Nyquist Didn't Say That

Tim,

I hope you'll be very careful with the "interesting" part. Too often I see it referred to as "frequency of interest" - which is very misleading. I think "interesting" means "has enough energy to have measurable aliases" and "of interest" may mean, to some, "the only part of the signal that I care about" to the exclusion of higher frequency components of significant energy.

This thread is so long that I can't really tell if anyone touched on this.....

For others: one must sample at a frequency that is greater than 2X the highest frequency *content* - where "content" is a subjective term indicating there is significant enough energy to cause measurable/objectionable aliasing.

Fred

Fred

In article , Tim Wescott wrote: [....]

That is not true, if you allow a filter to have an infinit delay. It is only if you ever want to see the middle of the output that you have to have a response extending before the input.

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This is an awfully complex subject, with ramifications that are easy to overlook. Even the last statement, "... *content* is a subjective term indicating there is significant enough energy to cause measurable/objectionable aliasing" needs qualifying. To be strict, one needs to add "measurable/objectionable aliasing *into the band of real interest*". Other aliases can be filtered out. Accurate and definitive statements not subject to nit picking are exceedingly hard to make (at least without Wescott's tortured syntax). :-) That's why it's an art.

Jerry

Jerry,

If it wasn't "into the band of real interest" then it wouldn't be measurable/objectionable. To say more seems like "quantifying" rather than "qualifying" - e.g. "this band vs. that band, etc."

I don't know that math and hard work are really artful. One does need to know what one is doing. Knowing what is too big to ignore is calculable most of the time.

Fred

Phase locked loop. Communication systems do it all the time.

Eric Jacobsen Minister of Algorithms, Intel Corp. My opinions may not be Intel's opinions.

Not really. You can be sampling at exactly Fs/2 but be off phase. If the detector and loop are decent it'll figure out which way to steer the phase without additional samples. It won't always be on the right phase, that's the point of letting it lock, but you don't, theoretically, need more samples to do it. The detector may be hard to sort out depending on the signal, but you don't need more samples.

It's common to do this with PSK/QAM signals, where there is only one sample per symbol. You don't ever need to sample higher than that from just a sampling requirement perspective and common detectors will lock the sampling clock quickly to the symbol peaks. Eric Jacobsen Minister of Algorithms, Intel Corp. My opinions may not be Intel's opinions.

Andor: If your really interested you might look into the Muntz Polynomial theorems. They provide a way to approximate L2 waveforms. Resolving the equations would allow an almost arbitrary exponential basis, with the coefficients determined by sampled points. I also have some papers on irregular sampling around if your interested. My goal was determining "sampling" points for spectral analysis of IR absorbtions so it doesn't actually match up with this discussion smoothly. Sorry for the side issue, but I really think the Muntz theorems are neat and underutilized.

Ray

OK lets try it slowly now..... W h o ' s o n f i r s t ?

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I guess you're right if you read "measurable/objectionable" as "measurable _and_ objectionable". I read it as "measurable _or_ objectionable" without thinking to find a benefit of doubt.

Jerry

How would a phase locked loop lock without any additional samples (or equivalent information or measurements) prior to or during the exactly spaced Fs sampling? How would you know a PLL was in lock with no data other than fixed frequency samples? (even a "lock" signal would constitute one more sample, thus raising your sample rate to Fs+1)

IMHO. YMMV.

Unless you're willing to extend the meaning of 'Transfer Function' to be something like Y = H(s, X) instead of Y(s) = H(s) X(s), then no, strictly speaking, you can't. Are you indeed doing this?

You _can_ often make approximations that are more than good enough for many applications, however.

I'll agree with you, but only if you'll agree that that's a nit.

If you don't hook the detector and loop up to the input signal than it won't lock. If you do hook it up, then that is equivalent to taking at least one more sample (to do a phase comparison or something). So your sample rate is now greater than Fs by whatever measurements the detector made in order to convince you that the PLL is locked.

Of if you just looked at the output of the PLL, you would need a lock flag to know whether or not the PLL was locked or not. The lock flag constitutes at least one bit of information which increases the sample rate to Fs + 1 bit > Fs

If you don't look at the lock flag, the you won't have any idea what phase the samples were taken at. It could have been a DC level that the PLL couldn't lock to.

IMHO. YMMV.

Makes sense.

The practical consideration is that sometimes the anti-aliasing filter is chosen prior to knowing what might be connected downstream. For example, if some customer comes along wanting to hook up their 10 samples/sec receiver to the output, then they are probably going to be unhappy with the result.

Sometimes it happens in legacy systems where the ADC stage was built for a certain use, and later on other customers want to hook their receivers up to it. For everything to work best you'd need to modify all fielded units.

If I understand you correctly, the downstream system would need to receive ALL samples, then interpolate them for use at the slower 50 sample/sec rate. So in effect they'd still have to act on the data (interpolate) at the faster original rate. This makes sense to me, but the customer may balk at this. There's no ideal solution to this problem.

This discussion clarifies some things for me... thanks to all repliers.

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The high-rate signal must be filtered and decimated before it can become low rate. Those process can be performed by what you call the up-stream system, by the down-stream system, or split between them. Although there may be modules, there is only one signal-flow path. What is non-ideal?

Jerry

Thanks for the tip, Ray. Googling quickly revealed a paper titled "The Full M=FCntz Theorem in Lp[0,1] for 0 < p < inf" by Erd=E9lyi and Johnson. This is the first time I've heard about the M=FCntz theorem - very interesting! It might indeed pose the basis for approximating peridodic signals in Lp norm.

What else can you say about irregular sampling?

Regards, Andor

In message , dated Mon, 28 Aug 2006, glen herrmannsfeldt writes

The metric prefix femto- (10^-15) is named after the Danish word 'femten' - fifteen, not FeRmi

Yes, but the unit of length approximately the diameter of the nucleus was named after Fermi, and is abbreviated to fm. That happens to be 1e-15m. Someone was lucky.

-- glen