Nyquist Didn't Say That

No, once it's aliased it's indistinguishable.

Do you mean where the signal has been resampled at each step?

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Much of the paper is going to be the explanation necessary for me to make just that assertion -- plus explaining what "effectively disappeared" might mean in different systems.

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Tim Wescott
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Or to put it another way: for Fs = (2 + epsilon)F your observation interval is something like

1 t = --------- F*epsilon

(more or less -- there's probably a factor of 2 in here that I'm missing). The closer epsilon gets to zero the longer you have to wait. How patient are you?

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Tim Wescott
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Bingo. Yes. I'll be making just that point.

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Tim Wescott
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Can english write? maybe (not)

Actually designing for the sin x / x rolloff isn't too bad as long as you keep your eyes open -- in older digital video systems it was just done with a peaky 2nd-order LC circuit (in newer digital video systems the sampling rate is way higher than the effective resolution of the phosphor, which simplifies things).

But you can't avoid the issue of providing sufficiently steep skirts on your filters, both in and out. As you get closer and closer to Nyquist in a 'simple' system your filter complexity goes through the roof, as does the difficulty of actually realizing the filters in analog hardware. This is why many systems that must store or transmit data at close to Nyquist (like music on a CD) have A/D and D/A sample rates that are significantly higher than the internal transmission rate, with digital decimation and interpolation coupled with simplified analog anti-alias and reconstruction filters.

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Tim Wescott
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... snip ...

IIRC the sin x / x business only applies to sample and hold filtering. The impulse function avoids that. A further point is that the thing that counts in an end to end system, such as telephony, is the net transfer function. You can distribute this in various way with compensating input and output filters. This is generally known as equalization.

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Chuck F (cbfalconer@yahoo.com) (cbfalconer@maineline.net)
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Somewhere in the Nyquist discussion, you might mention that Nyquist didn't attend MIT or Stafford. He went to a small obscure school in North Dakota.

Robert, Did I miss anything?

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FOAD. It was a short, to-the-point comment. The only possible rational argument that can be made to his post is that he didn't trim the quoted text.

Tim

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HIO4?

Tim

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well, then maybe they should be moved the the "mathematical basis", then. you don't need the convolution with the Dirac comb in the frequency domain thingie if you can show by some other means (i think simpler means) that the spectrum is copied and shifted at all multiples of the sampling frequency. we can show that by showing that the Dirac comb is a periodic function with identical coefficients (all 1 if you scale it right) and then using the frequency shifting theorem.

i think so. this kind of AM modulation is called SSB. at least that's what we called it when i was a ham radio kid 38 years ago.

duh, i dunno. i think, if you do it the Hilbert way (we didn't have DSP in them olden days of the Heathkit HW100) you can have a carrier frequency of whatever you want. you can separate the positive and negative parts of the original baseband signal and move the positive up or down any amount with the negative part doing the mirror image and moving down or up the opposite amount.

r b-j

I'm not an expert in this area, so maybe you can clarify something. Does the system handle multiple carrier frequencies? If the carrier frequency is fixed, I would expect that the bandwidth of the PLL would be narrow enough to exclude the aliased frequencies.

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Thad

True, but the "information" on a periodic waveform will be all garnered one helluva lot quicker at 2.1x than at 2.000000001x

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.

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Tim Wescott
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"Applied Control Theory for Embedded Systems" came out in April.
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[...]

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.

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