Why no flat topped windows?

I'm looking at comparing FFT's of a batch of signals.
Right now I'm choosing how to window the input data.
I noticed that none of the _common_ windows have flat top except for the 
"rectangular window".

Is there a reason other than the simplicity of calculating the window 
function? I was thinking about piecewise continuous with continuous 
first derivative.

On 28 Okt, 14:37, Richard Owlett <rowl...@atlascomm.net> wrote:
> I'm looking at comparing FFT's of a batch of signals.
> Right now I'm choosing how to window the input data.
> I noticed that none of the _common_ windows have flat top except for the
> "rectangular window".
>
> Is there a reason other than the simplicity of calculating the window
> function? I was thinking about piecewise continuous with continuous
> first derivative.

There is the Tukey window, which I think has a "flat" top.

FIR filter design by window functions is all but obsolete
after methods like Parks-McClellan were developed. So the
detailed analysis of window functions is not all that
interesting these days...

Rune

Rune Allnor wrote:
> On 28 Okt, 14:37, Richard Owlett <rowl...@atlascomm.net> wrote:
> 
>>I'm looking at comparing FFT's of a batch of signals.
>>Right now I'm choosing how to window the input data.
>>I noticed that none of the _common_ windows have flat top except for the
>>"rectangular window".
>>
>>Is there a reason other than the simplicity of calculating the window
>>function? I was thinking about piecewise continuous with continuous
>>first derivative.
> 
> 
> There is the Tukey window, which I think has a "flat" top.
> 
> FIR filter design by window functions is all but obsolete
> after methods like Parks-McClellan were developed. So the
> detailed analysis of window functions is not all that
> interesting these days...
> 
> Rune
> 

OK? or perhaps ???? ;)
I understood windowing data prior to doing an fft was required to reduce 
spectral leakage.

What did I say that brought filters to mind?

On Oct 28, 9:37 am, Richard Owlett <rowl...@atlascomm.net> wrote:
> I'm looking at comparing FFT's of a batch of signals.
> Right now I'm choosing how to window the input data.
> I noticed that none of the _common_ windows have flat top except for the
> "rectangular window".
>
> Is there a reason other than the simplicity of calculating the window
> function? I was thinking about piecewise continuous with continuous
> first derivative.

a window i suggested (in an AES article in 1995) for use in audio
splicing (i like to think of a window as the concatination of two
splicing functions, a fade up followed by a fade down) is:

   w(t) = (1/2)*( 1 + (9/8)*cos(pi*t) - (1/8)*cos(3*pi*t) )

   for   -1  <=  t  <=  +1    ( w(t) = 0  for |t| > 1 )


it's a sorta extension to the Hann window,:

   w(t) = (1/2)*( 1 + cos(pi*t) )

   for   -1  <=  t  <=  +1    ( w(t) = 0  for |t| > 1 )

the Hann window is continuous everywhere in the 0th and 1st derivative
with a discontinuity in the 2nd derivative at the two points where the
operative part of the window is appended to silence (at t=-1 and t=
+1).

this "flattened Hann" (what i call it until somebody points to a prior
reference) is continuous everywhere through the 2nd derivative and has
discontinuity at the 3rd and higher derivatives at +1 and -1.


r b-j

Richard Owlett wrote:
> I'm looking at comparing FFT's of a batch of signals.
> Right now I'm choosing how to window the input data.
> I noticed that none of the _common_ windows have flat top except for the 
> "rectangular window".
> 
> Is there a reason other than the simplicity of calculating the window 
> function? I was thinking about piecewise continuous with continuous 
> first derivative.

A continuous and *finite* first derivative? Think what that implies. 
Points of sharp curvature -- i.e., large second derivative -- aren't 
helpful.

A window is almost always computed off line. Simplicity isn't a major
consideration.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

On Oct 28, 6:37 am, Richard Owlett <rowl...@atlascomm.net> wrote:
> I'm looking at comparing FFT's of a batch of signals.
> Right now I'm choosing how to window the input data.
> I noticed that none of the _common_ windows have flat top except for the
> "rectangular window".

I've also seen trapezoidal windows mentioned.  The flatter
the top in proportion to the total, the steeper the sides
need to be for a side of similar shape.  Steeper sides
usually correspond to more high frequency content in the
transform, which is often what one doesn't want from a window
function.


IMHO. YMMV.
--
rhn A.T nicholson d.0.t C-o-M

Richard Owlett wrote:

> I'm looking at comparing FFT's of a batch of signals.
> Right now I'm choosing how to window the input data.
> I noticed that none of the _common_ windows have flat top except for the
> "rectangular window".
> 
> Is there a reason other than the simplicity of calculating the window
> function?

I played around with this idea years ago and my experimentation showed
that all the the flat top windows I could find or think of suffered 
from worse spectral leakage effects that all of the traditional windows.

Erik
-- 
-----------------------------------------------------------------
Erik de Castro Lopo
-----------------------------------------------------------------
Complex problems have simple easy to understand wrong answers.

Erik de Castro Lopo wrote:
> Richard Owlett wrote:
> 
>> I'm looking at comparing FFT's of a batch of signals.
>> Right now I'm choosing how to window the input data.
>> I noticed that none of the _common_ windows have flat top except for the
>> "rectangular window".
>>
>> Is there a reason other than the simplicity of calculating the window
>> function?
> 
> I played around with this idea years ago and my experimentation showed
> that all the the flat top windows I could find or think of suffered 
> from worse spectral leakage effects that all of the traditional windows.

That shouldn't be too surprising. Windows suppress ringing in filters, 
while they reduce spillover in FTs. The two actions are related, so one 
can reason about both from either vantage point. In filters, steeper 
sides make for more ringing. In windows, steeper sides make for less 
suppression of ringing. The more the top stays flat, the steeper the 
sides need to be.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

On Oct 28, 12:34 pm, robert bristow-johnson
<r...@audioimagination.com> wrote:
> On Oct 28, 9:37 am, Richard Owlett <rowl...@atlascomm.net> wrote:
>
> > I'm looking at comparing FFT's of a batch of signals.
> > Right now I'm choosing how to window the input data.
> > I noticed that none of the _common_ windows have flat top except for the
> > "rectangular window".
>
> > Is there a reason other than the simplicity of calculating the window
> > function? I was thinking about piecewise continuous with continuous
> > first derivative.
>
> a window i suggested (in an AES article in 1995) for use in audio
> splicing (i like to think of a window as the concatination of two
> splicing functions, a fade up followed by a fade down) is:
>
>    w(t) = (1/2)*( 1 + (9/8)*cos(pi*t) - (1/8)*cos(3*pi*t) )
>
>    for   -1  <=  t  <=  +1    ( w(t) = 0  for |t| > 1 )
>
> it's a sorta extension to the Hann window,:
>
>    w(t) = (1/2)*( 1 + cos(pi*t) )
>
>    for   -1  <=  t  <=  +1    ( w(t) = 0  for |t| > 1 )
>
> the Hann window is continuous everywhere in the 0th and 1st derivative
> with a discontinuity in the 2nd derivative at the two points where the
> operative part of the window is appended to silence (at t=-1 and t=
> +1).
>
> this "flattened Hann" (what i call it until somebody points to a prior
> reference) is continuous everywhere through the 2nd derivative and has
> discontinuity at the 3rd and higher derivatives at +1 and -1.
>
> r b-j

For a quick comparison of Hann and flattened Hann:

Hann
Transform Size:  256 Bins         Coherent Gain:   0.5
Eff. Noise BW:   1.5 Bins         Scalloping/Wcae: -1.4236 dB
-3 dB BW:        1.45 Bins        Mainlobe Width:  4 Bins
-6 dB BW:        2.02 Bins        Max. Sidelobe:   -31.47 dB
-60 dB BW:       13.3 Bins        Max. SSL BW:     3.75 Bins
SL Falloff:     -18.06 dB/Octave  Bin 1 SL Intcpt:-9.92 dB

flattenedHann
Transform Size:  256 Bins         Coherent Gain:   0.5
Eff. Noise BW:   1.6406 Bins      Scalloping/Wcae: -1.1789 dB
-3 dB BW:        1.58 Bins        Mainlobe Width:  4 Bins
-6 dB BW:        2.19 Bins        Max. Sidelobe:   -21.81 dB
-60 dB BW:       9.66 Bins        Max. SSL BW:     3.56 Bins
SL Falloff:     -30.1 dB/Octave   Bin 1 SL Intcpt: 9.43 dB

The flattened version rolls off at -30 dB/oct versus -18 dB/oct. for
Hann, but in the region from 2 to 4 bin widths from the bin center,
response of the flattened version can be almost 20 dB higher than the
Hann response; the typical/near far window tradeoff.

Dale B. Dalrymple
http://dbdimages.com
http://stores.lulu.com/dbd

"Ron N." <rhnlogic@yahoo.com> wrote in message 
news:1193607019.493926.234540@z24g2000prh.googlegroups.com...
> On Oct 28, 6:37 am, Richard Owlett <rowl...@atlascomm.net> wrote:
>> I'm looking at comparing FFT's of a batch of signals.
>> Right now I'm choosing how to window the input data.
>> I noticed that none of the _common_ windows have flat top except for the
>> "rectangular window".
>
> I've also seen trapezoidal windows mentioned.  The flatter
> the top in proportion to the total, the steeper the sides
> need to be for a side of similar shape.  Steeper sides
> usually correspond to more high frequency content in the
> transform, which is often what one doesn't want from a window
> function.
>

heh ..... that would be the convolution of two gate functions.  When they 
are the same length then you get a triangular result.  When one is narrower 
than the other, you get a flat top.  When one is much narrower than the 
other you get quite a wide flat top.

Now let's look in frequency:
Multiply the ffts of those gate functions - which are sinc functions.

- If the gates are the same length and you have a triangular function then 
the fft is a sin^2(x)/x^2.  The decay is then proportional to 1/f^3 as I 
recall ... instead of 1/f for the sinc.  The decay is related to the order 
of the zero at the edge in time.

- If the gates are very different in length then you have a long flat top 
and linearly tapered ends - half a triangle.  In this case the fft of the 
narrow one is wide and the fft of the wide one is narrow.  Those are 
multiplied together so you will have the narrow sinx/x shape but while with 
more rapidly decaying tails than the sinc and about the same first sidelobes 
as the narrow sinc.

Another variation is to use a cosine taper at the edges.  The widest version 
is a raised cosine with DFT of [1/4 1/2 1/4].  Narrowing the cosine taper at 
the ends and adding a flat top gets the flat-topped versions.  Maybe not as 
good in frequency as the trapezoidal one.

You might ask a couple of questions:
1) Why window at all?
2) Why care about a flat top if you do?

Fred