"Ron N." <rhnlogic@yahoo.com> wrote in message news:1141190264.422694.294190@u72g2000cwu.googlegroups.com...> What is the difference in usefulness between a finite set of > samples taken from a perfectly bandlimited-signal, and a set > of samples taken from a signal that only has support within > the limits of that sample sets domain (thus not perfectly > bandlimited), but happens to be identical to the bandlimited > signal in that range?Well, "usefulness" is in the eye of the beholder. So that's too subjective for me! For example: Q: How useful is a perfectly bandlimited signal? A: Not at all useful in the real world because it doesn't exist. A: Very useful in the analytical world. Ignoring that then: A finite set of samples taken from a perfectly bandlimited signal doesn't represent the perfectly bandlimited signal any more because of the window. A reconstruction will have errors - although they may be small. Then, "a set of samples taken from a signal that only has support within the limits of that sample sets domain (thus not perfectly bandlimited" could mean samples taken of the windowed signal above - which would make both sets of samples the same and the difference between them zero in all respects. You can't tell the difference. Fred
Online · Live + on demand
February 9-11, 2027
Price goes up Nov 1: register now to lock in today's rate
Register Now →
Reconstruction and interpolation from irregularly spaced data
Started by ●February 28, 2006
Reply by ●March 1, 20062006-03-01
Reply by ●March 1, 20062006-03-01
Fred Marshall wrote:> "Ron N." <rhnlogic@yahoo.com> wrote in message > news:1141190264.422694.294190@u72g2000cwu.googlegroups.com... > > >>What is the difference in usefulness between a finite set of >>samples taken from a perfectly bandlimited-signal, and a set >>of samples taken from a signal that only has support within >>the limits of that sample sets domain (thus not perfectly >>bandlimited), but happens to be identical to the bandlimited >>signal in that range? > > > Well, "usefulness" is in the eye of the beholder. So that's too subjective > for me! > For example: > Q: How useful is a perfectly bandlimited signal? > A: Not at all useful in the real world because it doesn't exist. > A: Very useful in the analytical world. > > Ignoring that then: > > A finite set of samples taken from a perfectly bandlimited signal doesn't > represent the perfectly bandlimited signal any more because of the window. > A reconstruction will have errors - although they may be small. > > Then, "a set of samples taken from a signal that only has support within the > limits of that sample sets domain (thus not perfectly bandlimited" could > mean samples taken of the windowed signal above - which would make both sets > of samples the same and the difference between them zero in all respects. > You can't tell the difference.Shall we have another brouhaha? Suppose that we have one period of the bandlimited signal. (Not every band-limited signal is periodic, but some are.) Then the reconstruction of that signal will be precise. The reconstruction of the signal with limited support will necessarily be identical to it, and therefore incorrect. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●March 1, 20062006-03-01
"Jerry Avins" <jya@ieee.org> wrote in message news:6dKdnUG8hsK7W5jZRVn-qA@rcn.net...> Fred Marshall wrote: >> "Ron N." <rhnlogic@yahoo.com> wrote in message >> news:1141190264.422694.294190@u72g2000cwu.googlegroups.com... >> >> >>>What is the difference in usefulness between a finite set of >>>samples taken from a perfectly bandlimited-signal, and a set >>>of samples taken from a signal that only has support within >>>the limits of that sample sets domain (thus not perfectly >>>bandlimited), but happens to be identical to the bandlimited >>>signal in that range? >> >> >> Well, "usefulness" is in the eye of the beholder. So that's too >> subjective for me! >> For example: >> Q: How useful is a perfectly bandlimited signal? >> A: Not at all useful in the real world because it doesn't exist. >> A: Very useful in the analytical world. >> >> Ignoring that then: >> >> A finite set of samples taken from a perfectly bandlimited signal doesn't >> represent the perfectly bandlimited signal any more because of the >> window. A reconstruction will have errors - although they may be small. >> >> Then, "a set of samples taken from a signal that only has support within >> the limits of that sample sets domain (thus not perfectly bandlimited" >> could mean samples taken of the windowed signal above - which would make >> both sets of samples the same and the difference between them zero in all >> respects. You can't tell the difference. > > Shall we have another brouhaha? Suppose that we have one period of the > bandlimited signal. (Not every band-limited signal is periodic, but some > are.) Then the reconstruction of that signal will be precise. The > reconstruction of the signal with limited support will necessarily be > identical to it, and therefore incorrect. > > JerryJerry, Maybe precise and maybe not. In the analytical world you always need an infinite number of samples for a strictly bandlimited signal unless, of course, that it's also defined to be periodic in which case samples over a single period are OK - *but* don't they have to divide the period into an integer number of segments? You didn't mention that. I think Jean Baptiste did it that way on purpose in his series. Fred
Reply by ●March 1, 20062006-03-01
Fred Marshall wrote: ...> Maybe precise and maybe not. In the analytical world you always need an > infinite number of samples for a strictly bandlimited signal unless, of > course, that it's also defined to be periodic in which case samples over a > single period are OK - *but* don't they have to divide the period into an > integer number of segments? You didn't mention that. I think Jean Baptiste > did it that way on purpose in his series.Hello Fred, did you read that last thread I initiated titled "DSP riddle"? In it I explained how to perfectly reconstruct non-bandlimited periodic functions (with known period and bandwidth, if it is finite) from a sampling with arbitrary low sample rate. I thought it was quite neat :-). Part of the solution to the riddle was that bandlimited periodic functions can also be perfectly reconstructed from the same number of samples as would be needed by critically sampling one period of the function, however with arbitrarily low sampling rate. The trick was to chose the sampling period incommensurate with the function period. Regards, Andor
Reply by ●March 1, 20062006-03-01
Fred Marshall wrote:> A finite set of samples taken from a perfectly bandlimited signal doesn't > represent the perfectly bandlimited signal any more because of the window. > A reconstruction will have errors - although they may be small.I seem to recall some of my analysis textbooks use limits instead of infinities to describe some things. Perhaps something like: if, for a large enough N=Delta of samples, the reconstruction error can be made smaller than E=Epsilon, then the signal can be represented. Perhaps, in the real world, E is not measureable above the noise (quantum, thermal, jitter, quantization, numeric precision, etc.) for some very realistic sized values of N (1k, 1 CD, MTBF of human civilization, etc.)? If one can't measure whether a signal isn't perfectly bandlimited, why not assume that it is for most practical purposes? Or the OP needs to state whether the samples and FIR filter can be infinite in precision as well as length. I'd hate to wait that long for the filter output delay. IMHO. YMMV. -- rhn A.T nicholson d.0.t C-0-M
Reply by ●March 1, 20062006-03-01
Andor wrote:> The trick was to chose the sampling period incommensurate with the > function period.And to notice the requirement that the signal be continuous as well as periodic. IMHO. YMMV. -- rhn A.T nicholson d.0.t C-o-M
Reply by ●March 1, 20062006-03-01
Ron N. schrieb:> Andor wrote: > > The trick was to chose the sampling period incommensurate with the > > function period. > > And to notice the requirement that the signal be continuous as well > as periodic.Indeed. BTW, non-continous periodic functions can also not be reconstructed at their point of discontinuity from the Fourier series. However, it turns out that if all you have is continuity of the periodic function (as I required for my riddle), you still cannot necessarily reconstruct it as the limit of truncated Fourier series. There are examples where this procedure does not converge (for certain points). To guarantee reconstruction from the Fourier series you have to average the first N sinusoids of the truncated series and let N go to infinity (C=E8saro convergence). For discontinuous functions, this procedure will arrive at the arithmetic mean of the left and right limit at points of discontinuity. Regards, Andor
Reply by ●March 1, 20062006-03-01
Fred Marshall wrote:> "Jerry Avins" <jya@ieee.org> wrote in message > news:6dKdnUG8hsK7W5jZRVn-qA@rcn.net...>> ... Suppose that we have one period of the >>bandlimited signal. (Not every band-limited signal is periodic, but some >>are.) Then the reconstruction of that signal will be precise. The >>reconstruction of the signal with limited support will necessarily be >>identical to it, and therefore incorrect....> Maybe precise and maybe not. In the analytical world you always need an > infinite number of samples for a strictly bandlimited signal unless, of > course, that it's also defined to be periodic in which case samples over a > single period are OK - *but* don't they have to divide the period into an > integer number of segments? You didn't mention that. I think Jean Baptiste > did it that way on purpose in his series.I originally wrote "Suppose that we have exactly one period ...", but edited "exactly" for (ahem) brevity. I should have left it. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●March 1, 20062006-03-01
Andor wrote:> Fred Marshall wrote: > ... > >>Maybe precise and maybe not. In the analytical world you always need an >>infinite number of samples for a strictly bandlimited signal unless, of >>course, that it's also defined to be periodic in which case samples over a >>single period are OK - *but* don't they have to divide the period into an >>integer number of segments? You didn't mention that. I think Jean Baptiste >>did it that way on purpose in his series. > > > Hello Fred, > > did you read that last thread I initiated titled "DSP riddle"? In it I > explained how to perfectly reconstruct non-bandlimited periodic > functions (with known period and bandwidth, if it is finite) from a > sampling with arbitrary low sample rate. I thought it was quite neat > :-). > > Part of the solution to the riddle was that bandlimited periodic > functions can also be perfectly reconstructed from the same number of > samples as would be needed by critically sampling one period of the > function, however with arbitrarily low sampling rate. > > The trick was to chose the sampling period incommensurate with the > function period.But some bandlimited signals have no period. An example is sin(t) + sin(e*t). Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●March 1, 20062006-03-01
"Jerry Avins" <jya@ieee.org> wrote in message news:Js2dnbSdIr5IbJjZRVn-qA@rcn.net...> Fred Marshall wrote: >> "Jerry Avins" <jya@ieee.org> wrote in message >> news:6dKdnUG8hsK7W5jZRVn-qA@rcn.net... > > >>> ... Suppose that we have one period of the >>> bandlimited signal. (Not every band-limited signal is periodic, but some >>> are.) Then the reconstruction of that signal will be precise. The >>> reconstruction of the signal with limited support will necessarily be >>> identical to it, and therefore incorrect. > > ... > >> Maybe precise and maybe not. In the analytical world you always need an >> infinite number of samples for a strictly bandlimited signal unless, of >> course, that it's also defined to be periodic in which case samples over >> a single period are OK - *but* don't they have to divide the period into >> an integer number of segments? You didn't mention that. I think Jean >> Baptiste did it that way on purpose in his series. > > I originally wrote "Suppose that we have exactly one period ...", but > edited "exactly" for (ahem) brevity. I should have left it. > > JerryWell, for me that's not an issue because I assumed you meant it was continuous. Then, I thought, you need an integral number of sample intervals within it. Andor suggests differently so I'll have to ponder his assertion. Fred
