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https://en.wikipedia.org/wiki/Discrete_Fourier_transform

"Trigonometric interpolation polynomial" Section.

Image of Wiki Article

Shouldn't the middle term in the second line be?

$$ \cdots + X_{[(N-1)/2]} e^{i\ ( (N-1)/2)2 \pi t} + X_{[(N+1)/2]} e^{-i((N-1)/2)2 \pi t} + \cdots $$

Not that N/2 on an odd N is bad or anything.

Mixed indexing conventions here. Indexing subscripts from 0 to N-1, and exponents 0 to L and -L to -1. Nothing wrong with that. Split Nyquist on the evens, not because you have to in a bandlimited sense, just because it keeps things real when needed.


Update:

This is not an error as pointed out by V.V.T., just my misperception or poor eyesight. The symbols in the image are the greatest integer brackets, aka floor:

$$ \cdots + X_{ \lfloor N/2 \rfloor } e^{i\ \lfloor N/2 \rfloor 2 \pi t} + X_{ \lfloor N/2 \rfloor + 1} e^{-i \lfloor N/2 \rfloor 2 \pi t} + \cdots $$

Therefore it becomes a matter of opinion of which expression is better.

$$ L = (N-1)/2 = \lfloor N/2 \rfloor $$

The first is a tighter definition and more procedurally descriptive, the second is more conceptually splitting at the Nyquist. "$L$" is what I've been using as I am really starting to favor odd sized centered DFTs.

Since VVT didn't answer, just commented, I'm going to change the question to whether it should be changed?

I still think so.

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    $\begingroup$ If you agree this is an error, please upvote this comment. If it's wrong, please add answer with why and earn a check. If you are a Wiki moderator (the target audience here), please add a comment when fixed (assuming it needs it) and I'll delete this question. $\endgroup$ – Cedron Dawg Aug 5 '20 at 23:46
  • $\begingroup$ AFAIK [N/2] denotes the floor of its argument, i.e., the max integer less than N/2. Therefore, the second line makes sense. Maybe I upvoted you comment in haste. Notice also that it is a "Trigonometric interpolation polynomial" section, not a DFT definition. $\endgroup$ – V.V.T Aug 6 '20 at 4:12
  • $\begingroup$ That section is supposed to represent the inverse interpreted as a Fourier series. The subscripts need to be integers, as do the exponents. I'm finding it hard to believe that such a simple mistake is in wiki. Conceptually N/2 is Nyquist, odd or even, but with odd DFTs we don't hit Nyquist. $\endgroup$ – Cedron Dawg Aug 6 '20 at 4:44
  • $\begingroup$ Either way, the integer part notation ([·]) is legitimate for use any place in expressions. This notation is used both in indices and in the exp arguments of the second series. You can only deny the right to use [·] in indices, but IMO you are overzealous. The expressions for series (and the Wiki article author's intent) are clear and can be easily grasped. $\endgroup$ – V.V.T Aug 6 '20 at 5:22
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    $\begingroup$ Looking closer into your question, I've understood your confusion. You may be seeing the integer part notation ([·]) in the index expressions as the bracketed form of indexing. Notice the math expression convention of showing indices in the form of subscripts. Also, the "[N/2] + 1" expression in a subscript of X is a clear indication that the Wiki author use subscripts rather than brackets to denote indices. The brackets in subscripts and in exponent args are values in the integer part notation, so the indices in the second series are integers and also the values of these indices are correct. $\endgroup$ – V.V.T Aug 6 '20 at 7:06

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