Talk:List of Fourier-related transforms

"frequency transform"

Note: this discussion pertains to the old name of this page, frequency transform, before it was moved

I've never seen this terminology used outside of Wikipedia; e.g. Oppenheim and Schafer doesn't use it, Google turns up no usages of frequency transform as a generic descriptor, etcetera. (I do find several references, both online and in the literature, to the phrase time-frequency transform; perhaps a simple renaming is in order.) Steven G. Johnson

Maybe you should email the original author User:Rade Kutil and ask. He's a postdoc who's written some papers on video encoding with wavelets, so it's quite possible he knows what he's talking about. -- Tim Starling 06:34 25 Jul 2003 (UTC)

Hi. You're right. It's not standard terminology. I chose this term for the following reasons:

• There should be a term describing it (since the article makes sense, doesn't it).
• I didn't come across a better term.
• The term seems to be a natural combination of existing terms, just as needle cutter would be the natural term for something that cuts needles ...
• Since time-frequency transform is used, it shouldn't be too made up.

Note that time-frequency transform is not a proper name for it because it excludes Fourier and cosine transforms and others lacking time locality of basis functions. If someone finds a better term, I'd be glad. If not, I'd be sorry if we decide to delete the article - as I find it an important category. Another solution might be an article function transform or so, which lists frequency transforms, time-frequency transforms and all the others. --Rade

I sympathize; it's nice to have a page that lists a bunch of related transformations, but the idea of inventing new terminology in an encyclopedia makes me uneasy. We should be descriptive, not prescriptive. I think it would be better to make a page that is clearly just a Wikipedia organizational page, like the many List of pages that we already have. How about List of Fourier-related transforms or List of topics related to Fourier analysis for DFT, DCT, DHT, MDCT, FFT, harmonic analysis, etcetera? (There could be a sentence pointing to wavelets, time-frequency transforms, and other related topics at the bottom.) (A page just listing invertible function transforms, i.e. "all" invertible linear operators on Hilbert spaces, seems too unfocused to be useful.) Steven G. Johnson

I've moved the page to "List of Fourier-related transforms" and changed the linking articles. Wikipedia should not attempt to coin new technical terms, and this coinage was spreading through more and more articles. Steven G. Johnson 00:44, 12 Feb 2004 (UTC)

Hilbert transform question

Hello. I think that the list of Fourier-related transforms is lacking of the Hilbert transformation. Please tell me why this transformation for analytic signals isn't here. Andrés Páez

If the domain of a function is time (t), the Fourier transform produces a function of ordinary frequency. The Hilbert transform produces another function of time.

--Bob K (talk) 14:01, 3 December 2010 (UTC)

Laplace transforms

I deleted an assertion that the Fourier transform is a special case of the allegedly more general Laplace transform. That would mean every Fourier transform is a Laplace transform but not every Laplace transform is a Fourier transform. If we define the Fourier transform by

$({\mathcal {F}}f)(t)=\int _{-\infty }^{\infty }e^{-itx}f(x)\,dx$

and the Laplace transform by

$({\mathcal {L}}f)(t)=\int _{-\infty }^{\infty }e^{-tx}f(x)\,dx$

then, proceeding formally ("formally" means, in effect, not worrying about convergence, integrability, etc.), we have

$({\mathcal {F}}f)(t)=({\mathcal {L}}f)(it)$

and

$({\mathcal {L}}f)(t)=({\mathcal {F}}f)(-it).$

This might make them appear equally general. But if f is Lebesgue-integrable (i.e., the integral from −∞ to ∞ of the absolute value of f is finite) then the Fourier transform of f clearly exists, whereas the Laplace transform may blow up—either diverge to ∞ or be only conditionally integrable. This might make the Fourier transform appear more general, not less.

But then we get complications from the fact that there are ways to define Fourier transforms for quadratically integrable functions that are not Lebesgue integrable, and also for tempered distributions, etc.

Could the person who asserted that the Laplace transform is strictly more general please clarify? Michael Hardy 22:33, 18 May 2004 (UTC)

I didn't write that assertion, but note that anything you say about the Laplace transform here probably also applies to the z Transform, which is mentioned similarly in the discrete case. (I suspect that the thinking of the person who wrote it was that normally with Fourier transforms you don't even raise the case of arbitrary complex ω.) —Steven G. Johnson 23:18, May 18, 2004 (UTC)

The statement is (too) naive; but perhaps some remarks could be added. For example, the simplest functions one might ask about are polynomials; their FTs don't exist in the simple-minded sense, but LTs are a different matter. Charles Matthews 08:19, 19 May 2004 (UTC)

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Discrete-time Fourier transform (DTFT)

By the way, Wikipedia needs an entry for the DTFT:

$F(\omega )=\sum _{k=-\infty }^{\infty }e^{-i\omega k}f_{k}$

$f_{k}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }e^{i\omega k}F(\omega )d\omega$

when someone gets around to it. (This is also a closer analogue to the z transform than the DFT.) (Of course, this is just the dual of the Fourier series representation, but it is important enough to deserve a page in its own right.)

Done. Discrete-time Fourier transform.