Nyquist Didn't Say That

You should discuss the question of whether it is possible to remove unwanted aliased-in noise by clever digital filtering in a downstream calculation. In my understanding this is not possible. But maybe I slept through that part of the class.

You should discuss what happens to a signal that is filtered and sampled in one system at rate X, but is transmitted to a receiving system at update rate Y, then used by that receiving system at rate Z. How should one select the analog anti-aliasing filter in this situation?

mw

Reply to
mw
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Consider anything *other than* a pure sine wave at x Hz. Consider say a square wave at x Hz, sampled at 2x Hz. What do *you* envisage those sample will let you reconstruct?

Reply to
rebel

..

Sometimes. If it's a closed-loop servo, maybe 5X oversampling is called for. I've written about why before. It's enough to say here that one sample delay is 180 degrees phase shift at the sampling frequency. Anti-alias filters have delays of their own. Sampling at 10 or 20 x can avoid the need for an anti-alias filter altogether. "It depends."

Jerry

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Engineering is the art of making what you want from things you can get.
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Reply to
Jerry Avins

And the frequency of a square wave is what? Hint, read up on Fourier series.

Sigh. A square wave has infinite frequency, so what sample rate do you propose?

All real signals are composites of sine waves in theory. In practice, they usually don't have infinite numbers of composite waves at infinite bandwidth.

BTW, a square wave can usually be expressed in four or six bytes. just encode "squarewave, 10hz, 2 volt" and you are done.

Reply to
Pat Farrell

Does it?

No.

The answers are

a) Sample at Fs > 2X Hz b) Cut-off at Fc < X/2 Hz

Note no equality signs here.

The sampling theorem states a *lower*bound* on the relation between sampling frequency and the highest significant frequency component in the signal.

There is nothing in the sampling theorem to suggest that sampling at 2X Hz is *sufficient*.

Tiny detail in phrasing; huge difference in practice.

Rune

Reply to
Rune Allnor

Just tell them that they've got to make sure that they sample BELOW the Nyquist frequency of the HIGHEST frequency present in the signal, and that the cutoff frequency of a filter isn't the frequency at which the output is effectively disappeared.

Reply to
Paul Burke

How about a few observable facts. Like a signal at frequency F1 can be sampled at a rate F2 and the net is the phase difference if these frequencies are *exactly* the same, or if the ratio is exactly 1:2 or 2:1 or any other integer ratio. If there is a slight difference in the ratio F1/F2 or F2/F1, that the difference frequency is observable but no clue as to which one is the least stable with short term measurements.

Reply to
Robert Baer

Do consider this interesting (atleast for me) example

Consider pure Sin wave at X Hz. I start sample it at 2X. Unfortunately i start sampling from time = 0. What would i get? Aint i statisifying Nyquist here?

Regards

Tim Wescott wrote:

Reply to
mobi

No you are not. You seemed to have missed Rune's post in this thread about '=' vs ' >'.

--
Stef    (remove caps, dashes and .invalid from e-mail address to reply by mail)
Reply to
Stef

Please, please, PLEASE stop using maximum frequency as a proxy for bandwidth. I wish people would stop using this 'simplification' because the less mathematically astute take it as true regardless because it seems so reasonable. But reasonableness doesn't make things true. Its bandwidth, bandwidth, BANDWIDTH that matters.

Now for the case at point. A pure sine wave has zero bandwidth by definition. As such the lower bound on sampling rate is 0. Note that this is the rate i.e samples per second. However, a few samples are needed to fix starting phase, amplitude, and if a variable, the frequency. But this is not a per-second requirement.

There is one complication, which is if the frequency is completely free then the number of samples needed to determine the frequency is infinite (because you could be sub-sampling). But if the frequency were completely free then the 2X sampling frequency in the quote would also be infinite. Given an upper bound on the frequency three (perfect) samples should be enough to fix phase, amplitude and frequency.

So, again, remeber - bandwidth.

Peter

Reply to
Peter Dickerson

Does Wikipedia have a posting mode that allow only original author or a "approved" contributor to modify an article. I heard a recent story of how the Wikipedia article about an Arkansas city had derogatory comments inserted.

Might the best approach be using "External links"? Tim keeps control. The "world" gets the information. If Tim gets paid for the article, the publisher gets site exposure.

Reply to
Richard Owlett

Well, if you can guarantee that the cos has no phase shift, then you may have a cos term at Nyquist frequency in discrete periodic sequences without introducing aliasing ambiguity. OTH, any periodic, discrete sequence with Nyquist frequency will be interepreted as a cos (zero phase shift) by the discrete Fourier sum (aka DFT).

For example, the sequence

..., 1, -1, 1, -1, ...

will be interpreted as a cos with amplitude 1 by any (finite) DFT. The sequence

..., 1/sqrt(2), -1/sqrt(2), 1/sqrt(2), ...

will be interpreted as a cos with amplitude 1/sqrt(2) as opposed to a unit amplitude cos with pi/4 phase shift. By defintion, the imaginary part of the Nyquist DFT coefficient is always zero for real sequences (just as for the DC coefficient, but we don't want to discuss phase shifts for DC signals again :-).

Regards, Andor

Reply to
Andor

In theory you get nothing. In practice you get a good indication of just how non-linear and inaccurate your signal and sampling system really are.

-jim

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Reply to
jim

More quantitatively, the various questions about anti-alias and sampling can be answered by reconstructing the signal from the proposed signal system and them computing the error for antcipated input signals by taking the difference (in simple systems). Put another way, model the signal processing path and compare it to what you want, to see if the approximations you make in your implementation matter. This provides guidance for sampling rates and anti-aliasing; vesus various input spectra/signals. In signal processing we typically approximate perfection (which is sometimes impossible) by various means; the adequacy depends upon the errors that we allow. Given a description of what we want and a proposed implementation the errors should be calculable. Nyquist moerely talks about what can be made to wrk given perfect resources; reconstruction of an incoming signal of a certain type. If you feed >2X signals or don't reconstruct/use the data optimally, you have to do the error analysis to see how much you are paying for not being perfect. In other words, you allways have to do an error calculation for an proposed design and enviroment.

Ray

Ray

Reply to
RRogers

Well yes, but that is only due to the fact if you sample at exactly 2x you might sample at the zero points of the the sin wave, and not be able to reproduce the signal, but most people write =2x because of convenience, but if 2.0000000000001 is how you like to write it, then ok.

Reply to
steve

After reading some of the contributions to this thread, I can see that you were right.

Reply to
pomerado

...

No; it's more than that. It means (among other problems) that there's no way to determine the component in phase with the sample clock (sine component), so the amplitude remains unknown.

That's the least of the problems, though. To resolve a frequency of f Hz, one must sample on the order of 1/f seconds. Frequencies in the sampled domain lie on a circular scale, so that it is also necessary to sample on the order of 1/f seconds to resolve a frequency of Fs/2 - f.

We can no more sample frequencies close to Fs/2 in a reasonably short time than we can those close to DC.

So many misconceptions, so little time. Tim: are you tuned in?

To the person who wondered if he had been asleep in class when the way to remove aliases after sampling was explained: you didn't miss a thing. Think of the original components as sticks of varying lengths. (The lengths are proportional to frequency.) The sampling process chops up any length greater than Fs/2 into pieces of length Fs/2 which it discards, and leaves the remainder in the pile. The result is that all the sticks are shorter than Fs/2, even though some *were originally* part of longer sticks. There is absolutely no way to tell the original length after the ax falls.

Jerry

--
Engineering is the art of making what you want from things you can get.
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Reply to
Jerry Avins

Yes, you are. Your example shows that while satisfying the Nyquist sampling criterion may be a necessary condition, it certainly isn't sufficient. That's what some of us have been trying to get across.

...

Jerry

--
Engineering is the art of making what you want from things you can get.
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Reply to
Jerry Avins

Equality is enough to avoid aliasing. The inequality is needed to enable reconstruction. Don't ignore the needed sampling duration in the "almost equal" case.

Jerry

--
Engineering is the art of making what you want from things you can get.
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Reply to
Jerry Avins

The other one I run into is that N. really applies to the bandwidth, not the highest frequency as is commonly thought. Harmonic mixers make use of this all the time, using the equivalence of the sampled interval to the fundamental interval [-f_s/2, f_s/2), and alias down to some lower frequency in the process. If you really reconstruct with impulses, you can use a bandpass filter to get back the original signal at the original carrier frequency.

People also routinely neglect the to account for the zero-order hold in their DAC circuits--if you take a signal, run it through an A/D and a D/A, you don't wind up with the original signal, but one with an additional sinc function rolloff.

Cheers,

Phil Hobbs

Reply to
Phil Hobbs

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