[Vorbis-dev] Window function (again)

[Tuomo Latto]

bender at doanstar.de wrote:
> I'm searching some information about the Vorbis window function. I have 
> read the following post from the mailing list archive (July 2003) where the 
> function and its proof was explained:
> 
> http://lists.xiph.org/pipermail/vorbis-dev/2003-July/007588.html
> 
> Everything apart from one small passage was clear to me.
> 
> f^2(x) + f^2(n/2+x) = 1
> <=> f^2(x) + f^2(n/2-x) = 1 <-- here in this line
> <=> sin^2(pi/2*pi*g(z)) + sin^2(pi/2*pi*g(1-z)) = 1
> 
> Well, I do not know how to eliminate the Variable n in the second line.

Well, there was this in the original post (the link you gave):
--- clip -->
If you substitute z=2(x+0.5)/n, you get
     f(x(z)) = sin(1/2*pi*sin^2(z*pi/2))
<-- pilc ---

So, z = (2x + 1) / n.
If we then substitute x = n/2 - x we get
z = (2(n/2 - x) + 1) / n = 1 - (2x - 1)/n
=> 1 - z = (2x - 1)/n
which seems wrong.

But if we forget the second line and try n/2+x in the first line,
we get
z = (2(n/2 + x) + 1) / n = 1 + (2x + 1)/n
=> z - 1 = (2x + 1)/n
which seems wrong too, but if we applied the symmetry here
we get 1 - z = z - 1 = (2x + 1)/n, which looks about right.

I could be on a completely wrong track, though.

-- 
Tuomo

... It's Ensign Flintstone - he's Fred, Jim