You didn't quote context, so I don't know what you're driving at. Whatever, I don't get it; can you add some meat to the bones?
Jerry
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
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
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
sampling at 2.000001X solves that problem, there are no frequencies in phase with the sample clock anymore, the point I was making
There is no additional information obtained by sampling at a higher rate.
doesn't make any sense to me, so to resolve a frequency of 10 hz one must sample on the order of 1/10 seconds? Is that what you are saying, or am I reading it wrong?
Tim is making many assumptions (unfairly in my opinion) beforehand about the signal and anti-alias filter in his original post, and then saying this and that statement is not correct. Is he assuming frequencies higher then the desired signal exist, I think so, but I don't know, is he assuming a non-brick wall anti-alias filter? I think so but who knows. Nyquist assumes the ideals, you can't have a theorem otherwise.
"steve" wrote in news:1156354799.705801.226400 @b28g2000cwb.googlegroups.com:
No additional information, but its certainly easier to look at your data when there's more than one point in each half cycle.
This last paragraph seems worth emphasizing, particularly on the subject of sampling rates, as it points out a reason why rather more than 2.00...01 X sampling may be important. I'm not sure how a practical reconstruction filter to compensate for ZOH could be arranged, causal or acausal, otherwise. You need some margin for the skirts, don't you?
Jon
Yes, easier to reconstruct a signal with more samples
Wikipedia articles often have external links, which people frequently leave alone (IE don't vandalize). Just contribute *something* useful to the wikipedia article, then link to your own area for in-depth coverage. Win-win scenario.
I don't think Wikipedia's going away. I find myself using it more and more as a first step to getting any info on some new subject -- even before google actually.
Now here's a thought -- if Wikipedia can become financially viable in its own right (currently it depends on donations) maybe a business model can appear where based on number of "views" of pages, the contributing authors can get some $$$ sent their way.
Yes -- it's viable!
#1) Suggest the possibility #2) ??? #3) Profit!
-Dave
David Ashley said the following on 23/08/2006 19:28:
In some areas, Wikipedia is great, in others it's dire (no disrepect intended to anyone that contributes, myself included). Articles about comms and signal processing (as relevant examples) are on the whole scant, badly written and error-prone. However, I'm sure this will change over time.
Nice idea, but I don't think it's ever going to happen. For one, the Wikipedia administrators are already working hard to reduce the systematic bias that exists in Wikipedia (see
I used to work in an FFT factory, and we typically sampled at 2.56 x BW.
Oh, in that case I'm not clear on where there's so much confusion that needs an entire article to clear up. I was hoping you'd hit the idea that Nyquist really said 2x the bandwidth of interest (as others have already mentioned). Clarifying that doesn't lose the context of baseband sampling, does address where the most common pitfalls lie, and provide a full treatment of the issue as well as covers what Nyquist really said.
That's a fine notion to address, that all systems are essentially bandwidth limited by nature or can be made so easily. Tying that to the sampling rate is a fundamental issue, but I'm not certain that it can't be cleared up in a few well-written paragraphs with an illustration or two.
But maybe I'm too optimistic... Eric Jacobsen Minister of Algorithms, Intel Corp. My opinions may not be Intel's opinions.
Right or wrong, that's what I meant.* What's more, to resolve Fs/2 - 10 Hz, you also need to to sample for a time in the order of 1/10 second. Why does it seem strange?
It seems to me that Tim is assuming anti-alias filters that produce results sooner than next week, and signals that would have components above Fs/2 without them. I don't think those assumptions are unfair.
The Nyquist criterion does indeed assume ideal conditions. Tim will show that the assumption is rarely justified for real work.
Jerry ___________________________________________
steve wrote: > Jerry Avins wrote: >> steve wrote: >
In theory only. To resolve a signal at 2.00000X would require 1,000,000 seconds. (OK: maybe only 150 hours.)
True, but you can get that information a lot faster.
Right or wrong, that's what I meant.* What's more, to resolve Fs/2 - 10 Hz, you also need to to sample for a time in the order of 1/10 second. Why does it seem strange?
It seems to me that Tim is assuming anti-alias filters that produce results sooner than next week, and signals that would have components above Fs/2 without them. I don't think those assumptions are unfair.
The Nyquist criterion does indeed assume ideal conditions. Tim will show that the assumption is rarely justified for real work.
Jerry ___________________________________________
it's not a howler, but the sampling frequency, Fs, must be strictly greater than twice the highest frequency, B, at least if that highest frequency is sinusoidal resulting in two dirac spikes at +/- B on the spectrum. the simplest way to say it is that Fs > 2*B
the other thing i was gonna say is that at Wikipedia we are stuggling with some of this same stuff (what the Sampling Theorem, as commonly depicted in textbooks really says, the historic sampling theorem from Shannon is a bit different) and, perhaps, to avoid duplication of effort, you might want to jump into that fray instead. it's at:
i think there is some writing that craps up the article, but that is the lot and legacy of Wikipedia. an encyclopedia written by committee (the biggest, most inclusive committee possible). so "design by committee" is a problem.
Bitte.
r b-j
150 hours? Why?
1/f is twice the nyquist time (1/2 the rate, undersampling) yet you seem to be implying oversampling
but, Tim, the point is that you have to sample *faster* than 2X. sampling at 2X ain't good enough, even theoretically. sampling at
2.000001X might be good enough theoretically (the reconstruction filter will be a bitch) if acausality (or a long delay for the causal case) ain't a problem. the other thing to think about is that no D/A really outputs dirac impulses, so then something like a zero-order hold (ZOH) might have to be modeled for reasons of practicallity.lastly, even though we fight about a bunch of other things, i was surprized at the support i had at Wikipedia to include that "T factor" in the dirac comb sampling operator. putting it there and not in the passband gain of the reconstruction filter is dimensionally most appropriate and help set up the ZOH model without dropping the T factor (or going through a contorted argument for how to include it).
i haven't read through this thread yet, so i apologize in advance if i am repeating someone else's words.
r b-j
i just came upon this. as one who has recently jumped into that fray, alls i can say is "HELP!".
even if you lock horns with me, i would really like it if more comp.dspers would come to Wikipedia and contribute. someday, that can be the new FAQ, for *any* newsgroup.
r b-j
... snip ...
There is a world of difference between the output filters needed after a sample and hold, and after a quasi impulse function. Also in the gain needed.
The impulse function has the advantage that several can be mixed. I took advantage of this in a PABX years ago to provide call merging. The actual pulses were about 1% of the repetition rate period. The accumulated DC components limited the merging to three calls.
A bad assumption on my part. 2.000001X isn't Hz; it needs to be normalized. The result is not a million seconds, but a million sample times. That's still a long time. Most of the time, there's pretty good resolution at half that, in this case, 500,000 sample times.
I don't see what you mean. Could you explain with an equation or two?
Jerry
Have something to add? Share your thoughts — no account required.
Ask the community — no account required