Nyquist Didn't Say That

Jani Huhtanen wrote:

> robert bristow-johnson wrote:
> 
-- comments about bandpass sampling snipped --
> 
> 
> I don't know if he is talking about bandpass sampling. In fact, I have to
> admit that I'm not even sure of the exact definition of bandpass sampling. 
> 
> However, consider wavelet transform and particularly signals produced by the
> wavelet synthesis. Such signals have theoretically infinite bandwidth
> assuming that the scaling function has finite support. This is true also
> when signals are synthesized only in truncated resolution (i.e. from scale
> u to v where u and v are finite). It's true even when synthesized in single
> resolution. Here synthesis means:
> 
>         f(t) = sum_s sum_n x_s[n]*phi_n_s(t), where
> 
> phi_n_s(t) is the scaling function with translation n and scale s and x_s[n]
> are the samples from scale (or resolution) s.
> 
> Even though these signals have infinite bandwidth, they can be sampled and
> when given the correct scaling function these samples correspond to the
> original samples used in wavelet synthesis. Here sampling means 
> 
>         x_s[n] = <f,phi_n_s>, 
> 
> where f is the analyzed (sampled) function, phi is the scaling function with
> translation n and scale s. It is obvious that the signal can be later
> perfectly reconstructed from the samples by wavelet synthesis (assuming the
> scaling function matched the scaling function used in the original
> synthesis). 
> 
> In fact, sinc function is just one possible scaling function (in which case
> one talks about shannon wavelets). This makes traditional sampling just a
> special case of wavelet transform (in single resolution). Note that the
> previous comment about infinite bandwidth does not obviously apply to
> shannon wavelets.
> 
> Any comments, or corrections?
> 
I'm not a wavelet guru, but if I'm reading your math right you are 
constraining yourself to sampling a finite number of wavelets, each of 
which may have frequency content going out to infinity, and being able 
to perfectly reconstruct the signal.

I'll believe you that this is true*.

Where your observation falls down, however, is that there are no 
perfect, physically realizable 'wavelet filters' to use -- you can 
_synthesize_ a function in some mathematical domain, but you can't go to 
and from physical reality with arbitrary wavelets.

I suspect that even if you could make some magic wavelet filter bank 
with a finite number of wavelets that you would have to constrain your 
signal to be composed only of the wavelets in your set, and any 
additional wavelets in the actual signal would 'alias' into your set of 
realized wavelets**.

* Gee, my syntax is really fractured this evening.

** Really, really fractured.  I think it's the subject matter, but maybe 
it's just been a long day.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

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Andy Peters wrote:
> Jerry Avins wrote:
> 
>> The Nyquist criterion does indeed assume ideal conditions.
> 
> The Nyquist criterion tells you if your closed-loop feedback system
> will be stable or not.  It has nothing to do with sampling.

The Nyquist sampling theorem. We weren't carrying on about enclosing the 
point -1, 0 on a Nyquist plot in the s plane. That guy Nyquist had more 
than one feather in his cap. Spirule, anyone?

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

Tim Wescott wrote:
> Andy Peters wrote:
>> Jerry Avins wrote:
>>
>>
>>> The Nyquist criterion does indeed assume ideal conditions.
>>
>>
>> The Nyquist criterion tells you if your closed-loop feedback system
>> will be stable or not.  It has nothing to do with sampling.
>>
>> -a
>>
> You're thinking of the Barkhausen criterion, which gives a necessary, 
> but not sufficient, condition for oscillation.  While it's useful for 
> building oscillators, it doesn't help you tell if your control system is 
> stable or not -- and having built plenty of type III control systems I 
> can assure you that 180 degrees of phase shift and gain >> 1 doesn't 
> mean you're oscillating.
> 
> The Nyquist rate is about sampling, and while I haven't heard it called 
> a "criterion", it's still about sampling.

But see http://www.atp.ruhr-uni-bochum.de/rt1/syscontrol/node43.html
http://www.atp.ruhr-uni-bochum.de/rt1/syscontrol/node45.html and 
http://www.facstaff.bucknell.edu/mastascu/eControlHTML/Freq/Nyquist2.html

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

CBFalconer wrote:
> vasile wrote:
>> Tim Wescott wrote:
>>
> ... snip ...
>>> But you can't avoid the issue of providing sufficiently steep
>>> skirts on your filters, both in and out.  As you get closer and
>>> closer to Nyquist in a 'simple' system your filter complexity
>>> goes through the roof, as does the difficulty of actually
>>> realizing the filters in analog hardware.
>>   I'm not entirely agree with that. There are a lot of analog
>> antialising filters, which are quite good near the Nyquist. One
>> of them frequently used is the Cebashev filter (eliptical
>> filter) which design and implementation is easy up to quite
>> high frequencies (say 100-200Mhz, at least tested by myself).
> 
> That's fine if you don't care about phase linearity (time delay). 
> Chebychev filters are notoriously poor at preserving phase, or
> having constant delay characteristics.  This results in heavy
> distortion of analog waveforms, and will manifest itself as such
> effects as overshoot and ringing.  A Bessel filter is designed to
> minimize this effect, but has much more gentle rejection slopes.
> 
I wonder if he really means Chebyshev or elliptic (Cauer)? They both 
ring badly, but the Cauer rings like a bell if you design one for any 
reasonably steep cutoff slope. In a particular application you might not 
care directly about the ringing. However it badly compromises the 
anti-aliasing qualities of the filter, since it can allow bursts of out 
of band energy through.

Steve

Oli Filth wrote:
> David Ashley said the following on 23/08/2006 19:28:
>> I don't think Wikipedia's going away. I find myself using it more and
>> more as a first step to getting any info on some new subject -- even
>> before google actually.
> 
> In some areas, Wikipedia is great, in others it's dire (no disrepect 
> intended to anyone that contributes, myself included).  Articles about 
> comms and signal processing (as relevant examples) are on the whole 
> scant, badly written and error-prone.  However, I'm sure this will 
> change over time.

Do you think it will change for the better or the worse? It seems 
numerous articles on Wikipedia start out pretty good, but editors much 
less knowledgeable than the original author gradually scramble them.

>> Now here's a thought -- if Wikipedia can become financially
>> viable in its own right (currently it depends on donations) maybe a
>> business model can appear where based on number of "views" of
>> pages, the contributing authors can get some $$$ sent their
>> way.
>>
>> Yes -- it's viable!
>>
>> #1) Suggest the possibility
>> #2) ???
>> #3) Profit!
> 
> Nice idea, but I don't think it's ever going to happen.  For one, the 
> Wikipedia administrators are already working hard to reduce the 
> systematic bias that exists in Wikipedia (see 
> http://en.wikipedia.org/wiki/Wikipedia:WikiProject_Countering_systemic_bias), 
> introducing a financial incentive to writing good articles could only 
> make this worse.

Why would that be? If people are paid to do a good job writing articles, 
they might possibly do so. Right now, any financial or other personal 
gain from contributing lies outside Wikipedia, leading to blatant 
agendas in the writing. I think giving prominence to the names of valued 
authors might be a solid incentive to good work.

Someone pointed me to the Wikipedia articles on a couple of porn stars. 
 From there you can link to many others. They seem to have been written 
with a genuine affection and interest for the subject matter. I was 
amazed to see how much effort people will put into that. If they could 
only encourage a similar level of dedication in the technical articles 
Wikipedia could become outstanding. :-)

Steve

Tim Wescott wrote:

> >
> What really happens is that you get a signal at Fs/2 that is sometimes
> big and sometimes small, and you have no clue if it's _actually_ a
> signal at Fs/2 that's sometimes big and sometimes small, or a signal
> that's big and sometimes at Fs/2 and sometimes slightly off.

OK.

> 
> Which is a bad thing.

Maybe. There are situations where it doesn't matter if its a fluctuation
in the amplitude or frequency. Like a 44khz sound recording where your
ears can't discern the difference anyway.
	Bad or good always depends on what you are attempting to accomplish. I
never said it was a good thing or a bad thing. What I did say was
sampling a sine wave at Fs/2 mo matter what you think the phase is
produces a good indication of how non-linear and inaccurate your signal
and sampling system really are. 


-jim



> 
> Which is why you don't want to do it.
> 
> --
> 
> Tim Wescott
> Wescott Design Services
> http://www.wescottdesign.com
> 
> Posting from Google?  See http://cfaj.freeshell.org/google/
> 
> "Applied Control Theory for Embedded Systems" came out in April.
> See details at http://www.wescottdesign.com/actfes/actfes.html

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>> You should discuss what happens to a signal that is filtered and 
>> sampled in one system at rate X, but is transmitted to a receiving 
>> system at update rate Y, then used by that receiving system at rate 
>> Z.  How should one select the analog anti-aliasing filter in this 
>> situation?
>>
>> mw
> 
> Do you mean where the signal has been resampled at each step?
> 
No, not re-sampled (from analog to digital) by each system, but instead 
sent digitally to the next system such that the receiving system only 
uses some of the data, not every sample.  Let's say the initial ADC step 
has a 1000 samples/sec conversion rate, then the signal is broadcast 
out, and a receiver system receives at a rate of 200 samples/sec.  Then 
the processing inside that system only has time to perform 50 
calculations/sec.

Would the analog anti-aliasing filter selection be dependent on the 50 
calc/sec?  If that's true you'd have to select the aliasing filter based 
on the slowest end user of the data.  It seems odd that if you design an 
ADC stage, you'd have to choose analog filtering based on the slowest 
performing "weakest link" in the eventual design. Opinions?

mw wrote:
>>> You should discuss what happens to a signal that is filtered and 
>>> sampled in one system at rate X, but is transmitted to a receiving 
>>> system at update rate Y, then used by that receiving system at rate 
>>> Z.  How should one select the analog anti-aliasing filter in this 
>>> situation?
>>>
>>> mw
>>
>> Do you mean where the signal has been resampled at each step?
>>
> No, not re-sampled (from analog to digital) by each system, but instead 
> sent digitally to the next system such that the receiving system only 
> uses some of the data, not every sample.  Let's say the initial ADC step 
> has a 1000 samples/sec conversion rate, then the signal is broadcast 
> out, and a receiver system receives at a rate of 200 samples/sec.  Then 
> the processing inside that system only has time to perform 50 
> calculations/sec.
> 
> Would the analog anti-aliasing filter selection be dependent on the 50 
> calc/sec?  If that's true you'd have to select the aliasing filter based 
> on the slowest end user of the data.  It seems odd that if you design an 
> ADC stage, you'd have to choose analog filtering based on the slowest 
> performing "weakest link" in the eventual design. Opinions?

It's not a matter of opinion; this is well charted territory. Look up 
interpolation and decimation, or up- and down converting. Digital 
filtering is usually required at each stage.

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

jim wrote:
> robert bristow-johnson wrote:
>
> > sounds to me that expecting a sampler to be phase locked to what we
> > would normally think is an unknown signal (if it were known, why bother
> > to sample it to determine its amplitude?) is having one's cake and
> > eating it too.
>
> Is there an echo in here. The above is exactly what I just said.

well, you said this:
>
> What is what happens? Do you actually know what happens if you actually
> try this in a real world context? Set up a speaker generating the Fs/2
> signal. Set up a microphone and and ADC to record the sound at Fs. Are
> you claiming that you can adjust the sampling phase to produce a digital
> recording of either full scale or zero? That's what in theory should
> happen - right? But can you do that in real life?
>

and you said this:
>
> 	If you are going to trigger the sample timing to twice the highest
> frequency component, then you should have no trouble measuring the
> amplitude of that frequency component.

i didn't realize you were being facetious here.

> So apparently you are now saying
> that those who say you need to sample at more than twice the rate are
> completely wrong, since there is a practical way to overcome the
> perceived difficulty.

...

> The question was asked - What really happens when you sample a frequency
> at Fs/2.

we know what happens when you sample something at precisely Nyquist.
it only matters what relative phase the sampling is done on and the
rest is unremarkable.  there is nothing else that happens.

r b-j

Jerry Avins wrote:

> Tim Wescott wrote:
> 
>> Andy Peters wrote:
>>
>>> Jerry Avins wrote:
>>>
>>>
>>>> The Nyquist criterion does indeed assume ideal conditions.
>>>
>>>
>>>
>>> The Nyquist criterion tells you if your closed-loop feedback system
>>> will be stable or not.  It has nothing to do with sampling.
>>>
>>> -a
>>>
>> You're thinking of the Barkhausen criterion, which gives a necessary, 
>> but not sufficient, condition for oscillation.  While it's useful for 
>> building oscillators, it doesn't help you tell if your control system 
>> is stable or not -- and having built plenty of type III control 
>> systems I can assure you that 180 degrees of phase shift and gain >> 1 
>> doesn't mean you're oscillating.
>>
>> The Nyquist rate is about sampling, and while I haven't heard it 
>> called a "criterion", it's still about sampling.
> 
> 
> But see http://www.atp.ruhr-uni-bochum.de/rt1/syscontrol/node43.html
> http://www.atp.ruhr-uni-bochum.de/rt1/syscontrol/node45.html and 
> http://www.facstaff.bucknell.edu/mastascu/eControlHTML/Freq/Nyquist2.html
> 
> Jerry

Y'know, I use that all the time, but I totally forgot it's name.

Whadda ya know.

AFAIK Nyquist got his first fame with the analysis of negative feedback 
in vacuum tube amplifiers back in the '20s when it was all magic. 
_Then_ he got into cahoots with Shannon to make his rate.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Posting from Google?  See http://cfaj.freeshell.org/google/

"Applied Control Theory for Embedded Systems" came out in April.
See details at http://www.wescottdesign.com/actfes/actfes.html