Glen: [snip]> For a signal of length T, sampled at sampling rate f, there > are N=T f samples. > > If you add an additional sample on each end with a value of zero, > at T=0 and T=(N+1)/f only solutions of the form sin(w T)=0 > will be allowed, so w(n)=n pi f/(N+1). >[snip]Such a signal is theoretical and definitely not physical, e.g. it does not satisfy the Paley-Weiner criteria, etc... That said, I wonder what the OP meant when he asked about reconstructing the original signal? The OP did not state or specify how the original signal was produced. His problem may be different than what is being addressed here. For example: What if his original signal is a real physical signal produced by say a physical sensor, a seismometer or a hydrophone for instance, subsequently passed through a practical real physical analog bandlimiting filter, say a passive LC low pass Cauer parameter filter filter of order N = 10 with maximum reflection coefficient magnitude in the pass-band of 0.01, before sampling. Now, an interesting question might be asked: Apart from quantization errors, is it possible to later reconstruct that signal exactly at all points between the sample points, or is it just possible to do so at the sample points? How would that reconstruction be done? With a sinc kernel? Or is it possible at all? Food for thought... -- Peter
Sampling theorem revisited...
Started by ●March 29, 2004
Reply by ●March 30, 20042004-03-30
Reply by ●March 30, 20042004-03-30
Peter O. Brackett wrote:> Glen: > > [snip] > >>For a signal of length T, sampled at sampling rate f, there >>are N=T f samples. >> >>If you add an additional sample on each end with a value of zero, >>at T=0 and T=(N+1)/f only solutions of the form sin(w T)=0 >>will be allowed, so w(n)=n pi f/(N+1). >>[snip] > > > Such a signal is theoretical and definitely not physical, e.g. it does not > satisfy the > Paley-Weiner criteria, etc... > > That said, I wonder what the OP meant when he asked about reconstructing the > original signal? > > The OP did not state or specify how the original signal was produced. His > problem > may be different than what is being addressed here. > > For example: > > What if his original signal is a real physical signal produced by say a > physical sensor, > a seismometer or a hydrophone for instance, subsequently passed through a > practical real > physical analog bandlimiting filter, say a passive LC low pass Cauer > parameter filter > filter of order N = 10 with maximum reflection coefficient magnitude in the > pass-band of > 0.01, before sampling. > > Now, an interesting question might be asked: > > Apart from quantization errors, is it possible to later reconstruct that > signal exactly at all points > between the sample points, or is it just possible to do so at the sample > points?Assuming reconstruction is possible, it is clearly the signal after filtering, not the original signal as defined above that is reconstructed. If the signal after the filter has no aliasing and the signal after the reconstruction filter likewise has no components outside the original band, how could the reconstruction not be exact? Any departures from the pre-sampled waveform would have to be caused by frequencies explicitly disallowed. What? You claim that real reconstructions are not exact? That merely shows that disallowing and removing aren't necessarily the same thing. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●March 30, 20042004-03-30
Jerry: Hey, hello... No, what I meant was the reconstruction of the signal at the output of the physical continuous time passive low pass filter. e.g. can a reconstruction of that signal be theoretically exact such that if one were to do a continuous signal subtraction, say with a precision difference amplifier the error signal becomes zero for all t, not just at the sample points? Or perhaps bounded in a mini-max sense? etc... Clearly using the Sinc kernel you can only do a reconstruction in the the minimum mean squared error sense at the sample points. The actual error between sample points can become very very large! In fact it can be unbounded in theory! Is it possible to do better than this? There are quite a few applications where the reconstruction between the sample points is important. The whole universe is not digital. -- Peter Professional Consultant - Signal Processing and Analog Electronics Indialantic By-the-Sea, FL "Jerry Avins" <jya@ieee.org> wrote in message news:40698c1f$0$3072$61fed72c@news.rcn.com...> Peter O. Brackett wrote: > > > Glen: > > > > [snip] > > > >>For a signal of length T, sampled at sampling rate f, there > >>are N=T f samples. > >> > >>If you add an additional sample on each end with a value of zero, > >>at T=0 and T=(N+1)/f only solutions of the form sin(w T)=0 > >>will be allowed, so w(n)=n pi f/(N+1). > >>[snip] > > > > > > Such a signal is theoretical and definitely not physical, e.g. it doesnot> > satisfy the > > Paley-Weiner criteria, etc... > > > > That said, I wonder what the OP meant when he asked about reconstructingthe> > original signal? > > > > The OP did not state or specify how the original signal was produced.His> > problem > > may be different than what is being addressed here. > > > > For example: > > > > What if his original signal is a real physical signal produced by say a > > physical sensor, > > a seismometer or a hydrophone for instance, subsequently passed througha> > practical real > > physical analog bandlimiting filter, say a passive LC low pass Cauer > > parameter filter > > filter of order N = 10 with maximum reflection coefficient magnitude inthe> > pass-band of > > 0.01, before sampling. > > > > Now, an interesting question might be asked: > > > > Apart from quantization errors, is it possible to later reconstruct that > > signal exactly at all points > > between the sample points, or is it just possible to do so at the sample > > points? > > Assuming reconstruction is possible, it is clearly the signal after > filtering, not the original signal as defined above that is > reconstructed. If the signal after the filter has no aliasing and the > signal after the reconstruction filter likewise has no components > outside the original band, how could the reconstruction not be exact? > Any departures from the pre-sampled waveform would have to be caused by > frequencies explicitly disallowed. > > What? You claim that real reconstructions are not exact? That merely > shows that disallowing and removing aren't necessarily the same thing. > > Jerry > -- > Engineering is the art of making what you want from things you can get. > ����������������������������������������������������������������������� >
Reply by ●March 30, 20042004-03-30
In article <P8oac.5277$NL4.2210@newsread3.news.atl.earthlink.net>, Peter O. Brackett <no_such_address@ix.netcom.com> wrote:>No, what I meant was the reconstruction of the signal at the output of >the physical continuous time passive low pass filter.Given that any physically realizable passive filter will not have a perfectly flat response in the passband and zero leakage outside, there's you answer.>e.g. can a reconstruction of that signal be theoretically exact such that >if one were to do a continuous signal subtraction, say with a precision >difference amplifier the error signal becomes zero for all t, not just >at the sample points? Or perhaps bounded in a mini-max sense? etc...If your bounds are sufficiently large as related to both the filter flatness and the quantization, and you have some a priori knowledge of the input signal as it relates to aliasing and roll-off, perhaps.>Clearly using the Sinc kernel you can only do a reconstruction in the >the minimum mean squared error sense at the sample points.Not sure I understand your reconstruction algorithm. At the sample points, a Sinc kernel seems like it should reproduce the samples exactly as presented to it, since the Sinc function is exactly 1 there, and 0 at all other sample points. Or are you talking about the effects of I/O quantization and sample time jitter?>The whole universe is not digital.There seems to be some controversy about whether this is true of the "real" universe at Planck space and time dimensions (10^-43). IMHO. YMMV. -- Ron Nicholson rhn AT nicholson DOT com http://www.nicholson.com/rhn/ #include <canonical.disclaimer> // only my own opinions, etc.
Reply by ●March 30, 20042004-03-30
Ronald: [snip]> >Clearly using the Sinc kernel you can only do a reconstruction in the > >the minimum mean squared error sense at the sample points. > > Not sure I understand your reconstruction algorithm. At the sample > points, a Sinc kernel seems like it should reproduce the samples exactly > as presented to it, since the Sinc function is exactly 1 there, and 0 > at all other sample points. Or are you talking about the effects of > I/O quantization and sample time jitter?[snip] Sinc interpolation can only match exactly at the sample points for waveforms of limited time extent. TW points and all that stuff. And this kind of theoretical/mathematical match can be calulated exactly by a finite system of linear equations as was stated by another poster a little further back along this thread. If the waveform is of infinite time extent then a Sinc interpolation can only match at the sample points in a mean square sense, i.e an SVD solution rather than a direct inverse. Sinc interpolation cannot exactly match at the infiintiy of sample points of an actual physical waveform such as the output of an IIR or analog filter. I am asking about closely matching and limiting the error everywhere on a continuous time infinite extent signal. There is a difference. Is it possible to do what I suggest, and if so what is the algorithm, "best" interpolation algorithm. This is a very practical problem, not a theorem proof! -- Peter
Reply by ●March 31, 20042004-03-31
In article QSdac.5483$yN6.1555@newsread2.news.atl.earthlink.net, Peter O. Brackett at no_such_address@ix.netcom.com wrote on 03/30/2004 07:38:> The OP did not state or specify how the original signal was produced. His > problem > may be different than what is being addressed here. > > For example: > > What if his original signal is a real physical signal produced by say a > physical sensor, > a seismometer or a hydrophone for instance, subsequently passed through a > practical real > physical analog bandlimiting filter, say a passive LC low pass Cauer > parameter filter > filter of order N = 10 with maximum reflection coefficient magnitude in the > pass-band of > 0.01, before sampling. > > Now, an interesting question might be asked: > > Apart from quantization errors, is it possible to later reconstruct that > signal exactly at all points > between the sample points, or is it just possible to do so at the sample > points?assuming the seismometer or hydrophone signal had content above Nyquist (requiring the bandlimiting filter), then no, the signal cannot be reconstructed "exactly" because the Cauer filter, although very sharp, is not a brickwall filter and frequencies at or above Nyquist will leak through and alias.> How would that reconstruction be done? With a sinc kernel? Or is it > possible at all?if it were possible, with a sinc() kernal (which is also not possible since it is not time-limited). r b-j
Reply by ●March 31, 20042004-03-31
Hi all, Thanks for the replies... things are getting clearer now... But I was also wondering if there is any limitation on the sinc function used.. as in with respect to the highest frequency of the signal. If my signal is a very high frequency signal containing sharp edges (for obvious reasons).. then would our sinc function be able to reconstruct it. The sinc function has got smooth edges while our original signal has now sharp edges... then how come this reconstruction happen so easily... how is this problem tackled? Till what frequency does the sampling theorem hold good. I believe some factor should surely appear while dealing wiht high frequencies.... Am I diverting too much from the main stream? -Anoop Deoras
Reply by ●March 31, 20042004-03-31
"Peter O. Brackett" <no_such_address@ix.netcom.com> wrote in message news:P8oac.5277> Clearly using the Sinc kernel you can only do a reconstruction in the the > minimum mean squared > error sense at the sample points. The actual error between sample points > can become very very > large! In fact it can be unbounded in theory!If the sample points are limited in magnitude and actually samples of a signal that strongly meets the Nyquist criteria, i.e., contains no frequencies over fs/2 - c, for some constant c > 0, then this will not happen, and the sinc interpolation will converge to the actual signal value everywhere. How fast it is guaranteed to converge depends entirely on |c|. If you know that the condition holds for some particular value of c, then you can even design a perfect reconstruction filter with amplitude that falls off as 1/t^2, instead of 1/t, and will produce precisely the same result as a sinc filter, because it differs only in the excluded band.
Reply by ●March 31, 20042004-03-31
(previously snipped question regarding reconstruction from sampled data) EC-AKD wrote:> Thanks for the replies... things are getting clearer now... > But I was also wondering if there is any limitation on the sinc > function used.. as in with respect to the highest frequency of the > signal. If my signal is a very high frequency signal containing sharp > edges (for obvious reasons).. then would our sinc function be able to > reconstruct it. > The sinc function has got smooth edges while our original signal has > now sharp edges... then how come this reconstruction happen so > easily... > how is this problem tackled?If you have sharp edges then you will have a higher Nyquist frequency. In that case, the samples will be closer together and the sinc() functions will be able to reconstruct it. If you mean really sharp edges, such as square waves, sawtooth waves, or triangle waves, ones that might be described as pointy, then they are not band limited and have an infinite Nyquist frequency, the sinc will turn into a Dirac delta function, and you will be able to reconstruct with an infinite number of such functions. -- glen
Reply by ●March 31, 20042004-03-31
In article <xVpac.6090$yN6.1774@newsread2.news.atl.earthlink.net>, Peter O. Brackett <no_such_address@ix.netcom.com> wrote:>I am asking about closely matching and limiting the error everywhere on >a continuous time infinite extent signal.That brings up an interesting question. How, in practical terms, can one actually determine whether or not a continuous time infinite extent signal meets or strongly meets the Nyquist criteria for the sample rate used?>This is a very practical problem, not a theorem proof!Sorry, I couldn't tell whether this was a serious question or a poorly worded rephasing of a homework assignment question... IMHO. YMMV. -- Ron Nicholson rhn AT nicholson DOT com http://www.nicholson.com/rhn/ #include <canonical.disclaimer> // only my own opinions, etc.
