Raymond Toy wrote:>>>>>>"Fred" == Fred Marshall <fmarshallx@remove_the_x.acm.org> writes: > > > Fred> If you don't like to use infinite sincs then I bet you will find that > Fred> suitably windowed sincs will support the same arguments. > > > Is there not a theorem that says you can't have both a time-limited > and a band-limited signal? Something about the time-bandwidth product > being constant? > > RayDoesn't the development of that theorem assumes that the signals are finite? If a clever mathematician can chop a sphere into an infinite number of pieces, then assemble those pieces to make two spheres the same size and density as the original, anything goes. If course, the scheme to do this with gold spheres founders on the indivisibility of gold atoms. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Band limited function zero crossings, how many?
Started by ●February 18, 2004
Reply by ●February 19, 20042004-02-19
Reply by ●February 19, 20042004-02-19
"Jerry Avins" <jya@ieee.org> wrote in message news:4034dc9e$0$3099$61fed72c@news.rcn.com...> Raymond Toy wrote: > > >>>>>>"Fred" == Fred Marshall <fmarshallx@remove_the_x.acm.org> writes: > > > > > > Fred> If you don't like to use infinite sincs then I bet you willfind that> > Fred> suitably windowed sincs will support the same arguments. > > > > > > Is there not a theorem that says you can't have both a time-limited > > and a band-limited signal? Something about the time-bandwidth product > > being constant? > > > > Ray > > Doesn't the development of that theorem assumes that the signals are > finite? If a clever mathematician can chop a sphere into an infinite > number of pieces, then assemble those pieces to make two spheres the > same size and density as the original, anything goes. If course, the > scheme to do this with gold spheres founders on the indivisibility of > gold atoms.Jerry, Do you mean TW product analyses? Well, if I get your meaning and interpret it a little, Yes. There's a measure of T and a measure of W that has some meaning in the analysis. For example, a CW pulse has T=length of the pulse and W=some measure of the corresponding sinc width which is like 1/T so that TW=1 for a CW pulse. Obviously W isn't infinite although the sinc has infinite extent in the analytical sense. And the real world waveform might well possibly have greater "significant" bandwidth than W depending on your definition of what is significant. e.g. if you are trying to use two frequencies at once that are close together..... then significant bandwidth could well be greater than W in order to prevent cross coupling / spectral overlap. So, W is a scalar width measure of an infinitely wide function - and if you were to plot it as a box in the right place in frequency then it would "look" finite. But I would be at least a little careful about calling that "finite"..... Same thing for an FM pulse. T is the length of the pulse and W is something like the range of the frequency sweep - which is "most of" the bandwidth but not all of it. The whole thing is about time/frequency resolution. The higher the TW product, the better the combined resolution in some sense - and I will leave that "sense" to the experts.... I don't know about spheres.... Fred
Reply by ●February 19, 20042004-02-19
Matt Timmermans wrote:> For a lightish introduction to the good stuff, see: > See http://www.math.ucdavis.edu/%7Esaito/courses/ACHA/slepian83.pdf. >Lightish, eh? :-) Nonetheless fascinationg. I particularly like his discussion of fuzzy models and his views on models in general. It seems to me that a lot of our discussion here gets hung up on what he calls the "secondary constructs" and the way he sweeps them under the rug in favor of the salient features is particularly appropriate to engineering. Do you know of applications wherein the prolate spheroid decomposition has been used? Bob -- "Things should be described as simply as possible, but no simpler." A. Einstein
Reply by ●February 19, 20042004-02-19
Raymond Toy wrote:>>>>>>"Fred" == Fred Marshall <fmarshallx@remove_the_x.acm.org> writes: > > > Fred> If you don't like to use infinite sincs then I bet you will find that > Fred> suitably windowed sincs will support the same arguments. > > > Is there not a theorem that says you can't have both a time-limited > and a band-limited signal? Something about the time-bandwidth product > being constant? >Check out this paper that Matt just posted a link to for a beautiful view of that situation. http://www.math.ucdavis.edu/%7Esaito/courses/ACHA/slepian83.pdf Bob -- "Things should be described as simply as possible, but no simpler." A. Einstein
Reply by ●February 19, 20042004-02-19
On Thu, 19 Feb 2004 06:59:37 -0800, "Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote:>> >>"Greg Berchin" <Bank-to-Turn@mchsi.com> wrote in message >>news:u5h930lgfso7o04up2lvh0ait9rnnjqck9@4ax.com... >>> Do not forget that the zeroes of a function are not necessarily >>> real. For zeroes in the complex plane, the concepts of "positive" >>> and "negative" are not so well defined. >>> >> >>NB: I was very careful to say that we were talking about zero crossings (of >>a real function) and *not* the zeros of a function of a complex variable. >>So, something like f(t) and not H(s). OK?OK. There may be a semantic problem here, though. A real function can have zeroes that are complex: y(t) = x(t) + x(t-1) + x(t-2) for example, has a complex nontrivial zeroes. I am having some trouble distinguishing between what you mean and what this example represents. The upper limit established in the remainder of that message still applies, though. Greg Berchin
Reply by ●February 19, 20042004-02-19
"Greg Berchin" <Bank-to-Turn@mchsi.com> wrote in message news:4via30he48712mcd8msv1u7oom2h5lt7l5@4ax.com...> On Thu, 19 Feb 2004 06:59:37 -0800, "Fred Marshall" > <fmarshallx@remove_the_x.acm.org> wrote: > > >> > >>"Greg Berchin" <Bank-to-Turn@mchsi.com> wrote in message > >>news:u5h930lgfso7o04up2lvh0ait9rnnjqck9@4ax.com... > >>> Do not forget that the zeroes of a function are not necessarily > >>> real. For zeroes in the complex plane, the concepts of "positive" > >>> and "negative" are not so well defined. > >>> > >> > >>NB: I was very careful to say that we were talking about zero crossings(of> >>a real function) and *not* the zeros of a function of a complexvariable.> >>So, something like f(t) and not H(s). OK? > > OK. There may be a semantic problem here, though. A real > function can have zeroes that are complex: > y(t) = x(t) + x(t-1) + x(t-2)OK - so I guess I'd have to learn how to express the thought. Real zeros of y? Zeros of y evaluated on the real x axis? Observable zeros of y in the reals? I hope the meaning is clear now. How is it best expressed? Fred
Reply by ●February 19, 20042004-02-19
"Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message news:9YKdnT4TL8KyRandRVn-uQ@centurytel.net...> Meaning that the larger the family or set of them you use, the closer you > get..... The Fourier Series still has its Gibbs phenomenon. Do the PSWFs > have some equally nice way to characterise the remaining error? Iprobably> should know..... but don't. > (I'm not sure everyone knows what "complete" means.....)I don't know either -- that's an interesting question. Just from looking at a few graphs, I would guess that inside the interval, it looks a lot like Gibbs, but getting more pronounced near the interval's ends. Outside the interval, I have no idea.
Reply by ●February 19, 20042004-02-19
"Bob Cain" <arcane@arcanemethods.com> wrote in message news:c12t9o0ddg@enews3.newsguy.com...> Do you know of applications wherein the prolate spheroid > decomposition has been used?The most common context in which I see them mentioned is in designing linear predcitors for bandlimited signals. Others, I would have to google for. See, for example: http://www.ece.nmsu.edu/~rlyman/flatpred3.pdf And that is how I found them when I was looking for a way to design better feedback filters for sigma-delta modulators, because I got it into my head the the feedback filter is actually a predictor. I'll just cut and paste my explanation of that from a post I wrote just a few days ago, but which was buried in an early thread -- aplolgies to those who have to see it twice: Let me borrow a nice S-D modulator from an old post of Randy's: q[n] | |----| | + | + w[n] + V | x[n] ----> + -----1/z--> + --------> y[n] ^ | - | v[n] | | | | -------- | |-| H(z) |<--------- -------- This models the quantizer as a noise input, and I put in the 1/z so you could see the delay through the latch. We do the standard analysis to get Y(z) = Q(z) + W(z)/z =Q(z) + X(z)/z - V(z)/z =Q(z) + X(z)/z - H(z)Q(z)/z =X(z)/z + (1-H(z)/z)Q(z) And so we can see that the signal goes through a simple delay, while the noise gets shaped by (1-H(z)/z). One of the major design goals is to minimize (1-H(z)/z) in the target bandwidth, i.e, for z of interest, we want (~ is approximately equal to): 1-H(z)/z ~ 0 <=> H(z)/z ~ 1 <=> H(z) ~ z That is to say, H(z) is a negative delay of one sample period in the band of interest, and that's a very simple way to look at the whole S-D modulation business -- *after* it sees an error impulse, it compensates by generating an opposing impuse, and shifting its in-band components back in time by one sample to cancel the error. Matt
Reply by ●February 19, 20042004-02-19
On Thu, 19 Feb 2004 16:26:50 -0800, "Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote:>>OK - so I guess I'd have to learn how to express the thought. Real zeros of >>y? Zeros of y evaluated on the real x axis? Observable zeros of y in the >>reals? I hope the meaning is clear now. How is it best expressed?Thanks; now I understand. Kay and Sudhaker call them "real (physical) zeros". Any of your suggestions are good, as long as one understands that the imaginary part is zero. Greg Berchin
Reply by ●February 20, 20042004-02-20
Greg Berchin wrote:> On Thu, 19 Feb 2004 08:21:03 +0100, Bernhard Holzmayer > <holzmayer.bernhard@deadspam.com> wrote: > >>>Following is required for the signal then: >>>1. Every second zero crossing (ZC) has opposite direction: >>> neg->pos, pos->neg, neg->pos, ... >>>2. Derivate dA/dt (slope) changes sign between consequent ZCs. >>>3. Second derivate has ZCs between consequent ZCs >>> of the signal. > > Do not forget that the zeroes of a function are not necessarily > real. For zeroes in the complex plane, the concepts of "positive" > and "negative" are not so well defined. >Thanks for the hint. I thought that a function, if it is complex, it can be described by a real part and an imaginary part, and that, because both are orthogonal, they can be treated independently. But I guess, you're right, it's too much simplification. But now a question arises: if we talk of complex solutions or even complex functions, would 0+5i or 5+0j be called a zero crossing? Or only the value 0+0i ? Bernhard
