RE: [SI-LIST] : Nyquist Sampling Rate

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From: Dagostino, Tom (tom_dagostino@mentorg.com)
Date: Tue Apr 03 2001 - 16:00:31 PDT


I spent a lot of time designing DSO's, so I've become very familiar with
sampling rates.

The Nyquist criteria states that a signal must be sampled AT LEAST twice the
rate of the highest frequency of interest in the input waveform. Twice does
not work. For a sine wave you need AT LEAST twice its frequency. Since it
is at least twice its frequency you do not need to worry about sampling at
the zero crossing, if one sample is at the zero crossing the next will be
just before the next zero crossing of the opposite slope and the third will
be just before the third zero crossing of the original slope. From that
information you can construct the original waveform.

If you sample at a rate of 2X the input frequency or less you will have
aliasing.

If you just have the two samples then there are any number of waveforms that
can be drawn through them but in reality you need 2+ samples per cycle.
Once you collect more than a few samples you will find that there is a limit
on the number of waveforms that are band limited, not aliased, and can fit
the data. As you collect more and more samples you will find a unique
waveform that will fit the data. I don't think the theorem ever applied to
just 2 samples taken any where in time from a continuous waveform. The
theorem applies to how fast one needs to sample data to recover information.

The factor of 10X of sample rate to input frequency was first used to
describe how much over sampling is required to visually describe a sine
wave. I did this in about 1978 at Tek. We sampled sine waves and used
different display technologies to reconstruct the waveforms. With a dot
display it took about 25 samples per cycle to represent a sine wave on a
CRT. If you connected the dots there is an improvement to 10 samples per
cycle to represent the sine wave. If you use a SinX/X interpolation then
you can get to about 2.5 samples per cycle to reconstruct the sine wave and
recognize it as such.

Most signals are described by an infinite sum of sinusoids. But once the
amplitude of the nth harmonic gets below the resolution of your sampling
system going higher in sampling rate will buy you little if any additional
information.

Tom Dagostino
IBIS and Tau Modeling Manager
SDD
Mentor Graphics Corp.
503-685-1613
tom_dagostino@mentor.com

-----Original Message-----
From: Ken Cantrell [mailto:Ken.Cantrell@srccomp.com]
Sent: Tuesday, April 03, 2001 2:38 PM
To: Shoba Rao; AA
Cc: si-list@silab.eng.sun.com
Subject: RE: [SI-LIST] : Nyquist Sampling Rate

Oops! Normal sample rate is Nyq*10, min is Nyq/2.

-----Original Message-----
From: owner-si-list@silab.eng.sun.com
[mailto:owner-si-list@silab.eng.sun.com]On Behalf Of Shoba Rao
Sent: Tuesday, April 03, 2001 2:47 PM
To: AA
Cc: si-list@silab.eng.sun.com
Subject: Re: [SI-LIST] : Nyquist Sampling Rate

Hi,
  My two cents worth..

I dont think your assumption is true. Sampling theory
holds good for all type of signals and not just a true
sine waveform signal which includes aperiodic signals.
For aperiodic signals, since practically you cannot take
infinite number of samples,we assume the signal to be
periodic after N samples.In the fourier domain, you get
a periodic spectra of the fourier transform of your
orginal signal and it can be mathematically proven that
you will be able to recover your signal from the fourier
domain as long as you sample at the Nyquist rate and dont
create aliasing.
    Example, if you have a ramp signal, you will be able to
sample at all frequencies and still able to retrieve the
original waveform.

-Shoba.

AA wrote:
>
> Thanks all for responding to my inquiry on the Nyquest
> sampling rate. I guess the important take away is that
> this sampling theory applies only to a true sine
> waveform signal not "Periodic Signals" as I mentioned
> in my first mail. I always remembered Fourier series
> form my freshman year but did not establish the mental
> correlation between "periodic" and "band limited".
>
> I appreciate all you input.
> Adam
>
> --- Ken Cantrell <Ken.Cantrell@srccomp.com> wrote:
> > Gaussian, Hamming, and Hanning windows can be used
> > to approximate a square
> > wave.
> >
> > -----Original Message-----
> > From: owner-si-list@silab.eng.sun.com
> > [mailto:owner-si-list@silab.eng.sun.com]On Behalf Of
> > Thomas Jackson
> > Sent: Monday, April 02, 2001 9:38 AM
> > To: 'AA'; Thomas Jackson; si-list@silab.eng.sun.com
> > Subject: RE: [SI-LIST] : Nyquist Sampling Rate
> >
> >
> > Adam.
> >
> > Fourier analysis shows that anything like a
> > square-wave, trapezoidal-wave,
> > triangular-wave, sawtooth-wave, etc. has frequency
> > components going up to
> > infinity. Therefore, these signals are not
> > band-limited.
> >
> > The only kinds of signals that can be recovered from
> > discrete samples are
> > those that can be constructed from a band-limited
> > set of sinusoids.
> > Luckily, these include or approximate many useful
> > real-world signals. The
> > highest frequency one of these is a sinewave at 1/2
> > the sampling rate.
> >
> > Tom
> >
> > -----Original Message-----
> > From: AA [mailto:alokbya@yahoo.com]
> > Sent: Friday, March 30, 2001 5:43 PM
> > To: Thomas Jackson; si-list@silab.eng.sun.com
> > Subject: RE: [SI-LIST] : Nyquist Sampling Rate
> >
> >
> > Tom,
> > Thanks for the feedback. I know that the sampling
> > rate has to be at least twice that hight frequency
> > component in the signal. I.e to recover a 60 HZ
> > sinwave it needed to be sampled by 120sample/sec
> > min.
> > How do we know a sine wave produced these samples
> > not
> > a triangulare wave or other periodic wave form.
> >
> > Thanks
> >
> >
> > --- Thomas Jackson <tjackson@fmi.fujitsu.com> wrote:
> > > Adam,
> > >
> > > The sampling theorem assumes that you are sampling
> > a
> > > band-limited signal.
> > > Therefore, the highest possible frequency signal
> > > through any two points
> > > would be a sinewave at 1/2 the sampling rate.
> > > Anything else would have
> > > frequency components above the Nyquist rate and
> > that
> > > violates the first
> > > assumption.
> > >
> > > By the way, it should be obvious that the two
> > > samples cannot occur at the
> > > zero crossings.
> > >
> > > Tom
> > >
> > > Thomas L. Jackson, P.E.
> > > Staff VLSI Design Engineer
> > > Network Access Development
> > > Systems Solutions Group
> > > FUJITSU MICROELECTRONICS, INC.
> > > 3545 North First Street
> > > San Jose, CA 95134-1804
> > > telephone: (408) 922-9574
> > > facsimile: (408) 922-9618
> > > http://www.fujitsumicro.com
> > >
> > >
> > > -----Original Message-----
> > > From: AA [mailto:alokbya@yahoo.com]
> > > Sent: Friday, March 30, 2001 4:43 PM
> > > To: si-list@silab.eng.sun.com
> > > Subject: [SI-LIST] : Nyquest Sampling Rate
> > >
> > >
> > > DEAR SI list subscribers,
> > > Can any one explain to me how you can recover a
> > > periodic signal form only 2 samples. I can
> > > understand
> > > the math but I am having difficulty visualizing
> > > this.
> > > Draw me any 2 points in the time domain and I can
> > > make
> > > endless number of periodic signal go through them?
> > >
> > > I know I am missing a key point but I can quite
> > put
> > > my
> > > finger on it.
> > >
> > > Your input is very well appreciated.
> > >
> > > Adam
> > >
> > >
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> > > > tech/recruit/jobreq_optic
> > > > si-list@silab.Eng.Sun.COM
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> === message truncated ===
>
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-- 
/**************************************************/
Shoba Rao                        Phone numbers    */
NATIONAL SEMICONDUCTORS,         408-721-2056(W)  */
2900,semiconductor Drive,        408-260-8567(H)  */
SantaClara,CA-95052.                              */
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