Talk:Window function

Perfect windowing function!

1. Double 'k'
2. Create a B–spline window with 'k'

rinse and repeat

This procedure reduces sidelobe levels each time it is done!

Cosine sum optimization

cos(2 pi n) can be turned into cos(4 pi n) by (cos(2 pi n))²×2-1 so that another cosine won't have to be calculated. 2A01:119F:21D:7900:54E1:9029:DEED:43A1 (talk) 09:44, 8 August 2018 (UTC)

There's more to it. cos (6×π×n) is also (cos (2×π×n))³×4-(cos (2×π×n))×3. And cos (8×π×n) is also (cos (2×π×n))⁴×8-(cos (2×π×n))²×8+1. So we have t=cos (2×π×n), and five cosine sum terms:

• cos (0×π×n)=1
• cos (2×π×n)=t
• cos (4×π×n)=2t²-1
• cos (6×π×n)=4t³-3t
• cos (8×π×n)=8t⁴-8t²+1

Five cosine sum terms with one cosine computation.79.185.231.241 (talk) 17:59, 2 May 2021 (UTC)

Discrete convention is bad idea at all for purpose of this page

Essential purposes of window functions are not related to their discretisations within particular applications. Discretisation of window functions, where it needed, is separate problem. Lets not mix that all into common messy heap. Lets define window function as function defined on real range [-1,1] instead. Almost all windows have simplest definition in this form, and simplest method to get discrete version from it is trivial and common, but more advanced methods are also may be useful, and they may relate on rather technical application-specific details, and not related to selection of particular window function.

There are two well-known windows which can be complicated to understand in this context, Dolph-Chebyshev and ultraspheric. Here problem can arise because of specific way how they defined mathematically. They are defined by periodic functions in frequency domain, with explicit parameter N in definition, hence they intrinsically discrete in time domain. From such definition, we have natural prescription how to calculate window in discrete form only, but it is not prescription how to understand and use such calculated window, we can still understand it as discrete samples of some non-discrete function to use it same way as any other window function. — Preceding unsigned comment added by Lexey73 (talk • contribs) 12:29, 19 December 2021 (UTC)

The separate articles Hann_function and Kaiser_window are examples of what you suggest except for the parameter $L,$ which you suggest should be simply 2. Both the zero-phase forms and the lagged forms are equally simple functions, but someone decided both windows deserve their own article, which is fine. And that is the approach you should take for your two examples. Let's see what you come up with, and then revisit how it affects this article, or not.

--Bob K (talk) 23:26, 19 December 2021 (UTC)

That "L" is trivial scaling factor, it only unnecessary complicates equations, and makes illusion that it is essential parameter of window function. Anyone can apply such trivial scaling where needed. Lets simply say something like "support bounds of windows here assumed to be [-1,1]" in "convention" section,

The continuous Hann function is not defined as duration 2. In these examples, it is length T :  https://www.sciencedirect.com/topics/engineering/hanning-window and http://www.vibrationdata.com/tutorials_alt/Hanning_compensation.pdf

And in this one it is T_Span :  https://www.dataphysics.com/downloads/technical/Effects-of-Windowing-on-the-Spectral-Content-of-a-Signal-by-P.-Wickramarachi.pdf

And here is one with length L (see eq.(3)): https://zenodo.org/record/1280930

It is easier for someone to set L=1 (or 2) in a general version of a formula, or a subsequent result of the formula, than to reinsert a missing parameter in all the correct places. For instance these results in Hann_function:

$W_{0}(f)={\tfrac {1}{2}}{\frac {\sin(\pi Lf)}{\pi f(1-L^{2}f^{2})}}$

$W_{0}(f)={\tfrac {1}{2}}{\frac {\sin(\pi Lf)}{\pi f}}+{\tfrac {1}{4}}{\frac {\sin(\pi L(f-1/L))}{\pi (f-1/L)}}+{\tfrac {1}{4}}{\frac {\sin(\pi L(f+1/L))}{\pi (f+1/L)}}.$

Not so trivial, I would say.

--Bob K (talk) 01:07, 27 December 2021 (UTC)

and remove all that annoying "N" and "L" from where they not really necessary.

That is how you do the transformation from continuous to discrete, which is your original suggestion.

I think it is bad idea to make separate page for every simple thing like Hann_function and Kaiser_window Lexey73 (talk) 10:36, 26 December 2021 (UTC)

But those articles are able to go into greater depth, if someone cares to do so. Window function is already long enough.

--Bob K (talk) 23:25, 26 December 2021 (UTC)

$W_{0}(L,f)={\frac {L}{2}}W_{0}({\frac {L}{2}}f)$ - it is common formula for any window to convert it from [-1,1] convention to [-L/2,L/2]  --Lexey73

The fact that it's correct does not make it common. Much more common to set T or T_Span or L to whatever specific length an application calls for. I've given you several online examples. Where are yours? --Bob K (talk) 21:14, 27 December 2021 (UTC)

$W_{0}(f)={\frac {\sin(2\pi f)}{2\pi f}}+{\tfrac {1}{2}}{\frac {\sin(2\pi (f-1/2))}{2\pi (f-1/2)}}+{\tfrac {1}{2}}{\frac {\sin(2\pi (f+1/2))}{2\pi (f+1/2)}}.$ for [-1,1] convention (simply replace L by 2 everywhere in formulas where [-L/2,L/2] convention was used)  --Lexey73

Yeah... it is simple... which is my point. Thank you. --Bob K (talk) 21:14, 27 December 2021 (UTC)

Note that your first formula is wrong Lexey73 (talk) 14:57, 27 December 2021 (UTC)

$W_{0}(f)={\tfrac {1}{2}}{\frac {\sin(\pi Lf)}{\pi f}}+{\tfrac {1}{4}}{\frac {\sin(\pi L(f-1/L))}{\pi (f-1/L)}}+{\tfrac {1}{4}}{\frac {\sin(\pi L(f+1/L))}{\pi (f+1/L)}}$

Here you are caught by illusion that L is essential parameter of function, because L not appeared everywhere near f, but it exactly equals to:

$W_{0}(f)=L\left({\tfrac {1}{2}}{\frac {\sin(\pi Lf)}{\pi Lf}}+{\tfrac {1}{4}}{\frac {\sin(\pi (Lf-1))}{\pi (Lf-1)}}+{\tfrac {1}{4}}{\frac {\sin(\pi (Lf+1))}{\pi (Lf+1)}}\right)$

Lexey73 (talk) 18:56, 27 December 2021 (UTC)

Declaring a mathematical equivalence wrong is not a credible way to make your point. But I'm OK with either expression, as long as they are parametric in L.

BTW, what I like about $L(f\pm 1/L)$ vs $(Lf\pm 1)$ is that the frequency offset between terms is more readily apparent (at least to me).

--Bob K (talk) 21:38, 27 December 2021 (UTC)

It is clear that two formulations of same function are not equivalent, and I see that second is exact, hence something wrong with first expression. Maybe it is approximation? than you will say clearly that it is approximate equation. And it is unclear reason to introduce approximate equation which is only slightly simpler than exact equation. So I suggest to simply remove that suspicious first equation from that page if you can't clarify this question in any other way. --Lexey73 (talk) 08:46, 28 December 2021 (UTC)

And yesterday you said: "it exactly equals to:

$W_{0}(f)=L\left({\tfrac {1}{2}}{\frac {\sin(\pi Lf)}{\pi Lf}}+{\tfrac {1}{4}}{\frac {\sin(\pi (Lf-1))}{\pi (Lf-1)}}+{\tfrac {1}{4}}{\frac {\sin(\pi (Lf+1))}{\pi (Lf+1)}}\right)$"

which I agree with. Apparently all you want is the last word, so I'm probably done with this thread.

--Bob K (talk) 15:59, 28 December 2021 (UTC)

I'm about this pair:

$W_{0}(f)={\tfrac {1}{2}}{\frac {\sin(\pi Lf)}{\pi f(1-L^{2}f^{2})}}$

$W_{0}(f)={\tfrac {1}{2}}{\frac {\sin(\pi Lf)}{\pi f}}+{\tfrac {1}{4}}{\frac {\sin(\pi L(f-1/L))}{\pi (f-1/L)}}+{\tfrac {1}{4}}{\frac {\sin(\pi L(f+1/L))}{\pi (f+1/L)}}.$

they are far not equivalent to each other --Lexey73 (talk) 16:56, 28 December 2021 (UTC)

Nuttal - Window - Text shortened

For some reason, the test "Nuttal" is shortend in the image's headline. The "N" is not visible. The graphic will have to be changed, i think. 85.190.194.180 (talk) 14:36, 3 February 2022 (UTC)

Thanks. But you're only looking at a "thumb" representation of the actual figure. Thumbs are just approximations. To see the actual, click on the thumb and then click on the Original file link.

--Bob K (talk) 22:52, 3 February 2022 (UTC)

Nulls on chart are wrong

See the WikiCommons discussion: https://commons.wikimedia.org/wiki/File_talk:Window_functions_in_the_frequency_domain.png — Preceding unsigned comment added by 92.40.200.3 (talk) 12:55, 29 April 2022 (UTC)

I replaced the figure with a better one. --Bob K (talk) 13:04, 1 May 2022 (UTC)