Forums

Nyquist????

Started by krish_dsp October 26, 2008
Greg Berchin wrote:
> 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.
No loss of generality, but a significant loss of simplicity. BTW, are those sines and cosines waves, or particles? :-) 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;
On Mon, 27 Oct 2008 12:53:46 -0400, Jerry Avins <jya@ieee.org> wrote:

>BTW, are those sines and cosines waves, or particles? :-)
Yes. -- Greg
Fred Marshall wrote:
> 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.
I think that Tim had in mind the possibility that the sampled signal was not baseband. The sampled signal, represented by (close enough) deltas without any filtering, has a baseband component and images at higher frequencies, some of which are spectrally inverted. Provided the original signal was suitably bandlimited for the sampling regime, *any one* of those images might have been the original signal. I say to Tim that none of the signal is discarded, but I agree with him that details of sampling process must be known in order to affect reconstruction. 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;
Greg Berchin wrote:
> On Mon, 27 Oct 2008 12:53:46 -0400, Jerry Avins <jya@ieee.org> wrote: > >> BTW, are those sines and cosines waves, or particles? :-)
I was ahead of you there, Greg. Note the (very significant) comma. 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;
Jerry Avins wrote:
> Fred Marshall wrote: >> 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. > > I think that Tim had in mind the possibility that the sampled signal > was not baseband. The sampled signal, represented by (close enough) > deltas without any filtering, has a baseband component and images at > higher frequencies, some of which are spectrally inverted. Provided > the original signal was suitably bandlimited for the sampling regime, > *any one* of those images might have been the original signal. I say > to Tim that none of the signal is discarded, but I agree with him > that details of sampling process must be known in order to affect > reconstruction. > Jerry
Jerry, If that's the case then there'd at least have to be I and Q sequences wouldn't there? In that case the *original* center frequency has been thrown out and can be made to be anything one likes. It's like a "don't care" parameter. Well ... at least that's an argument. Fred
Jerry Avins wrote:
> Fred Marshall wrote: >> 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. > > I think that Tim had in mind the possibility that the sampled signal was > not baseband. The sampled signal, represented by (close enough) deltas > without any filtering, has a baseband component and images at higher > frequencies, some of which are spectrally inverted. Provided the > original signal was suitably bandlimited for the sampling regime, *any > one* of those images might have been the original signal. I say to Tim > that none of the signal is discarded, but I agree with him that details > of sampling process must be known in order to affect reconstruction. > > Jerry
Actually Tim did have that in mind, as well as the possibility that the original signal _wasn't_ adequately bandlimited, or that perhaps it was intentionally allowed to alias around the Nyquist rate because that was a frequency band that was unimportant (imagine a signal where you only care about the energy from 300-3000Hz; you sample at 8kHz, with significant content up to but not beyond 5kHz, then you low-pass in the digital world down to 3kHz). Tim's point being that if he gave you his hypothetical sampled signal and said "the original signal had no content beyond XYZ" then you could reconstruct it. But if he gave you the same thing and he don't _tell_ you that the signal that he was sampling goes from 7.1203MHz to 7.123MHz, then you have no clue of how to go about reconstructing what he sampled. -- 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
Fred Marshall wrote:
> 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.
But given just the sampled signal you could not say if, and to what degree, it was bandlimited, nor could you say if it was downsampled. So, for the purpose of arguing that information is _always_ lost in the sampling process, I would say that you could _not_ guarantee a reconstruction of the original signal from the sampled one without having some information handed to you alongside the bare sampled signal. -- 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
Fred Marshall wrote:
> Jerry Avins wrote: >> Fred Marshall wrote: >>> 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. >> I think that Tim had in mind the possibility that the sampled signal >> was not baseband. The sampled signal, represented by (close enough) >> deltas without any filtering, has a baseband component and images at >> higher frequencies, some of which are spectrally inverted. Provided >> the original signal was suitably bandlimited for the sampling regime, >> *any one* of those images might have been the original signal. I say >> to Tim that none of the signal is discarded, but I agree with him >> that details of sampling process must be known in order to affect >> reconstruction. >> Jerry > > Jerry, > > If that's the case then there'd at least have to be I and Q sequences > wouldn't there? In that case the *original* center frequency has been > thrown out and can be made to be anything one likes. It's like a "don't > care" parameter. Well ... at least that's an argument.
Fred, I made two assertions. For which one are the quadrature pairs needed? In either case, if quadrature sampling had been done, wouldn't that be evident? 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;
On Oct 25, 11:26&#2013266080;pm, "krish_dsp" <nmkr...@gmail.com> wrote:
> Hi everyone, > What is the dirrerence between nyquist frequency, nyquist rate and nyquist > bandwidth? I am sure that they dont point to the same concept. I know what > is a nyquist frequency. But I am not able to understand the concept of the > other two. > I came across these terms while learning about pulse shaping filters > (raised cosine filters). I tried to understand, but still I am not able to > completely do it. > Can anyone explain me or say the difference between these terms and what > they mean?
i haven't come across the term "Nyquist bandwidth" before. we could complicate the issue with discussion of sampling bandpass (or "non- baseband) signals, but if it's regular old baseband signals we're sampling, "Nyquist rate" is pretty much always twice the bandlimit (or bandwidth) of the signal. the open lower bound of theoretically sufficient sampling rates for a given signal or signal class. there are disagreements about precisely the meaning of "Nyquist frequency". the good guys say it's always 1/2 of the sampling rate, an open upper bound of acceptable frequency content in the signal to be sampled to avoid aliasing. in normalized radian frequency, Nyquist frequency = pi. but some authors (notably O&S, which i found disappointing) define it as 1/2 the Nyquist rate. essentially the bandwidth or bandlimit of the signal to be sampled. Rick Lyons' book didn't take a side on the issue (which might have been smart). r b-j
On Mon, 27 Oct 2008 17:31:08 -0700 (PDT), robert bristow-johnson
<rbj@audioimagination.com> wrote:

>the good guys say it's always 1/2 of the sampling rate,
The "good guys"? I guess basing the Nyquist Frequency on characteristics of the signal to be sampled must therefore be "elitist". :-)