# RE: [SI-LIST] : Nyquist Sampling Rate

From: Dagostino, Tom ([email protected])
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
[email protected]

-----Original Message-----
From: Ken Cantrell [mailto:[email protected]]
Sent: Tuesday, April 03, 2001 2:38 PM
To: Shoba Rao; AA
Cc: [email protected]
Subject: RE: [SI-LIST] : Nyquist Sampling Rate

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

-----Original Message-----
From: [email protected]
[mailto:[email protected]]On Behalf Of Shoba Rao
Sent: Tuesday, April 03, 2001 2:47 PM
To: AA
Cc: [email protected]
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.
>
> --- Ken Cantrell <[email protected]> wrote:
> > Gaussian, Hamming, and Hanning windows can be used
> > to approximate a square
> > wave.
> >
> > -----Original Message-----
> > From: [email protected]
> > [mailto:[email protected]]On Behalf Of
> > Thomas Jackson
> > Sent: Monday, April 02, 2001 9:38 AM
> > To: 'AA'; Thomas Jackson; [email protected]
> > Subject: RE: [SI-LIST] : Nyquist Sampling Rate
> >
> >
> >
> > 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:[email protected]]
> > Sent: Friday, March 30, 2001 5:43 PM
> > To: Thomas Jackson; [email protected]
> > 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 <[email protected]> wrote:
> > >
> > > 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:[email protected]]
> > > Sent: Friday, March 30, 2001 4:43 PM
> > > To: [email protected]
> > > 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.
> > >
> > >
> > >
> > > --- Heinz Blennemann <[email protected]> wrote:
> > > > tech/recruit/jobreq_optic
> > > > [email protected]
> > > > OPENING: Juniper Networks: Electrical/optical
> > > > signal-integrity engineer
> > > >
> > > > POSITION:
> > > > The engineer would be responsible for
> > technology
> > > > and signal integrity
> > > > for both electrical and optical
> > > > hardware/interfaces, including
> > > > providing design rules to meet high speed
> > timing
> > > > and noise budgets
> > > > optical and electrical technology/component
> > > > selection
> > > > - understand the communication standards
> > > (Sonet
> > > > spec's)
> > > > - drive technology issues necessary for
> > > > selecting electro-optical
> > > > transmitter/receiver modules (critical to
> > > > Juniper's router
> > > > interfaces)
> > > > - test and verify in lab
> > > >
> > > > In the course of the project, to initially
> > > perform
> > > > the necessary analysis,
> > > > suggest and evaluate competing technologies.
> > By
> > > > developing noise and
> > > > timing budgets, to propose routing rules.
> > During
> > > > the design phase,
> > > > to run field solvers and SPICE to update the
> > > > budgets and the resulting
> > > > rules. Finally, in the lab, to perform
> > > > measurements using
> > > > high-speed scopes TDRs, and jitter analyzers
> > to
> > > > verify operation
> > > > and to verify assumptions made in earlier
> > parts
> > > of
> > > > the design.
> > > >
> > > > Separately, a position for an electrical
> > signal
> > > > integrity engineer
> > > > is also available.
> > > >
> > > > EXPERIENCE:
> > > > Requires
> > > > - BS in EE (MS or Ph.D. preferred)
> > > > - a solid understanding of circuits, analog
> > > > behavior, and
> > > > transmission line fundamentals, and optics
> > > > - demonstrated practical expertise in optical
> > and
> > > > electrical
> > > > measurements/design and signal integrity,
> > > > in such areas as interconnect technolology,
> > > > packaging,
> > > > measurement techniques
> > > > - ideally, experience in high-speed system
> > design
> > > > in communications
> > > >
> > > > COMPANY DESCRIPTION
> > > > Juniper Networks Inc., is a leading provider of
> > > > next-generation Internet
> > > > infrastructure systems designed to meet the
> > > > scalability, performance,
> > > > density, and compatibility requirements of
> > rapidly
> > > > evolving,
> > > > optically-enabled IP networks. The company's
> > > > purpose-built M160 Internet
> > > > backbone router uses JUNOS software, ASICs, and
> > > > wire-rate performance to
> > > > provide new Internet infrastructure solutions
> > for
> > > > service providers. The
> > > > M160 router is used by some of the world's
> > > > service providers.
> >
> === message truncated ===
>
> __________________________________________________
> Do You Yahoo!?
> Get email at your own domain with Yahoo! Mail.
> http://personal.mail.yahoo.com/?.refer=text
>
> **** To unsubscribe from si-list or si-list-digest: send e-mail to
> [email protected] In the BODY of message put: UNSUBSCRIBE
> si-list or UNSUBSCRIBE si-list-digest, for more help, put HELP.
> si-list archives are accessible at http://www.qsl.net/wb6tpu
> ****

```--
/**************************************************/
Shoba Rao                        Phone numbers    */
NATIONAL SEMICONDUCTORS,         408-721-2056(W)  */
2900,semiconductor Drive,        408-260-8567(H)  */
SantaClara,CA-95052.                              */
/**************************************************/
/*  The difficult I do immediately,..             */
/*    the impossible takes a little longer...     */
/**************************************************/
**** To unsubscribe from si-list or si-list-digest: send e-mail to
[email protected] In the BODY of message put: UNSUBSCRIBE
si-list or UNSUBSCRIBE si-list-digest, for more help, put HELP.
si-list archives are accessible at  http://www.qsl.net/wb6tpu
****
**** To unsubscribe from si-list or si-list-digest: send e-mail to
[email protected] In the BODY of message put: UNSUBSCRIBE
si-list or UNSUBSCRIBE si-list-digest, for more help, put HELP.
si-list archives are accessible at  http://www.qsl.net/wb6tpu
****
**** To unsubscribe from si-list or si-list-digest: send e-mail to
[email protected] In the BODY of message put: UNSUBSCRIBE
si-list or UNSUBSCRIBE si-list-digest, for more help, put HELP.
si-list archives are accessible at  http://www.qsl.net/wb6tpu
****
```

This archive was generated by hypermail 2b29 : Thu Jun 21 2001 - 10:11:26 PDT