FPGA Central

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" <[email protected]> wrote in message
news:[email protected]..
> 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...001_2-220.html

http://citeseer.ist.psu.edu/cache/pa...0mastering.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" <[email protected]>
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
<[email protected]> 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 <[email protected]>,
r.lyons@_BOGUS_ieee.org (Rick Lyons) wrote:
>On Fri, 20 Aug 2004 10:13:10 -0500, Richard Owlett
><[email protected]> 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
> <[email protected]> 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/s...ws/Windows.pdf
http://mathworld.wolfram.com/ApodizationFunction.html

Tim Wescott wrote:
....
> 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 just remembered that I once had to create a function as homework
with the following properties:

1. Defined and infinitely often differentiable for all real numbers.
2. Zero outside the interval [0,1].
3. For any arbitrarily small but positive epsilon, it is required that
f(x) = 1, for x in [0-epsilon, 1-epsilon], and monotonous in the
boundary regions.

I guess this would be such a window which Chris Bores talked about. I
have no idea how its Fourier transform looks like (probably very
similar to a sinc for small epsilon :-).

Regards,
Andor

"Richard Owlett" in news:[email protected]..
> Rick Lyons wrote:
> > ...
> > 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.
> >
>
> I don't have access to _Proceedings of the IEEE_ , but using
> article title I found two WEB pages with excellent graphics. . . .


There are probably several good relevant documents online now, but I wanted
to second what Rick Lyons said -- Harris's overview paper is so definitive
that I have not seen anything approaching a substitute. (I've recommended
it on newsgroups too, since the 1980s.) Like Watson on Bessel functions,
Strunk and White on basic writing style, Inose and Yasuda on delta-sigma
modulation, or A. E. Housman on scholarly putdowns. It is also likely
available, or soon will be, for download via ieee.org:

http://ieeexplore.ieee.org

Regardless, its value and impact are such that obstacles in getting it will
probably seem unimportant once it is in use. Most people who work in DSP
research or related specialties have copies of it, most engineering
libraries at universities etc. have the IEEE Proceedings, so it may be
possible to find a copy not far from hand but anyway, to paraphrase an
athletic-shoe maker, just get it.


-- MH


--------
If a man will comprehend the richness and variety of the universe, and
inspire his mind with a due measure of wonder and awe, he must contemplate
the human intellect not only on its heights of genius but in its abysses of
ineptitude; and it might be fruitlessly debated to the end of time whether
Richard Bentley or Elias Stoeber was the more marvellous work of the
Creator: Elias Stoeber, whose reprint of Bentley's text, with a commentary
intended to confute it, saw the light in 1767 at Strasbourg, a city still
famous for its geese. This commentary is a performance in comparison with
which the _Aetna_ of Mr S. Sudhaus is a work of science and of genius.
Stoeber's mind, though that is no name to call it by, was one which turned
as unswervingly to the false, the meaningless, the unmetrical, and the
ungrammatical, as the needle to the pole.

-- A. E. Housman (Preface, Manilius _Astronomicon_ I, 1903) -- one example
of many