What Nyquist Didn't Say

["Followup-To:" header set to sci.electronics.design.]
On Fri, 29 Sep 2006 22:35:46 GMT,
  John Herman <John_W_Herman@yahoo.com> wrote
  in Msg. <6dhTg.12524$rE5.8259@tornado.socal.rr.com>

> In the first sentence of your papere, you say that if the signal is band 
> limited to fo or less, then a sample frequency of 2fo or more is adequate to 
> contain completely all the information necessary to recreate the signal.  My 
> understanding from water cooler conservation is that the signal bandwidth must 
> be strictly less than fo for a sample frequency of 2fo and that the signal 
> must be infinite in extent to allow perfect reconstruction.  

I see it that way as well. A pure sine signal of exactly f_0, sampled at
exactly 2f_0, might come out as all-zero, or as a pulse train with
alternating polarity and an amplitude of anything between 0 and V_p. 
There is not enough information to reconstruct the signal in this case.

However, if you sample the same signal at 2f_0+f_e (f_e being very
small, think epsilon), it will come out as a pulse train with
alternating polarity and the amplitude modulated by a beat frequency
f_e. This would seem to be not enough information to reconstruct the
signal, but in fact that's not true: A f_0 sine is the only possible
input signal because the modulated pulse train contains frequency
components greater than f_0 which couldn't have been in the input signal
(which, as a prerequsitite to Nyquist, is brick-walled at f_0).

So, the way I see it is that the sampling frequency must be strictly 
greater than the highest frequency in the input.

Thanks, Tim, for a great write-up on the subject!

robert

["Followup-To:" header set to sci.electronics.design.]
On Sat, 30 Sep 2006 09:22:43 -0700,
  Tim Wescott <tim@seemywebsite.com> wrote
  in Msg. <wZKdnd6yVvFUBIPYnZ2dnUVZ_qydnZ2d@web-ster.com>

> I have to say that this paper rather turned into a monster while I was 
> writing it.  I thought it was going to be between 2000 and 3000 words, 
> with almost no math and very little real work.  Instead it's about 5500 
> words, and I've got about a man-week into all those pretty charts and 
> graphs (I should publish an appendix with "the making of..." along with 
> all the math underneath).
>
> As a reaction to this I haven't done my usual stage of letting it rest 
> and getting back to it -- I was afraid I'd never do the "getting back to 
> it" step.  Instead I've put it outside without giving it time to get 
> it's coat and boots on.  If I can figure out how to tighten it up I 
> certainly will -- assuming that I don't run away screaming at the 
> thought of doing even _more_ work on it.

Let me tell you that this article is so high-class from an educational
point of view that it /deserves/ coat and boots. Next time anybody comes
up to me and wants something explained about Nyquist I can just point
them at your page, and I'm sure many others will do likewise.

The only thing I'd want is the whole thing as a pretty, nicely printable 
PDF document. But don't listen to me.

robert

Ban wrote:
> miso@sushi.com wrote:
> >
> > You really need to look at theory versus what spice predicts. The
> > input is an ideal step. As you do the convolution, the negative
> > region of the impulse response subtracts from the result, making the
> > signal drop in value, then the postive regions make the result
> > increase in value, hence ringing.. The Gaussian impulse response is
> > always positive, hence the convolution output can't decrease as time
> > increases.
> >
> > Linear phase alone is not enough to stop ringing.
> >
> > The discrete time situation is easier to understand, especially if you
> > consider a finite impulse response filter. The response of a FIR
> > filter to a step input is the the sum of the tap coefficients from
> > one to N. That is, the first output is the first tap. The second
> > output is the sum of the first two taps, etc. If no tap is negative,
> > then the output always rises, hence no ringing.
> >
> > No circuits were simulated in writing this post. ;-)
>
> There are differences between analog and digital filters. Digital means an
> approximation of the desired characteristic in the passband, but the poles
> and zeros are modified to compensate for the modulation effects.

Modulation effect? All you have are two different domains (s and z).

 aliasing is
> always happening, but it can be reduced below the noise floor with the
> analog input filter.
> Linear phase has a very undesirable side effect, it rings *before and aft=
er*
> the step, supposed to be more audible.

Ring before the signal arrives? That sound non-causal to me.

>
> And Audio is a very forgiving m=E9tier, because the ears themselves funct=
ion
> as reconstruction filters, suppressing all those high frequency artifacts=
.=2E
> If you want to display the signal with a digital scope or ECG, you better
> start with 5 to 10 times the upper frequency rolloff, look at the specs
> there, nobody even considers Nyquist adequate, exept programmers having no
> idea of reality.
> --=20
> ciao Ban
> Apricale, Italy

Mike Monett wrote:
> "Ban" <bansuri@masterweb.it> wrote:
> >
> > Bessel filters do have a small overshoot, which decreases at higher
> > orders. 2nd    .43%
> > 4th    .84%
> > 6th    .64%
> > 8th    .34%
> > 10th    .06%
> > note the *very low* value for higher orders. In fact the frequency
> > response gets hardly better then.
> > The Gauss filter has indeed zero ringing, but a much larger transition
> > band.
>
> Ban,
>
> Thanks for the clarification. In most references, the Bessel is considered
> to have low, negligible, or no overshoot, especially when compared to
> Butterworth, Chebyshev and other types of filters. Your numbers confirm
> this.
>
> In practise, it is difficult to obtain the exact component values needed
> for the theoretical performance. Not only are the values non-standard, but
> it may be difficult to get the inductor "Q" values used in most
> calculations. So we can assume there will be some deviation from the
> theoretical performance, and the overshoot will probably increase slightly.
> However, it is still low enough to be difficult to measure, and the terms
> "low", "neglible" or "no overshoot" are quite descriptive.
>
> Regards,
>
> Mike Monett
>
> Antiviral, Antibacterial Silver Solution:
> http://silversol.freewebpage.org/index.htm
> SPICE Analysis of Crystal Oscillators:
> http://silversol.freewebpage.org/spice/xtal/clapp.htm
> Noise-Rejecting Wideband Sampler:
> http://www3.sympatico.ca/add.automation/sampler/intro.htm

For audio signal processing, the filters are generally active, so no
inductors are used. [speaker crossovers excepted.]

In many applications, ringing cannot be tolerated. Scales for instance.

On 2 Oct 2006 13:27:06 -0700, in sci.electronics.design miso@sushi.com
wrote:

>
>Ban wrote:
>> miso@sushi.com wrote:
>> >

>> Linear phase has a very undesirable side effect, it rings *before and after*
>> the step, supposed to be more audible.
>
>Ring before the signal arrives? That sound non-causal to me.
>
ISTR Roger Lagadec at Studer (and Sony) pointing this out in the early
1980's, after listening tests on early digital systems with  piano
music showed that something was amiss. Probably in AES and IEEE
archives.




martin

miso@sushi.com wrote:
> Ban wrote:
> 
>>miso@sushi.com wrote:
>>
>>>You really need to look at theory versus what spice predicts. The
>>>input is an ideal step. As you do the convolution, the negative
>>>region of the impulse response subtracts from the result, making the
>>>signal drop in value, then the postive regions make the result
>>>increase in value, hence ringing.. The Gaussian impulse response is
>>>always positive, hence the convolution output can't decrease as time
>>>increases.
>>>
>>>Linear phase alone is not enough to stop ringing.
>>>
>>>The discrete time situation is easier to understand, especially if you
>>>consider a finite impulse response filter. The response of a FIR
>>>filter to a step input is the the sum of the tap coefficients from
>>>one to N. That is, the first output is the first tap. The second
>>>output is the sum of the first two taps, etc. If no tap is negative,
>>>then the output always rises, hence no ringing.
>>>
>>>No circuits were simulated in writing this post. ;-)
>>
>>There are differences between analog and digital filters. Digital means an
>>approximation of the desired characteristic in the passband, but the poles
>>and zeros are modified to compensate for the modulation effects.
> 
> 
> Modulation effect? All you have are two different domains (s and z).
> 
>  aliasing is
> 
>>always happening, but it can be reduced below the noise floor with the
>>analog input filter.
>>Linear phase has a very undesirable side effect, it rings *before and after*
>>the step, supposed to be more audible.
> 
> 
> Ring before the signal arrives? That sound non-causal to me.
> 
Ring before the main part of the signal arrives.  FIR filter responses 
are often shown as if zero delay were the middle of the filter, 
effectively subtracting out the constant delay added by the filter.

-- 

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

miso@sushi.com wrote:
> 
> Ban wrote:
>
... snip ...
>> 
>> aliasing is always happening, but it can be reduced below the
>> noise floor with the analog input filter.
>> Linear phase has a very undesirable side effect, it rings
>> *before and after* the step, supposed to be more audible.
> 
> Ring before the signal arrives? That sound non-causal to me.

Don't think of a single signal.  An impulse (or a step) has a wide
range of frequency components.  Some are delayed more than others
when passing through the filter.

-- 
 Some informative links:
   <news:news.announce.newusers
   <http://www.geocities.com/nnqweb/>
   <http://www.catb.org/~esr/faqs/smart-questions.html>
   <http://www.caliburn.nl/topposting.html>
   <http://www.netmeister.org/news/learn2quote.html>
   <http://cfaj.freeshell.org/google/>

miso@sushi.com wrote:

> Ring before the signal arrives? That sound non-causal to me.

Please read more carefully. The filter rings before the main part of the 
output step *emerges* but after the step arrives at the input. The 
filter's inherent delay makes that quite possible.

Jerry
-- 
        "The rights of the best of men are secured only as the
        rights of the vilest and most abhorrent are protected."
            - Chief Justice Charles Evans Hughes, 1927
���������������������������������������������������������������������

miso@sushi.com wrote:
> Ring before the signal arrives? That sound non-causal to me.

Its only non-causal if it arrives before it was sent. :-)

There is no reason why the main signal should not arrive latter than 
some of the crud which accompanies it. Its perfectly normal in a 
dispersive medium.

Steve

Hi Tim,

Tim Wescott wrote:
> I've seen a lot of posts over the last year or so that indicate a lack 
> of understanding of the implications of the Nyquist theory, and just 
> where the Nyquist rate fits into the design of sampled systems.
> 
> So I decided to write a short little article to make it all clear.
> 
> It's a little longer than 'short', and it took me way longer than I 
> thought it would, but at least it's done and hopefully it's clear.
> 
> You can see it at 
> http://www.wescottdesign.com/articles/Sampling/sampling.html.
> 
> If you're new to this stuff, I hope it helps.  If you're an expert and 
> you have the time, please feel free to read it and send me comments or 
> post them here.
> 

May I ask what software you used to render the maths on that page? It 
looks clearer than the stuff I produce. MathML is getting into browsers 
now, but the rendering of that looks so bad with anything I have tried, 
that inserted images in HTML pages still seems the only practical approach.

I'd still like to see a web page I can point people to when they say a 
10kHz sine wave on a CD will come out as a square wave/triangular 
wave/some other weird notion.

Steve