Forums

Nyquist????

Started by krish_dsp October 26, 2008
On Sun, 26 Oct 2008 19:56:17 -0500, Greg Berchin wrote:

> On Sun, 26 Oct 2008 11:38:12 -0700, Tim Wescott <tim@seemywebsite.com> > wrote: > >>I'll have to take a look at that; I don't know that I'll change any >>wording though -- the audience is intended to extend to people who have >>not yet grokked complex signals, which pretty much excludes >>understanding the concept of a negative frequency (other than as >>something that's identical to a positive frequency). > > I am sympathetic, but if you ignore the negative frequencies for > baseband signals, then you are faced with having to explain their > unexpected appearance when you discuss modulated signals. > >>I would contend that the difference is semantic -- if the Nyquist >>criterion is satisfied then the information being thrown away is >>redundant. I gather that you would say (and perhaps be more >>theoretically correct) that if it's redundant it isn't really >>information. Once again, I'm trying to give tools to beginners so they >>can wrap their brains around the subject. > > The single most common question that I receive about sampling is, "What > if something important happens between the samples?" The answer is in > the statements above, interpreted either your way or mine: "Bandlimiting > of the signal so that the Nyquist Criterion is satisfied GUARANTEES that > nothing of importance can possibly occur between the samples." >
-- snip -- I guess that I'm using the phrase to illustrate what happens in general (i.e., something is always thrown away). In the _completely_ bandlimited (and physically impossible) case, nothing _unique_ is thrown away, in the _adequately_ bandlimited (and subject to interpretation) case, nothing _important_ is thrown away, and in the _inadequately_ bandlimited case (such as trying to sample speech at 8kHz without an anti-alias filter) something important is _definitely_ being thrown away. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" gives you just what it says. See details at http://www.wescottdesign.com/actfes/actfes.html
On Sun, 26 Oct 2008 07:13:57 -0500, Greg Berchin wrote:

> On Sat, 25 Oct 2008 22:35:20 -0500, Tim Wescott > <tim@justseemywebsite.com> wrote: > >>See if this helps. >> >>http://www.wescottdesign.com/articles/Sampling/sampling.html > > Tim, > > There are a few statements in that with which I disagree: >
-- snip --
> > "By ignoring anything that goes on between samples the sampling process > throws away information about the original signal." > > NOT if the Nyquist Criterion is satisfied! There is NO loss of > information. >
-- snip again -- I contend that yes, information is lost, absolutely and positively. How can I tell? Easy. I give you a sequence of numbers and their sampling instants, and ask you to reconstruct the continuous-time signal that they are a sample of. You immediately come back to me and say "Tim, what was the bandwidth of the signal and what was it centered around". If no information was lost, you wouldn't have to ask. Now, If I give you that sequence of numbers AND the limits of the band of the source signal, then you can either hand me back a reconstruction of my original signal or you can tell me that I hopelessly mangled it in the sampling process. So information is always lost. QED. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" gives you just what it says. See details at http://www.wescottdesign.com/actfes/actfes.html
Tim Wescott wrote:
> On Sun, 26 Oct 2008 07:13:57 -0500, Greg Berchin wrote: > >> On Sat, 25 Oct 2008 22:35:20 -0500, Tim Wescott >> <tim@justseemywebsite.com> wrote: >> >>> See if this helps. >>> >>> http://www.wescottdesign.com/articles/Sampling/sampling.html >> >> Tim, >> >> There are a few statements in that with which I disagree: >> > -- snip -- >> >> "By ignoring anything that goes on between samples the sampling >> process throws away information about the original signal." >> >> NOT if the Nyquist Criterion is satisfied! There is NO loss of >> information. >> > -- snip again -- > > I contend that yes, information is lost, absolutely and positively. > How can I tell? Easy. I give you a sequence of numbers and their > sampling instants, and ask you to reconstruct the continuous-time > signal that they are a sample of. > > You immediately come back to me and say "Tim, what was the bandwidth > of the signal and what was it centered around". > > If no information was lost, you wouldn't have to ask. >
If the samples were equally spaced, I wouldn't have asked. I'd interpolate with sincs or reasonable facsimile thereof that have regular zeros at the samples - or maybe some other interpolation scheme. The point is that regularly spaced samples can be treated "as if" the band limit was at 1/2T=Fs/2 Now, if the signal band limit is at less than that, no harm done. A nearly perfect lowpass, even if not as low as it might be, is a good reconstructor. What's the harm in passing a band segment with zero energy in it? Same as asking: "what's the harm in sampling at a higher frequency than necessary - and then reconstructing with a perfect lowpass matched to 1/2 the sampling frequency?" Then we could argue about whether I did a "good enough" job in interpolating - just as we could argue about whether the original signal was "well enough" bandlimited to support the resulting sampling interval. Fred
Jerry Avins wrote:
> glen herrmannsfeldt wrote:
>> So the Nyquist bandwidth is half the sampling bandwidth? >> That doesn't sound right.
> I think I understand sampling rate. What is sampling bandwidth?
The OP was asking about Nyquist frequency, Nyquist rate, and Nyquist bandwidth. If the Nyquist frequency is half the sampling frequency, then shouldn't the Nyquist bandwidth be half the sampling bandwidth? Otherwise, yes, what is the sampling bandwidth? -- glen
Tim Wescott wrote:
(snip)

> I contend that yes, information is lost, absolutely and positively. How > can I tell? Easy. I give you a sequence of numbers and their sampling > instants, and ask you to reconstruct the continuous-time signal that they > are a sample of.
> You immediately come back to me and say "Tim, what was the bandwidth of > the signal and what was it centered around".
> If no information was lost, you wouldn't have to ask.
> Now, If I give you that sequence of numbers AND the limits of the band of > the source signal, then you can either hand me back a reconstruction of > my original signal or you can tell me that I hopelessly mangled it in the > sampling process.
> So information is always lost. QED.
I am not sure it is quite fair not to include the sampling rate, but maybe the center frequency. For an infinite time signal, you now require infinity+2 values. However, (infinity+2)/(infinity)=1 so it doesn't require any more information. For finite number of samples on a periodic signal, (N+2)/N is not 1. For a finite length non-periodic signal you need one more data value. -- glen
Greg Berchin wrote:
(snip)

> I am sympathetic, but if you ignore the negative frequencies for > baseband signals, then you are faced with having to explain their > unexpected appearance when you discuss modulated signals.
It isn't that you ignore them, but they don't count toward the bandwidth. Maybe it should be non-redundant bandwidth. Consider a signal with a given bandwidth, but in that bandwidth contains three copies of the spectrum. The non-redundant bandwidth is then one third the actual bandwidth. (Though as Tim says, that information will be needed for reconstruction.) Also, consider sampling of AM-DSB signals. But the sidebands come from 2sin(a)sin(b)=cos(a-b)-cos(a+b) they are created in the modulation, not moved up from positive and negative frequencies. -- glen
On Mon, 27 Oct 2008 01:23:38 -0800, glen herrmannsfeldt
<gah@ugcs.caltech.edu> wrote:

>Also, consider sampling of AM-DSB signals. > >But the sidebands come from 2sin(a)sin(b)=cos(a-b)-cos(a+b) > >they are created in the modulation, not moved up from >positive and negative frequencies.
I'm not so sure I'd agree with that. Let's use cosine modulation instead of sine, because cos(0)=1 ... cos(a)cos(b) = &#2013266109;[cos(a-b)+cos(a+b)] ... and pretend that a baseband signal is modulated by DC. Assume that the signal being modulated is composed of several sines and cosines of various frequencies: cos(a)[cos(b)+sin(c)+...+cos(y)+sin(z)] = cos(0)[cos(b)+sin(c)+...+cos(y)+sin(z)] = cos(b)+sin(c)+...+cos(y)+sin(z) [the original signal] Now let's assume that the signal is composed of several cosines and sines of various frequencies, all of which are GREATER than the carrier frequency (associated with "cos(a)"): cos(a)[cos(b)+sin(c)+...+cos(y)+sin(z)] = cos(a)cos(b)+cos(a)sin(c)+...+cos(a)cos(y)+cos(a)sin(z) = &#2013266109;[cos(a-b)+cos(a+b)] + &#2013266109;[sin(a+c)-sin(a-c)] + ... + &#2013266109;[cos(a-y)+cos(a+y)] + &#2013266109;[sin(a+z)-sin(a-z)] In this case the lower sideband of the modulated positive frequencies falls at negative frequencies, and the upper sideband of the modulated negative frequencies falls at positive frequencies. We might consider this to be analogous to aliasing, since the envelopes of the spectra overlap. Now let's assume that the signal is composed of several cosines and sines of various frequencies, all of which are LESS than the carrier frequency (associated with "cos(a)"). Again: cos(a)[cos(b)+sin(c)+...+cos(y)+sin(z)] = cos(a)cos(b)+cos(a)sin(c)+...+cos(a)cos(y)+cos(a)sin(z) = &#2013266109;[cos(a-b)+cos(a+b)] + &#2013266109;[sin(a+c)-sin(a-c)] + ... + &#2013266109;[cos(a-y)+cos(a+y)] + &#2013266109;[sin(a+z)-sin(a-z)] In this case, however, there is no overlap between the lower sideband of the modulated positive frequencies and the upper sideband of the negative modulated frequencies. In other words, there is no loss of generality in viewing a baseband signal as a two-sided signal modulated by DC.
It seems to me that someone here recently said that the "Nyquist frequency" 
is often used to refer to two different things:

1) The Nyquist sampling frequency (lower limit)
2) The Nyquist bandwidth / frequency (upper limit)

So, I'm surprised that nobody has mentioned this yet .. if my recollection 
is correct.

In the end, Nyquist defined some "things" that I believe we all understand. 
So this is purely about semantics and it would be nice to have a definitive 
reference.

Not suggesting that it is such a reference but
From Wikipedia:
http://en.wikipedia.org/wiki/Nyquist_frequency

"Nyquist rate, as commonly used with respect to sampling, is a property of a 
continuous-time signal, not of a system, whereas Nyquist frequency is a 
property of a discrete-time system, not of a signal. "

I can understand the first half of the sentence - it makes some sense 
although I'd have to say that the same criterion could be applied to 
discrete time signals when doing things like decimation, etc.
I conclude that the distinction between continuous time and discrete time 
here is unfortunate and uncessary.

I don't understand the second half of the sentence at all.  Apparently it 
stems from
"The Nyquist frequency, named after the Swedish-American engineer Harry 
Nyquist or the Nyquist-Shannon sampling theorem, is half the sampling 
frequency of a discrete signal processing system"

...which I dont' agree with.  After all, one can sample at a rate higher 
than the Nyquist rate - always do.

Fred 


Tim Wescott wrote:

> ... in the _inadequately_ bandlimited case > (such as trying to sample speech at 8kHz without an anti-alias filter) > something important is _definitely_ being thrown away.
Worse. It's still there, but scrambled in as contamination. Jerry -- Engineering is the art of making what you want from things you can get. &macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;
glen herrmannsfeldt wrote:
> Greg Berchin wrote: > (snip) > >> I am sympathetic, but if you ignore the negative frequencies for >> baseband signals, then you are faced with having to explain their >> unexpected appearance when you discuss modulated signals. > > It isn't that you ignore them, but they don't count > toward the bandwidth. Maybe it should be non-redundant > bandwidth. > > Consider a signal with a given bandwidth, but in that > bandwidth contains three copies of the spectrum. > The non-redundant bandwidth is then one third the > actual bandwidth. (Though as Tim says, that information > will be needed for reconstruction.) > > Also, consider sampling of AM-DSB signals. > > But the sidebands come from 2sin(a)sin(b)=cos(a-b)-cos(a+b) > > they are created in the modulation, not moved up from > positive and negative frequencies.
There's the rub. It can be seen either way, and sometimes one way is more convenient or enlightening than the other. Then we start to argue about which way is more "fundamental". Jerry -- Engineering is the art of making what you want from things you can get. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;