Sampling, Again -- Updates

(snip)

I do wonder how many people who discuss Nyquist have ever actually read his paper. I used to have a copy of it. It isn't hard to find in most university libraries, (at least ones old enough to have been around).

-- glen

The title is actually a bit of a misnomer, in that it's really the Nyquist-Shannon theorem, and if you get some continental input there's a Russian guy who figured out the same stuff.

It was motivated by a bunch of folks who all came onto comp.dsp, comp.arch.embedded, and sci.electronics.design and said "Nyquist says" followed by something naive and incorrect. I actually replied to a few posts with "Nyquist _didn't_ say that" before I realized that I had a title and paper concept staring me in the face.

Anyway, he only sort-of states his criterion, without proof. It was Claude Shannon who actually gave a -remarkably short- demonstration in his 1949 paper "Communication in the presence of noise".

Neither were first, by any means. For example, V. Kotelnikov stated basically the same thing, much more verbosely, in the context of interpolation theory. And before that, in 1915, E.T Whittaker wrote something similar, and before that there was this Japanese fellow... You get the idea.

Jeroen Belleman

it was mentioned, and because it was there. It's still

look good when it's printed (and I'm lazy about

scanning across the line. It certainly looks better in

upstairs.

I hope that if it doesn't fully answer your

infinite rate) it does explain what I'm thinking more

if you see something that I still haven't caught! I

critique my work.

Tim,

Appearance-wise this looks EXCELLENT to my eyes! The fonts are now vector, at least all the ones I checked.

FYI, a nice sans font I like to use is palatino. Simply place \usepackage{palatino} near the top, e.g.:

\documentclass[english,11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{babel} \usepackage{palatino}

I'm reading the content now...

I think that for the moment I'm going to register any comments, and unless someone comes up with something really tremendously bad, I'm going to move on to all the _other_ grotty papers I have that need to be improved. They're all old OpenOffice documents that have been printed out as HTML, with all sorts of problems.

So -- I'd rather have a bunch of pretty good documents than one really excellent one.

I'm trying to decide if I want to do a paper next, or if I want to start in on all the homework problems I did for the first 1/2 of "Applied Control Theory for Embedded Systems".

I made a comedy of errors with that statement! Palatino is serif, not sans serif! Also, the psnfss document (texdoc psnfss), which is itself some six years old, states that palatino is obsolete and instead suggests to use mathpazo.

Sorry!

Looks great with Foxit.

John

Please indulge a re-newbie question. Where is the paper under discussion?

Jerry

(snip)

Still, it is interesting to read the older papers and see how they thought about things differently, or not so much differently, than we do today.

It also reminds me of the introduction to an ISDN book explaining the first time that the electronic communications system was all digital. Nyquist was trying to get digital signals through an analog channel, now we put (digitized) analog signals through a digital channel. Same math, though.

-- glen

if

e,

first one here I assume:

-Lasse

2) Use the same link as from the previous thread:

Eric Jacobsen Minister of Algorithms Abineau Communications

Oi! Where's the emoticon for the forehead-slap?

:-/

Duh! I found it right away, but Google groups wouldn't let me respond for over an hour.

Jerry

What Nyquist didn't say: I vonder if they will noteece that I stole all zis crap from other authors like Whittacker and zat russian chappy..

Actually, as near as I can tell without meticulously going through a bunch of papers from the late 40's, the use of the Nyquist rate as a limit on PCM modulation was articulated by Shannon, who was just very precise about crediting Nyquist with being _his_ source of information.

I think that the various sampling theorems must have arisen from the zeitgeist of the early 20th century -- everyone was working on similar problems, and were using similar mathematical tools, so they all arrived at similar results at (roughly) similar times. Even though the 'similar' times were spread out by years, the barriers of language, commercial interests, and separate sub-disciplines means that the results were still arrived at independently (probably).

rif

ll

ce,

Tim, Thanks for making available a clear and informative paper. I would like to make a comment at this late stage about the section on signal reconstruction, mainly about terminology.

You explain that reconstruction is done by interpolating the output signal but unfortunately I feel that you have not clearly identified where the interpolaton occurs.

You say that the first step in interpolation is the generation of a stepped waveform by the zero-hold D/A and you mention the stepped waveform of Figure 5 as being a picture of an interpolated signal. In my view this should be changed to say that generating the stepped waveform is the first step in the reconstruction process, interpolaton being the second step.

Interpolation, the generation of intermediate values between known data points, occurs when the stepped analog signal reaches the analog output reconstruction filter. It is the analog components that generate the intermediate data values, on a continuous basis. You have mentioned the output filter, and I think it would be useful to identify this filter as the spot where interpolation occurs.

Regards, John

The ideal reconstruction is an impulse generator (not the ZOH) followed by an ideal lowpass filter. That avoids the sinc frequency error thingie you get from the ZOH.

John

I thought about that before I used the term, and I like my way. The reason I used it is because, in the context of that paper (and as I generally think about sampling), the sampled time signal _does not exist_ in between the sampling instants. That's a rather strict way to consider things, but I find it keeps me from miss-applying continuous-time wisdom to sampled time systems. Since the signal _doesn't_ exist, the action of a ZOH is, indeed, to interpolate.

The usage is different from what some people would use, but I (a) don't think it's all that far off, (b) think it's more accurate, and (c) think that if someone can't handle Author A using different terminology from Author B, then they can't handle engineering.