seeking formula for Fourier xform of Hamming family of windowing functions

On Feb 6, 1:50 am, Rune Allnor <all...@tele.ntnu.no> wrote:
...
> While these are interesting facts, I wonder if they are relevant
> from a practical point of view?
>
> Wouldn't it make a lot more sense to go straight for the
> Parks-McClellan method if one has an application where
> textbook (simplified) formulas come up short?
>
> Rune

P-M makes sense if:
1) Time, code and calculation resources are available to run the
method after the window size is determined.
AND
2) Storage is available for the P-M derived coefficients
AND
3) Windowing can be performed (efficiently for the application) in the
time domain.
AND
4) There are no conflicting requirements (such as monotonicity)

Often, these constraints are met and P-M is useful. Often they aren't.
A good designer should always have P-M in the toolkit AND know when
not to use it.

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

On 6 Feb, 17:01, dbd <d...@ieee.org> wrote:
> On Feb 6, 1:50 am, Rune Allnor <all...@tele.ntnu.no> wrote:
> ...
>
> > While these are interesting facts, I wonder if they are relevant
> > from a practical point of view?
>
> > Wouldn't it make a lot more sense to go straight for the
> > Parks-McClellan method if one has an application where
> > textbook (simplified) formulas come up short?
>
> > Rune
>
> P-M makes sense if:
> 1) Time, code and calculation resources are available to run the
> method after the window size is determined.
> AND
> 2) Storage is available for the P-M derived coefficients
> AND
> 3) Windowing can be performed (efficiently for the application) in the
> time domain.
> AND
> 4) There are no conflicting requirements (such as monotonicity)
>
> Often, these constraints are met and P-M is useful. Often they aren't.
> A good designer should always have P-M in the toolkit AND know when
> not to use it.

Those are valid considerations regarding PM, but I still can't see
why or how a modified Hamming type window would be useful?
That goes for comparisions with either PM or the 'usual' window
functions.

Rune

On Feb 6, 8:13 am, Rune Allnor <all...@tele.ntnu.no> wrote:
> On 6 Feb, 17:01, dbd <d...@ieee.org> wrote:
>
>
>
> > On Feb 6, 1:50 am, Rune Allnor <all...@tele.ntnu.no> wrote:
> > ...
>
> > > While these are interesting facts, I wonder if they are relevant
> > > from a practical point of view?
>
> > > Wouldn't it make a lot more sense to go straight for the
> > > Parks-McClellan method if one has an application where
> > > textbook (simplified) formulas come up short?
>
> > > Rune
>
> > P-M makes sense if:
> > 1) Time, code and calculation resources are available to run the
> > method after the window size is determined.
> > AND
> > 2) Storage is available for the P-M derived coefficients
> > AND
> > 3) Windowing can be performed (efficiently for the application) in the
> > time domain.
> > AND
> > 4) There are no conflicting requirements (such as monotonicity)
>
> > Often, these constraints are met and P-M is useful. Often they aren't.
> > A good designer should always have P-M in the toolkit AND know when
> > not to use it.
>
> Those are valid considerations regarding PM, but I still can't see
> why or how a modified Hamming type window would be useful?
> That goes for comparisions with either PM or the 'usual' window
> functions.
>
> Rune

Rectangular, Von Hann and Hamming windows are all 'usual' windows that
fall in the modified Hamming family. Modified Hamming windows can be
appropriately applied to trade off between mainlobe width and sidelobe
region rejections. The generalized Hamming family allows a finer
grained tradeoff. It's easy to 'see' how they could be useful. I've
just never had a reason to use any but the 3. By the same token, I've
never seen an RFQ for a system including windowing in spectrum
analysis that allowed for PM. I think the 'usual' windows were
considered the only ones suitable to generate data for comparison to
data previously generated by 'accepted' methods. That's an historical
rather than a technological artifact, but a real one nonetheless.

Dale B. Dalrymple

Dale B. Dalrymple

On Feb 5, 1:38&#4294967295;pm, robert bristow-johnson <r...@audioimagination.com>
wrote:
>
> i'm curious what is meant by "family". &#4294967295;would that be different sizes
> of the "pedestal" that the Hann part of the Hamming sits upon?
>

If you hold your mouth just right, you can see that a Hamming window
is in the same "family" as the Blackman window and the Blackman-Harris
windows.  They are all derived from the same basic notion of
cancelling Gibbs by adding out-of-phase ripples.

On 6 Feb, 18:11, dbd <d...@ieee.org> wrote:
> On Feb 6, 8:13 am, Rune Allnor <all...@tele.ntnu.no> wrote:
>
>
>
>
>
> > On 6 Feb, 17:01, dbd <d...@ieee.org> wrote:
>
> > > On Feb 6, 1:50 am, Rune Allnor <all...@tele.ntnu.no> wrote:
> > > ...
>
> > > > While these are interesting facts, I wonder if they are relevant
> > > > from a practical point of view?
>
> > > > Wouldn't it make a lot more sense to go straight for the
> > > > Parks-McClellan method if one has an application where
> > > > textbook (simplified) formulas come up short?
>
> > > > Rune
>
> > > P-M makes sense if:
> > > 1) Time, code and calculation resources are available to run the
> > > method after the window size is determined.
> > > AND
> > > 2) Storage is available for the P-M derived coefficients
> > > AND
> > > 3) Windowing can be performed (efficiently for the application) in the
> > > time domain.
> > > AND
> > > 4) There are no conflicting requirements (such as monotonicity)
>
> > > Often, these constraints are met and P-M is useful. Often they aren't.
> > > A good designer should always have P-M in the toolkit AND know when
> > > not to use it.
>
> > Those are valid considerations regarding PM, but I still can't see
> > why or how a modified Hamming type window would be useful?
> > That goes for comparisions with either PM or the 'usual' window
> > functions.
>
> > Rune
>
> Rectangular, Von Hann and Hamming windows are all 'usual' windows that
> fall in the modified Hamming family. Modified Hamming windows can be
> appropriately applied to trade off between mainlobe width and sidelobe
> region rejections. The generalized Hamming family allows a finer
> grained tradeoff. It's easy to 'see' how they could be useful. I've
> just never had a reason to use any but the 3.

OK, let me rephrase: What would the justification be for
using an 'unusual' Hamming-type window and not one of
the 'usual' ones or a PM filter?

> By the same token, I've
> never seen an RFQ for a system including windowing in spectrum
> analysis that allowed for PM. I think the 'usual' windows were
> considered the only ones suitable to generate data for comparison to
> data previously generated by 'accepted' methods. That's an historical
> rather than a technological artifact, but a real one nonetheless.

Agreed.

Rune

M.Aramini@verizon.net wrote:
> 
> On second thought, there is supposed to be a null at alpha=5*pi
> (*not*, as I erroneously
> previously posted, at 5*pi/2) when a=25/46.  Substituting 5*pi for
> alpha and 25/46 for a
> in the formula for F(alpha) in my previous posting does in fact yield
> 0.
> 

If by Fourier transform you are asking about DFT of your Window function
where the DFT length and the Window length are the same and the window
function consists of the sum of a constant and a sinusoid (that sinusoid
has a period that is the same as the window length and DFT length) then
the result of your DFT better be all zeroes except the 3 bins centered at
f0. 

-jim

On Feb 6, 9:50 am, Rune Allnor <all...@tele.ntnu.no> wrote:
> On 6 Feb, 18:11, dbd <d...@ieee.org> wrote:

> ...
> OK, let me rephrase: What would the justification be for
> using an 'unusual' Hamming-type window and not one of
> the 'usual' ones or a PM filter?
> ...
> Rune

A system requirement for a mode with sidelobe rejection that didn't
match a 'usual' and a desire to minimize the mainlobe width subject to
that constraint in an implementation with identical structure to the
'usual' implementation structure. B&K did this at the 5 coefficient
point with some instruments by implementing all windows, including a
'Kaiser'  window by approximating with a 5 coefficient kernel,
including user specified windows.

Dale B. Dalrymple

On Feb 6, 9:26 am, Jubilation_T_Cornpone_...@hotmail.com wrote:
> On Feb 5, 1:38 pm, robert bristow-johnson <r...@audioimagination.com>
> wrote:

> ...

> If you hold your mouth just right, you can see that a Hamming window
> is in the same "family" as the Blackman window and the Blackman-Harris
> windows.  They are all derived from the same basic notion of
> cancelling Gibbs by adding out-of-phase ripples.

If I open my mouth far enough I would see these as examples of the odd
number of coefficient subset of the 'small frequency domain
kernel' (also called 'cosine-summed') windows. Von Hann set the
response at the adjacent bin center to 0.0 with two coefficients.
Blackman extended the bin center zeroing to 2 and more adjacent bins.
Hamming did an optimization to minimize the peak sidelobe level with 2
coefficients. Harris considered optimizations to minimize peak
sidelobe level and to maximize sidelobe rolloff rate. As Nuttal
pointed out, the max rolloff case has a closed form solution and
harris's 4 term sidelobe minimizing result was off by about 6dB. Other
authors have minimized sidelobes with more terms. Every Tom Dick and
Harry who comes along feels free to apply some mix of names to ever
more general families. I try to limit the blame to what the named
person actually did, and use functionally related terms if I have the
choice. While many people call 0.42, 0.5, 0.08 the Blackman
coefficients, I try to think of them as '2 digit rounded three term
Blackman', but try to sell that. Tukey and Blackman published these
coefficients in the Bell System Technical Journal in 1958 as 'R.
Blackman's not very serious proposal'. Harris misstated 3 term exact
Blackman sidelobe rejection as 17 dB worse than the real value and
people have taken Blackman's joke seriously ever since. Some authors
have noted that von Hann and Blackman's choice of bin center was not
the peak of the sinc response being canceled and improved on
Blackman's initial ad hoc choice by canceling at the peak. That's
still ad hoc, not optimal.  But sidelobe canceling is not the only
application of cosine-summed windows. There are also flattop window
designs and others in this form.

Dale B. Dalrymple

On Wed, 6 Feb 2008 09:26:06 -0800 (PST),
Jubilation_T_Cornpone_CSA@hotmail.com wrote:

>On Feb 5, 1:38&#4294967295;pm, robert bristow-johnson <r...@audioimagination.com>
>wrote:
>>
>> i'm curious what is meant by "family". &#4294967295;would that be different sizes
>> of the "pedestal" that the Hann part of the Hamming sits upon?
>>
>
>If you hold your mouth just right, you can see that a Hamming window
>is in the same "family" as the Blackman window and the Blackman-Harris
>windows.  They are all derived from the same basic notion of
>cancelling Gibbs by adding out-of-phase ripples.

Hello Jubilation,
   don't be too hard on Robert B-J.  He's 
a DSP expert, pure and simple, and he deserves some respect.

As for holding our mouths just right, we generally
hold our mouths in just the right position in which 
to pour beer.

The Hamming window is in the "class" of windows formally 
called "cos^a(x)" windows, which includes Hanning (von Hann),
Hamming, and Blackman.

Are you a fan of Al Capp?  I'll bet that %95 of the 
guys here do not know who Jubilation_T_Cornpone is.
(Of course, Jerry Avins will know.)

[-Rick-]

Rick Lyons wrote:

   ...

> Are you a fan of Al Capp?  I'll bet that %95 of the 
> guys here do not know who Jubilation_T_Cornpone is.
> (Of course, Jerry Avins will know.)
> 
> [-Rick-]

When we fought the Yankees and annihilation was near,
Who was there to lead the charge, that took us safe to the rear?
Why it was Jubilation T. Cornpone.
Old "Toot your own horn pone",
Jubilation T. Cornpone, a man who knew no fear.

When we almost had 'em but the issue still was in doubt,
Who suggested the retreat that turned it into a rout?
Why it was Jubilation T. Cornpone.
Old "Tattered and Torn Pone",
Jubilation T. Cornpone, he kept us hidin' out.

With our ammunition gone and faced with utter defeat,
Who was it that burned the crops and left us with nothin' to eat?
Why it wuz Jubilation T. Cornpone.
Old "September Morn-pone",
Jubilation T. Cornpone, the pants blown off his seat.

Thank you, Ray Charles.

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