Nyquist Didn't Say That

Jani Huhtanen wrote:
>
> 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.

*exact* definition, i dunno either (we argue about exact definitions).
but i do know that there are circumstances of a bandpass signal  that
has zero spectrum except for (-f0-B  -f0) and (f0  f0+B) being
sufficiently sampled even when Fs < 2*(f0+B).  i think this is what
"bandpass sampling" refers to.

> 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
>

i'm rewriting this to something that is more visable (monospaced font):


      x(t) =  SUM{  SUM{ x_s[n]*phi_s(t-n) }   }
               s     n

i have "t" normalized time so it's phi_s(t-n) instead of phi_s(t-nT).

is this a faithful renotation of what you're saying, Jani?

> phi_s(t-n) 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] = <x(t),phi_s(t-n)>,

> where x(t) is the analyzed (sampled) function, phi_s(t-n) 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).

yeah, but those guys are all bandlimited and there need be only one
scaling factor so there is no SUM_s above, only the SUM_n

> 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.

i guess i did notice.

> Any comments, or corrections?

the only thing i would say is that it's about the wavelet transform and
is not about the Nyquist-Shannon sampling and reconstruction theorem.

r b-j

robert bristow-johnson wrote:

> Jani Huhtanen wrote:
>>
>> 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.
> 
> *exact* definition, i dunno either (we argue about exact definitions).
> but i do know that there are circumstances of a bandpass signal  that
> has zero spectrum except for (-f0-B  -f0) and (f0  f0+B) being
> sufficiently sampled even when Fs < 2*(f0+B).  i think this is what
> "bandpass sampling" refers to.
> 
>> 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
>>
> 
> i'm rewriting this to something that is more visable (monospaced font):
> 
> 
>       x(t) =  SUM{  SUM{ x_s[n]*phi_s(t-n) }   }
>                s     n
> 
> i have "t" normalized time so it's phi_s(t-n) instead of phi_s(t-nT).
> 
> is this a faithful renotation of what you're saying, Jani?

Yep.

> 
>> phi_s(t-n) 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] = <x(t),phi_s(t-n)>,
> 
>> where x(t) is the analyzed (sampled) function, phi_s(t-n) 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).
> 
> yeah, but those guys are all bandlimited and there need be only one
> scaling factor so there is no SUM_s above, only the SUM_n

See below.

> 
>> 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.
> 
> i guess i did notice.
> 
>> Any comments, or corrections?
> 
> the only thing i would say is that it's about the wavelet transform and
> is not about the Nyquist-Shannon sampling and reconstruction theorem.

OK, consider

      f(t) = SUM_n x[n]*phi(t-n)  (1)
      x[n] = <f(t), phi(t-n)>     (2)

Do we agree that if phi(t) = sinc(t), then (2) equals to Nyquist-Shannon
sampling of bandlimited signal and (1) to reconstruction of the said
signal? In a way the dirac-comb and anti-aliasing filter are combined into
(2).

Now consider 
               { 1, when 0 < t < 1
      phi(t) = {
               { 0, otherwise

which corresponds to the scaling function of the simplest possible wavelet
trasform: Haar transform. In this case, (1) corresponds to synthesis
(reconstruction) at scale 0 and (2) corresponds to analysis (sampling) at
scale 0. Again, as I previously stated, if f(t) is synthesized by (1) it
can be sampled by (2) and subsequently perfectly reconstructed by (1).

What I'm getting at is that, there is no need for SUM_s in the case of
wavelet transform either, but _only_ if the analyzed signal is synthesized
in single resolution. Surely you see that both, Nyquist-Shannon and Haar
seem very alike? Difference is in the requirements for f(t). In case of
Nyquist-Shannon, f(t) has to be bandlimited (sinc) and in case of Haar,
f(t) has to be piecewise constant.

So perhaps this guy, talking about the converse of the Nyquist-Shannon
sampling theorem not being true, was referring to something similar I
described? Or did I competely misread you?

-- 
Jani Huhtanen
Tampere University of Technology, Pori

Jani Huhtanen wrote:
>
> So perhaps this guy, talking about the converse of the Nyquist-Shannon
> sampling theorem not being true, was referring to something similar I
> described?

i doubt it.  i think he was talking about "bandpass sampling".

> Or did I competely misread you?

not me.  i think you and i have completely coherent communication.  i
just don't think that this guy at Wikipedia was talking about this
wavelet transform thing but about cases where Fs < 2(f0+B) (the highest
frequency) and someone (using bandpass sampling) they are still able to
reconstruct.

r b-j

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.

-- 
Chuck F (cbfalconer@yahoo.com) (cbfalconer@maineline.net)
   Available for consulting/temporary embedded and systems.
   <http://cbfalconer.home.att.net> USE maineline address!

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. 

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

	Set up a speaker generating the Fs/2 signal. Set up a microphone and
ADC to sample the sound at Fs. What really happens?  If you adjust the
phase of the sampling can you record silence? This was not a theoretical
question. We all know how it should work in a perfect world. How does it
work in the real world?

	 No locking the ADC to the signal allowed since that would be a
completely different question that no one asked and no one is interested
in.  

-jim

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Jerry Avins wrote:
> jim wrote:
> >
> > Robert Baer wrote:
> >> mobi wrote:
> >>
> >>> Do consider this interesting (atleast for me) example
> >>>
> >>> Consider pure Sin wave at X Hz. I start sample it at 2X. Unfortunately
> >>> i start sampling from time = 0. What would i get? Aint i statisifying
> >>> Nyquist here?
> >>>
> >>    Yup!
> >>    Also try sampling at a constant delay from the sine zero crossing.
> >>    That is what happens when people blindly follow a "criteria" without
> >> knowing the full reason and background.
> >
> > 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?
>
> Of course you can lock the sampler to the sampled waveform or one of its
> harmonics. Google for "PLL".

However the phase information fed to the PLL to allow it to lock
would constitute additional samples, thus raising the total sample
rate of all information coming into the system above Fs/2.  You
have to count all the samples, not just the ones you label as
"samples".


IMHO. YMMV.
-- 
rhn A.T nicholson d.0.t C-o-M

Ron N. wrote:
> Jerry Avins wrote:
>> jim wrote:
>>> Robert Baer wrote:
>>>> mobi wrote:
>>>>
>>>>> Do consider this interesting (atleast for me) example
>>>>>
>>>>> Consider pure Sin wave at X Hz. I start sample it at 2X. Unfortunately
>>>>> i start sampling from time = 0. What would i get? Aint i statisifying
>>>>> Nyquist here?
>>>>>
>>>>    Yup!
>>>>    Also try sampling at a constant delay from the sine zero crossing.
>>>>    That is what happens when people blindly follow a "criteria" without
>>>> knowing the full reason and background.
>>> 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?
>> Of course you can lock the sampler to the sampled waveform or one of its
>> harmonics. Google for "PLL".
> 
> However the phase information fed to the PLL to allow it to lock
> would constitute additional samples, thus raising the total sample
> rate of all information coming into the system above Fs/2.  You
> have to count all the samples, not just the ones you label as
> "samples".

That's an interesting assertion. Can you justify it?

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

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

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.

-- 

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

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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. 
> 
> The question was asked - What really happens when you sample a frequency
> at Fs/2. 
> 
> 	Set up a speaker generating the Fs/2 signal. Set up a microphone and
> ADC to sample the sound at Fs. What really happens?  If you adjust the
> phase of the sampling can you record silence? This was not a theoretical
> question. We all know how it should work in a perfect world. How does it
> work in the real world?
> 
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.

Which is a bad thing.

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