# Nyquist Didn't Say That

Started by August 22, 2006
```Rune Allnor wrote:
> steve wrote:
> > Tim Wescott wrote:
> >
> > > One, "I need to monitor a signal that happens at X Hz, so I'm going to
> > > sample it at 2X Hz".
> > >
> > > Two, "I can sample at X Hz, so I'm going to build an anti-alias filter
> > > with a cutoff of X/2 Hz".
> > >
> > looks ok to me
>
> Does it?
>
> >and Mr Nyquist, I suspect,
>
> No.
>
> > ...what do you think the
> > relationships should be
>
> The answers are
>
> a) Sample at Fs > 2X Hz
> b) Cut-off at Fc < X/2 Hz
>
> Note no equality signs here.
>
Well yes, but that is only due to the fact if you sample at exactly 2x
you might sample at the zero points of the the sin wave, and not be
able to reproduce the signal, but most people write =2x because of
convenience, but if 2.0000000000001 is how you like to write it, then
ok.

> The sampling theorem states a *lower*bound* on the relation
> between sampling frequency and the highest significant frequency
> component in the signal.
>
> There is nothing in the sampling theorem to suggest that
> sampling at 2X Hz is *sufficient*.
>
> Tiny detail in phrasing; huge difference in practice.
>
> Rune

```
```
Tim Wescott wrote:

>>
> Because I'm a mercenary.
>
> Posting to wikipedia doesn't pay directly, nor does it give me public
> credit.  While it benefits the world it doesn't lead to me getting any
> checks in the mail.

I share that pure cynical point of view and consider myself as a kind of
whore also :)))) Good luck.

Here is another tip:

What is generally referred as "sampling" actually consists of two
different processes rather then one:

1. Continuos time non-linear quantization of the amplitude
2. Linear quantization in time

This representation clarifies many issues such as "quantization noise",
why the high sample rates are required, etc.

DSP and Mixed Signal Design Consultant

http://www.abvolt.com

```
```Tim Wescott wrote:
> Kinda off topic --
>
> A month or two ago there was a spate of postings on these groups
> displaying a profound misunderstanding of how to apply Nyquist's theorem
> to problems of setting sampling or designing anti-alias filters.

After reading some of the contributions to this thread, I can see that
you were right.

```
```steve wrote:

...

> Well yes, but that is only due to the fact if you sample at exactly 2x
> you might sample at the zero points of the the sin wave, and not be
> able to reproduce the signal, but most people write =2x because of
> convenience, but if 2.0000000000001 is how you like to write it, then
> ok.

No; it's more than that. It means (among other problems) that there's no
way to determine the component in phase with the sample clock (sine
component), so the amplitude remains unknown.

That's the least of the problems, though. To resolve a frequency of f
Hz, one must sample on the order of 1/f seconds. Frequencies in the
sampled domain lie on a circular scale, so that it is also necessary to
sample on the order of 1/f seconds to resolve a frequency of Fs/2 - f.

We can no more sample frequencies close to Fs/2 in a reasonably short
time than we can those close to DC.

So many misconceptions, so little time. Tim: are you tuned in?

To the person who wondered if he had been asleep in class when the way
to remove aliases after sampling was explained: you didn't miss a thing.
Think of the original components as sticks of varying lengths. (The
lengths are proportional to frequency.) The sampling process chops up
any length greater than Fs/2 into pieces of length Fs/2 which it
discards, and leaves the remainder in the pile. The result is that all
the sticks are shorter than Fs/2, even though some *were originally*
part of longer sticks. There is absolutely no way to tell the original
length after the ax falls.

Jerry
--
Engineering is the art of making what you want from things you can get.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
```
```jacko wrote:
> sample and hold circuit needs sample interval more than twice frequency
> of sample, but lower frequency sampling can be done if more than two
> samplings is happening.
>
> eg 1024Hz and 125HzHz => max freq descrimination = 2^10*5^3 Hz

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
--
Engineering is the art of making what you want from things you can get.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
```
```mobi wrote:
> Do consider this interesting (atleast for me) example
>
> Consider pure Sin wave at X Hz. I start sample it at 2X. Unfortunately
> i start sampling from time = 0. What would i get? Aint i statisifying
> Nyquist here?

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
--
Engineering is the art of making what you want from things you can get.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
```
```Stef wrote:
> In comp.arch.embedded,
> mobi <mobien@gmail.com> wrote:
>> Do consider this interesting (atleast for me) example
>>
>> Consider pure Sin wave at X Hz. I start sample it at 2X. Unfortunately
>> i start sampling from time = 0. What would i get? Aint i statisifying
>> Nyquist here?
>
> No you are not. You seemed to have missed Rune's post in this thread
> about '=' vs ' >'.

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
--
Engineering is the art of making what you want from things you can get.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
```
```Tim Wescott wrote:

> Kinda off topic --
>
> A month or two ago there was a spate of postings on these groups
> displaying a profound misunderstanding of how to apply Nyquist's theorem
> to problems of setting sampling or designing anti-alias filters.  I
> helped folks out as much as I could, but it really demands an article,
> which I am currently working on.
>
> The misconceptions that I noticed pretty much boiled down to the
> following two:
>
> One, "I need to monitor a signal that happens at X Hz, so I'm going to
> sample it at 2X Hz".
>
> Two, "I can sample at X Hz, so I'm going to build an anti-alias filter
> with a cutoff of X/2 Hz".
>
> I estimate that answering these misconceptions will only take 3-4k
> words, but I don't want to miss any other big ones.
>
> Have you seen any other real howlers that relate to Nyquist, and what
> you should really be thinking about when you're pondering sampling
> rates, anti-aliasing filters and/or reconstruction filters?
>
> Danke.
>

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
```
```Jerry Avins wrote:
> steve wrote:

> No; it's more than that. It means (among other problems) that there's no
> way to determine the component in phase with the sample clock (sine
> component), so the amplitude remains unknown.
>
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.

> That's the least of the problems, though. To resolve a frequency of f
> Hz, one must sample on the order of 1/f seconds.

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?

> So many misconceptions, so little time. Tim: are you tuned in?
>
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" <bungalow_steve@yahoo.com> wrote in news:1156354799.705801.226400