Talk:Discrete Fourier series

Merge?

Regarding the merge suggestion, my reservation is that the DFT article is already very "heavy" with detail that probably should be exported to one or more other articles. I hesitate to pile more on. And any new needle we add might just get lost in the haystack.

All of this information can also be found at Fourier analysis

--Bob K (talk) 17:13, 3 November 2010 (UTC)

Correctness

This article might actually add to the confusion by claiming that the DFS is both:

• a Fourier series, and
• "very similar to" a DFT.

But a Fourier series is not a transform... its domain is time, not frequency. It is actually an inverse transform of a frequency domain function (I will call X(f)) constructed by assigning the Fourier series coefficients to discrete frequency values... essentially using them to modulate a Dirac comb. X(f) is often referred to as a Fourier transform because:

• It can be derived from the Fourier transform integral if one ignores the problem of non-existence at the harmonic frequencies. (They diverge to infinity, but they diverge at different "amplitudes", which preserves the unique information about the signal.)
• Its inverse transform, x(t), is the original periodic function from which the FS coefficients were derived. x(t) can therefore be represented as a summation of sinusoids, weighted by the FS coefficients, and that representation is what's known as a Fourier series. The sequence of FS coefficients has no similar handy name that I am aware of. So it is tempting to call it a Fourier series, but that will confuse some students who have been taught otherwise.

I think this article has a worthy goal, but it needs to be more meticulous about definitions and semantics if it is actually going to clarify a semantically muddled situation.

--Bob K (talk) 15:26, 2 November 2010 (UTC)

I just figured out that it was Bob K that wrote essentially what the article is today. Bob K, the DFS is more than "very similar" to the DFT. They are one and the same. Their equations are the same, their theorems are the same (particularly those that involve shifting of data in one domain or the other). Either the DFT periodically extends the finite set of data passed to it or, if you prefer not thinking of it that way, applying the modulo N operation to the indices (which you must do to any shifted sequence, unless you accept the inherent periodic extension) is operationally the same thing as periodically extending the data.

The DFT is the DFS. 70.109.187.2 (talk) 21:04, 18 January 2011 (UTC)

Anonymous user of IP 70.109.187.2: No the DFT and DFS do not have the exact same formulas. --Alej27 (talk) 23:27, 21 June 2021 (UTC)

Who wrote this article??

It's full of factual inaccuracies. Like "But we must again note that a Fourier series is a time-domain representation, not a frequency domain transform. So DFS is a potentially confusing substitute for DFT."

Both statements are completely false. I might suggest that someone buy a good book, like O&S.

I would also concur with any opinion that this article gets redirected to the DFT article. 70.109.187.2 (talk) 20:56, 18 January 2011 (UTC)

Change the article structure

The current version of the article is not optimal. The leading paragraph defines the DFS by defining it as a case of the inverse DFT, which means the reader must first know what is the DFT.

Also, the main section of the article should not include formulas, only an introduction without much details and without using formulas. A new section should be created to include the current formulas and possibly more details.

And, I've seen the DFS called as DTFS (discrete-time Fourier series) in two textbooks, including Oppenheim's textbook, not as the DFS (discrete Fourier series). --Alej27 (talk) 23:37, 21 June 2021 (UTC)

Update: I've decided to be bold. I made the previous suggested edits. --Alej27 (talk) 20:39, 22 June 2021 (UTC)

I've been thinking more about this article (thanks to you), and I realized that an inverse DFT is a form of DFS, but a DFS is not necessarily an inverse DFT. It depends on where the X[k] coefficients came from. That is the key distinction. So I'm working on a revision.

--Bob K (talk) 16:48, 27 June 2021 (UTC)