Unusual DFT Windows

I have come across two Window functions for use in a DSP
application.  I wonder if anyone has seen them before and
can name them.  I have tested them using Harris' two-tone
tests and their performance is comparable to
Blackman-Harris-67db windows.  Both have the property that
the function value and its first two derivatives go to zero
at the endpoints.  The tails of the DFT of the windows tend
to decrease monotonically and smoothly to below -100 db.
faster than the ubiquitous Hann window and without the
"sniglets" common to the Blackman and B-H windows.  The DFT
of Window 2 is much broader than that of Window 1 but both
gave similar performance in the two-tone discrimination
tests.  Can anyone shed any light on them and how they might
be best used?

Window 1:

   W(x) = (1/2)*((3/4) - cos(x) + (1/4) cos(2x))

   where 0 <= x < 2Pi

Window 2:

   W(x) = 64 * (x^3) * ((1-x)^3)

   where 0 <= x < 1

-jeh

John E. Hadstate wrote:

> The tails of the DFT of the windows tend
> to decrease monotonically and smoothly to below -100 db.
> faster than the ubiquitous Hann window ...

??? The DFT of the Hann window only has three (consecutive) 
non-zero coeficients.


Bob
-- 

"Things should be described as simply as possible, but no 
simpler."

                                              A. Einstein

"Bob Cain" <arcane@arcanemethods.com> wrote in message
news:csub0u014ub@enews4.newsguy.com...
>
>
> John E. Hadstate wrote:
>
> > The tails of the DFT of the windows tend
> > to decrease monotonically and smoothly to below -100 db.
> > faster than the ubiquitous Hann window ...
>
> ??? The DFT of the Hann window only has three
(consecutive)
> non-zero coeficients.
>
>
> Bob

Yes.  I tripped over my own feet.  I started out describing
the DFT's of these two new windows and somehow wandered into
comparing the DFT's of the windowed two-tone data.  That's
what happens when you edit something and then don't go back
and carefully re-read what you changed.  My apologies.

Qualitatively, the DFT of the trig-based window (Window 1)
is about as sharply defined as any of the well-known windows
(including the Hann).  The DFT of the polynomial-based
window (Window 2) tails-off slowly, smoothly and
monotonically with increasing frequency.  In spite of these
differences, both perform about the same as the BH67 window
on Harris' two-tone tests.

Again, sorry for the confusion.

John E. Hadstate wrote:
> I have come across two Window functions for use in a DSP
> application.  I wonder if anyone has seen them before and
> can name them.  I have tested them using Harris' two-tone
> tests and their performance is comparable to
> Blackman-Harris-67db windows.  Both have the property that
> the function value and its first two derivatives go to zero
> at the endpoints.  The tails of the DFT of the windows tend
> to decrease monotonically and smoothly to below -100 db.
> faster than the ubiquitous Hann window and without the
> "sniglets" common to the Blackman and B-H windows.  The DFT
> of Window 2 is much broader than that of Window 1 but both
> gave similar performance in the two-tone discrimination
> tests.  Can anyone shed any light on them and how they might
> be best used?
>
> Window 1:
>
>    W(x) = (1/2)*((3/4) - cos(x) + (1/4) cos(2x))
>
>    where 0 <= x < 2Pi
>
> Window 2:
>
>    W(x) = 64 * (x^3) * ((1-x)^3)
>
>    where 0 <= x < 1
>
> -jeh

I asked a question about window functions here a few months
ago. The answers tended towards suggesting that window
functions for filter designs are obsolete. Computer-aideid
methods like the Parks/McClellan algorithm give more
efficient filters, in terms of performance per computational
complexity. Of course, window functions are still very
useful in nonparametric spectrum estimation.

Having said that, there seems to be two papers available
that discuss window functions:

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

Albert H. Nuttall, "Some Windows with Very Good Sidelobe
Behaviour", IEEE Transactions of Acoustics, Speech, and
Signal Processing, Vol. ASSP-29, No. 1, February 1981,
pp 84-91

Chances are that wyour windows will be listed and named
in one of these articles.

Rune

"Rune Allnor" <allnor@tele.ntnu.no> wrote in message
news:1106478053.911419.46160@f14g2000cwb.googlegroups.com...
>
>
> Having said that, there seems to be two papers available
> that discuss window functions:
>
> Fredric J. Harris, "On the Use of Windows for Harmonic
> Analysis with the Discrete Fourier Transform,"
> Proceedings of the IEEE, Vol. 66, No. 1, January 1978.
>
> Albert H. Nuttall, "Some Windows with Very Good Sidelobe
> Behaviour", IEEE Transactions of Acoustics, Speech, and
> Signal Processing, Vol. ASSP-29, No. 1, February 1981,
> pp 84-91
>
> Chances are that wyour windows will be listed and named
> in one of these articles.
>
> Rune
>

Thanks for the pointers.  I still have the "Proceedings"
issue containing Harris' paper (it's yellow and brittle
after 33 years of abuse).  Neither of the windows is
mentioned there.  I'll try one of the local college
libraries for Nuttall's paper.

Rune Allnor wrote:
> John E. Hadstate wrote:
> 
>>I have come across two Window functions for use in a DSP
>>application.  I wonder if anyone has seen them before and
>>can name them.  I have tested them using Harris' two-tone
>>tests and their performance is comparable to
>>Blackman-Harris-67db windows.  Both have the property that
>>the function value and its first two derivatives go to zero
>>at the endpoints.  The tails of the DFT of the windows tend
>>to decrease monotonically and smoothly to below -100 db.
>>faster than the ubiquitous Hann window and without the
>>"sniglets" common to the Blackman and B-H windows.  The DFT
>>of Window 2 is much broader than that of Window 1 but both
>>gave similar performance in the two-tone discrimination
>>tests.  Can anyone shed any light on them and how they might
>>be best used?
>>
>>Window 1:
>>
>>   W(x) = (1/2)*((3/4) - cos(x) + (1/4) cos(2x))
>>
>>   where 0 <= x < 2Pi
>>
>>Window 2:
>>
>>   W(x) = 64 * (x^3) * ((1-x)^3)
>>
>>   where 0 <= x < 1
>>
>>-jeh
> 
> 
> I asked a question about window functions here a few months
> ago. The answers tended towards suggesting that window
> functions for filter designs are obsolete. Computer-aideid
> methods like the Parks/McClellan algorithm give more
> efficient filters, in terms of performance per computational
> complexity. Of course, window functions are still very
> useful in nonparametric spectrum estimation.
> 
> Having said that, there seems to be two papers available
> that discuss window functions:
> 
> Fredric J. Harris, "On the Use of Windows for Harmonic
> Analysis with the Discrete Fourier Transform,"
> Proceedings of the IEEE, Vol. 66, No. 1, January 1978.
> 
> Albert H. Nuttall, "Some Windows with Very Good Sidelobe
> Behaviour", IEEE Transactions of Acoustics, Speech, and
> Signal Processing, Vol. ASSP-29, No. 1, February 1981,
> pp 84-91
> 
> Chances are that wyour windows will be listed and named
> in one of these articles.
> 
> Rune
> 
John,

Window #1 is essentially in Nuttall's paper as equation (26).  The 
difference between his and yours is that he defines the window over -PI 
to +PI (that is why your middle coefficient is negative and his is 
postive).

Window #1 was derived by assuming a 3-term cosine series and choosing 
the coefficients so that the window and its 2'nd derivative are zero at 
the endpoints (the 1'st derivative is also zero there but that comes for 
free).  As a result, the frequency response of this window decays as 1/f 
to the fifth power which is the reason for the low sidelobes.

Your second window is unfamiliar to me.  It is not addressed in 
Nuttall's paper because he considers only windows which can be 
represented as sums of cosines.  If you would like insight into why the 
filter performs so well, it might be helpful to express your window in 
terms of Chebyshev polynomials (no guarantees, just a thought).  They 
are discussed in many places but for your problem, a good reference 
would be Richard Hamming's book "Digital Filters" which I think is now 
published by Dover (I have the Prentice-Hall 3'rd edition).

Mike

"Michael Soyka" <msoyka_nospam@remove_nospam.fctvplus.net>
wrote in message news:3EPId.145$6k.1759@eagle.america.net...

> They
> are discussed in many places but for your problem, a good
reference
> would be Richard Hamming's book "Digital Filters" which I
think is now
> published by Dover (I have the Prentice-Hall 3'rd
edition).
>
> Mike

Thanks for the reference!  Pointers to good books and papers
are always appreciated.

To elaborate on my real question: why would one choose
either of these windows in preference to Hann or
Blackman-Harris windows (or vice-versa for that matter)?

"John E. Hadstate" <jh113355@hotmail.com> wrote in message 
news:1iTId.7927$Gj.1606@bignews3.bellsouth.net...
>
> "Michael Soyka" <msoyka_nospam@remove_nospam.fctvplus.net>
> wrote in message news:3EPId.145$6k.1759@eagle.america.net...
>
>> They
>> are discussed in many places but for your problem, a good
> reference
>> would be Richard Hamming's book "Digital Filters" which I
> think is now
>> published by Dover (I have the Prentice-Hall 3'rd
> edition).
>>
>> Mike
>
> Thanks for the reference!  Pointers to good books and papers
> are always appreciated.
>
> To elaborate on my real question: why would one choose
> either of these windows in preference to Hann or
> Blackman-Harris windows (or vice-versa for that matter)?

Because the window *you* choose has some properties that *you* like or need:
- the width of the main lobe is narrower
- the sidelobe level is lower in a variety of ways:
 .. the first sidelobe is smaller
 .. the sidelobe energy overall is smaller
 .. etc.
- the time domain unit sample response has certain properties (e.g. it's a 
Nyquist filter)
- it's easier to compute or to work with

So, it's not up to anyone else to say why you would choose one window over 
another unless you have some very specific requirements.  All one can do is 
to compare various windows with respect to these characteristics - which is 
what many of the references do.

Fred 

Michael Soyka wrote:

> a good reference
> would be Richard Hamming's book "Digital Filters" which I think is
now
> published by Dover (I have the Prentice-Hall 3'rd edition).

It is indeed published by Dover, it is available for less than $11
at amazon.com. To me, the (Dover) book just about re-paid itself by
giving a very short, concise recipe for computing the modified Bessel
function that goes into the Kaiser window. The computed values checked
to within numerical accuracy with the tables listed by Abramowitz and
Stegun. 

Rune

Rune Allnor wrote:
...

> To me, the (Dover) book just about re-paid itself by
> giving a very short, concise recipe for computing the modified Bessel
> function that goes into the Kaiser window. The computed values checked
> to within numerical accuracy with the tables listed by Abramowitz and
> Stegun. 
> 
> Rune

I posted this reference [1] once already, a second time won't hurt. The 
author describes how to compute the Fourier-transform of the Kaiser 
window family. The Fourier-transform is also a one-parameter family of 
windows with similar trade-off properities in the parameter alpha, but 
can be computed using the exp-function only (no Bessel).

I got this article from the IEEE signal processing e-library. It is also 
very well spent money:
http://www.ieee.org/organizations/society/sp/SPeL.html

Saves many trips to the library!

Regards,
Andor


[1] John J Knab, "An Alternate Kaiser Window"
IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol.
ASSP-27, No.5, October 1979