Talk:Sine and cosine transforms

Cosine Transform (CT) has been underestimated. Since virtually all physical quantities do not actually require description by negative values but each of them has a natural zero, CT is the tailor made transform if one asks for primary physical reletionships, not the complex-valued Fourier transform (FT). There is no negative distance. Radius, elapsed time, energy, frequency, temperature, charge of an electron, mass, wave length, etc. are also always positive. Cartesian coordinates and the ordinary time demand arbitrary reference points. This redundant information on linear phase is the only one that makes the FT of a function of elapsed time more 'complete' as compared to CT by various immediate and subsequent notorius worries: required double redundancy, non-causality, ambiguity, misinterpretation of apparent symmetry, the need for arbitrary windows in signal processing, etc. Our ears have no knownledge of the zero of time exactly related to Christ's birth midnight New Year in Greenwich. They just relate to the very moment. Performing CT they are able to add a one-way rectification to the motion of basilar membrane according to CT. This would be impossible with FT. Therefore we could not distiguish by ear between rarefaction and condensation clicks. Fortunately, hearing is still much better than the FT and Ohm's law of acoustics allow. What about the outcome of the second expensive in history physical experiment, I am curious. If Higg's boson is an artifact of improper use of FT, then it will never be found.

For more details see: http://iesk.et.uni-magdeburg.de/~blumsche/M283.html The paper "Adaptation of spectral analysis to reality" has been amended following suggestions by R. Fritzius. The old version is available via IEEE.

Blumschein 13:25, 10 September 2007 (UTC)

Yes, I think you are right (apart from what you speculate about the Higgs). However, unless you can produce (other) mainstream publications beside your own that support your view, the statement constitutes original research and cannot be included in the article.--149.217.1.6 (talk) 22:32, 14 January 2009 (UTC)

Symmetry around the origin?

Integrating from -inf to inf can't be considered symmetric, because the origin can be arbitrarily chosen. So unless I misunderstand something, the ground of the discussion is not sound. —Preceding unsigned comment added by Michael Litvin (talk • contribs) 21:26, 26 November 2010 (UTC)

the integral is indeed symmetric because its value does not depend on where you put the origin. You can change coordinates and change which point is considered the zero, and always you get the same answer. That is the definition of symmetry.98.109.240.7 (talk) 16:13, 16 November 2012 (UTC)

Restricting the Integration Range to the Positive Real Axis

It appears to me that the integration range of the cosine transform should run from $-\infty$ to $\infty$ for general functions. In the post above from November 2010 Michael Litvin already pointed out that the symmetry applies only to particular functions. To verify this simply assume the function
$f(x)=\left\{{\begin{array}{cc}0&x>0\\\exp(x)&x<0\end{array}}\right.$.
Obviously the fourier integral over the positive half axis is zero, whilst over the negative half axis it is nonzero.

A short research on google books shows that many authors are rather sloppy regarding the integration range. A reference where it is stated correctly is Wolfram Mathworld: http://mathworld.wolfram.com/FourierCosineTransform.html

--Pia novice (talk) 13:48, 18 July 2012 (UTC)

You are quite right. In fact, this article is ridiculous, and the one source quoted is just a random textbook by a nobody. Why not consult Whittaker--Watson, the gold standard in this kind of field? The article only defines the sine transform for odd functions, and only defines the cosine transform for even functions, which is not the usual definition at all. Since your example is neither odd nor even, the premisses of this article forbid you to take its transform.98.109.240.7 (talk) 06:23, 16 November 2012 (UTC)

I have rewritten the article almost completely to address your concerns. Also, the discrete cosine transform is not defined using only the positive axis, usually. So integrating over the whole real axis, as is usual, not only defines the transform for all functions, but is more analogous to the discrete case. But some engineering texts do restrict the definition of the cosine transform to even functions and if you do this, you can simplify the formula using the symmetry about zero of cosine and f, and get a formula with the lower limit of the integral's being zero. I will add some more reliable references soon.98.109.240.7 (talk) 16:11, 16 November 2012 (UTC)

Inline comment from anonymous user removed from article

The following text was taken out of the article by me:

Looking at the example on http://wiki.riteme.site/wiki/Fourier_transform

ƒ(t) = cos(6πt) e^(-πt^2)

with \nu =3

Fourier (3) = Int(cos(6πt)*e^(-πt^2)*cos(6*pi), -inf, inf)

Fourier(3)~.5

While for the formula on this page

Fourier(3)~2*.5~1.

I do not see where this 2 factor comes from and seems to give a different result. The cosine can come from Euler's formula, but there is no 2 outside the integral.

http://www.wolframalpha.com/input/?i=int%28cos%286*pi*t%29*e%5E%28-pi*t%5E2%29*cos%282*3*pi*t%29%2Ct%2C-inf%2Cinf%29

To answer the question, I suspect the extra factor comes from "normalizing". If you change the factor inside, it changes the integral of the basis functions, so you need to compensate for that.--84.161.174.248 (talk) 21:38, 11 December 2012 (UTC)

Removed factors 1/2 in the final line of "Relation with complex exponentials"

I do not see where it is coming from, and it is clearly wrong.

Removed paragraphs from Fourier transform

I removed the following paragraph from Fourier transform, feeling that such detail is more suited to this article, but I don't really think it is needed. Here it is, if anyone wants to use it:

The operational properties with respect to convolution, differentiation, etc., are awkward to express in this setting, but were well known nevertheless.

The relations between ${\hat {f}}$, $a$, and $b$ are obvious.

The real part of ${\hat {f}}$ is ${\frac {a}{2}}$, and its imaginary part is $b \over 2$. (This is because the contribution of the frequency $\lambda$ to $f$ is divided evenly between ${\hat {f}}(\lambda )$ and ${\hat {f}}(-\lambda )$---remember, since $f$ is real-valued, ${\hat {f}}(-\xi )$ is simply the complex conjugate of ${\hat {f}}(\xi )$ and so does not contain any new information.)

If $f$ is even, then its sine transform vanishes, and so does the imaginary part of ${\hat {f}}$, and so ${\hat {f}}={\frac {a}{2}}$.

If $f$ is odd, then its cosine transform vanishes, and so does the real part of ${\hat {f}}$, and so ${\hat {f}}={\frac {b}{2}}$.

--Sławomir Biały (talk) 12:51, 23 December 2014 (UTC)

Suggest to add new section of an example function to decompose with sine & cosine transform

I'm thinking an example function like $e^{x}$ would be great as an new section to show the decomposition of a particular function, because that function is not too complicated nor not too trivial nor is odd or even, but is very common. First step would be show that it is composed of the even function cosh(x) and the odd function sinh(x). Then a quick derivation of the cosine transform of cosh(x) and a separate quick derivation of the sine transform of sinh(x). And include graphs of those and the result. But before I go through that effort I'm checking for feedback or maybe if there is another similar common function to decompose. The Heaviside step function would be another candidate for a really simple function that an ordinary reader would have trouble imagining as composed of sine & cosine. Em3rgent0rdr (talk) 04:07, 11 September 2024 (UTC)

These functions aren't integrable, so they make poor candidates. A better choice would be the transform pair $e^{-a|x|}$ and ${\frac {a}{a^{2}+4\pi ^{2}x^{2}}}$. Also, $1/(\cosh x)$ and Gaussians. Tito Omburo (talk) 13:21, 11 September 2024 (UTC)

Oh, good point. Transforming from Gaussian to seems like a useful even function example. (edit: the transform of the gaussian is a gaussian as I now recall). But then I'd also want a good odd function example to add to the even gaussian. Em3rgent0rdr (talk) 14:59, 11 September 2024 (UTC)

Probably the simplest example with a non-trivial sine transform is the indicator function of an interval. Also, you can get non-even functions from the basic ones by doing a time translation. Another nice example is a time-limited sine wave. Also, $e^{-x}\theta (x)$ has sine transform $2\pi u/(4\pi ^{2}u+1)$ where theta is the heaviside function. Tito Omburo (talk) 16:24, 11 September 2024 (UTC)

Thanks. I think I settled on gaussian for even example, and am using it when talk about even. A time-limited sine wave that only exists from -pi to +pi, or something like that (I'll try to figure out what scaling of for that looks easiest) seems like a good odd example.

Gaussian functions have the form $e^{-\alpha t^{2}}$ and are even functions (their left half is a mirror image of their right half). Interestingly, their cosine transform:
${\hat {f}}^{c}(\xi )={\sqrt {\tfrac {\pi }{\alpha }}}\,e^{-{\frac {(\pi \xi )^{2}}{\alpha }}}$
also has a Gaussian form. For the particular case of α=π (used in this plot), the Gaussian is own cosine transform. Like all even functions, the sine transform of a Gaussian is entirely 0.

Em3rgent0rdr (talk) 18:50, 11 September 2024 (UTC)

Actually, I was thinking of a time limited sine wave between 0 and a, where a is a free parameter. Then one can see in a couple of plots how the sine transform tends to a delta function concentrated at the frequency of the sine wave. Tito Omburo (talk) 18:57, 11 September 2024 (UTC)

The truncated sine example you describe sounds like something interesting to show, but maybe in a new own section. I was trying to make something even simpler, and ended up using and odd function that is -1 from t between -1 and 0 and is 1 between 0 and 1, and otherwise zero, in which case the sine transform integral is simple enough that it could be done inside the image's caption. Though feel free to add another example. I'll take a break from editing. Em3rgent0rdr (talk) 21:42, 11 September 2024 (UTC)

Well I came to the realization that was a little too complicated to fit its sine transform inside the caption. So I went with two time-shifted Dirac delta functions:

Odd functions are unchanged if rotated 180 degrees about the origin. Their cosine transform is entirely zero. The above odd function contains two half-sized time-shifted Dirac delta functions. Its sine transform is simply $\sin(a\xi ).$ Likewise, the sine transform of $\sin(a\xi )$ is the above plot. The two functions form a transform pair.

Em3rgent0rdr (talk) 02:34, 12 September 2024 (UTC)

I added a couple more images and am will try to stop myself from making more edits, as I likely introduced some errors. A truncated sine example does sound interesting to show, but I don't think I will get around to it, maybe even as an animated gif whose transform tends towards a delta function, but I'll leave that for someone else whose interested. In the meantime, I did notice there was an example in Fourier_transform#Example that must have been added recently, and as the function was even, would have been a good example for employing the cosine transform. Em3rgent0rdr (talk) 04:39, 13 September 2024 (UTC)