sampling a perfect sinusoid at Nyquist rate?

Let's say you have a perfect sine wave at frequency w.
According to Nyquist, in order to be able to recover the sine wave, you
need to have a sampling rate of at least 2w.  
So if you decide to sample at 2w, you end up with 2 samples for each cycle
of this sine wave.

If you sample at the peaks and troughs (90 and 270 degrees) of the sine
wave in time (or spatial) domain, you indeed preserve the amplitude.
But what happens when the sampling happens at 0 and 180 degrees?  You end
up with only zeros.

Am I missing something about sampling theorem here?  Is a shifted sampling
not the same as the origianl unshifted sampling?

Let's say you have a perfect sine wave at frequency w.
According to Nyquist, in order to be able to recover the sine wave, you
need to have a sampling rate of at least 2w.  
So if you decide to sample at 2w, you end up with 2 samples for each cycle
of this sine wave.

If you sample at the peaks and troughs (90 and 270 degrees) of the sine
wave in time (or spatial) domain, you indeed preserve the amplitude.
But what happens when the sampling happens at 0 and 180 degrees?  You end
up with only zeros.

Am I missing something about sampling theorem here?  Is a shifted sampling
not the same as the origianl unshifted sampling?

jjmai wrote:
> Let's say you have a perfect sine wave at frequency w.
> According to Nyquist, in order to be able to recover the sine wave, you
> need to have a sampling rate of at least 2w.  
> So if you decide to sample at 2w, you end up with 2 samples for each cycle
> of this sine wave.
> 
> If you sample at the peaks and troughs (90 and 270 degrees) of the sine
> wave in time (or spatial) domain, you indeed preserve the amplitude.
> But what happens when the sampling happens at 0 and 180 degrees?  You end
> up with only zeros.
> 
> Am I missing something about sampling theorem here?  Is a shifted sampling
> not the same as the origianl unshifted sampling?

How does "equal" come to replace "greater than" in so many people's 
imagination? 'Taint so. If it's infinitesimally greater, it takes 
infinite time to resolve the waveform. The sampling frequency must be 
enough greater so that it slips about half a cycle in the time you're 
willing to spend sampling.

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

Jerry Avins wrote:
> jjmai wrote:
> > Let's say you have a perfect sine wave at frequency w.
> > According to Nyquist, in order to be able to recover the sine wave, you
> > need to have a sampling rate of at least 2w.
> > So if you decide to sample at 2w, you end up with 2 samples for each cycle
> > of this sine wave.
> >
> > If you sample at the peaks and troughs (90 and 270 degrees) of the sine
> > wave in time (or spatial) domain, you indeed preserve the amplitude.
> > But what happens when the sampling happens at 0 and 180 degrees?  You end
> > up with only zeros.
> >
> > Am I missing something about sampling theorem here?  Is a shifted sampling
> > not the same as the origianl unshifted sampling?
>
> How does "equal" come to replace "greater than" in so many people's
> imagination? 'Taint so. If it's infinitesimally greater, it takes
> infinite time to resolve the waveform. The sampling frequency must be
> enough greater so that it slips about half a cycle in the time you're
> willing to spend sampling.
>
>
well actually a pure sine wave has zero bandwidth so if you know the
frequency, you don't need to sample it at all????

Mark

jjmai wrote:
> Let's say you have a perfect sine wave at frequency w.
> According to Nyquist, in order to be able to recover the sine wave, you
> need to have a sampling rate of at least 2w.  

This is wrong.

If you know that the signal is a perfect sine wave, all you need is 3 
samples to find the amplitude, the phase and the frequency.

VLV

Vladimir Vassilevsky wrote:
> jjmai wrote:
> > Let's say you have a perfect sine wave at frequency w.
> > According to Nyquist, in order to be able to recover the sine wave, you
> > need to have a sampling rate of at least 2w.
>
> This is wrong.
>
> If you know that the signal is a perfect sine wave, all you need is 3
> samples to find the amplitude, the phase and the frequency.

Still not quite correct.  You need 3 non-overlapping samples
spaced closer together than 2w (or other non-exact-multiples
of 2w or w).  Otherwise you could get all zeros, or a DC value,
for any sine wave.


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

jjmai wrote:
> Let's say you have a perfect sine wave at frequency w.
> According to Nyquist, in order to be able to recover the sine wave, you
> need to have a sampling rate of at least 2w.
> So if you decide to sample at 2w, you end up with 2 samples for each cycle
> of this sine wave.

However the time it takes to recover the amplitude is proportional
to the reciprocal of how close your sampling rate is to 2w.  So if
you sample at 2w you will only need an infinite number of samples.

With infinite samples beyond the life of the universe, any Planck-size
clock jitter, drift or galactic decay in any universal time constants
such as the speed of light (etc.) will eventually expose the amplitude
of your sine wave, if you are willing to wait till "past" the end of
time.
(maybe... :)


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

jjmai wrote:

> Let's say you have a perfect sine wave at frequency w.
> According to Nyquist, in order to be able to recover the sine wave, you
> need to have a sampling rate of at least 2w.  
> So if you decide to sample at 2w, you end up with 2 samples for each cycle
> of this sine wave.
> 
> If you sample at the peaks and troughs (90 and 270 degrees) of the sine
> wave in time (or spatial) domain, you indeed preserve the amplitude.
> But what happens when the sampling happens at 0 and 180 degrees?  You end
> up with only zeros.
> 
> Am I missing something about sampling theorem here?  Is a shifted sampling
> not the same as the origianl unshifted sampling?
> 
> 
This question, and more, are answered in my article "Sampling: What 
Nyquist Didn't Say, and What to Do About It", to be found at 
http://www.wescottdesign.com/articles/Sampling/sampling.html.

I hope it helps.

-- 

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

"Tim Wescott" <tim@seemywebsite.com> wrote in message
news:64ednf87dpLYVxXYnZ2dnUVZ_tadnZ2d@web-ster.com...
> jjmai wrote:
>
> > Let's say you have a perfect sine wave at frequency w.
> > According to Nyquist, in order to be able to recover the sine wave, you
> > need to have a sampling rate of at least 2w.
> > So if you decide to sample at 2w, you end up with 2 samples for each
cycle
> > of this sine wave.
> >
> > If you sample at the peaks and troughs (90 and 270 degrees) of the sine
> > wave in time (or spatial) domain, you indeed preserve the amplitude.
> > But what happens when the sampling happens at 0 and 180 degrees?  You
end
> > up with only zeros.
> >
> > Am I missing something about sampling theorem here?  Is a shifted
sampling
> > not the same as the origianl unshifted sampling?
> >
> >
> This question, and more, are answered in my article "Sampling: What
> Nyquist Didn't Say, and What to Do About It", to be found at
> http://www.wescottdesign.com/articles/Sampling/sampling.html.
>
> I hope it helps.
>
> -- 
>
> 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

Actually I doubt Nyquist said much at all - it was Shannon wasn't it?


S.



-- 
Posted via a free Usenet account from http://www.teranews.com

Sanctus wrote:

> "Tim Wescott" <tim@seemywebsite.com> wrote in message
> news:64ednf87dpLYVxXYnZ2dnUVZ_tadnZ2d@web-ster.com...
> 
>>jjmai wrote:
>>
>>
>>>Let's say you have a perfect sine wave at frequency w.
>>>According to Nyquist, in order to be able to recover the sine wave, you
>>>need to have a sampling rate of at least 2w.
>>>So if you decide to sample at 2w, you end up with 2 samples for each
> 
> cycle
> 
>>>of this sine wave.
>>>
>>>If you sample at the peaks and troughs (90 and 270 degrees) of the sine
>>>wave in time (or spatial) domain, you indeed preserve the amplitude.
>>>But what happens when the sampling happens at 0 and 180 degrees?  You
> 
> end
> 
>>>up with only zeros.
>>>
>>>Am I missing something about sampling theorem here?  Is a shifted
> 
> sampling
> 
>>>not the same as the origianl unshifted sampling?
>>>
>>>
>>
>>This question, and more, are answered in my article "Sampling: What
>>Nyquist Didn't Say, and What to Do About It", to be found at
>>http://www.wescottdesign.com/articles/Sampling/sampling.html.
>>
>>I hope it helps.
>>
>>-- 
>>
>>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
> 
> 
> Actually I doubt Nyquist said much at all - it was Shannon wasn't it?
> 
yea yea yea.  Officially it's the Nyquist-Shannon theorem, or the 
Shannon Sampling Theorem.

The title (the whole article, in fact) was a response to a spate of 
newsgroup postings that started with "Nyquist says" and ended with some 
naive conclusion that comes from taking an absolute limit as a design guide.

-- 

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