Talk:Linear canonical transformation

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)

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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

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

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

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

when b ≠ 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

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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)

Something smells rotten

The product of the matrix [[-1 0][0 -1]] with itself equals the identity matrix, but apparently, when using LCT's, this isn't true.

Proof: 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 different from x(t).

I guess LCT's aren't like matrices after all! Doubledork (talk) 00:15, 31 August 2011 (UTC)