Nyquist Didn't Say That

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?

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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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Spice models continuous systems. Isn't the iteration interval dynamically adjusted to be at least as small as needed?

Jerry

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.

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

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.

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.

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:

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

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.

I'm fully aware of that, but thanks for passing the tip on for others. That WAS why I posed the question that way.

Of course they don't, but the fourier series illustrates the point - the need to sample at least twice per period of the highest frequency component present (in a significant enough amplitude to matter wrt the sampling step)

For a sampling oscilloscope looking at an analog waveform, that isn't really much help.

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

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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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

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.

sample and hold circuit needs sample interval more than twice frequency of sample, but lower frequency sampling can be done if more than two samplings is happening.

eg 1024Hz and 125HzHz => max freq descrimination = 2^10*5^3 Hz

cheers

I share that pure cynical point of view and consider myself as a kind of w**re also :)))) Good luck.

Here is another tip:

What is generally referred as "sampling" actually consists of two different processes rather then one:

1. Continuos time non-linear quantization of the amplitude
2. Linear quantization in time

This representation clarifies many issues such as "quantization noise", why the high sample rates are required, etc.

Vladimir Vassilevsky

DSP and Mixed Signal Design Consultant

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

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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