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From: Fred Marshall on 3 Jul 2010 18:21
robert bristow-johnson wrote:
> On Jul 1, 10:59 am, Fred Marshall <fmarshallx(a)remove_the_xacm.org>
> wrote:
>> r b-j,
>>
>> Well, maybe I've had it wrong all these years but I'd say that the
>> windowing method starts with N frequency samples where N is the length
>> of the filter you want.
>
> so then, since the DFT and iDFT are bijective (i love using fancy-
> pants words), why window? if h[n] has N samples and N degrees of
> freedom, so does H[k]. you specify your N frequency samples and you
> can hit it perfectly with no windowing.
>
> so, that seems curious to me.
>
> r b-j

You window because the frequency response between those points can be
nasty. Either you don't window and convolve those samples with a
matching sinc er.. Dirichlet or you window and convolve those samples
with something else.

The trade is that the Dirichlet matches all the samples exactly. Other
windows don't. Some match them all except the adjacent two as in the
1/2 1 1/2 sum of adjacent Dirichlets. It still has zeros in the sum
at all the other sample points - so the convolution doesn't perturb
their values. And, the values between points are better behaved.

And making N larger to begin with doesn't affect any of this in my way
of looking at it.

Fred
From: Fred Marshall on 5 Jul 2010 12:54
Fred Marshall wrote:
> robert bristow-johnson wrote:
>> On Jul 1, 10:59 am, Fred Marshall <fmarshallx(a)remove_the_xacm.org>
>> wrote:
>>> r b-j,
>>>
>>> Well, maybe I've had it wrong all these years but I'd say that the
>>> windowing method starts with N frequency samples where N is the length
>>> of the filter you want.
>>
>> so then, since the DFT and iDFT are bijective (i love using fancy-
>> pants words), why window? if h[n] has N samples and N degrees of
>> freedom, so does H[k]. you specify your N frequency samples and you
>> can hit it perfectly with no windowing.
>>
>> so, that seems curious to me.
>>
>> r b-j
>
> You window because the frequency response between those points can be
> nasty. Either you don't window and convolve those samples with a
> matching sinc er.. Dirichlet or you window and convolve those samples
> with something else.
>
> The trade is that the Dirichlet matches all the samples exactly. Other
> windows don't. Some match them all except the adjacent two as in the
> 1/2 1 1/2 sum of adjacent Dirichlets. It still has zeros in the sum
> at all the other sample points - so the convolution doesn't perturb
> their values. And, the values between points are better behaved.
>
> And making N larger to begin with doesn't affect any of this in my way
> of looking at it.
>
> Fred

And, I'm sure you know ..

If you use the basis 1/2 1 1/2 sum of Dirichlets (raised cosine in time)
which I will refer to as "RC" then you do have to convolve and can't
solve a system of equations that matches the samples exactly at each
point. If you did that, then it would revert to using a single
Dirichlet I do believe.

Since the stopbands are zero then all you get in them are the
fast-decaying tails of RC components from the passbands by 1/f^3 [rather
than 1/f for a single Dirichlet].

Fred
From: robert bristow-johnson on 5 Jul 2010 13:18
On Jul 3, 6:21 pm, Fred Marshall <fmarshallx(a)remove_the_xacm.org>
wrote:
> robert bristow-johnson wrote:
> > On Jul 1, 10:59 am, Fred Marshall <fmarshallx(a)remove_the_xacm.org>
> > wrote:
> >> r b-j,
>
> >> Well, maybe I've had it wrong all these years but I'd say that the
> >> windowing method starts with N frequency samples where N is the length
> >> of the filter you want.
>
> > so then, since the DFT and iDFT are bijective (i love using fancy-
> > pants words), why window?  if h[n] has N samples and N degrees of
> > freedom, so does H[k].  you specify your N frequency samples and you
> > can hit it perfectly with no windowing.
>
> > so, that seems curious to me.
>
> > r b-j
>
> You window because the frequency response between those points can be
> nasty.  Either you don't window and convolve those samples with a
> matching sinc er.. Dirichlet or you window and convolve those samples
> with something else.

you're always windowing it (unless your impulse response is an
infinitely-repeating periodic function. you draw your N (or more)
discrete points in the frequency domain, you iDFT it and what you have
there is still a periodic sequence. if you use just one cycle of that
(and zero the rest, as you would for an FIR), then you've applied the
rectangular window.

> The trade is that the Dirichlet matches all the samples exactly.  Other
> windows don't.  Some match them all except the adjacent two as in the
> 1/2  1  1/2 sum of adjacent Dirichlets.  It still has zeros in the sum
> at all the other sample points - so the convolution doesn't perturb
> their values.  And, the values between points are better behaved.

the values between points exhibit less "high frequency" behavior or
are less smooth. that's normally thought of as "better". and the
reason is that in the other domain (where the window is applied),
you've reduced the amplitude of the points further from h[0] (the
"high frequency" points, but they're really the high time-displacement
points that fuel the "high frequency" wiggling in the frequency
domain.

> And making N larger to begin with doesn't affect any of this in my way
> of looking at it.

it allows you to draw the frequency response more densely. so you can
explicitly specify how "wiggly" you want it between the sparser points
you have with your smaller "N". but to do so, the impulse response is
very large, too large. so you have to shorten it and that is
windowing. then, starting with the ideal (but too long) impulse
response, you can try shortening it in a variety of different ways
(using different windows) and see how well you do. you can also try
shortening it to different lengths (using whatever window that looks
best) to make a tradeoff decision on FIR length and its performance.

it's very similar in philosophy to what we do with the windowed-sinc
LPF design. start with the ideal (which is too long) and see what
happens when you settle for something shorter.

r b-j


From: Fred Marshall on 5 Jul 2010 16:30
robert bristow-johnson wrote:
> On Jul 3, 6:21 pm, Fred Marshall <fmarshallx(a)remove_the_xacm.org>
> wrote:
>> robert bristow-johnson wrote:
>>> On Jul 1, 10:59 am, Fred Marshall <fmarshallx(a)remove_the_xacm.org>
>>> wrote:
>>>> r b-j,
>>>> Well, maybe I've had it wrong all these years but I'd say that the
>>>> windowing method starts with N frequency samples where N is the length
>>>> of the filter you want.
>>> so then, since the DFT and iDFT are bijective (i love using fancy-
>>> pants words), why window? if h[n] has N samples and N degrees of
>>> freedom, so does H[k]. you specify your N frequency samples and you
>>> can hit it perfectly with no windowing.
>>> so, that seems curious to me.
>>> r b-j
>> You window because the frequency response between those points can be
>> nasty. Either you don't window and convolve those samples with a
>> matching sinc er.. Dirichlet or you window and convolve those samples
>> with something else.
>
> you're always windowing it (unless your impulse response is an
> infinitely-repeating periodic function. you draw your N (or more)
> discrete points in the frequency domain, you iDFT it and what you have
> there is still a periodic sequence. if you use just one cycle of that
> (and zero the rest, as you would for an FIR), then you've applied the
> rectangular window.
>
>> The trade is that the Dirichlet matches all the samples exactly. Other
>> windows don't. Some match them all except the adjacent two as in the
>> 1/2 1 1/2 sum of adjacent Dirichlets. It still has zeros in the sum
>> at all the other sample points - so the convolution doesn't perturb
>> their values. And, the values between points are better behaved.
>
> the values between points exhibit less "high frequency" behavior or
> are less smooth. that's normally thought of as "better". and the
> reason is that in the other domain (where the window is applied),
> you've reduced the amplitude of the points further from h[0] (the
> "high frequency" points, but they're really the high time-displacement
> points that fuel the "high frequency" wiggling in the frequency
> domain.
>
>> And making N larger to begin with doesn't affect any of this in my way
>> of looking at it.
>
> it allows you to draw the frequency response more densely. so you can
> explicitly specify how "wiggly" you want it between the sparser points
> you have with your smaller "N". but to do so, the impulse response is
> very large, too large. so you have to shorten it and that is
> windowing. then, starting with the ideal (but too long) impulse
> response, you can try shortening it in a variety of different ways
> (using different windows) and see how well you do. you can also try
> shortening it to different lengths (using whatever window that looks
> best) to make a tradeoff decision on FIR length and its performance.
>
> it's very similar in philosophy to what we do with the windowed-sinc
> LPF design. start with the ideal (which is too long) and see what
> happens when you settle for something shorter.
>
> r b-j
>
>

r b-j,

OK. That interesting - the approaches have the same underlying
ingredients and are only different by virtue of how the starting points
are viewed.

In the "infinite-N" case you would be seeking N based on responses and,
I guess would decide on a non-rectangular window (or windows) to use in
the process. As soon as you do the first test case it's the same as the
"fixed-N" approach. It rather assumes that you don't know N very well,
right?

In the "fixed-N" case you would be seeking acceptable response by
choosing non-rectangular window functions of length N. And then, if you
aren't satisfied, iterate N. Just as the "infinite-N" case does in the end.

I think of both as the "windowing method". How you approach it depends
on how closely you figure you know N and your own preference. Either
way, just like using P-M, you have to choose an N at some point.

My way of roughly choosing N is to look at the minimum transition band
width W and spec N so that the temporal length of the filter is approx
1.5/W. And, for even rougher estimates just 1/W which is what I usually
use when discussing the "physics". You get to pick based on where you
put the break points in the transition - so that choice usually gets
mixed up with band ripple - particularly with minimax criterion. The
lower the ripple, the wider the transition and the longer the filter.
But, it's sure that the filter can't be any *shorter*!! :-)

And, then, if you like you can use one of those length-estimating formulas.

Fred
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