Nyquist Didn't Say That

Richard Henry wrote:
> Jerry Avins wrote:
>> Of course you can lock the sampler to the sampled waveform or one of its
>> harmonics. Google for "PLL".
> 
> Not all signals are periodic waveforms.

We are evidently dealing with a periodic waveform in a discussion of 
sampling at constant phase. Why drag in a non sequitur?

Jerry
-- 
Engineering is the art of making what you want from things you can get.
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Rick Lyons wrote:
...
> I thought after all the criticism that Fonte
> received regarding his 1993 EDN that we'd
> heard the last from Mr. Fonte.   Not so.
> He wrote another titled "Breaking Nyquist"
> in the October 1998 issue of the Circuit Cellar
> magazine.  Again he claimed that the
> Nyquist sampling theorem is not valid
> and that it can be "broken" without
> causing "problems".   Using vague, ambiguous,
> undefined terminology, Fonte again claimed that
> he can tell the difference between an Fo
> (F sub zero) discrete spectral component of
> an analog sinewave whose Fo frequency was less
> than Fs/2 and an Fo discrete spectral component
> of an aliased analog sinewave whose frequency was
> greater than Fs/2.  In other words, he claims that
> "aliasing" (violating Nyquist) does NOT cause
> spectral ambiguity in the frequency domain.
> I can hardly wait for Fonte's next article.
>
> (I'm not being hateful here...Fonte's probably
> a nice guy whom his family loves.)
>
> My guess is, again, Fonte is using software to
> model the process of periodic sampling, and the
> signal he is "sampling" is a pure sinewave.
> Such modeling is very risky in my opinion because
> it's easy for a beginner in the field of
> DSP to misinterpret/misunderstand the results
> of such modeling.
>
> Concerning sampling, Bonnie Baker wrote an
> article titled "Turning Nyquist Upside Down
> by Undersampling" in the May 12th 2005 issue
> of EDN magazine.  The article discusses bandpass
> sampling.  However, I think the article's
> title is unfortunate because bandpass
> sampling does NOT "turn Nyquist upside down"
> ---bandpass sampling is included in the Nyquist
> Sampling Theorem.

are any of these online?

Rick, i fear that similar stuff is being done in the Wikipedia article.
 this guy (whose name is very similar to yours and claims to have Alan
Oppenheim and Ron Schafer as friends) would say to you that "the
converse of the Nyquist-Shannon sampling theorem is not true", meaning
that there are cases where frequency components at or above the Nyquist
frequence can be validly reconstructed under some circumstances.   i
think he's talking about bandpass sampling.

Anyway, Wikipedia is so selective in the qualifications of editors, i
am not sure how this experiment will turn out.  sometimes very well
written articles get "improved" by some editor that comes in who knows
something about the topic (a little bit of knowledge is a dangerous
thing) but makes edits that interrupt the flow of concept of the older
version.  there is no guarantee that the articles will improve in time.
 sometimes they get worse.

r b-j

Jerry Avins wrote:

> I don't get it. What have I written that makes it seem that I believe
> the amplitude of a component f can be determined by sampling at 2f? We
> both know it can't be done, and why.

It certainly can be done if your sampling points are locked to the
signal. As usual your arguments consist of having you cake and eating it
too.

-jim

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jim wrote:
> 
> Jerry Avins wrote:
> 
>> I don't get it. What have I written that makes it seem that I believe
>> the amplitude of a component f can be determined by sampling at 2f? We
>> both know it can't be done, and why.
> 
> It certainly can be done if your sampling points are locked to the
> signal. As usual your arguments consist of having you cake and eating it

It can only be done if the sampling clock is known to be in quadrature 
with the second harmonic on the signal; the sampling occurs on the 
peaks. The cases discussed were about sampling at or near the zero 
crossings.

Prior knowledge of the sampling conditions and the signal can lead to 
systems of equations much simpler than the general cases that were under 
discussion. Knowing that the samples are taken at the peak od a sine of 
known frequency allows complete characterization of the signal with a 
single sample. I don't find such simplifications interesting.

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

Jerry Avenues wrote:
 Knowing that the samples are taken at the peak od a sine of
> known frequency allows complete characterization of the signal with a
> single sample. I don't find such simplifications interesting.

So why did you introduce it into the discussion in the first place?

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jim wrote:
> 
> Jerry Avenues wrote:
>  Knowing that the samples are taken at the peak od a sine of
>> known frequency allows complete characterization of the signal with a
>> single sample. I don't find such simplifications interesting.
> 
> So why did you introduce it into the discussion in the first place?

Because knowing the relative phase of the sampler to the signal's second 
harmonic is the only condition that makes possible the determination of 
amplitude when sampling at at 2f, a scenario that you introduced.

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

robert bristow-johnson wrote:
> 
> Rick Lyons wrote:
> ...
>
<snip>
>>
>> Concerning sampling, Bonnie Baker wrote an
>> article titled "Turning Nyquist Upside Down
>> by Undersampling" in the May 12th 2005 issue
>> of EDN magazine.  The article discusses bandpass
>> sampling.  However, I think the article's
>> title is unfortunate because bandpass
>> sampling does NOT "turn Nyquist upside down"
>> ---bandpass sampling is included in the Nyquist
>> Sampling Theorem.
> 
> are any of these online?
> 
> Rick, i fear that similar stuff is being done in the Wikipedia article.
>  this guy (whose name is very similar to yours and claims to have Alan
> Oppenheim and Ron Schafer as friends) would say to you that "the
> converse of the Nyquist-Shannon sampling theorem is not true", meaning
> that there are cases where frequency components at or above the Nyquist
> frequence can be validly reconstructed under some circumstances.   i
> think he's talking about bandpass sampling.

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?

-- 
Jani Huhtanen
Tampere University of Technology, Pori

In message <EP8Hg.410$wo2.164@newsfe05.lga>, dated Wed, 23 Aug 2006, Tim 
Williams <tmoranwms@charter.net> writes
>"Jim Stewart" <jstewart@jkmicro.com> wrote in message
>news:UaidnVPZ369OFXbZnZ2dnUVZ_t-dnZ2d@omsoft.com...
>> I'd guess he wants the word "periodic" in there somewhere (:
>
>HIO4?
>
Yes, unionised.
-- 
OOO - Own Opinions Only. Try www.jmwa.demon.co.uk and www.isce.org.uk
2006 is YMMVI- Your mileage may vary immensely.

John Woodgate, J M Woodgate and Associates, Rayleigh, Essex UK

Tim Wescott wrote:
> Jonathan Kirwan wrote:
>
> > On Wed, 23 Aug 2006 12:13:33 -0400, Phil Hobbs
> > <pcdh@SpamMeSenseless.pergamos.net> wrote:
> >
> >
> >>Tim Wescott wrote:
> >>
> >>
> >>>Kinda off topic --
> >>>
> >>>A month or two ago there was a spate of postings on these groups
> >>>displaying a profound misunderstanding of how to apply Nyquist's theorem
> >>>to problems of setting sampling or designing anti-alias filters.  I
> >>>helped folks out as much as I could, but it really demands an article,
> >>>which I am currently working on.
> >>>
> >>>The misconceptions that I noticed pretty much boiled down to the
> >>>following two:
> >>>
> >>>One, "I need to monitor a signal that happens at X Hz, so I'm going to
> >>>sample it at 2X Hz".
> >>>
> >>>Two, "I can sample at X Hz, so I'm going to build an anti-alias filter
> >>>with a cutoff of X/2 Hz".
> >>>
> >>>I estimate that answering these misconceptions will only take 3-4k
> >>>words, but I don't want to miss any other big ones.
> >>>
> >>>Have you seen any other real howlers that relate to Nyquist, and what
> >>>you should really be thinking about when you're pondering sampling
> >>>rates, anti-aliasing filters and/or reconstruction filters?
> >>>
> >>>Danke.
> >>
> >>The other one I run into is that N. really applies to the bandwidth, not
> >>the highest frequency as is commonly thought.  Harmonic mixers make use
> >>of this all the time, using the equivalence of the sampled interval to
> >>the fundamental interval [-f_s/2, f_s/2), and alias down to some lower
> >>frequency in the process.  If you really reconstruct with impulses, you
> >>can use a bandpass filter to get back the original signal at the
> >>original carrier frequency.
> >>
> >>People also routinely neglect the to account for the zero-order hold in
> >>their DAC circuits--if you take a signal, run it through an A/D and a
> >>D/A, you don't wind up with the original signal, but one with an
> >>additional sinc function rolloff.
> >
> >
> > This last paragraph seems worth emphasizing, particularly on the
> > subject of sampling rates, as it points out a reason why rather more
> > than 2.00...01 X sampling may be important.  I'm not sure how a
> > practical reconstruction filter to compensate for ZOH could be
> > arranged, causal or acausal, otherwise.  You need some margin for the
> > skirts, don't you?
> >
> > Jon
> Actually designing for the sin x / x rolloff isn't too bad as long as
> you keep your eyes open -- in older digital video systems it was just
> done with a peaky 2nd-order LC circuit (in newer digital video systems
> the sampling rate is way higher than the effective resolution of the
> phosphor, which simplifies things).
>
> 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).

greetings,
Vasile

jim wrote:
> Jerry Avins wrote:
>
> > I don't get it. What have I written that makes it seem that I believe
> > the amplitude of a component f can be determined by sampling at 2f? We
> > both know it can't be done, and why.
>
> It certainly can be done if your sampling points are locked to the
> signal.

works really good when the sampling points are locked to the
zero-crossings of the Nyquist frequency signal.

> As usual your arguments consist of having you cake and eating it too.

(snicker)

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.

r b-j