Talk:Linear canonical transformation

Use i=(-1) instead of j
I have corrected two simple details:
 * Use of i = sqrt(-1) is better than 'j'. i is used in Mathematics instead of j
 * In matrix for fourier transform the minus sign was wrong in the upper value -1. It's 1.


 * In the applications section there are so many i's already, and the figures use i as subscripts, so that they should be redone without the i. Without changing figure j is better. In any event, $$i=\sqrt{-1}$$ or $$j=\sqrt{-1}$$ must appear in the text. Either i or j is acceptable, but they must be defined in the text.--Rmoba (talk) 20:39, 1 August 2009 (UTC)

Formula for spherical lens is incorrect
user cems2 says: I beleive the formula used for the spherical lens is wrong. specifically the x in the exponent numerator I think should be a constant, probably unity. as it stands the formula is not symmetric in x and y. —Preceding unsigned comment added by 192.12.184.2 (talk) 22:11, 21 April 2009 (UTC)


 * Correct. The x should be the k defined in the previous section. I have corrected this - jhealy 14:05 (GMT), May 20, 2009

Problem with definition
I have two issues with the article as is, both related to the equation


 * {| border="0"


 * $$X_{(a,b,c,d)}(u) = \sqrt{-i} \cdot e^{-i \pi \frac{d}{b} u^{2}} \int_{-\infty}^\infty e^{-i 2 \pi \frac{1}{b} ut}e^{i  \pi \frac{a}{b} t^2} x(t) \; dt \,, $$  || when b ≠ 0,
 * $$X_{(a,0,c,d)}(u) = \sqrt{d} \cdot e^{-i \pi cdu^{2}} x(du) \, ,$$ || when b = 0.
 * }
 * $$X_{(a,0,c,d)}(u) = \sqrt{d} \cdot e^{-i \pi cdu^{2}} x(du) \, ,$$ || when b = 0.
 * }

First, when I do a search of articles on the LCT, most use a kernel that does not have the factor of $$2\pi$$ that occurs in the kernel here. This difference is a bit like the choice of kernel for the Fourier transform, where some authors put the $$2\pi$$ in the exponent of the kernel, and some put it as a constant outside the exponential. (For my purposes, the form here is better! But the majority of the literature seems to have the other form.)

Second, I think that there is a factor of $$1/\sqrt{b}$$ missing from the kernel. It should be


 * {| border="0"


 * $$X_{(a,b,c,d)}(u) = \sqrt{\frac{1}{ib}} \cdot e^{-i \pi \frac{d}{b} u^{2}} \int_{-\infty}^\infty e^{-i 2 \pi \frac{1}{b} ut}e^{i  \pi \frac{a}{b} t^2} x(t) \; dt \,, $$  || when b ≠ 0,
 * }
 * }

If anyone is following this page, please check that this is correct --- I'm reluctant to make a change without confirmation. PiperArrow123 July 19, 2010


 * In the equation for Electromagnetic Wave, how does lambda z get in the denominator in front? I think the previous poster might be right. In the Equation for Spherical Lens, there is no integral. At best, this is confusing; at worst, wrong. In the graphic for Satellite Antenna (which should have another name such as Parabolic Antenna since they are used in many applications besides satellite communications), the figure is confusing. It is hard to see the correct perspective, and the labelled quantity R does not in fact indicate the "disk" diameter ("dish" diameter would be a better term). —Preceding unsigned comment added by Oscarruitt (talk • contribs) 23:44, 13 September 2010 (UTC)


 * Yes, I agree there is missing 1/sqrt(b) factor in the equation for X(a,b,c,d).

Because when I apply the LCT corresponding to [a b c d] to a delta function and then apply the inverse LCT, I get the spurious factor: -i|b|. If 1/sqrt(b) is included in the formula, then my spurious factor becomes -i|b|/(sqrt(b)*sqrt(-b)) = -1, which is better but still not right :( Doubledork (talk) 19:33, 31 August 2011 (UTC)

LCD does not generalize the Laplace Transform
I think that the LCD does not generalize the Laplace Transform, in fact, the LCT does not have the $$e^{-\sigma t}$$ factor with real $$\sigma$$ thefourlinestar (talk) 20:03, 1 November 2010 (UTC)

LCT's do not work like 2x2 matrices
I played a bit with the transform formulas and it appears that LCT's do not work like SL2(R), i.e. 2x2 matrices with unit determinant. The result can sometimes be off by a factor of unit magnitude, such as -1, and there doesn't seem to be any way to rectify this other than to just overlook the issue.

It would be nice if the article mentioned this issue instead of misleading people (like me).

Example 1: a) For matrices, the product of the matrix -1 0][0 -1 with itself equals the identity matrix. b) In the case of LCT's, let a = d = -1 and b = c = 0. Then using the suggested formula on the function x(t) results in X(a,0,c,d)(t) = ix(-t). Applying it again results in -x(t). This is not the same as x(t).

Example 2:

For this case you need to include the 1/sqrt(b) factor that's missing in the incorrect formula for nonzero b given here.

Apply LCT (a,b,c,d) followed by LCT (alpha,beta,gamma,delta) to a delta function delta(t-T) and compare the result to what you get applying a single LCT representing to the product of the matrices. The result may or may not be the same (possibly off by factor of -1) depending on whether b, beta, and alpha*b+beta*d are each positive or negative. It seems there are no magical factors that can be included in the transform to fix the problem (but perhaps *major* changes to the definition of the transform could fix it). Doubledork (talk) 23:24, 2 September 2011 (UTC)


 * I came here to ask what vector the 2x2 matrices are acting upon. Presumably it's some vector like [x,p], [x,k], or [x,FT(x)]. However, based on your comment, maybe I am misunderstanding how the LCT is supposed to be applied.2001:480:91:3304:0:0:0:658 (talk) 16:23, 25 June 2021 (UTC)

Real or complex matrices?
The text says that the matrices belong to SL2(R), but then there are matrices with imaginary elements. Shouldn't it thus be SL2(C)? --Md2perpe (talk) 18:03, 25 April 2013 (UTC)

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Merger with quadratic Fourier transform
I think the LCT is another name for the quadratic Fourier transform, but I haven't looked at the details. If that's correct, I think all that would need to be done is to mention "quadratic Fourier transform" as a synonym (perhaps with a reference to the book by Maurice Gosson, referenced on that page, as proof that this name is used) and to create a redirect at quadratic Fourier transform to point to this page. Jess_Riedel (talk) 20:14, 16 June 2021 (UTC)


 * This isn't correct. The quadratic Fourier transform is even more general than the LCT. Convolutional Network (talk) 00:21, 31 July 2021 (UTC)
 * Thanks for noting. It would be good if both articles mentioned the other's concept, mentioning the distinction between the two. Especially QFT, which is greatly lacking as an article, if it's a more general concept. Can you recommend any improvements, or section(s) to add to either/both? Thanks. &emsp;—&#8239;sbb&#8239;(talk) 01:11, 31 July 2021 (UTC)
 * I think the LCT needs a section on its computation and perhaps a note on 2D generalizations. QFT it would be helpful to show how it recovers various integral transforms (including the LCT). I am planning to improve both articles. Convolutional Network (talk) 18:04, 31 July 2021 (UTC)