It applies in spades to zero-order holds on the output, AKA garden-variety DACs.
And where aliasing is a problem, there's more to it than the end-to-end transfer function -- strictly speaking you can't formulate a laplace-domain transfer function for a time-varying system, such as a system that incorporates sampled data.
-- Tim Wescott Wescott Design Services http://www.wescottdesign.com Posting from Google? See http://cfaj.freeshell.org/google/ "Applied Control Theory for Embedded Systems" came out in April. See details at http://www.wescottdesign.com/actfes/actfes.html
[...]
The sampling rate is an entirely practical limit in systems which embrace the aliases, rather than trying to eliminate them. In those cases the "practical limit" is not the sampling rate aspect of the sampling theorem, but how well you can approximate the ideal sampler.
Steve
Yup! Also try sampling at a constant delay from the sine zero crossing. That is what happens when people blindly follow a "criteria" without knowing the full reason and background.
...then re-state with sampling at 2X+delta where delta is (say) 1Hz!
As mentioned earlier, the information rate about a sampled waveform is proportional to the rate above the 2x limit. If you sample at 2.5 the highest frequency iof interest (speaking in a bandwidth sense), you will get sufficient information about said signal faster than if you sample at 2.1x.
That obviously impacts the reconstruction filter (as has also been mentioned). I seem to recall (it's been a long time, but makes sense) that the time required to properly train to a reconstructed signal is inversely proportional to the normalised sample rate above the 2x limit.
This may seem obvious, but as noted a lot of people don't think through the effect of the sampling or the theory behind it. If I sample at
2.1x, I need more full output cycles at the x rate for full reconstruction than I would need if I sampled at 2.5x.My rule of thumb is to sample at 2.5x at a minimum . There are times I sample at 10x or more. A number of people want a fixed answer for all applications, where there isn't any such panacea. 'It depends' is probably the most common engineering term ;)
Something else that might be usefully mentioned in this context is the ADC type used at the input - a Delta Sigma converter inherently decimates the signal, reducing the requirements on the front end anti-aliasing filter. A SAR gives no such assistance. The same consideration of DAC type might also be useful.
Cheers
PeteS
... snip ...
In the PABX telephony example I gave earlier, we initially sampled at 10 kHz to get essentially flat response to 3.5 kHz with constant delay. We upped the sample rate to 12 kHz later to ease the requirements on the filters and equalization.
--
Chuck F (cbfalconer@yahoo.com) (cbfalconer@maineline.net)
Available for consulting/temporary embedded and systems.
USE maineline address!
Hi Tim, Writing about the effects of "periodic sampling" is an interesting and educational thing to do.
My guess is that you'll have to address the controversial notion of "negative frequency", as well as why it is valid to show spectral replications (spaced Fs hertz apart) when we draw a freq-domain picture of the spectrum of a discrete (digital) signal.
One interesting aspect of periodic sampling is that it's easy to misinterpret the results of software modeling of the process of periodic sampling.
That's (I think) what happened when J. L. Smith wrote his "Breaking the Nyquist Barrier" in the July 1995 issue of the IEEE Sig. Processing magazine. I believe Smith misunderstood his software-generated plots when he wrote his embarrassing article. Smith claimed that he could violate the Nyquist Theorem and not lose any information (and avoid any ambiguous information) regarding some time domain signal. Smith's article resulted in a flurry of "Letters To the Editor" that debunked the article (See the Nov. 1995, Jan. 1996, and the May 1996 issues for examples of the letters.) How embarrassing that must have been for both Smith, and the Editors of the magazine who should have known better.
Another very "misguided" sampling article was "Apply Fundamentals To Avoid Surprises With Sampled Systems" written by Gerard Fonte and printed in the June 24th 1993 issue of EDN magazine. Fonte also claimed that you could violate Nyquist and not lose any information. Almost every paragraph of that article contains misconceptions and ambiguities regarding the Nyquist sampling theorem. It's truly a "ghastly" article --- and it also caused a deluge of "Letters To the Editor" pointing out all the errors in the article. (See page 25 of the Sept. 30th 1993 issue of the EDN magazine for example.)
I thought after all the criticism that Fonte received regarding his 1993 EDN that we'd heard the last from Mr. Fonte. Not so. He wrote another titled "Breaking Nyquist" in the October 1998 issue of the Circuit Cellar magazine. Again he claimed that the Nyquist sampling theorem is not valid and that it can be "broken" without causing "problems". Using vague, ambiguous, undefined terminology, Fonte again claimed that he can tell the difference between an Fo (F sub zero) discrete spectral component of an analog sinewave whose Fo frequency was less than Fs/2 and an Fo discrete spectral component of an aliased analog sinewave whose frequency was greater than Fs/2. In other words, he claims that "aliasing" (violating Nyquist) does NOT cause spectral ambiguity in the frequency domain. I can hardly wait for Fonte's next article.
(I'm not being hateful here...Fonte's probably a nice guy whom his family loves.)
My guess is, again, Fonte is using software to model the process of periodic sampling, and the signal he is "sampling" is a pure sinewave. Such modeling is very risky in my opinion because it's easy for a beginner in the field of DSP to misinterpret/misunderstand the results of such modeling.
Concerning sampling, Bonnie Baker wrote an article titled "Turning Nyquist Upside Down by Undersampling" in the May 12th 2005 issue of EDN magazine. The article discusses bandpass sampling. However, I think the article's title is unfortunate because bandpass sampling does NOT "turn Nyquist upside down"
---bandpass sampling is included in the Nyquist Sampling Theorem.
I tell the students in my DSP class that, "Periodic Sampling is one of the most misunderstood topics in DSP." I think I'm justified in making that claim.
Hey Tim, I think in any dissertation on "sampling" it would be a good idea to discuss bandpass sampling. Bandpass sampling is not only an interesting topic, but it's a very practical topic in these days of digital communications. (Just my two cents.)
See Ya', [-Rick-]
(snipped)
Hi, I just finished posting a long rant about sampling articles written by Gerard Fonte.
I just noticed (on the web) that the San Fernando Valley Engineers' Council Inc. has awarded Gerard Fonte an "Outstanding Engineering Achievement Merit Award" for 2006.
See:
Another issue, Isnt it important to always keep into mind what the recustruction filter is?
Lets consider this situation. I have an output signal from a ZOH. I want to sample it again and reconstruct it back. Now i think i need only one sample per ZOH symbol. Why? Cos my reconstruction filter can construct my signal exactly from one sample. Certainly i am not satisfying Nyquist in this case. Or maybe i have not been sleeping to well :o)
Jerry Av>
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What is what happens? Do you actually know what happens if you actually try this in a real world context? Set up a speaker generating the Fs/2 signal. Set up a microphone and and ADC to record the sound at Fs. Are you claiming that you can adjust the sampling phase to produce a digital recording of either full scale or zero? That's what in theory should happen - right? But can you do that in real life?
-jim
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Rick Lyons wrote: [...]
And complex sampling. You must include that, otherwise people won't understand the solid unshakeable reality of negative frequencies. :-)
Regards, Steve
The only possible rational argument? You really are a complete f****ng moron.
Tim
It has been my experience with local engineers' groups that the people who get awards are the ones who show up regularly.
Of course you can lock the sampler to the sampled waveform or one of its harmonics. Google for "PLL".
Jerry
-- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
_______________________________________________________________________ Engineering is the art of making what you want from things you can get.
-- Jerry
Oh, yes: please don't top post. It makes sequence hard to follow.
implications of the sampling theorem needs broadening. pair, or, as in your example, magnitude and derivative. Your view of the cycle. They can be individual samples, they can be a real/quadrature The sampling theorem requires at least two pieces of information per that the derivative is zero at the instant that the sample is taken. case, that its derivative is zero /almost everywhere/. In particular, You are taking advantage of additional knowledge of the signal; in this
That again? I can show you the trig that explicates "complex" sampling without resort to negative frequencies, but so what? :-)
I think a generalization needs to be hammered home. *Any* two independent measurements will do. Value and derivative, for example, as I mentioned by way of explaining why only one sample per cycle is needed to reconstruct the output of a zero-order hold.
Jerry
-- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
-- snip --
Enough people have mentioned this that I'm going to have to give it serious consideration, but I think I may just point out the existence of bandpass sampling (and complex sampling) then write a follow-on article.
I certainly agree with your statement about the sampling process being so often misunderstood. I think this is because sampling seems so simple, yet there is a ton of unintuitive results flowing just below the surface.
-- Tim Wescott Wescott Design Services http://www.wescottdesign.com Posting from Google? See http://cfaj.freeshell.org/google/ "Applied Control Theory for Embedded Systems" came out in April. See details at http://www.wescottdesign.com/actfes/actfes.html
If the signal constellation has N different phases, then you will have to multiply the phase error by N in order to suppress the influence of the data. The simplest example of that is the squaring Costas loop for BPSK. Any kind of non data-aided PLL will have to do this multiplication of phase. It is typical for modems that the carrier PLL operates once when the current symbol is strobed. Thus if the carrier offset is higher then +/- Baudrate/(2*N), the PLL will not lock properly.
It is not always possible, especially if the carrier is RF and if the incoming SNR is low.
Vladimir Vassilevsky
DSP and Mixed Signal Design Consultant
I can google PLL, but I wasn't aware that this extra timing circuitry that you are belatedly introducing to the discussion was included in the theory being discussed. Just exactly how does it fit? You already have a switch on the microphone you could just switch it off if you are going to resort to introducing a phony solution - why would you need to go to all the trouble of a PLL?
-jim
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You introduced what you claimed was a practical difficulty of phase lock in the real world into a theoretical discussion about sampling at a fixed phase offset. I pointed out one practical way to overcome the perceived difficulty. If you knew it all along, what was your cavil about?
Jerry
-- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
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