Nyquist Didn't Say That

Jerry Avenues wrote: Knowing that the samples are taken at the peak od a sine of

So why did you introduce it into the discussion in the first place?

----== Posted via Newsfeeds.Com - Unlimited-Unrestricted-Secure Usenet News==----

----= East and West-Coast Server Farms - Total Privacy via Encryption =----

Because knowing the relative phase of the sampler to the signal's second harmonic is the only condition that makes possible the determination of amplitude when sampling at at 2f, a scenario that you introduced.

Jerry

I don't know if he is talking about bandpass sampling. In fact, I have to admit that I'm not even sure of the exact definition of bandpass sampling.

However, consider wavelet transform and particularly signals produced by the wavelet synthesis. Such signals have theoretically infinite bandwidth assuming that the scaling function has finite support. This is true also when signals are synthesized only in truncated resolution (i.e. from scale u to v where u and v are finite). It's true even when synthesized in single resolution. Here synthesis means:

f(t) = sum_s sum_n x_s[n]*phi_n_s(t), where

phi_n_s(t) is the scaling function with translation n and scale s and x_s[n] are the samples from scale (or resolution) s.

Even though these signals have infinite bandwidth, they can be sampled and when given the correct scaling function these samples correspond to the original samples used in wavelet synthesis. Here sampling means

x_s[n] = ,

where f is the analyzed (sampled) function, phi is the scaling function with translation n and scale s. It is obvious that the signal can be later perfectly reconstructed from the samples by wavelet synthesis (assuming the scaling function matched the scaling function used in the original synthesis).

In fact, sinc function is just one possible scaling function (in which case one talks about shannon wavelets). This makes traditional sampling just a special case of wavelet transform (in single resolution). Note that the previous comment about infinite bandwidth does not obviously apply to shannon wavelets.

Any comments, or corrections?

In message , dated Wed, 23 Aug 2006, Tim Williams writes

Yes, unionised.

I'm not entirely agree with that. There are a lot of analog antialising filters, which are quite good near the Nyquist. One of them frequently used is the Cebashev filter (eliptical filter) which design and implementation is easy up to quite high frequencies (say 100-200Mhz, at least tested by myself).

greetings, Vasile

works really good when the sampling points are locked to the zero-crossings of the Nyquist frequency signal.

(snicker)

sounds to me that expecting a sampler to be phase locked to what we would normally think is an unknown signal (if it were known, why bother to sample it to determine its amplitude?) is having one's cake and eating it too.

r b-j

... snip ...

That's fine if you don't care about phase linearity (time delay). Chebychev filters are notoriously poor at preserving phase, or having constant delay characteristics. This results in heavy distortion of analog waveforms, and will manifest itself as such effects as overshoot and ringing. A Bessel filter is designed to minimize this effect, but has much more gentle rejection slopes.

Is there an echo in here. The above is exactly what I just said.

The question was asked - What really happens when you sample a frequency at Fs/2.

Set up a speaker generating the Fs/2 signal. Set up a microphone and ADC to sample the sound at Fs. What really happens? If you adjust the phase of the sampling can you record silence? This was not a theoretical question. We all know how it should work in a perfect world. How does it work in the real world?

No locking the ADC to the signal allowed since that would be a completely different question that no one asked and no one is interested in.

-jim

----== Posted via Newsfeeds.Com - Unlimited-Unrestricted-Secure Usenet News==----

----= East and West-Coast Server Farms - Total Privacy via Encryption =----

However the phase information fed to the PLL to allow it to lock would constitute additional samples, thus raising the total sample rate of all information coming into the system above Fs/2. You have to count all the samples, not just the ones you label as "samples".

IMHO. YMMV.

That's an interesting assertion. Can you justify it?

Jerry

The Nyquist criterion tells you if your closed-loop feedback system will be stable or not. It has nothing to do with sampling.

-a

You're thinking of the Barkhausen criterion, which gives a necessary, but not sufficient, condition for oscillation. While it's useful for building oscillators, it doesn't help you tell if your control system is stable or not -- and having built plenty of type III control systems I can assure you that 180 degrees of phase shift and gain >> 1 doesn't mean you're oscillating.

The Nyquist rate is about sampling, and while I haven't heard it called a "criterion", it's still about sampling.

What really happens is that you get a signal at Fs/2 that is sometimes big and sometimes small, and you have no clue if it's _actually_ a signal at Fs/2 that's sometimes big and sometimes small, or a signal that's big and sometimes at Fs/2 and sometimes slightly off.

Which is a bad thing.

Which is why you don't want to do it.

The Nyquist sampling theorem. We weren't carrying on about enclosing the point -1, 0 on a Nyquist plot in the s plane. That guy Nyquist had more than one feather in his cap. Spirule, anyone?

Jerry

But see

Jerry

I wonder if he really means Chebyshev or elliptic (Cauer)? They both ring badly, but the Cauer rings like a bell if you design one for any reasonably steep cutoff slope. In a particular application you might not care directly about the ringing. However it badly compromises the anti-aliasing qualities of the filter, since it can allow bursts of out of band energy through.

Steve

OK.

Maybe. There are situations where it doesn't matter if its a fluctuation in the amplitude or frequency. Like a 44khz sound recording where your ears can't discern the difference anyway. Bad or good always depends on what you are attempting to accomplish. I never said it was a good thing or a bad thing. What I did say was sampling a sine wave at Fs/2 mo matter what you think the phase is produces a good indication of how non-linear and inaccurate your signal and sampling system really are.

-jim

----== Posted via Newsfeeds.Com - Unlimited-Unrestricted-Secure Usenet News==----

----= East and West-Coast Server Farms - Total Privacy via Encryption =----

No, not re-sampled (from analog to digital) by each system, but instead sent digitally to the next system such that the receiving system only uses some of the data, not every sample. Let's say the initial ADC step has a 1000 samples/sec conversion rate, then the signal is broadcast out, and a receiver system receives at a rate of 200 samples/sec. Then the processing inside that system only has time to perform 50 calculations/sec.

Would the analog anti-aliasing filter selection be dependent on the 50 calc/sec? If that's true you'd have to select the aliasing filter based on the slowest end user of the data. It seems odd that if you design an ADC stage, you'd have to choose analog filtering based on the slowest performing "weakest link" in the eventual design. Opinions?

It's not a matter of opinion; this is well charted territory. Look up interpolation and decimation, or up- and down converting. Digital filtering is usually required at each stage.

Jerry