# Linear canonical transform

The linear canonical transform (LCT) is a three-parameter family of linear integral transforms of one variable that generalizes the [Fourier transform](https://www.edgepedia.org/fourier-transform), the fractional Fourier transform, the Fresnel transform, scaling operations, and complex Laplace-type transforms within a single framework.<sup>[1](https://www.ee.bilkent.edu.tr/~haldun/publications/ozaktas312.pdf)</sup><sup> • </sup><sup>[2](https://google.iopscience.iop.org/article/10.1088/2040-8986/aa9e20)</sup><sup> • </sup><sup>[3](https://www.sciencedirect.com/science/article/abs/pii/S0165168405002756)</sup> Each transform in the family is labeled by a real two-by-two matrix with unit determinant, and the family forms a group under composition, so an optical system or a signal-processing operation can be analyzed by multiplying its matrices.<sup>[4](https://ar5iv.labs.arxiv.org/html/1206.1123)</sup> This unified view lets one treat lenses, free-space propagation, and their signal-processing analogs with the same algebra and the same fast algorithms.<sup>[1](https://www.ee.bilkent.edu.tr/~haldun/publications/ozaktas312.pdf)</sup>

| Key fact | Detail |
|---|---|
| Parameter count (1D) | Three real parameters, arranged as a matrix \( M = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) with \( a \cdot d - b \cdot c = 1 \)<sup>[4](https://ar5iv.labs.arxiv.org/html/1206.1123)</sup> |
| Special cases | Fourier transform at \( (a,b,c,d) = (0,1,1,0) \); fractional Fourier transform at \( (\cos\theta, \sin\theta, -\sin\theta, \cos\theta) \); also Fresnel transform, scaling, chirp multiplication<sup>[5](https://www.atlantis-press.com/article/21515.pdf)</sup><sup> • </sup><sup>[1](https://www.ee.bilkent.edu.tr/~haldun/publications/ozaktas312.pdf)</sup> |
| Group structure | \( \mathrm{Sp}(2,\mathbb{R}) = \mathrm{SL}(2,\mathbb{R}) \), isomorphic to \( \mathrm{SU}(1,1) \); the integral form is the metaplectic double cover \( \mathrm{Mp}(2,\mathbb{R}) \)<sup>[4](https://ar5iv.labs.arxiv.org/html/1206.1123)</sup> |
| Optical meaning | Models first-order optical systems: thin lenses, free space in the Fresnel approximation, quadratic graded-index media, and their concatenations<sup>[1](https://www.ee.bilkent.edu.tr/~haldun/publications/ozaktas312.pdf)</sup> |
| Computational cost | Direct integration costs \( O(N^{2}) \) for \( N \) samples; fast algorithms achieve \( O(N \log N) \)<sup>[1](https://www.ee.bilkent.edu.tr/~haldun/publications/ozaktas312.pdf)</sup> |
| Accuracy of fast algorithms | Radix-2 fast LCT shows peak phase errors of order \( 10^{-3} \) rad and peak magnitude errors of order 0.1% of the output dc value<sup>[6](https://doi.org/10.1364/josaa.27.000021)</sup> |
| 2D extension | Nonseparable 2D LCTs form the ten-parameter symplectic group \( \mathrm{Sp}(4,\mathbb{R}) \)<sup>[1](https://www.ee.bilkent.edu.tr/~haldun/publications/ozaktas312.pdf)</sup> |

## How it works

For a signal \( f(x') \), the LCT with parameter matrix \( M \) of unit determinant is the integral transform with kernel<sup>[4](https://ar5iv.labs.arxiv.org/html/1206.1123)</sup>

\[ C_{M}(x, x') = \frac{1}{\sqrt{2\pi i b}} \exp\!\left( \frac{i}{2b} \left( d \cdot x^{2} - 2 x \cdot x' + a \cdot x'^{2} \right) \right), \]

so the output is a chirp-modulated, scaled Fourier-type integral. The quadratic phase factors in \( x \) and \( x' \) and the linear cross term are what generalize the pure Fourier kernel. Because \( a \cdot d - b \cdot c = 1 \), the set of matrices closes under multiplication, and composing two LCTs amounts to multiplying their matrices; the transform is therefore a group action generated by the [Lie algebra](https://www.edgepedia.org/lie-algebra) of quadratic Hamiltonian operators.<sup>[4](https://ar5iv.labs.arxiv.org/html/1206.1123)</sup>

The group \( \mathrm{Sp}(2,\mathbb{R}) \) equals the group of real \( 2 \times 2 \) unit-determinant matrices \( \mathrm{SL}(2,\mathbb{R}) \) and is isomorphic to \( \mathrm{SU}(1,1) \); the integral form of the LCT represents its double cover, the metaplectic group \( \mathrm{Mp}(2,\mathbb{R}) \).<sup>[4](https://ar5iv.labs.arxiv.org/html/1206.1123)</sup> The metaplectic phase, the sign ambiguity that distinguishes the two covers, originates in the Lie-algebraic relation of the transform with second-order differential operators.<sup>[7](https://www.fis.unam.mx/~bwolf/Book%27s/Sketch.pdf)</sup> In optics, a system of lenses and free-space sections is characterized in the paraxial approximation by an ABCD or ray-transfer matrix, which is symplectic and relates ray position and angle at the input and output<sup>[6](https://doi.org/10.1364/josaa.27.000021)</sup>; the same matrix labels the LCT kernel.

## How it is done

Direct numerical evaluation of the integral costs \( O(N^{2}) \) for \( N \) input samples.<sup>[1](https://www.ee.bilkent.edu.tr/~haldun/publications/ozaktas312.pdf)</sup> Fast algorithms reach \( O(N \log N) \) by decomposing the matrix \( M \) into a product of special-case symplectic matrices, each of which defines scaling, a Fourier transform, or chirp multiplication, and then composing those operations.<sup>[1](https://www.ee.bilkent.edu.tr/~haldun/publications/ozaktas312.pdf)</sup><sup> • </sup><sup>[8](https://proceedings.jacow.org/ipac2022/papers/thpopt068.pdf)</sup> Koc, Ozaktas, Candan, and Kutay gave two such decompositions in 2008: one built from chirp multiplication, Fourier transformation, and scaling, the other from a fractional Fourier transform followed by scaling and chirp multiplication; both run in time proportional to the time-bandwidth product and achieve FFT-like speed and accuracy.<sup>[9](https://doi.org/10.1109/tsp.2007.912890)</sup> The only deviation from exactness comes from approximating the continuous Fourier transform by the discrete Fourier transform, so performance matches that of the FFT in computing the continuous Fourier transform.<sup>[9](https://doi.org/10.1109/tsp.2007.912890)</sup>

Two implementation styles exist, analogous to the DFT/FFT distinction: a discrete LCT (DLCT) defined directly on samples, from which a fast LCT (FLCT) is derived, and hybrid approaches.<sup>[1](https://www.ee.bilkent.edu.tr/~haldun/publications/ozaktas312.pdf)</sup><sup> • </sup><sup>[10](https://pubmed.ncbi.nlm.nih.gov/15898553/)</sup> Chirp multiplication raises the resolution needed to represent the signal, so resampling is added as a fourth basic operation alongside scaling, Fourier transformation, and chirp multiplication.<sup>[8](https://proceedings.jacow.org/ipac2022/papers/thpopt068.pdf)</sup> For complex-parameter transforms, a decomposition into real and complex chirp multiplications and Fourier transforms computes output samples in about \( N \log N \) time, with a space-bandwidth product tracking formalism that keeps the sample count information-theoretically sufficient without redundancy.<sup>[11](https://opg.optica.org/josaa/abstract.cfm?uri=josaa-27-9-1896)</sup>

## Origin

The quantum-mechanical form of the transform was introduced by Moshinsky and Quesne, who reported "Linear Canonical Transformations and Their Unitary Representations" in the [Journal of Mathematical Physics](https://www.edgepedia.org/journal-of-mathematical-physics) in 1971, printed there as volume 12, issue 8, pages 1772 to 1780.<sup>[12](https://doi.org/10.1063/1.1665805)</sup><sup> • </sup><sup>[13](https://onlinelibrary.wiley.com/doi/10.1002/mma.9920)</sup> Their motivation was quantum mechanical: the unitary integral transforms that preserve the basic Heisenberg uncertainty relation, studied while they worked on alpha clustering and decay of radioactive nuclei at the [Institute of Physics](https://www.edgepedia.org/institute-of-physics) of the Universidad Nacional Autónoma de México.<sup>[7](https://www.fis.unam.mx/~bwolf/Book%27s/Sketch.pdf)</sup> The fast algorithms for digital computation of the transform were introduced by Aykut Koc and colleagues in 2008 in IEEE Transactions on Signal Processing.<sup>[9](https://doi.org/10.1109/tsp.2007.912890)</sup>

## Variants

The transform carries many names in the literature, including quadratic-phase integrals, quadratic-phase systems, generalized Huygens integrals, generalized Fresnel transforms, special affine Fourier transforms, extended fractional Fourier transforms, and Moshinsky–Quesne transforms.<sup>[9](https://doi.org/10.1109/tsp.2007.912890)</sup>

Discretization was undertaken in one formulation, which remains the accepted one in practice but presents practical inconveniences connected with sampling; a newer DLCT definition with its own sampling methodology addresses those issues.<sup>[6](https://doi.org/10.1364/josaa.27.000021)</sup> Unlike the discrete fractional Fourier transform, whose definition is considered satisfactory and well recognized, the definition of the DLCT is far from established.<sup>[9](https://doi.org/10.1109/tsp.2007.912890)</sup>

In two dimensions, the nonseparable LCT (NS-LCT) is a unitary linear integral transform relating the input and output monochromatic paraxial scalar wave fields of optical systems characterized by a \( 4 \times 4 \) ray-tracing matrix; it represents nonaxially symmetric systems such as the gyrator transform and image rotation, and a sampling theorem for it generalizes the published 1D theorems.<sup>[14](https://opg.optica.org/josaa/abstract.cfm?uri=josaa-31-12-2631)</sup> Its matrix has 16 parameters with six constraints, leaving ten independent parameters, forming the ten-parameter group \( \mathrm{Sp}(4,\mathbb{R}) \).<sup>[1](https://www.ee.bilkent.edu.tr/~haldun/publications/ozaktas312.pdf)</sup> Complex LCTs (CLCTs) model Gaussian ducts, complex graded-index media, lossless thin lenses, and free-space sections, with complex-ordered fractional Fourier transforms as a special case.<sup>[11](https://opg.optica.org/josaa/abstract.cfm?uri=josaa-27-9-1896)</sup> Adding two more parameters to the three-parameter integral yields the five-parameter special affine Fourier transform (SAFT).<sup>[15](https://mural.maynoothuniversity.ie/id/eprint/8724/1/BH-Generalizing-2005.pdf)</sup>

## Applications

At optical frequencies, LCTs model first-order optical systems, and they represent solutions of the wave equation in electromagnetic, acoustic, and other wave-propagation problems, with uses including scattering from periodic potentials, laser cavities, multilayered structures, and fast filtering in LCT domains.<sup>[1](https://www.ee.bilkent.edu.tr/~haldun/publications/ozaktas312.pdf)</sup> An LCT library built on the decomposition into scaling, Fourier transforms, and chirp multiplication, with resampling, is used for fast coherent X-ray wavefront propagation in accelerator physics.<sup>[8](https://proceedings.jacow.org/ipac2022/papers/thpopt068.pdf)</sup> An edited Springer volume surveys sampling theory and fast algorithms for the family, with application chapters ranging from digital holography to speckle metrology.<sup>[16](https://link.springer.com/book/10.1007/978-1-4939-3028-9)</sup>

## Limitations and alternatives

Sampling is the main practical constraint. Sampling in the output domain places a further requirement on the input sampling rate beyond input-domain considerations; one analysis established an upper bound on the necessary input rate together with an appropriate reconstruction method.<sup>[17](https://doi.org/10.1364/ol.33.002599)</sup> The output sampling period may need to be not less than \( 2\hat{B} \) so that a low-pass filter can separate the central spectral copy from its replicas.<sup>[17](https://doi.org/10.1364/ol.33.002599)</sup> A naive application of the Nyquist sampling theorem to set the rate would produce an excessively large number of samples and inefficient computation; careful rate management keeps the sample count close to the time-bandwidth product.<sup>[9](https://doi.org/10.1109/tsp.2007.912890)</sup> Discrete and finite analogs of the LCT remain an area with open problems in their construction.<sup>[7](https://www.fis.unam.mx/~bwolf/Book%27s/Sketch.pdf)</sup>

The relation to other representations is exact at the parameter level. With \( (a,b,c,d) = (0,1,1,0) \) the LCT coincides with the conventional Fourier transform, and with \( (\cos\theta, \sin\theta, -\sin\theta, \cos\theta) \) it coincides with the fractional Fourier transform, so the fractional transform is the one-parameter rotation subgroup of the three-parameter LCT family.<sup>[5](https://www.atlantis-press.com/article/21515.pdf)</sup> On the [Wigner distribution function](https://www.edgepedia.org/wigner-distribution-function), a Cohen-class pseudodistribution, an LCT acts as an affine remapping of the time-frequency plane, \( W(x,k) \to W(x',k') \).<sup>[17](https://doi.org/10.1364/ol.33.002599)</sup>

## References

1. [Fast Algorithms for Digital Computation of Linear Canonical Transforms](https://www.ee.bilkent.edu.tr/~haldun/publications/ozaktas312.pdf)
2. [Simulating first order optical systems, algorithms for and composition of discrete linear canonical transforms](https://google.iopscience.iop.org/article/10.1088/2040-8986/aa9e20)
3. [Sampling of linear canonical transformed signals](https://www.sciencedirect.com/science/article/abs/pii/S0165168405002756)
4. [A Top-Down Account of Linear Canonical Transforms (arXiv:1206.1123)](https://ar5iv.labs.arxiv.org/html/1206.1123)
5. [The Linear Canonical Transform and Time-Frequency Representations](https://www.atlantis-press.com/article/21515.pdf)
6. [Fast linear canonical transforms](https://doi.org/10.1364/josaa.27.000021)
7. [Development of Linear Canonical Transforms: A Historical Sketch](https://www.fis.unam.mx/~bwolf/Book%27s/Sketch.pdf)
8. [Linear Canonical Transform Library for Fast Coherent X-Ray Wavefront Propagation](https://proceedings.jacow.org/ipac2022/papers/thpopt068.pdf)
9. [Aykut Koc and colleagues (2008). Digital Computation of Linear Canonical Transforms. IEEE Transactions on Signal Processing.](https://doi.org/10.1109/tsp.2007.912890)
10. [Fast numerical algorithm for the linear canonical transform](https://pubmed.ncbi.nlm.nih.gov/15898553/)
11. [Fast and accurate algorithm for the computation of complex linear canonical transforms](https://opg.optica.org/josaa/abstract.cfm?uri=josaa-27-9-1896)
12. [M. Moshinsky, C. Quesne (1971). Linear Canonical Transformations and Their Unitary Representations. Journal of Mathematical Physics.](https://doi.org/10.1063/1.1665805)
13. [Lp-theory of linear canonical transforms and related uncertainty principles (Chen, 2024, Mathematical Methods in the Applied Sciences)](https://onlinelibrary.wiley.com/doi/10.1002/mma.9920)
14. [Two-dimensional nonseparable linear canonical transform: sampling theorem and unitary discretization](https://opg.optica.org/josaa/abstract.cfm?uri=josaa-31-12-2631)
15. [Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms](https://mural.maynoothuniversity.ie/id/eprint/8724/1/BH-Generalizing-2005.pdf)
16. [Linear Canonical Transforms: Theory and Applications (Springer book)](https://link.springer.com/book/10.1007/978-1-4939-3028-9)
17. [Additional sampling criterion for the linear canonical transform](https://doi.org/10.1364/ol.33.002599)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Analysis and mathematical models › Harmonic analysis, transforms, and integral equations*

*Initially written Sep 29, 2026 · Reviewed: Sep 30, 2026 · Edited: Oct 15, 2026 · Last review: Sep 30, 2026*

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