"Ron N." <rhnlogic@yahoo.com> wrote in message news:1141236114.221866.286590@u72g2000cwu.googlegroups.com...> 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-MBoth perspectives are useful. In contrast, I'd hate to have to carry around noise and such things when doing signal analysis - *unless* it were necessary to the discussion. I'd not want to suggest that sinwt all by itself isn't a worthy term for purposes of disccussion. We can go from: - a signal that is strictly bandlimited is of infinite extent according to the math to: - all signals are pretty much bandlimited and pretty much time limited ... whatever "pretty much" means to whoever. "Pretty much" also varies according to how much noise there is. These are generally a little harder to deal with if we're using the math from above or below. It takes a more sophisticated treatment as you've suggested. and to: - a signal that is strictly time limited occupies infinite bandwidth according to the math What I think we do mentally is to use the perfect cases / math and then in the back of our minds say: "well, it really doesn't matter much if we truncate in some appropriate way - both in time and in frequency" Fred
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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:1141236114.221866.286590@u72g2000cwu.googlegroups.com... > >>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 > > > Both perspectives are useful. In contrast, I'd hate to have to carry around > noise and such things when doing signal analysis - *unless* it were > necessary to the discussion. I'd not want to suggest that sinwt all by > itself isn't a worthy term for purposes of disccussion. > > We can go from: > - a signal that is strictly bandlimited is of infinite extent according to > the math > to: > - all signals are pretty much bandlimited and pretty much time limited ... > whatever "pretty much" means to whoever. "Pretty much" also varies > according to how much noise there is. These are generally a little harder > to deal with if we're using the math from above or below. It takes a more > sophisticated treatment as you've suggested. > and to: > - a signal that is strictly time limited occupies infinite bandwidth > according to the math > > What I think we do mentally is to use the perfect cases / math and then in > the back of our minds say: "well, it really doesn't matter much if we > truncate in some appropriate way - both in time and in frequency"It's a variation on a trick every practical engineer uses almost unconsciously: when a problem is too hard to solve, invent an easier one whose solution is almost the same. We run into trouble when easier solution becomes so familiar that we -- or the ones we teach -- come to believe that it's the true solution. But you and I know better than that. Right! Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●March 2, 20062006-03-02
Jerry Avins wrote: ...> But some bandlimited signals have no period. An example is sin(t) + > sin(e*t).Ok. How many samples at regular intervals will it take to reconstruct f(t) = sin(t) + sin(e*t) everywhere on the real line, and what is the corresponding sampling period? Obviously, using any period T > (2 e - 1) and countably infinite many samples suffice, by the sampling theorem. Hey, don't you want to pose a new riddle of this? Regards, Andor
Reply by ●March 2, 20062006-03-02
Andor wrote:> Jerry Avins wrote: > ... > > But some bandlimited signals have no period. An example is sin(t) + > > sin(e*t). > > Ok. How many samples at regular intervals will it take to reconstruct > > f(t) = sin(t) + sin(e*t)Sinusoids which are well spaced will allow the usual frequency estimation techniques to work with reasonable numeric precision. In the zero noise case, once you estimate the presense of two constant sinusoids, you should be able to do an exact curve fit with a very small subset of samples. IMHO. YMMV. -- rhn A.T nicholson d.0.t C-o-M
Reply by ●March 2, 20062006-03-02
"Andor" <andor.bariska@gmail.com> wrote in message news:1141235247.969514.151730@i40g2000cwc.googlegroups.com...> > 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.Andor, I went back and read the thread. Others touched on it: when you consider what is really going on then it's like saying "if you have a nonbandlimited, periodic continuous function then you can effectively increase the sample rate by having the sample interval an irrational fraction of the function's period" That's because the samples fold back onto one another onto the same function / period. The effective sample rate then becomes arbitrarily large. As a practical matter, at some point we say "enough!", the sample rate is high enough for our purposes. And, that's the sampling scope rationale. Of course, irrational but very close to the period won't do very well in the real world because you have to observe forever and use widely separated observations together, etc. etc. Ditto irrational but very close to some large integer multiple of the period. I dislike it when others jump into a theoretical / analytical discussion and throw practical considerations in - as if to discount the merit of the theoretical discussion. That's why above I said: "as a practical matter" to delimit the paragraph. Before somebody jumps in to suggest practical considerations are useful or important well ... I agree already! I'm not sure what you mean by "reconstruct" though.... How do you do that? What is the interpolator of choice? sinx/x doesn't seem to be an available choice here..... Somehow I got the impression that you were reconstructing by simply having lots of samples close together. That's not what I'd call "reconstruction" in a strict sense - maybe "dense reconstruction"?. I probably missed something. Fred
Reply by ●March 2, 20062006-03-02
"Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message news:9ICdnRM9I-QmzprZnZ2dnUVZ_t2dnZ2d@centurytel.net...> > "Andor" <andor.bariska@gmail.com> wrote in message > news:1141235247.969514.151730@i40g2000cwc.googlegroups.com... > >> >> 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. > > Andor, > > I went back and read the thread. Others touched on it: when you consider > what is really going on then it's like saying "if you have a > nonbandlimited, periodic continuous function then you can effectively > increase the sample rate by having the sample interval an irrational > fraction of the function's period" That's because the samples fold back > onto one another onto the same function / period. The effective sample > rate then becomes arbitrarily large. As a practical matter, at some point > we say "enough!", the sample rate is high enough for our purposes. And, > that's the sampling scope rationale. > > Of course, irrational but very close to the period won't do very well in > the real world because you have to observe forever and use widely > separated observations together, etc. etc. Ditto irrational but very > close to some large integer multiple of the period. I dislike it when > others jump into a theoretical / analytical discussion and throw practical > considerations in - as if to discount the merit of the theoretical > discussion. That's why above I said: "as a practical matter" to delimit > the paragraph. > > Before somebody jumps in to suggest practical considerations are useful or > important well ... I agree already! > > I'm not sure what you mean by "reconstruct" though.... How do you do > that? What is the interpolator of choice? sinx/x doesn't seem to be an > available choice here..... Somehow I got the impression that you were > reconstructing by simply having lots of samples close together. That's > not what I'd call "reconstruction" in a strict sense - maybe "dense > reconstruction"?. I probably missed something. > > FredOh, yes. And you *must* know the period absolutely or the folding back of the samples won't work. Fred
Reply by ●March 2, 20062006-03-02
Fred Marshall wrote: ...> I'm not sure what you mean by "reconstruct" though.... How do you do that? > What is the interpolator of choice? sinx/x doesn't seem to be an available > choice here..... Somehow I got the impression that you were reconstructing > by simply having lots of samples close together. That's not what I'd call > "reconstruction" in a strict sense - maybe "dense reconstruction"?. I > probably missed something.I think quite a few people missed that point. Let's recapitulate: Let x(t) be a bandlimited signal, and let x_n := x(n T), for n = ..., -1, 0 ,1, 2, ..., be the samples of the signal, such that the sampling period T is small enough (with respect to the bandlimit). Sinc-interpolation of the samples results in the interpolation function f(t), defined by f(t) := sum_{n= -infinity}^{infinity} x_n sinc(t - n T) Shannon's sampling theorem now claims: f(t) is equal to x(t) for all t. That means, we can perfectly reconstruct the value x(t) for any t on the real line just from the samples x_n, via sinc-interpolation. Note that perfect reconstruction is a limit process (the two infinities in the sum are just notation for the limit process). Now assume that x(t) is periodic with period 1, continuous but not bandlimited. We sample x(t) with a sampling period T that is irrational (it doesn't matter whether it is smaller or larger than 1), getting the samples x_n. It is possible to perfectly reconstruct x(t) for any t in the interval [0,1] (and therefore on the whole real line) by another limit process. However, we don't define an interpolating function, rather a two step process. First we have to find a sequence t_n which converges towards t, where t_n are the (modulo) sample times. This sequence exists because the sampling times lie dense in the interval (essentially because T is irrational). Second we compute the limit of the samples x(t_n), which is then equal to x(t). I proved all that in my follow-up here: http://groups.google.ch/group/comp.dsp/msg/9a794e06b0fba4d3 Regards, Andor
