R_{B}}}&1\end{bmatrix}}.$

Last, we need to consider the propagation in air, the LCT matrix looks like this:

${\begin{bmatrix}1&\lambda D\\0&1\end{bmatrix}}.$

If we put all the effects together, the LCT would look like this:

${\begin{bmatrix}a&b\\c&d\end{bmatrix}}={\begin{bmatrix}1&0\\{\frac {-1}{\lambda R_{B}}}&1\end{bmatrix}}{\begin{bmatrix}1&\lambda D\\0&1\end{bmatrix}}{\begin{bmatrix}1&0\\{\frac {-1}{\lambda R_{A}}}&1\end{bmatrix}}={\begin{bmatrix}1-{\frac {D}{R_{A}}}&-\lambda D\\{\frac {1}{\lambda }}(R_{A}^{-1}+R_{B}^{-1}-R_{A}^{-1}R_{B}^{-1}D)&1-{\frac {D}{R_{B}}}\end{bmatrix}}\,.$

See also

• Segal-Shale-Weil distribution, a metaplectic group of operators related to the chirplet transform

Other time-frequency transforms:

• Fractional Fourier transform
• continuous Fourier transform
• chirplet transform

Applications:

• Focus recovery based on the linear canonical transform

References

• J.J. Ding, "Time-frequency analysis and wavelet transform course note", the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2007.
• K.B. Wolf, "Integral Transforms in Science and Engineering", Ch. 9&10, New York, Plenum Press, 1979.
• S.A. Collins, "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Amer. 60, 1168–1177 (1970).
• M. Moshinsky and C. Quesne, "Linear canonical transformations and their unitary representations," J. Math. Phys. 12, 8, 1772–1783, (1971).
• B.M. Hennelly and J.T. Sheridan, "Fast Numerical Algorithm for the Linear Canonical Transform", J. Opt. Soc. Am. A 22, 5, 928–937 (2005).
• H.M. Ozaktas, A. Koç, I. Sari, and M.A. Kutay, "Efficient computation of quadratic-phase integrals in optics", Opt. Let. 31, 35–37, (2006).
• Bing-Zhao Li, Ran Tao, Yue Wang, "New sampling formulae related to the linear canonical transform", Signal Processing '87', 983–990, (2007).
• A. Koç, H.M. Ozaktas, C. Candan, and M.A. Kutay, "Digital computation of linear canonical transforms", IEEE Trans. Signal Process., vol. 56, no. 6, 2383-2394, (2008).