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From: pikey on 13 Apr 2008 11:03
Hi All,

I have searched high and low to no avail; I am trying to find the formula
for the 5-term Blackman-Harris. Certainly there is plenty of material on
the 3 and 4 terms but not the fifth.
Any help would be greatly appreciated.

Cheers

Pikey


From: bharat pathak on 13 Apr 2008 23:34
Hi,

Search for follwing paper. if you
dont get the paper over net send
me your personal email id.

my email id is bharat\at\arithos\com

A FAMILY OF COSINE-SUM WINDOWS
FOR HIGH-RESOLUTION MEASUREMENTS
Hans-Helge Albrecht
Physikalisch-Technische Bundesanstalt
Abbestraße 2-12, D-10587 Berlin, Germany
Phone: +49 30 3481 311 Fax: +49 30 3481 490
E-mail: hans-helge.albrecht(a)ptb.de

rgds
bharat pathak

Arithos Designs
www.Arithos.com

DSP design consultancy and Training Company.
From: dbd on 14 Apr 2008 01:15
On Apr 13, 8:03 am, "pikey" <pi...(a)amor888.fsnet.co.uk> wrote:
> Hi All,
>
> I have searched high and low to no avail; I am trying to find the formula
> for the 5-term Blackman-Harris. Certainly there is plenty of material on
> the 3 and 4 terms but not the fifth.
> Any help would be greatly appreciated.
>
> Cheers
>
> Pikey

Pikey

A better terminology might be "minimum sidelobe cosine-sum" window.

harris doesn't seem to have published past 4 terms and his 4 term
optimized window had 92 dB sidelobe rejection. Nuttall provided a
correction to the 4 term coefficients to yield 98 dB sidelobe
rejection. So, harris' 4 term wasn't minimum sidelobe. Nuttall doesn't
seem to have published past 4 terms either.

So, if you want a four term minimum sidelobe cosine-sum window you
could use:

acoef = [0.3232153788877343 0.4714921439576260
0.1755341299601972 ...
2.849699010614994e-2 1.261357088292677e-3];

from the Albrecht paper:

A family of cosine-sum windows for high-resolution measurements
Albrecht, H.H.
Acoustics, Speech, and Signal Processing, 2001. Proceedings. (ICASSP
apos;01). 2001 IEEE International Conference on
Volume 5, Issue , 2001 Page(s):3081 - 3084 vol.5

The paper gives up to 11 terms and -289 dB sidelobe rejection. At 12
terms there were numerical problems with running the optimization in
64 bit floating point. Albrecht's 4 term coefficients agree with
Nuttal's 4 term values.
The 7 term window appears in some 24-bit A-D converter application
notes published before Albrecht's paper. The values published were
consistent with Albrecht's. A number of people have successfully run
the optimization for various numbers of coefficients so I go with
"minimum sidelobe cosine-sum" window for the family.

Dale B. Dalrymple
http://dbdimages.com
From: Fred Marshall on 14 Apr 2008 16:45

"pikey" <pikie(a)amor888.fsnet.co.uk> wrote in message
news:h9-dnYu56pc2vZ_VnZ2dnUVZ_oqhnZ2d(a)giganews.com...
> Hi All,
>
> I have searched high and low to no avail; I am trying to find the formula
> for the 5-term Blackman-Harris. Certainly there is plenty of material on
> the 3 and 4 terms but not the fifth.
> Any help would be greatly appreciated.
>
> Cheers
>
> Pikey

The whole point of windows of this type is that they're simple. I'm
guessing but I believe the reason you don't see 5-term Blackman-Harris is
because there never was one formulated. By simple I mean that there are
just a few cosine terms summed to form the window.

For each cosine term there is a pair of sincs in frequency. The sincs add
to get the overall, low sidelobe, frequency response of the window. The
higher the "order" the larger the number of nonzero sincs. Outside the
range of nonzero sincs, the sidelobes have the same regular zeros as the
sincs have.

I developed a method similar to Parks-McClellan - at about the same time -
that can design windows using the same Remez algorithm but using sincs as
the basis set. This gets directly to what you're looking for. The
sidelobes can be minimax or weighted minimax and, if you need, can have
forced zeros. And, the sincs can be replaced with Dirichlets if you're
working with a periodic function / discrete in the transformed domain.

See: Temes, Barcilon & Marshall: "The Optimization of Bandlimited Sytems"
Proc IEEE Vol 61 No. 2 Feb 1973 pp 196-234.

Back to why simple windows?

A window is useful to reduce sidelobes / leakage at a bit of expense of the
main lobe width or resolution.

Some "windows" are actually implemented in their transform domain as a
convolution. For example the [1/4 1/2 /14] filter often used in video
processing for interpolation. It can be much more efficient and convenient
to convolve with very short filters than to multiply an entire array with a
window function.

Fred





From: dbd on 14 Apr 2008 17:04
On Apr 13, 10:15 pm, dbd <d...(a)ieee.org> wrote:
> On Apr 13, 8:03 am, "pikey" <pi...(a)amor888.fsnet.co.uk> wrote:
>
> > Hi All,
>
> > I have searched high and low to no avail; I am trying to find the formula
> > for the 5-term Blackman-Harris. Certainly there is plenty of material on
> > the 3 and 4 terms but not the fifth.
> > Any help would be greatly appreciated.
>
> > Cheers
>
> > Pikey
>
> Pikey
>
> A better terminology might be "minimum sidelobe cosine-sum" window.
>
> harris doesn't seem to have published past 4 terms and his 4 term
> optimized window had 92 dB sidelobe rejection. Nuttall provided a
> correction to the 4 term coefficients to yield 98 dB sidelobe
> rejection. So, harris' 4 term wasn't minimum sidelobe. Nuttall doesn't
> seem to have published past 4 terms either.
>
> So, if you want a four term minimum sidelobe cosine-sum window you
> could use:
>
> acoef = [0.3232153788877343 0.4714921439576260
> 0.1755341299601972 ...
> 2.849699010614994e-2 1.261357088292677e-3];
>
> from the Albrecht paper:
>
> A family of cosine-sum windows for high-resolution measurements
> Albrecht, H.H.
> Acoustics, Speech, and Signal Processing, 2001. Proceedings. (ICASSP
> apos;01). 2001 IEEE International Conference on
> Volume 5, Issue , 2001 Page(s):3081 - 3084 vol.5
>
> The paper gives up to 11 terms and -289 dB sidelobe rejection. At 12
> terms there were numerical problems with running the optimization in
> 64 bit floating point. Albrecht's 4 term coefficients agree with
> Nuttal's 4 term values.
> The 7 term window appears in some 24-bit A-D converter application
> notes published before Albrecht's paper. The values published were
> consistent with Albrecht's. A number of people have successfully run
> the optimization for various numbers of coefficients so I go with
> "minimum sidelobe cosine-sum" window for the family.
>
> Dale B. Dalrymplehttp://dbdimages.com

Please note, this is the 5-term coefficient set:

acoef = [0.3232153788877343 0.4714921439576260
0.1755341299601972 ...
2.849699010614994e-2 1.261357088292677e-3];

not the 4-term set, from the paper.

Dale B. Dalrymple
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