Forums

Nyquist frequency in FFTs

Started by Shafik September 24, 2004
"Jerry Avins" <jya@ieee.org> wrote in message 
news:4156eeab$0$4033$61fed72c@news.rcn.com...
> Fred Marshall wrote: > >> "Shafik" <shafik@u.arizona.edu> wrote in message >> news:1096146865.755417.30390@k17g2000odb.googlegroups.com... >> >>>How come F[n/2] must be real? >>> >>>I can have a sequence -1, 1, -1, 1, -1, 1, -1, 1 >>>or: 1, -1, 1, -1, 1, -1, 1, -1 >>> >>>those are nyquist frequencies that are 180 degrees out of phase....or >>>am I wrong? > > I can't figure out what's meant either. Is the offset between the two > lines significant? How? > >> I can't tell whether you mean the sequence is in time or frequency. >> **That's not too important because of duality.... >> >> I don't know what you mean by 180 degrees out of phase in the context of >> this sequence. >> **That *is* important. It's not possible to talk about phase at more >> than one frequency at a time without some sort of notation convention. >> One might imagine but that would be conjecture. >> >> I don't know what you mean by "these are Nyquist frequencies".. >> but the last comment implies the sequence is in frequency and that >> implies that the fifth element of the sequence is at fs/2 which is THE >> Nyquist frequency. >> >> The fifth element of the sequence is non-zero. So, that means the >> sequence represents one that is not of bandwidth limited to LESS THAN >> fs/2 - which is the Nyquist criterion. Further, the value at fs/2 isn't >> negligible. Accordingly, there can be problems with such a sequence in >> reconstruction. >> >> Fred > > One can go overboard with LESS THAN. A frequency at fs/2 doesn't meet > the Nyquist criterion because can't be accurately reconstructed, but > neither does it invalidate the sample set, because it doesn't alias.
Jerry, I'd be willing to go along with the latter perhaps but you'll have to give me an example. It's not clear to me what you mean here when you say "it doesn't alias". I think the Nyquist criterion says that the data being sampled has to have frequencies below fs/2. Are you suggesting that this really isn't necessary? [And, yes I'm talking about lowpass situations - not bandpass just to keep it simple]. We saw an example some time back of a series that, upon reconstruction, would blow up. It was a sine at fs/2 that switched to -sine at t=0 and sampled at its peaks . Well .... OK, it was an infinite sequence but it still blew up with a finite version of it. I was surprised by that result. Before that I'd figured that any sequence of samples must perforce represent a bandlimited signal - aliasing and all. But that example raised a new issue - one of the reconstruction not behaving. So, if you can be sure that the samples will reconstruct then OK. But how can you be sure? Fred
Fred Marshall wrote:

> "Jerry Avins" <jya@ieee.org> wrote
...
>>One can go overboard with LESS THAN. A frequency at fs/2 doesn't meet >>the Nyquist criterion because can't be accurately reconstructed, but >>neither does it invalidate the sample set, because it doesn't alias. > > > Jerry, > > I'd be willing to go along with the latter perhaps but you'll have to give > me an example. It's not clear to me what you mean here when you say "it > doesn't alias". > > I think the Nyquist criterion says that the data being sampled has to have > frequencies below fs/2. Are you suggesting that this really isn't > necessary? [And, yes I'm talking about lowpass situations - not bandpass > just to keep it simple]. We saw an example some time back of a series that, > upon reconstruction, would blow up. It was a sine at fs/2 that switched > to -sine at t=0 and sampled at its peaks . Well .... OK, it was an infinite > sequence but it still blew up with a finite version of it. > > I was surprised by that result. Before that I'd figured that any sequence > of samples must perforce represent a bandlimited signal - aliasing and all. > But that example raised a new issue - one of the reconstruction not > behaving. So, if you can be sure that the samples will reconstruct then OK. > But how can you be sure? > > Fred
Maybe it's only ignorance that makes me sure. A sine at Fs/2 contributes nothing to the sample, no matter how large it may be. The cosine at that frequency is accurately sampled. The upshot of that is that we can't know from the samples how large the Fs/2 signal might be. As for not aliasing, consider that a signal at Fs/2 + delta is aliased to Fs/2 - delta. When delta is zero, the alias is itself. Jerry (aka Jerry) -- ... they proceeded on the sound principle that the magnitude of a lie always contains a certain factor of credibility, ... and that therefor ... they more easily fall victim to a big lie than to a little one ... A. H. &#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;
Fred Marshall wrote:
...
> I think the Nyquist criterion says that the data being sampled has to have > frequencies below fs/2. Are you suggesting that this really isn't > necessary? [And, yes I'm talking about lowpass situations - not bandpass > just to keep it simple]. We saw an example some time back of a series that, > upon reconstruction, would blow up. It was a sine at fs/2 that switched > to -sine at t=0 and sampled at its peaks . Well .... OK, it was an infinite > sequence but it still blew up with a finite version of it. > > I was surprised by that result. Before that I'd figured that any sequence > of samples must perforce represent a bandlimited signal - aliasing and all. > But that example raised a new issue - one of the reconstruction not > behaving. So, if you can be sure that the samples will reconstruct then OK. > But how can you be sure? > > Fred > >
Fred, since when we just recently talked about this I am plagued by the same question. That sequence in question is (for all who just popped in): { ..., -1, 1, -1, 1, 1, -1, 1, -1, ...} (Olli's sequence) Ellipses mark an extension into infinity with repetitive +1 and -1. The crux is the middle (1,1) pair. If you assume rectangular bandlimitation for interpolation (ie. sinc kernel interpolation), the interpolation diverges almost everywhere. If you use triangular bandlimitation for interpolation which uses a kernel with magnitude proportional to |sinc|^2, the interpolation sum converges (though I don't know to what). It looks like there is some kind of Nyquist-Frequency catastrophe happening with the above sequence. I would guess it has a pole of first order at Nyquist if we could talk about its spectrum - which leads us to the second problem: To talk of a spectrum of on infinite sequence, the DTFT of this sequence needs to exist. This does not seem to be the case here. Furthermore, if you change only finitely many entries (by setting them zero for example) in Olli's sequence, the sinc-interpolation still diverges. Are there any other sequences which are not just a modification of Olli's sequence which do not converge with sinc-interpolation? Conversely, what is a necessary and sufficient condition under which sinc-interpolation converges? This should be covered in any text book about dsp - Rick? Regards, Andor
"Andor Bariska" <an2or@nospam.net> wrote in message 
news:41582f41$1@pfaff2.ethz.ch...
> Fred Marshall wrote: > ... >> I think the Nyquist criterion says that the data being sampled has to >> have frequencies below fs/2. Are you suggesting that this really isn't >> necessary? [And, yes I'm talking about lowpass situations - not bandpass >> just to keep it simple]. We saw an example some time back of a series >> that, upon reconstruction, would blow up. It was a sine at fs/2 that >> switched to -sine at t=0 and sampled at its peaks . Well .... OK, it was >> an infinite sequence but it still blew up with a finite version of it. >> >> I was surprised by that result. Before that I'd figured that any >> sequence of samples must perforce represent a bandlimited signal - >> aliasing and all. But that example raised a new issue - one of the >> reconstruction not behaving. So, if you can be sure that the samples >> will reconstruct then OK. But how can you be sure? >> >> Fred > > Fred, since when we just recently talked about this I am plagued by the > same question. That sequence in question is (for all who just popped in): > > { ..., -1, 1, -1, 1, 1, -1, 1, -1, ...} (Olli's sequence) > > Ellipses mark an extension into infinity with repetitive +1 and -1. The > crux is the middle (1,1) pair. If you assume rectangular bandlimitation > for interpolation (ie. sinc kernel interpolation), the interpolation > diverges almost everywhere. > > If you use triangular bandlimitation for interpolation which uses a kernel > with magnitude proportional to |sinc|^2, the interpolation sum converges > (though I don't know to what). It looks like there is some kind of > Nyquist-Frequency catastrophe happening with the above sequence. I would > guess it has a pole of first order at Nyquist if we could talk about its > spectrum - which leads us to the second problem: To talk of a spectrum of > on infinite sequence, the DTFT of this sequence needs to exist. This does > not seem to be the case here. > > Furthermore, if you change only finitely many entries (by setting them > zero for example) in Olli's sequence, the sinc-interpolation still > diverges. Are there any other sequences which are not just a modification > of Olli's sequence which do not converge with sinc-interpolation? > Conversely, what is a necessary and sufficient condition under which > sinc-interpolation converges? This should be covered in any text book > about dsp - Rick?
Andor, Well said. Fred
Fred Marshall wrote:
> "Andor Bariska" <an2or@nospam.net> wrote in message
...
>>Furthermore, if you change only finitely many entries (by setting them >>zero for example) in Olli's sequence, the sinc-interpolation still >>diverges. Are there any other sequences which are not just a modification >>of Olli's sequence which do not converge with sinc-interpolation? >>Conversely, what is a necessary and sufficient condition under which >>sinc-interpolation converges? This should be covered in any text book >>about dsp - Rick? > > > Andor, > > Well said. > > Fred
You guys have no pity. Jerry -- ... they proceeded on the sound principle that the magnitude of a lie always contains a certain factor of credibility, ... and that therefor ... they more easily fall victim to a big lie than to a little one ... A. H. &#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;&#2013266071;
In article <1096146865.755417.30390@k17g2000odb.googlegroups.com>,
Shafik <shafik@u.arizona.edu> wrote:
>How come F[n/2] must be real? > >I can have a sequence -1, 1, -1, 1, -1, 1, -1, 1 >or: 1, -1, 1, -1, 1, -1, 1, -1 > >those are nyquist frequencies that are 180 degrees out of phase.... >or am I wrong?
Assuming you meant: :I can have a sequence -1, 1, -1, 1, -1, 1, -1, 1, ... :or: 1, -1, 1, -1, 1, -1, 1, -1, ... They are 180 degrees out of phase, which is the same as negating a vector. So a real component would stay real (but negated). Moved 90 degrees out of phase, your sequence would become: : 0, 0, 0, 0, 0, 0, 0, 0, ... Which produces a nil Imaginary F[n/2] component. No need to store a guaranteed zero (for an all Real input sequence). IMHO. YMMV. -- Ron Nicholson rhn AT nicholson DOT com http://www.nicholson.com/rhn/ #include <canonical.disclaimer> // only my own opinions, etc.