What Nyquist Didn't Say

I've seen a lot of posts over the last year or so that indicate a lack 
of understanding of the implications of the Nyquist theory, and just 
where the Nyquist rate fits into the design of sampled systems.

So I decided to write a short little article to make it all clear.

It's a little longer than 'short', and it took me way longer than I 
thought it would, but at least it's done and hopefully it's clear.

You can see it at 
http://www.wescottdesign.com/articles/Sampling/sampling.html.

If you're new to this stuff, I hope it helps.  If you're an expert and 
you have the time, please feel free to read it and send me comments or 
post them here.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Posting from Google?  See http://cfaj.freeshell.org/google/

"Applied Control Theory for Embedded Systems" came out in April.
See details at http://www.wescottdesign.com/actfes/actfes.html

Hello Tim,


> I've seen a lot of posts over the last year or so that indicate a lack 
> of understanding of the implications of the Nyquist theory, and just 
> where the Nyquist rate fits into the design of sampled systems.
> 
> So I decided to write a short little article to make it all clear.
> 
> It's a little longer than 'short', and it took me way longer than I 
> thought it would, but at least it's done and hopefully it's clear.
> 
> You can see it at 
> http://www.wescottdesign.com/articles/Sampling/sampling.html.
> 
> If you're new to this stuff, I hope it helps.  If you're an expert and 
> you have the time, please feel free to read it and send me comments or 
> post them here.
> 

Very nice. Should be distributed to universities so the kids learn some 
real stuff.

Re "3.2 Nyquist and Signal Content": If Nyquist would have listened to 
some of today's content (digital radio etc.) he'd have said that it 
ain't worth sampling it :-)

I like the wording "line in the sand". Isn't that how Archimedes started 
studying his circles? They didn't need any white board with the smelly 
marker pens.

-- 
Regards, Joerg

http://www.analogconsultants.com

On Fri, 29 Sep 2006 12:37:04 -0700, Tim Wescott <tim@seemywebsite.com>
wrote:

>I've seen a lot of posts over the last year or so that indicate a lack 
>of understanding of the implications of the Nyquist theory, and just 
>where the Nyquist rate fits into the design of sampled systems.
>
>So I decided to write a short little article to make it all clear.
>
>It's a little longer than 'short', and it took me way longer than I 
>thought it would, but at least it's done and hopefully it's clear.
>
>You can see it at 
>http://www.wescottdesign.com/articles/Sampling/sampling.html.
>
>If you're new to this stuff, I hope it helps.  If you're an expert and 
>you have the time, please feel free to read it and send me comments or 
>post them here.


Pretty good. My only quibble is the various statements about what
Nyquist said and didn't say.

http://en.wikipedia.org/wiki/Nyquist-Shannon_sampling_theorem#Historical_background


"Exactly how, when, or why Nyquist had his name attached to the
sampling theorem remains obscure. The first known use of the term
Nyquist sampling theorem is in a 1965 book[4]. It had been called the
Shannon Sampling Theorem as early as 1954[5], but also just the
sampling theorem by several other books in the early 1950s."

It was actually Shannon (among others) that did the sampling theorem;
Nyquist made an observation. Your bibliography doesn't cite either of
them. It's probably correct to use "Nyquist rate" but not "Nyquist
theorem."

John

Hello John,

> 
>>I've seen a lot of posts over the last year or so that indicate a lack 
>>of understanding of the implications of the Nyquist theory, and just 
>>where the Nyquist rate fits into the design of sampled systems.
>>
>>So I decided to write a short little article to make it all clear.
>>
>>It's a little longer than 'short', and it took me way longer than I 
>>thought it would, but at least it's done and hopefully it's clear.
>>
>>You can see it at 
>>http://www.wescottdesign.com/articles/Sampling/sampling.html.
>>
>>If you're new to this stuff, I hope it helps.  If you're an expert and 
>>you have the time, please feel free to read it and send me comments or 
>>post them here.
> 
> 
> Pretty good. My only quibble is the various statements about what
> Nyquist said and didn't say.
> 
> http://en.wikipedia.org/wiki/Nyquist-Shannon_sampling_theorem#Historical_background
> 
> 
> "Exactly how, when, or why Nyquist had his name attached to the
> sampling theorem remains obscure. The first known use of the term
> Nyquist sampling theorem is in a 1965 book[4]. It had been called the
> Shannon Sampling Theorem as early as 1954[5], but also just the
> sampling theorem by several other books in the early 1950s."
> 
> It was actually Shannon (among others) that did the sampling theorem;
> Nyquist made an observation. Your bibliography doesn't cite either of
> them. It's probably correct to use "Nyquist rate" but not "Nyquist
> theorem."
> 

Nyquist published his paper about the minimum required sample rate in 
1928. Shannon was a kid of 12 years back then. The paper wasn't about 
ADCs or sampling in today's sense but about how many pulses per second 
could be passed through a telegraph channel of a given bandwidth.

-- 
Regards, Joerg

http://www.analogconsultants.com

Tim Wescott wrote:

> If you're new to this stuff, I hope it helps.  If you're an expert and 
> you have the time, please feel free to read it and send me comments or 
> post them here.
> 

"2.1 Aliasing

By ignoring anything that goes on between samples the sampling process 
throws away information about the original signal.  This information 
loss must be taken into account during system design."

This seems like something of an oversimplification.  If the orginal
signal is naturally or otherwise bandwidth-limited to well below
2x the sample rate, there may not be any useful information available
to throw away and the loss may not have to be taken into account
during system design.

John Larkin wrote:
> On Fri, 29 Sep 2006 12:37:04 -0700, Tim Wescott <tim@seemywebsite.com>
> wrote:
> 
> 
>>I've seen a lot of posts over the last year or so that indicate a lack 
>>of understanding of the implications of the Nyquist theory, and just 
>>where the Nyquist rate fits into the design of sampled systems.
>>
>>So I decided to write a short little article to make it all clear.
>>
>>It's a little longer than 'short', and it took me way longer than I 
>>thought it would, but at least it's done and hopefully it's clear.
>>
>>You can see it at 
>>http://www.wescottdesign.com/articles/Sampling/sampling.html.
>>
>>If you're new to this stuff, I hope it helps.  If you're an expert and 
>>you have the time, please feel free to read it and send me comments or 
>>post them here.
> 
> 
> 
> Pretty good. My only quibble is the various statements about what
> Nyquist said and didn't say.
> 
> http://en.wikipedia.org/wiki/Nyquist-Shannon_sampling_theorem#Historical_background
> 
> 
> "Exactly how, when, or why Nyquist had his name attached to the
> sampling theorem remains obscure. The first known use of the term
> Nyquist sampling theorem is in a 1965 book[4]. It had been called the
> Shannon Sampling Theorem as early as 1954[5], but also just the
> sampling theorem by several other books in the early 1950s."
> 
> It was actually Shannon (among others) that did the sampling theorem;
> Nyquist made an observation. Your bibliography doesn't cite either of
> them. It's probably correct to use "Nyquist rate" but not "Nyquist
> theorem."
> 
> John
> 
Thanks John, that's a good point.  I'll probably leave the titles intact 
because the paper is a reaction to all the posts that have the words 
"Nyquist says" followed by something _wrong_ -- but I should put in a 
disclaimer, or something.

I'm going to go off and do some web searching; in the mean time do you 
have any URLs that point to the seminal papers?

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Posting from Google?  See http://cfaj.freeshell.org/google/

"Applied Control Theory for Embedded Systems" came out in April.
See details at http://www.wescottdesign.com/actfes/actfes.html

Jim Stewart wrote:
> Tim Wescott wrote:
> 
>> If you're new to this stuff, I hope it helps.  If you're an expert and 
>> you have the time, please feel free to read it and send me comments or 
>> post them here.
>>
> 
> "2.1 Aliasing
> 
> By ignoring anything that goes on between samples the sampling process 
> throws away information about the original signal.  This information 
> loss must be taken into account during system design."
> 
> This seems like something of an oversimplification.  If the orginal
> signal is naturally or otherwise bandwidth-limited to well below
> 2x the sample rate, there may not be any useful information available
> to throw away and the loss may not have to be taken into account
> during system design.

I can squirm out of that objection:

If you have the continuous-time signal, then you _know_ there's nothing 
of note above Fs/2.  If you don't have the continuous-time signal, then 
you _can't_ know there's nothing of note above Fs/2, unless the sampled 
signal train comes with a Certificate of Limited Bandwidth.

Later on in the article I talk about signals that are, indeed, 
sufficiently bandlimited by their nature, and the fact that you probably 
don't want to do any explicit anti-alias filtering in such a case.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Posting from Google?  See http://cfaj.freeshell.org/google/

"Applied Control Theory for Embedded Systems" came out in April.
See details at http://www.wescottdesign.com/actfes/actfes.html

On Fri, 29 Sep 2006 13:40:33 -0700, Tim Wescott <tim@seemywebsite.com>
wrote:

>John Larkin wrote:
>> On Fri, 29 Sep 2006 12:37:04 -0700, Tim Wescott <tim@seemywebsite.com>
>> wrote:
>> 
>> 
>>>I've seen a lot of posts over the last year or so that indicate a lack 
>>>of understanding of the implications of the Nyquist theory, and just 
>>>where the Nyquist rate fits into the design of sampled systems.
>>>
>>>So I decided to write a short little article to make it all clear.
>>>
>>>It's a little longer than 'short', and it took me way longer than I 
>>>thought it would, but at least it's done and hopefully it's clear.
>>>
>>>You can see it at 
>>>http://www.wescottdesign.com/articles/Sampling/sampling.html.
>>>
>>>If you're new to this stuff, I hope it helps.  If you're an expert and 
>>>you have the time, please feel free to read it and send me comments or 
>>>post them here.
>> 
>> 
>> 
>> Pretty good. My only quibble is the various statements about what
>> Nyquist said and didn't say.
>> 
>> http://en.wikipedia.org/wiki/Nyquist-Shannon_sampling_theorem#Historical_background
>> 
>> 
>> "Exactly how, when, or why Nyquist had his name attached to the
>> sampling theorem remains obscure. The first known use of the term
>> Nyquist sampling theorem is in a 1965 book[4]. It had been called the
>> Shannon Sampling Theorem as early as 1954[5], but also just the
>> sampling theorem by several other books in the early 1950s."
>> 
>> It was actually Shannon (among others) that did the sampling theorem;
>> Nyquist made an observation. Your bibliography doesn't cite either of
>> them. It's probably correct to use "Nyquist rate" but not "Nyquist
>> theorem."
>> 
>> John
>> 
>Thanks John, that's a good point.  I'll probably leave the titles intact 
>because the paper is a reaction to all the posts that have the words 
>"Nyquist says" followed by something _wrong_ -- but I should put in a 
>disclaimer, or something.
>
>I'm going to go off and do some web searching; in the mean time do you 
>have any URLs that point to the seminal papers?

The wiki article has links to Nyquist's 1928 paper and to Shannon's
1949 paper. Few EEs that I've met have ever read either of them, and
have absorbed "the Nyquist theorem" mostly by hearsay, which I guess
is your point.

John

On Fri, 29 Sep 2006 20:02:21 GMT, Joerg
<notthisjoergsch@removethispacbell.net> wrote:


>I like the wording "line in the sand". Isn't that how Archimedes started 
>studying his circles? They didn't need any white board with the smelly 
>marker pens.

Don't you go knocking whiteboards. A big board with a nice fresh set
of markers will multiply my IQ by about 1.3 or so. Sand doesn't work
nearly as well.

Oh well, back to the drawing board, literally.

John

Hello Tim,


> I'm going to go off and do some web searching; in the mean time do you 
> have any URLs that point to the seminal papers?
> 

This one could be a start, written by my old communications theory 
professor (he was actually one of the really nice profs):

http://www.ee.technion.ac.il/courses/044130/00755459.pdf

-- 
Regards, Joerg

http://www.analogconsultants.com