Understanding window functions

1. Anyone have a web reference that shows a sis by side comparison of 
the various window functions -- shape, formula, strong/weak points.

2. While I searched with Google, I came across
http://www.bores.com/courses/intro/freq/3_window.htm

It states "Put mathematically, a window function has the property that 
its value and all its derivatives are zero at the ends."

Is this strictly true or true because sampling is being discussed and 
all samples outside a specific range are set to zero?

Richard Owlett wrote:

> 1. Anyone have a web reference that shows a sis by side comparison of 
> the various window functions -- shape, formula, strong/weak points.
> 
> 2. While I searched with Google, I came across
> http://www.bores.com/courses/intro/freq/3_window.htm
> 
> It states "Put mathematically, a window function has the property that 
> its value and all its derivatives are zero at the ends."
> 
> Is this strictly true or true because sampling is being discussed and 
> all samples outside a specific range are set to zero?
> 
>

1.  No, but that's a good question & I hope I'll see a good answer here. 
  I don't use the FFT much, and I can usually get away with collecting 
way more data than necessary and just using a raised cosine (whichever 
haXXX window that is :).

2.  Put mathematically, a function that is continuous in all it's 
derivatives cannot have it's value and _all_ of its derivatives go to 
zero all at the same point -- I'm not sure what the official theorem 
would be but if you _did_ have such a function then the Taylor's 
expansion theorem would not work.

I think the author meant that a windowing function who's value and 1st 
derivative go to zero at the ends of the window will be "good" from the 
standpoint of harmonic suppression.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Richard Owlett wrote:
> 1. Anyone have a web reference that shows a sis by side comparison of 
> the various window functions -- shape, formula, strong/weak points

Matlab has a nice window-selector box (called "wintool", I think). 
Admittedly, that's not a web reference, sorry.


> 
> 2. While I searched with Google, I came across
> http://www.bores.com/courses/intro/freq/3_window.htm
> 
> It states "Put mathematically, a window function has the property that 
> its value and all its derivatives are zero at the ends."

A trivial counter example to that claim is the rectangular window. Chris 
must be talking about a special class of windows (I didn't check the link).

Regards,
Andor

Andor Bariska wrote:
> Richard Owlett wrote:
> 
>> 1. Anyone have a web reference that shows a sis by side comparison of 
>> the various window functions -- shape, formula, strong/weak points
> 
> 
> Matlab has a nice window-selector box (called "wintool", I think). 
> Admittedly, that's not a web reference, sorry.
> 
> 

Ah but knowing that Matlab had such a selection helped.
Knowing that Scilab claims to be as powerful, I re-looked at help. I 
missed it before. It's not well documented for a beginner, but it's there.

"Richard Owlett" <rowlett@atlascomm.net> wrote in message
news:10ic518s239g7a6@corp.supernews.com...
> 1. Anyone have a web reference that shows a sis by side comparison of
> the various window functions -- shape, formula, strong/weak points.
>
> 2. While I searched with Google, I came across
> http://www.bores.com/courses/intro/freq/3_window.htm
>
> It states "Put mathematically, a window function has the property that
> its value and all its derivatives are zero at the ends."
>
> Is this strictly true or true because sampling is being discussed and
> all samples outside a specific range are set to zero?

No, it's not correct.  But, if you take it in context, there should have
been a modifier in there ... like "this type of" or "a good" or .....

If you look for "harris blackman kaiser nuttall comparison"you will get a
lot of hits on Google:

http://www.ece.cmu.edu/~ee762/hspice-docs/html/hspice_and_qrg/hspice_2001_2-220.html

http://citeseer.ist.psu.edu/cache/papers/cs/15464/ftp:zSzzSzftp.cg.tuwien.ac.atzSzpubzSzTRzSz00zSzTR-186-2-00-08Paper.pdf/hauser00mastering.pdf
(which didn't load properly for me but looked interesting from what I could
see).

and there is the comp.dsp FAQ with notable windowing articles:
http://www.bdti.com/faq/1.htm#114

Fred

"Andor Bariska" schrieb 
> > 
> > 2. While I searched with Google, I came across
> > http://www.bores.com/courses/intro/freq/3_window.htm
> > 
> > It states "Put mathematically, a window function has the
> > property that its value and all its derivatives are zero at 
> > the ends."
> 
> A trivial counter example to that claim is the rectangular 
> window. Chris  must be talking about a special class of 
> windows (I didn't check the link).
> 
Or the triangular window or the Hamming Window.

> Is this strictly true or true because sampling is being 
> discussed and all samples outside a specific range are 
> set to zero?
The basic idea is not so much to make the samples zero, 
but to make a smooth transition between
x[n-3],x[n-2],x[n-1],x[0],x[1],x[2],x[3]
because the FFT algorithm "assumes" that the signal 
x[0]...x[n] repeats indefinitely.

HTH
Martin

On Fri, 20 Aug 2004 19:28:08 +0200, "Martin Blume" <mblume@socha.net>
wrote:

>"Andor Bariska" schrieb 
>> > 
>> > 2. While I searched with Google, I came across
>> > http://www.bores.com/courses/intro/freq/3_window.htm
>> > 
>> > It states "Put mathematically, a window function has the
>> > property that its value and all its derivatives are zero at 
>> > the ends."
>> 
>> A trivial counter example to that claim is the rectangular 
>> window. Chris  must be talking about a special class of 
>> windows (I didn't check the link).
>> 
>Or the triangular window or the Hamming Window.
>
>> Is this strictly true or true because sampling is being 
>> discussed and all samples outside a specific range are 
>> set to zero?
>The basic idea is not so much to make the samples zero, 
>but to make a smooth transition between
>x[n-3],x[n-2],x[n-1],x[0],x[1],x[2],x[3]
>because the FFT algorithm "assumes" that the signal 
>x[0]...x[n] repeats indefinitely.
>
>HTH
>Martin

Hi Martin,

   for the FFT to make assumptions, it has to 
be alive!

[-Rick-]

On Fri, 20 Aug 2004 10:13:10 -0500, Richard Owlett
<rowlett@atlascomm.net> wrote:

>1. Anyone have a web reference that shows a sis by side comparison of 
>the various window functions -- shape, formula, strong/weak points.
>
>2. While I searched with Google, I came across
>http://www.bores.com/courses/intro/freq/3_window.htm
>
>It states "Put mathematically, a window function has the property that 
>its value and all its derivatives are zero at the ends."
>
>Is this strictly true or true because sampling is being discussed and 
>all samples outside a specific range are set to zero?

Hi,

   Here's my two cents.

Not every window function has the property that 
its value and all its derivatives are zero at the ends.

But those windows that have higher-order end-point 
derivatives equal to zero will have steeper freq-domain 
sidelobe level rolloff.

The *best* windows paper is:

Harris, F. "On the Use of Windows for Harmonic Analysis with the
Discrete Fourier Transform," Proceedings of the IEEE, Vol. 66, No. 1,
Jan. 1978.

fred compares just about every window function there is!!

[-Rick-]

The harris paper has a few examples from several useful classes of window 
functions.  

In article <41272d0b.1386839968@news.sf.sbcglobal.net>, 
r.lyons@_BOGUS_ieee.org (Rick Lyons) wrote:
>On Fri, 20 Aug 2004 10:13:10 -0500, Richard Owlett
><rowlett@atlascomm.net> wrote:

>Not every window function has the property that 
>its value and all its derivatives are zero at the ends.
>
>But those windows that have higher-order end-point 
>derivatives equal to zero will have steeper freq-domain 
>sidelobe level rolloff.
>
>The *best* windows paper is:
>
>Harris, F. "On the Use of Windows for Harmonic Analysis with the
>Discrete Fourier Transform," Proceedings of the IEEE, Vol. 66, No. 1,
>Jan. 1978.
>
>fred compares just about every window function there is!!
>
>[-Rick-]
>

Rick Lyons wrote:

> On Fri, 20 Aug 2004 10:13:10 -0500, Richard Owlett
> <rowlett@atlascomm.net> wrote:
> 
> 
>>1. Anyone have a web reference that shows a sis by side comparison of 
>>the various window functions -- shape, formula, strong/weak points.
>>
>>2. While I searched with Google, I came across
>>http://www.bores.com/courses/intro/freq/3_window.htm
>>
>>It states "Put mathematically, a window function has the property that 
>>its value and all its derivatives are zero at the ends."
>>
>>Is this strictly true or true because sampling is being discussed and 
>>all samples outside a specific range are set to zero?
> 
> 
> Hi,
> 
>    Here's my two cents.
> 
> Not every window function has the property that 
> its value and all its derivatives are zero at the ends.
> 
> But those windows that have higher-order end-point 
> derivatives equal to zero will have steeper freq-domain 
> sidelobe level rolloff.
> 
> The *best* windows paper is:
> 
> Harris, F. "On the Use of Windows for Harmonic Analysis with the
> Discrete Fourier Transform," Proceedings of the IEEE, Vol. 66, No. 1,
> Jan. 1978.
> 
> fred compares just about every window function there is!!
> 
> [-Rick-]
> 


I don't have access to _Proceedings of the IEEE_ , but using article 
title I found two WEB pages with excellent graphics.


http://ccrma.stanford.edu/software/scmp/Windows/Windows.pdf
http://mathworld.wolfram.com/ApodizationFunction.html