Matt: [snip]> frequencies over fs/2 - c, for some constant c > 0, then this will not > happen, and the sinc interpolation will converge to the actual signalvalue> everywhere. How fast it is guaranteed to converge depends entirely on |c|.[snip] Indeed! There are lots of functions which are "different" or shall we say "better" between sample points than a function interpolated with a Sinc kernel. It seems, probably becuse of pedagogy, that many folks have "Sinc on the brain". Sinc has theoretical uses and has played a central role in most presentations of the sampling theorems but for a variety of practical engineering reasons Sinc is a very impractical interpolation function. Of more interest in practice are "real" [IIR?] analog reconstruction filters or their oversampled equivalents. How does one go about designing such a filter? Even though non-Sinc cook book filters are always used in practice, few have offered detailed analyis of their design, performance, advantages, disadvantages, etc for precision applications. [There are lots of mainstream applications for which it just doesn't matter! I'm referring to those where it does!] One, of several that I know of, interesting applications requiring a deeper knowledge of interpolation in practice is that of synchronous data pulse signaling [say... just to make it real here... multi-level trellis coded PAM] where one ultimately needs to extract timing information from the synchronous "random data" pulse stream. The robustness of the sample timing recovery algorithms, i.e. convergence rate, phase jitter and wander, etc... are strong functions of the intersample values of the continuous bandlimited waveform. In particular for this application Sinc like pulses are *not* advised! Data is recovered from the sample values, but to find the sample times one needs a function that is "well behaved" between the sample values! The characteristics of a bandlimited waveform between the sample values determines the accuacy of the sample times in this application, and hence the sample values them selves. In other words system performance is not just dependent upon the sample values but also on how the continuous waveforms behave between samples. And... it has been found, by many in practice, that Sinc like behavior between samples is undesirable. So it is useful to ask... in not Sinc, then.... What is the best bandlimited interpolation filter aka pulse forming filter to use based upon sample timing recovery behavior i.e. based upon function behavior between samples? As I suggested up the therad, this is a practical engineering question that goes beyond mere theory and theorem proving and enters the realm of "design". For some views and techniques on *other* [i.e. non-Sinc] reconstruction filters and methods see for instance: R. A. Gibby and J. W. Smith, "Some extensions of Nyquist's telegraph transmission theory.", BSTJ, Vol. 44, No. 7, pp. 1487-1510, Sept. 1965. N. C. Beaulieu, C. C. Tan and M. O. Damen, "A 'Better than' Nyquist pulse" IEEE Communications Letters, Vol. 5, No. 9, ICLEF6, pp. 367-368, Sept. 2001. There is life beyond Sinc! -- Peter Professional Consultant - Signal Processing and Analog Electronics Indialantic By-the-Sea, FL.
Sampling theorem revisited...
Started by ●March 29, 2004
Reply by ●March 31, 20042004-03-31
Reply by ●March 31, 20042004-03-31
Ronald H. Nicholson Jr. wrote:> 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.Pass it through a low-pass filter for Fs/2 and compare input to output. If the output power is diminished, the signal didn't meet the criterion. Then pass it through a narrow filter at Fs/2. Output = failure. What do you mean by "strongly meets"? A component at Fs/2 fails. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●April 1, 20042004-04-01
In article <406b868c$0$3066$61fed72c@news.rcn.com>, Jerry Avins <jya@ieee.org> wrote:>Ronald H. Nicholson Jr. wrote: >> That brings up an interesting question. How, in practical terms, can one >> actually determine whether or not a continuous time infinite extent signal >> meets or strongly meets the Nyquist criteria for the sample rate used? >> IMHO. YMMV. > >Pass it through a low-pass filter for Fs/2 and compare input to output. >If the output power is diminished, the signal didn't meet the criterion.If the output power is diminished compared to what and measured when over what time interval?>Then pass it through a narrow filter at Fs/2. Output = failure.For any realizable filter which would have finite bandwidth B, a signal at Fs/2 - B/4 would fail the above test, but meet the Nyquist criteria.>What do you mean by "strongly meets"? A component at Fs/2 fails.A signal which passes the above narrow band test with a finite bandwidth filter and the low-pass filter test as measured after a finite delay over a finite time period would seem to more than meet the minimum Nyquist sampling criteria. IMHO. YMMV. -- Ron Nicholson rhn AT nicholson DOT com http://www.nicholson.com/rhn/ #include <canonical.disclaimer> // only my own opinions, etc.
Reply by ●April 1, 20042004-04-01
In article <9fLac.6195$NL4.2133@newsread3.news.atl.earthlink.net>, Peter O. Brackett <no_such_address@ix.netcom.com> wrote:>There are lots of functions which are "different" or shall we say >"better" between sample points than a function interpolated with a >Sinc kernel.So are you saying that there are multiple signal waveforms with exactly the same representation in sample values, and the one input to your system is probably different from the one identical to a convolution of the sample values with a Sinc kernel? Or that the actual input signal may resemble the convolution of the samples with a Sinc kernel, but you are looking for an approximation function which can be implemented with a lower computational complexity and delay, or a filter easier (or just possible) to physically realize?>IIRIIRC, IIR filters don't have a linear phase response. So for any IIR reconstruction filter, it seems like there will be a class of input functions which will not be reproduced exactly. IMHO. YMMV. -- Ron Nicholson rhn AT nicholson DOT com http://www.nicholson.com/rhn/ #include <canonical.disclaimer> // only my own opinions, etc.
Reply by ●April 2, 20042004-04-02
Ronald: [snip]> IIRC, IIR filters don't have a linear phase response. So for any IIR > reconstruction filter, it seems like there will be a class of input > functions which will not be reproduced exactly. > > > IMHO. YMMV. > -- > Ron Nicholson rhn AT nicholson DOT com http://www.nicholson.com/rhn/ > #include <canonical.disclaimer> // only my own opinions, etc.[snip] Except at the sample points. Different in between. :-) Exactly my point. An interesting question then is... is 'different in between' just a curiosity or is it useful for anything? The Gibby and Smith paper I quoted up the thread is very illuminating on this whole subject. i.e. non-Sinc, non-linear phase interpolation filters/functions. Gibby and Smith treat both arbitrary phase and magnitude [not just linear phase and even symmetry magnitudes] of reconstruction filters and interpolation waveforms. I have found that there are sometimes real practical Engineering implementation advantages to be gained when looking at interpolation and sampling in this more general way. Explore... there is life beyond Sinc. :-) See ya'll... -- Peter Professional Consultant - Signal Processing and Analog Electronics Indialantic By-the-Sea, FL.
