{ "id": "eps5rhce0x", "slug": "linear-canonical-transform", "title": "Linear canonical transform", "updated": "2026-10-15", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.analysis", "label": "Analysis and mathematical models", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.analysis" }, { "id": "physical.math.analysis.harmonic_analysis_and_transforms", "label": "Harmonic analysis, transforms, and integral equations", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.analysis.harmonic_analysis_and_transforms" } ], "geo": [ { "id": "geo.nongeo.t1946.physical.math.analysis.harmonic_analysis_and_transforms", "label": "Non-geographic · 1946 to 2000: Harmonic analysis, transforms, and integral equations", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.analysis.harmonic_analysis_and_transforms", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math" }, { "id": "geo.nongeo.t1946.physical.math.analysis", "label": "Analysis and mathematical models", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.analysis" }, { "id": "geo.nongeo.t1946.physical.math.analysis.harmonic_analysis_and_transforms", "label": "Harmonic analysis, transforms, and integral equations", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.analysis.harmonic_analysis_and_transforms" } ] } ], "excerpt": "The linear canonical transform (LCT) is a three-parameter family of integral transforms generalizing the Fourier, fractional Fourier, and Fresnel transforms, used to model first-order optical systems.", "snippet": "The linear canonical transform (LCT) is a three-parameter family of integral transforms generalizing the Fourier, fractional Fourier, and Fresnel transforms, used to model first-order optical systems.", "node": "physical.math.analysis.harmonic_analysis_and_transforms", "markdown": "# Linear canonical transform\n\nThe 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.[1](https://www.ee.bilkent.edu.tr/~haldun/publications/ozaktas312.pdf) • [2](https://google.iopscience.iop.org/article/10.1088/2040-8986/aa9e20) • [3](https://www.sciencedirect.com/science/article/abs/pii/S0165168405002756) 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.[4](https://ar5iv.labs.arxiv.org/html/1206.1123) This unified view lets one treat lenses, free-space propagation, and their signal-processing analogs with the same algebra and the same fast algorithms.[1](https://www.ee.bilkent.edu.tr/~haldun/publications/ozaktas312.pdf)\n\n| Key fact | Detail |\n|---|---|\n| 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 \\)[4](https://ar5iv.labs.arxiv.org/html/1206.1123) |\n| 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[5](https://www.atlantis-press.com/article/21515.pdf) • [1](https://www.ee.bilkent.edu.tr/~haldun/publications/ozaktas312.pdf) |\n| 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}) \\)[4](https://ar5iv.labs.arxiv.org/html/1206.1123) |\n| Optical meaning | Models first-order optical systems: thin lenses, free space in the Fresnel approximation, quadratic graded-index media, and their concatenations[1](https://www.ee.bilkent.edu.tr/~haldun/publications/ozaktas312.pdf) |\n| Computational cost | Direct integration costs \\( O(N^{2}) \\) for \\( N \\) samples; fast algorithms achieve \\( O(N \\log N) \\)[1](https://www.ee.bilkent.edu.tr/~haldun/publications/ozaktas312.pdf) |\n| 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[6](https://doi.org/10.1364/josaa.27.000021) |\n| 2D extension | Nonseparable 2D LCTs form the ten-parameter symplectic group \\( \\mathrm{Sp}(4,\\mathbb{R}) \\)[1](https://www.ee.bilkent.edu.tr/~haldun/publications/ozaktas312.pdf) |\n\n## How it works\n\nFor a signal \\( f(x') \\), the LCT with parameter matrix \\( M \\) of unit determinant is the integral transform with kernel[4](https://ar5iv.labs.arxiv.org/html/1206.1123)\n\n\\[ 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), \\]\n\nso 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.[4](https://ar5iv.labs.arxiv.org/html/1206.1123)\n\nThe 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}) \\).[4](https://ar5iv.labs.arxiv.org/html/1206.1123) 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.[7](https://www.fis.unam.mx/~bwolf/Book%27s/Sketch.pdf) 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[6](https://doi.org/10.1364/josaa.27.000021); the same matrix labels the LCT kernel.\n\n## How it is done\n\nDirect numerical evaluation of the integral costs \\( O(N^{2}) \\) for \\( N \\) input samples.[1](https://www.ee.bilkent.edu.tr/~haldun/publications/ozaktas312.pdf) 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.[1](https://www.ee.bilkent.edu.tr/~haldun/publications/ozaktas312.pdf) • [8](https://proceedings.jacow.org/ipac2022/papers/thpopt068.pdf) 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.[9](https://doi.org/10.1109/tsp.2007.912890) 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.[9](https://doi.org/10.1109/tsp.2007.912890)\n\nTwo 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.[1](https://www.ee.bilkent.edu.tr/~haldun/publications/ozaktas312.pdf) • [10](https://pubmed.ncbi.nlm.nih.gov/15898553/) 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.[8](https://proceedings.jacow.org/ipac2022/papers/thpopt068.pdf) 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.[11](https://opg.optica.org/josaa/abstract.cfm?uri=josaa-27-9-1896)\n\n## Origin\n\nThe 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.[12](https://doi.org/10.1063/1.1665805) • [13](https://onlinelibrary.wiley.com/doi/10.1002/mma.9920) 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.[7](https://www.fis.unam.mx/~bwolf/Book%27s/Sketch.pdf) The fast algorithms for digital computation of the transform were introduced by Aykut Koc and colleagues in 2008 in IEEE Transactions on Signal Processing.[9](https://doi.org/10.1109/tsp.2007.912890)\n\n## Variants\n\nThe 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.[9](https://doi.org/10.1109/tsp.2007.912890)\n\nDiscretization 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.[6](https://doi.org/10.1364/josaa.27.000021) Unlike the discrete fractional Fourier transform, whose definition is considered satisfactory and well recognized, the definition of the DLCT is far from established.[9](https://doi.org/10.1109/tsp.2007.912890)\n\nIn 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.[14](https://opg.optica.org/josaa/abstract.cfm?uri=josaa-31-12-2631) Its matrix has 16 parameters with six constraints, leaving ten independent parameters, forming the ten-parameter group \\( \\mathrm{Sp}(4,\\mathbb{R}) \\).[1](https://www.ee.bilkent.edu.tr/~haldun/publications/ozaktas312.pdf) 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.[11](https://opg.optica.org/josaa/abstract.cfm?uri=josaa-27-9-1896) Adding two more parameters to the three-parameter integral yields the five-parameter special affine Fourier transform (SAFT).[15](https://mural.maynoothuniversity.ie/id/eprint/8724/1/BH-Generalizing-2005.pdf)\n\n## Applications\n\nAt 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.[1](https://www.ee.bilkent.edu.tr/~haldun/publications/ozaktas312.pdf) 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.[8](https://proceedings.jacow.org/ipac2022/papers/thpopt068.pdf) An edited Springer volume surveys sampling theory and fast algorithms for the family, with application chapters ranging from digital holography to speckle metrology.[16](https://link.springer.com/book/10.1007/978-1-4939-3028-9)\n\n## Limitations and alternatives\n\nSampling 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.[17](https://doi.org/10.1364/ol.33.002599) 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.[17](https://doi.org/10.1364/ol.33.002599) 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.[9](https://doi.org/10.1109/tsp.2007.912890) Discrete and finite analogs of the LCT remain an area with open problems in their construction.[7](https://www.fis.unam.mx/~bwolf/Book%27s/Sketch.pdf)\n\nThe 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.[5](https://www.atlantis-press.com/article/21515.pdf) 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') \\).[17](https://doi.org/10.1364/ol.33.002599)\n\n## References\n\n1. [Fast Algorithms for Digital Computation of Linear Canonical Transforms](https://www.ee.bilkent.edu.tr/~haldun/publications/ozaktas312.pdf)\n2. [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)\n3. [Sampling of linear canonical transformed signals](https://www.sciencedirect.com/science/article/abs/pii/S0165168405002756)\n4. [A Top-Down Account of Linear Canonical Transforms (arXiv:1206.1123)](https://ar5iv.labs.arxiv.org/html/1206.1123)\n5. [The Linear Canonical Transform and Time-Frequency Representations](https://www.atlantis-press.com/article/21515.pdf)\n6. [Fast linear canonical transforms](https://doi.org/10.1364/josaa.27.000021)\n7. [Development of Linear Canonical Transforms: A Historical Sketch](https://www.fis.unam.mx/~bwolf/Book%27s/Sketch.pdf)\n8. [Linear Canonical Transform Library for Fast Coherent X-Ray Wavefront Propagation](https://proceedings.jacow.org/ipac2022/papers/thpopt068.pdf)\n9. [Aykut Koc and colleagues (2008). Digital Computation of Linear Canonical Transforms. IEEE Transactions on Signal Processing.](https://doi.org/10.1109/tsp.2007.912890)\n10. [Fast numerical algorithm for the linear canonical transform](https://pubmed.ncbi.nlm.nih.gov/15898553/)\n11. [Fast and accurate algorithm for the computation of complex linear canonical transforms](https://opg.optica.org/josaa/abstract.cfm?uri=josaa-27-9-1896)\n12. [M. Moshinsky, C. Quesne (1971). Linear Canonical Transformations and Their Unitary Representations. Journal of Mathematical Physics.](https://doi.org/10.1063/1.1665805)\n13. [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)\n14. [Two-dimensional nonseparable linear canonical transform: sampling theorem and unitary discretization](https://opg.optica.org/josaa/abstract.cfm?uri=josaa-31-12-2631)\n15. [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)\n16. [Linear Canonical Transforms: Theory and Applications (Springer book)](https://link.springer.com/book/10.1007/978-1-4939-3028-9)\n17. [Additional sampling criterion for the linear canonical transform](https://doi.org/10.1364/ol.33.002599)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Analysis and mathematical models › Harmonic analysis, transforms, and integral equations*\n\n*Initially written Sep 29, 2026 · Reviewed: Sep 30, 2026 · Edited: Oct 15, 2026 · Last review: Sep 30, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [], "url": "https://www.edgepedia.org/linear-canonical-transform", "markdown_url": "https://www.edgepedia.org/linear-canonical-transform.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Linear canonical transform\", Edgepedia (EdgeChat), https://www.edgepedia.org/linear-canonical-transform. Edgepedia Community License 1.0.", "credit_md": "\"[Linear canonical transform](https://www.edgepedia.org/linear-canonical-transform)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/linear-canonical-transform](https://www.edgepedia.org/linear-canonical-transform). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Linear canonical transform\", Edgepedia (EdgeChat), https://www.edgepedia.org/linear-canonical-transform. Edgepedia Community License 1.0.", "speakable": "The linear canonical transform is a three-parameter family of integral transforms generalizing the Fourier, fractional Fourier, and Fresnel transforms, used to model first-order optical systems." }