## Has any one seen this window function?

Started by 2 years ago●11 replies●latest reply 2 years ago●194 views\( w[n] = 4 \sin \left( \frac{n}{N}\pi \right) \sin \left( \frac{n+1}{N}\pi \right) \)

It's special, because it ensures that

$$ w[0] = w[N-1] = 0 $$

And it centers the window function on the samples (end to end) rather than the frame.

Why could window function help to get more accurate specific frequency amplitude?

I can't match it to any in the classic harris paper:

"On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform. FREDRIC J. HARRIS"

or here:

en.wikipedia.org/wiki/Window_function

Did I miss it as a special case of one of the parameterized equations?

Of course, the Sine terms could also be raised to powers for a wider stance on the DFT.

Thanks in advance.

maybe you could float this at the dsp.stackexchange

Follow the link, it sort of is. I wanted a more slow paced, discussion oriented forum.

it looks same (or very close) as hanning window scaled by 4

It is, very close. I would call it the Discrete VonHann (I consider the term hanning erroneous even if it is commonly used).

$$ \lim_{N \to \infty} w[n] = VonHann[n] $$

The scaling factor popped out of the derivation and I left it as it is immaterial.

It isn't new to me, implicit in my frequency formula.

And I assume any vector that goes up then comes down is defined as window but it needs a name. The only difference is the boxcar window which is not a window!!

Follow the derivation link at top. I framed it as a weighted average of DFT bins with a weighting of:

$$ ( -e^{i\omega}, e^{i\omega}+ e^{-i\omega},-e^{-i\omega} ) $$

and that is the window function when

$$ \omega = \frac{2\pi}{N} $$

The window function family can also be expressed in this near VonHonn form:

$$ \cos ( \omega) - \cos \left( \frac{n}{N}2\pi + \omega \right) $$

I'll whip up some numbers but I wanted nice charts and official numbers.

Hi kaz. Whether or not a boxcar (rectanglar) sequence is a window depends on your definition of the word "window".

Compare how the (1/N) DFT is related to its continuous counter parts, the FT and DTFT, are defined on the same interval.

The DFT:

\( \frac{1}{N} \sum_{n=0}^{N-1} x[n] e^{-i\frac{2\pi}{N}kn} \)

The FT:

\( \int_{-\infty}^{\infty} w(t) x(t) e^{-i2\pi t f } dt \)

The DTFT

\( \frac{1}{N} \sum_{n=-\infty}^{\infty} w[n] x[n] e^{-i\omega n} \)

\( \omega = \frac{k}{N}2\pi \)

Hi Cedron. I've not seen that w[n] window sequence before. It's frequency-domain behavior is super-similar to the Von Hann (hanning) window if we ignoring w[n]'s zero Hz gain factor. My guess is, in practice there will be very little difference between using your w[n] window compared to using a Von Hann window.

For any N more than fingers and a few toes, I totally agree, no functional difference beyond a gnat hair. This is more of a theoretical thing.

Perhaps, "The Discrete VonHann" is too strong of a claim. The VonHonn is at the base of a lot of families.

I am now thinking "Dipped and Shifted Von Hann" as that quite matches the cosine version. Thus, DSVH is my working acronym.

The most concise behavior description I can come up with is "The VonHann is slightly lowered and shifted by a half sample so $w[0]=w[N-1]=0$. This also shifts the peak of the window function one half sample to the mean of the endpoints not the center of the frame."

Here's a comparison of the two window functions for some fairly small N.

------------------------------------ VonHann DSVH Eleven 0 0.00000 0.00000 1 0.00722 0.01385 2 0.02657 0.03714 3 0.05192 0.06250 4 0.07522 0.08185 5 0.08907 0.08907 6 0.08907 0.08185 7 0.07522 0.06250 8 0.05192 0.03714 9 0.02657 0.01385 10 0.00722 0.00000 Twelve 0 0.00000 0.00000 1 0.00558 0.01078 2 0.02083 0.02946 3 0.04167 0.05103 4 0.06250 0.06971 5 0.07775 0.08049 6 0.08333 0.08049 7 0.07775 0.06971 8 0.06250 0.05103 9 0.04167 0.02946 10 0.02083 0.01078 11 0.00558 0.00000 ------------------------------------

A more formal analysis on the differences at low N might be more interesting to some one besides me.

Here's some starter code for those who want to to get started.

#---- Build the VonHann Windows theNormed = np.zeros( N ) theVonHann = np.zeros( N ) theDSVH = np.zeros( N ) for n in range( N ): theNormed[n] = RN theVonHann[n] = 0.5 * ( 1.0 - np.cos( theBeta * n ) )\ * RN theDSVH[n] = np.sin ( n * theBeta * 0.5 ) \ * np.sin ( ( n + 1 ) * theBeta * 0.5 ) \ * RN theDSVH2 = ( np.cos( theBeta * 0.5 ) \ - np.cos( theBeta * ( n + 0.5 ) ) ) \ * 0.5 * RN print( "%3d %8.5f %8.5f %8.5f" % \ ( n, theVonHann[n], theDSVH[n], theDSVH2 ) )I'm studying it more, works well in the math. I'll be putting that in a blog article and adding a link here (or not... for a while).

(All: I'm holding off any spoiliers I find, and if you have any of your own, please post only links here with a whispy description for those who wish to follow after taking a stab at it on their own.)

Thanks.

Okay, there was indeed a prize at the end of this search. If you want to find out what it is, follow this link to my latest article:

A huge trophy.