> Fred Marshall wrote:
>
>>"Jerry Avins" <jya@ieee.org> wrote in message
>>news:MpqdnUOlX4SW5M_ZnZ2dnUVZ_tydnZ2d@rcn.net...
>>
>>
>>>Aren't you impressed by the elegant simplicity of R.B-J.'s observation
>>>that the highest order harmonic generated by a soft limiter is the same as
>>>the order of the polynomial that represents it?
>
> ...
>
>>Yes, very.
>
>
> oh c'mon, guys. it's just ...

Hey, man: I meant it. It's one of those things I could have figured out
but didn't. It's obvious .. once it's said out loud. I'm _always_
impressed by "Gee, I should have thought of that!" insights. Give
yourself two pats on the back.
Jerry
--
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Reply by Fred Marshall●April 29, 20062006-04-29

"robert bristow-johnson" <rbj@audioimagination.com> wrote in message
news:1146280966.313879.11310@j33g2000cwa.googlegroups.com...

> Fred Marshall wrote:
>> "Jerry Avins" <jya@ieee.org> wrote in message
>> news:MpqdnUOlX4SW5M_ZnZ2dnUVZ_tydnZ2d@rcn.net...
>>
>> > Aren't you impressed by the elegant simplicity of R.B-J.'s observation
>> > that the highest order harmonic generated by a soft limiter is the same
>> > as
>> > the order of the polynomial that represents it?
> ...
>>
>> Yes, very.
>
> oh c'mon, guys.

hey, r b-j, I was serious. And, now that you've expounded on it, I really
like the treatment you've given it. Ever' now and then I bump into
something new (for me) that should have been obvious, etc. etc.... Well,
it's obvious now. Thanks!
...
Yeah, it's the signs of the wiggles in the sincs. So B strictly <fs/2 would
seem to be the lesson one can teach from this simple example.
Fred

Reply by robert bristow-johnson●April 29, 20062006-04-29

Fred Marshall wrote:

> "Jerry Avins" <jya@ieee.org> wrote in message
> news:MpqdnUOlX4SW5M_ZnZ2dnUVZ_tydnZ2d@rcn.net...
>
> > Aren't you impressed by the elegant simplicity of R.B-J.'s observation
> > that the highest order harmonic generated by a soft limiter is the same as
> > the order of the polynomial that represents it?

...

>
> Yes, very.

oh c'mon, guys. it's just a degeneration of knowing when you multiply
signals, you convolve their spectrums. the general case doesn't even
have to be a sinusoid. bandlimited signals have spectrums of finite
nonzero reach. squaring a signal doubles that reach. cubing it is
squaring it and multiplying by another one more time so it triples the
reach. we've had this before.
another way to look at it is to consider Tchebyshev polynomials (and an
arbitrary polynomial can be expressed as a sum of Tchebyshevs up to
that order). so when a cosine wave guzzinta an Nth order Tchebshev:
T_N( cos(w*t) ) = cos(N*arccos( cos(w*t) )) = cos(N*w*t)
that's what i think is cool. a nice frequency multiplying
non-linearity.
the trick is to limit the nature of the non-linearity to these finite
polynomials so you know how much to oversample. if it's has to be a
high order, you're just screwed.
another little trick is that you need not care about aliasing that
folds down to the area that you'll LPF out. so a 5th order polynomial
needs only to have an oversampling ratio of 3. those top 2 harmonics
might alias, but won't get back into the baseband. when downsampling,
you filter those aliased harmonics out. so i think the hard and fast
rule is
oversampling ratio = (polynomial order + 1)/2
what impresses me (because i haven't really figgered it out) is that
... 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 ...
blows up, if that does, and if
... 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 ...
reconstructs to a nice cosine at Nyquist (it does, doesn't it?), then
that means
... 0 0 0 0 0 0 0 0 0 0 0 2 -2 2 -2 2 -2 2 -2 2 -2 2 ...
is the component that blows up. but i would think it would just settle
to a nice steady state 2*cosine. it must be the transient that is hell
with all of those 1/n terms adding up to infinity (since the
alternating signs of the data cancels the alternating signs of the
wiggles in the sinc).
r b-j

Reply by Fred Marshall●April 29, 20062006-04-29

"Jerry Avins" <jya@ieee.org> wrote in message
news:pPGdnWqXycnoTM_ZnZ2dneKdnZydnZ2d@rcn.net...

> Fred Marshall wrote:
>
> ...
>
>> Well, might it be that the mystery flaw is that lower frequency
>> fundamentals still have their harmonics in-band and will sound "not so
>> good"?
>
> Actually, being harmonic, they don't sound very bad. If the clipping level
> is set right at the overmodulation level. the low-pass filter output can
> exceed it. Just think of a square wave out of the clipper.

Oh, yeah. Gibbs and all .....

Reply by Jerry Avins●April 28, 20062006-04-28

Fred Marshall wrote:
...

> Well, might it be that the mystery flaw is that lower frequency fundamentals
> still have their harmonics in-band and will sound "not so good"?

Actually, being harmonic, they don't sound very bad. If the clipping
level is set right at the overmodulation level. the low-pass filter
output can exceed it. Just think of a square wave out of the clipper.
Jerry
--
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Reply by Fred Marshall●April 28, 20062006-04-28

"Jerry Avins" <jya@ieee.org> wrote in message
news:MpqdnUOlX4SW5M_ZnZ2dnUVZ_tydnZ2d@rcn.net...

>> In some respects the specification contradicts itself.
>> To limit means to add harmonics and, thus, aliases in the sampled signal.
>> To lowpass filter means to remove some of those added frequencies and,
>> thus, to *not* really limit.
>
> Things are muddled. Limiting produces harmonics of the limited signal, but
> those harmonics needn't be aliases if the sample rate is high enough.
> Consider what hams do (did?) to get more "punch" in their transmissions.
> Crank the audio gain up to where overmodulation would be frequent, then
> clip the audio to preclude overmodulation. The resulting waveform is rich
> in harmonics that cause out-of-band sidebands ("splatter"), which are
> removed by a low-pass filter. The final signal (which can still have a
> mystery flaw: do you see it?) is obviously clipped -- a scope shows
> that -- but bandlimited.

Well, might it be that the mystery flaw is that lower frequency fundamentals
still have their harmonics in-band and will sound "not so good"?
Fred

Reply by Fred Marshall●April 28, 20062006-04-28

"Jerry Avins" <jya@ieee.org> wrote in message
news:MpqdnUOlX4SW5M_ZnZ2dnUVZ_tydnZ2d@rcn.net...

>
> Aren't you impressed by the elegant simplicity of R.B-J.'s observation
> that the highest order harmonic generated by a soft limiter is the same as
> the order of the polynomial that represents it? Most practical limiter
> polynomials will be symmetric, hence consist of only odd powers.
>
> Jerry
> --

Yes, very.
Fred

Reply by Martin Eisenberg●April 28, 20062006-04-28

robert bristow-johnson wrote:

> e.g. let your nonlinearity be a 5th order polynomial that
> represents some approximation to some curve for "soft" clipping.
> pass a sinusoid of frequency below Fs/2 and there will be no
> resulting frequency components with frequency greater than
> 5*(Fs/2). so if your new upsampled sampling frequency is 5*Fs
> before applying the upsampled input to the 5th order polynomial,
> there will be no aliasing.

You can get by with less. As long as the highest-frequency non-
negligible aliasing component stays in the decimation filter's
stopband, it does no harm. So, with a decimation filter whose
transition band is fully below Fs/2 (or assuming a brickwall filter
if you're fudging it), n-fold oversampling is good for (2n-1)-fold
expected spectrum expansion.
Martin
--
Sphinx of black quartz, judge my vow!
--David Lemon

Reply by Jerry Avins●April 28, 20062006-04-28

Fred Marshall wrote:

> "Jerry Avins" <jya@ieee.org> wrote in message
> news:84-dnQsCK-uTGszZnZ2dnUVZ_uWdnZ2d@rcn.net...
>
>>Fred Marshall wrote:
>>
>> ...
>>
>>
>>>I think this is an interesting question and I don't have an answer.
>>
>>I think I do, at least if I understand what you're driving at. Every
>>possible set of samples represents some bandlimited signal. To determine
>>what that is, just feed them to the reconstruction process and examine the
>>result. So a set of samples that originally represented a sinusoid but is
>>modified by zeroing out all negative samples represents *some* bandlimited
>>signal, but probably not the same signal that results from half-wave
>>rectifying followed by band limiting.
>>
>>Jerry
>
>
> Jerry and Mark,
>
> OK - I think we're (or I'm) getting somewhere.
>
> Jerry. We had this discussion some time back. Not every possible set of
> samples represents some bandlimited signal. I had thought so too but was
> stopped cold with the sequence:
> ......1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1
> 1 -1.......
> ****
> Note the flip in the sign sequence in the middle at ****.
> When this is reconstructed with sincs, it blows up. It demonstrates that
> not every sequence represents a bandlimited signal. It's the
> counter-example pathological case.

I do remember the discussion; it embarrasses me to have forgotten. I
hereby make another unfounded claim which has a better chance to be
true. Any set of actual samples, even if corrupted by aliasing,
represents a bandlimited signal (that may not closely resemble the
sampled signal).

> But, we are practical guys and I'm going to "remember it while forgetting
> about it" .. that is, "remember it while ignoring it - for practical
> situations".
>
> ----end of digression-----
>
> I emphasized and Jerry responded to the bandlimitedness question. But it
> seems I had missed the point because Mark is focused on aliasing and wants
> to reduce the resulting aliasing in selecting a limiting process. Gee, I
> didn't know that hard limiting *samples* would necessarily cause aliasing!
> So, I'm starting from scratch.
> Clearly if one takes a sinusoid and samples it at a very high rate then
> there will be harmonics.
> The question I had was "will be aliases from limiting as well"?
> A similar question would be "can one define a spectral character at all in
> this situation"?
>
> OK - so here's a way to look at it:
>
> Hard limiting of samples cause aliasing because the negative frequency
> components contribute harmonics in the negative direction (moving to lower
> frequencies from fs). This puts aliased components below fs/2.
> Simple but clear I hope. At least I learned something....
>
> So, how to mitigate this without lowpass filtering the results of limiting?
>
> Here's a flip answer:
> The "nonlinearity" can include linear components, thus a lowpass filter....
> But that isn't what Mark asks for.
>
> In some respects the specification contradicts itself.
> To limit means to add harmonics and, thus, aliases in the sampled signal.
> To lowpass filter means to remove some of those added frequencies and, thus,
> to *not* really limit.

Things are muddled. Limiting produces harmonics of the limited signal,
but those harmonics needn't be aliases if the sample rate is high
enough. Consider what hams do (did?) to get more "punch" in their
transmissions. Crank the audio gain up to where overmodulation would be
frequent, then clip the audio to preclude overmodulation. The resulting
waveform is rich in harmonics that cause out-of-band sidebands
("splatter"), which are removed by a low-pass filter. The final signal
(which can still have a mystery flaw: do you see it?) is obviously
clipped -- a scope shows that -- but bandlimited.

> Try this:
> Apply a sinusoid to a hard limiter such as a half-wave rectifier.
> Compute the Fourier Series of the resulting waveform.
> Eliminate all the higher order terms of the series.
> Construct the resulting waveform.
> [In general, there will be the Gibbs phenomenon due to truncation of the
> series.]

I don't like the stark example because most limiters for practical
purposes are symmetrical.

> This observation may lend some insight because you want to have the Gibbs
> phenomenon in order to meet your specification.
>
> Now, ask the question: what process will cause this new waveform to be
> generated?
> We know that a half-wave rectifier followed by a lowpass filter is one
> answer.
> Are there others?
> I think probably not because it would imply there is a multiplicity of
> processes that will result in the same waveform. But there aren't a
> multiplicity of lowpass filters of the exact same response - unless you get
> into implementation details and they aren't the point here.
> I suppose it might be argued that there may be a multiplicity of
> nonlinearities combined with filters that might have identical outputs ...
> and that leads to making a soft limiter that requires less filtering I
> suppose.
>
> Then there is the complication of input waveforms that aren't
> sinusoids...... I think this is the killer if the waveforms are complicated
> composites. Too much amplitude dependence, etc.
>
> I would imagine that power supply designers have pondered this kind of
> question in an attempt to make things less expensive.

Aren't you impressed by the elegant simplicity of R.B-J.'s observation
that the highest order harmonic generated by a soft limiter is the same
as the order of the polynomial that represents it? Most practical
limiter polynomials will be symmetric, hence consist of only odd powers.
Jerry
--
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Reply by Fred Marshall●April 28, 20062006-04-28

"Jerry Avins" <jya@ieee.org> wrote in message
news:84-dnQsCK-uTGszZnZ2dnUVZ_uWdnZ2d@rcn.net...

> Fred Marshall wrote:
>
> ...
>
>> I think this is an interesting question and I don't have an answer.
>
> I think I do, at least if I understand what you're driving at. Every
> possible set of samples represents some bandlimited signal. To determine
> what that is, just feed them to the reconstruction process and examine the
> result. So a set of samples that originally represented a sinusoid but is
> modified by zeroing out all negative samples represents *some* bandlimited
> signal, but probably not the same signal that results from half-wave
> rectifying followed by band limiting.
>
> Jerry

Jerry and Mark,
OK - I think we're (or I'm) getting somewhere.
Jerry. We had this discussion some time back. Not every possible set of
samples represents some bandlimited signal. I had thought so too but was
stopped cold with the sequence:
......1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1
1 -1.......
****
Note the flip in the sign sequence in the middle at ****.
When this is reconstructed with sincs, it blows up. It demonstrates that
not every sequence represents a bandlimited signal. It's the
counter-example pathological case.
But, we are practical guys and I'm going to "remember it while forgetting
about it" .. that is, "remember it while ignoring it - for practical
situations".
----end of digression-----
I emphasized and Jerry responded to the bandlimitedness question. But it
seems I had missed the point because Mark is focused on aliasing and wants
to reduce the resulting aliasing in selecting a limiting process. Gee, I
didn't know that hard limiting *samples* would necessarily cause aliasing!
So, I'm starting from scratch.
Clearly if one takes a sinusoid and samples it at a very high rate then
there will be harmonics.
The question I had was "will be aliases from limiting as well"?
A similar question would be "can one define a spectral character at all in
this situation"?
OK - so here's a way to look at it:
Hard limiting of samples cause aliasing because the negative frequency
components contribute harmonics in the negative direction (moving to lower
frequencies from fs). This puts aliased components below fs/2.
Simple but clear I hope. At least I learned something....
So, how to mitigate this without lowpass filtering the results of limiting?
Here's a flip answer:
The "nonlinearity" can include linear components, thus a lowpass filter....
But that isn't what Mark asks for.
In some respects the specification contradicts itself.
To limit means to add harmonics and, thus, aliases in the sampled signal.
To lowpass filter means to remove some of those added frequencies and, thus,
to *not* really limit.
Try this:
Apply a sinusoid to a hard limiter such as a half-wave rectifier.
Compute the Fourier Series of the resulting waveform.
Eliminate all the higher order terms of the series.
Construct the resulting waveform.
[In general, there will be the Gibbs phenomenon due to truncation of the
series.]
This observation may lend some insight because you want to have the Gibbs
phenomenon in order to meet your specification.
Now, ask the question: what process will cause this new waveform to be
generated?
We know that a half-wave rectifier followed by a lowpass filter is one
answer.
Are there others?
I think probably not because it would imply there is a multiplicity of
processes that will result in the same waveform. But there aren't a
multiplicity of lowpass filters of the exact same response - unless you get
into implementation details and they aren't the point here.
I suppose it might be argued that there may be a multiplicity of
nonlinearities combined with filters that might have identical outputs ...
and that leads to making a soft limiter that requires less filtering I
suppose.
Then there is the complication of input waveforms that aren't
sinusoids...... I think this is the killer if the waveforms are complicated
composites. Too much amplitude dependence, etc.
I would imagine that power supply designers have pondered this kind of
question in an attempt to make things less expensive.
Fred