# Nyquist????

Started by October 26, 2008
```Hi everyone,
What is the dirrerence between nyquist frequency, nyquist rate and nyquist
bandwidth? I am sure that they dont point to the same concept. I know what
is a nyquist frequency. But I am not able to understand the concept of the
other two.
I came across these terms while learning about pulse shaping filters
(raised cosine filters). I tried to understand, but still I am not able to
completely do it.
Can anyone explain me or say the difference between these terms and what
they mean?
bye.
```
```On Sat, 25 Oct 2008 22:26:21 -0500, krish_dsp wrote:

> Hi everyone,
> What is the dirrerence between nyquist frequency, nyquist rate and
> nyquist bandwidth? I am sure that they dont point to the same concept. I
> know what is a nyquist frequency. But I am not able to understand the
> concept of the other two.
> I came across these terms while learning about pulse shaping filters
> (raised cosine filters). I tried to understand, but still I am not able
> to completely do it.
> Can anyone explain me or say the difference between these terms and what
> they mean?
> bye.

See if this helps.

http://www.wescottdesign.com/articles/Sampling/sampling.html

--

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

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" gives you just what it says.
See details at http://www.wescottdesign.com/actfes/actfes.html
```
```On Sat, 25 Oct 2008 22:35:20 -0500, Tim Wescott
<tim@justseemywebsite.com> wrote:

>See if this helps.
>
>http://www.wescottdesign.com/articles/Sampling/sampling.html

Tim,

There are a few statements in that with which I disagree:

"The Nyquist theorem states that if you have a signal that is
perfectly band limited to a bandwidth of f0 then you can collect all
the information there is in that signal by sampling, as long as your
sample rate is 2f0 or more."

Two items here:
First, if you are using the colloquial definition of "band limited" to
mean "a baseband signal having finite amplitude between DC and f0,
inclusive, and zero amplitude at higher frequencies", then the actual
bandwidth of that signal is 2f0.  This generalizes to the case that
you later describe in section 3.4.
Second, if the signal is perfectly band limited to f0 (bandwidth of
2f0), then you cannot collect all the information if you sample AT
2f0; you must sample incrementally greater than 2f0.

"By ignoring anything that goes on between samples the sampling
process throws away information about the original signal."

NOT if the Nyquist Criterion is satisfied!  There is NO loss of
information.

"Interpolation is the process where the value of the continuous-time
signals between samples is constructed from previous values of the
discrete-time signal."

I'm being pedantic, but in the ideal case (using an ideal
reconstruction filter), the value between samples is constructed from
ALL values of the discrete-time signal.  In the more general case,
where delay is allowed, Interpolation may also include future values
of the discrete-time signal.

As for the OP's questions:  This was discussed here
and I don't know if the ultimate answer was ever found.  Two schools
of thought emerged in that discussion.  The first argues that the
definitions should be based upon the characteristics of the sampling,
while the second argues that the definitions should be based upon the
characteristics of the signal being sampled.

In the sampling-based point of view, the "Nyquist Frequency" is "half
the sampling frequency".  In the signal-based point of view, the
"Nyquist Frequency" is "the highest frequency of significant amplitude
contained within the baseband signal to be sampled", so that the
"Nyquist Rate" (the minimum frequency above which the signal must be
sampled) is "twice the Nyquist Frequency".

Both of these can easily be generalized to the baseband sampling
described in section 3.4 of Tim Wescott's reference, in which case the
"Nyquist Bandwidth" represents either the overall bandwidth that a
bandpass signal must not equal or exceed for the selected sampling
frequency, or the minimum sampling frequency above which the selected
bandpass signal must be sampled, respectively.
```
```Greg Berchin wrote:
> On Sat, 25 Oct 2008 22:35:20 -0500, Tim Wescott
> <tim@justseemywebsite.com> wrote:
>
>> See if this helps.
>>
>> http://www.wescottdesign.com/articles/Sampling/sampling.html
>
> Tim,
>
> There are a few statements in that with which I disagree:

You're the second person recently whose brought up similar issues.  I've
got this on my list to fix (time, time, time).

> "The Nyquist theorem states that if you have a signal that is
> perfectly band limited to a bandwidth of f0 then you can collect all
> the information there is in that signal by sampling, as long as your
> sample rate is 2f0 or more."

Agreed.  It should say "... rate is more than ...".  I guess I've taken
one too many limits to infinity.

> Two items here:
> First, if you are using the colloquial definition of "band limited" to
> mean "a baseband signal having finite amplitude between DC and f0,
> inclusive, and zero amplitude at higher frequencies", then the actual
> bandwidth of that signal is 2f0.  This generalizes to the case that
> you later describe in section 3.4.

I'll have to take a look at that; I don't know that I'll change any
wording though -- the audience is intended to extend to people who have
not yet grokked complex signals, which pretty much excludes
understanding the concept of a negative frequency (other than as
something that's identical to a positive frequency).

I used to be driven up the wall by instruction that purposely left stuff
like that out, or contained colloquial usage that wasn't "exactly
right".  But then I realized that there's probably only one in a million
first graders on whom you can start by positing multiplication and
addition, and derive the existence of the integers from that -- and
they've already learned to count, the little cheaters.

Certainly if I can make it correct _and_ understandable to the newbie I
will do so.

> Second, if the signal is perfectly band limited to f0 (bandwidth of
> 2f0), then you cannot collect all the information if you sample AT
> 2f0; you must sample incrementally greater than 2f0.

Same problem, same solution.

> "By ignoring anything that goes on between samples the sampling
> process throws away information about the original signal."
>
> NOT if the Nyquist Criterion is satisfied!  There is NO loss of
> information.

Yes and no.  I would contend that the difference is semantic -- if the
Nyquist criterion is satisfied then the information being thrown away is
redundant.  I gather that you would say (and perhaps be more
theoretically correct) that if it's redundant it isn't really
information.  Once again, I'm trying to give tools to beginners so they
can wrap their brains around the subject.

If I can word it to be clear _and_ correct I'll do so.

> "Interpolation is the process where the value of the continuous-time
> signals between samples is constructed from previous values of the
> discrete-time signal."
>
> I'm being pedantic, but in the ideal case (using an ideal
> reconstruction filter), the value between samples is constructed from
> ALL values of the discrete-time signal.  In the more general case,
> where delay is allowed, Interpolation may also include future values
> of the discrete-time signal.

Aieeee!  True.  But how to explain it to a beginner without making their
brain squirt out their ears in sheer self-defense?

> As for the OP's questions:  This was discussed here
> and I don't know if the ultimate answer was ever found.  Two schools
> of thought emerged in that discussion.  The first argues that the
> definitions should be based upon the characteristics of the sampling,
> while the second argues that the definitions should be based upon the
> characteristics of the signal being sampled.
>
> In the sampling-based point of view, the "Nyquist Frequency" is "half
> the sampling frequency".  In the signal-based point of view, the
> "Nyquist Frequency" is "the highest frequency of significant amplitude
> contained within the baseband signal to be sampled", so that the
> "Nyquist Rate" (the minimum frequency above which the signal must be
> sampled) is "twice the Nyquist Frequency".
>
> Both of these can easily be generalized to the baseband sampling
> described in section 3.4 of Tim Wescott's reference, in which case the
> "Nyquist Bandwidth" represents either the overall bandwidth that a
> bandpass signal must not equal or exceed for the selected sampling
> frequency, or the minimum sampling frequency above which the selected
> bandpass signal must be sampled, respectively.

When there is ambiguity about terms like this it's best to understand
the underlying theory; then you have a much better chance of figuring
out what form the author is using.

Of the three terms presented, the only one I recognized as being even
remotely unequivocal is "Nyquist rate", and there's enough folks that
get that one confused that you can't even trust it when you see it in
the general literature.

--

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

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" gives you just what it says.
See details at http://www.wescottdesign.com/actfes/actfes.html
```
```Tim Wescott wrote:
> Greg Berchin wrote:

...

>> Two items here:  First, if you are using the colloquial definition of
>> "band limited" to
>> mean "a baseband signal having finite amplitude between DC and f0,
>> inclusive, and zero amplitude at higher frequencies", then the actual
>> bandwidth of that signal is 2f0.  This generalizes to the case that
>> you later describe in section 3.4.

That's too pedantic (too math based?) for me. Granted, one can represent
sin(wt) as [exp(jwt) - exp(-jwt)]/2 and claim that in a sense the
spectrum runs from -w to +w. I think that the notation is a useful
convenience, but the double-wide spectrum is a bit hard to accept when
there is just one oscillator running or one wheel spinning.

> I'll have to take a look at that; I don't know that I'll change any
> wording though -- the audience is intended to extend to people who have
> not yet grokked complex signals, which pretty much excludes
> understanding the concept of a negative frequency (other than as
> something that's identical to a positive frequency).

Why must it be negative frequency? Negative-running time is no less
counterintuitive. :-)

> I used to be driven up the wall by instruction that purposely left stuff
> like that out, or contained colloquial usage that wasn't "exactly
> right".  But then I realized that there's probably only one in a million
> first graders on whom you can start by positing multiplication and
> addition, and derive the existence of the integers from that -- and
> they've already learned to count, the little cheaters.

Finite time forces us to leave stuff out. The real sin is simplifying to
the point of misrepresentation.

> Certainly if I can make it correct _and_ understandable to the newbie I
> will do so.

Good goal!

>> Second, if the signal is perfectly band limited to f0 (bandwidth of
>> 2f0), then you cannot collect all the information if you sample AT
>> 2f0; you must sample incrementally greater than 2f0.
>
> Same problem, same solution.

No. That's misrepresentation.

>> "By ignoring anything that goes on between samples the sampling
>> process throws away information about the original signal."
>>
>> NOT if the Nyquist Criterion is satisfied!  There is NO loss of
>> information.
>
> Yes and no.  I would contend that the difference is semantic -- if the
> Nyquist criterion is satisfied then the information being thrown away is
> redundant.  I gather that you would say (and perhaps be more
> theoretically correct) that if it's redundant it isn't really
> information.  Once again, I'm trying to give tools to beginners so they
> can wrap their brains around the subject.

Go with Greg here.

> If I can word it to be clear _and_ correct I'll do so.

Maybe that's one of the things you'd do well to leave out?

...

Jerry
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```
```On Sun, 26 Oct 2008 11:38:12 -0700, Tim Wescott <tim@seemywebsite.com>
wrote:

>I'll have to take a look at that; I don't know that I'll change any
>wording though -- the audience is intended to extend to people who have
>not yet grokked complex signals, which pretty much excludes
>understanding the concept of a negative frequency (other than as
>something that's identical to a positive frequency).

I am sympathetic, but if you ignore the negative frequencies for
baseband signals, then you are faced with having to explain their
unexpected appearance when you discuss modulated signals.

>I would contend that the difference is semantic -- if the
>Nyquist criterion is satisfied then the information being thrown away is
>redundant.  I gather that you would say (and perhaps be more
>theoretically correct) that if it's redundant it isn't really
>information.  Once again, I'm trying to give tools to beginners so they
>can wrap their brains around the subject.

"What if something important happens between the samples?"  The answer
is in the statements above, interpreted either your way or mine:
"Bandlimiting of the signal so that the Nyquist Criterion is satisfied
GUARANTEES that nothing of importance can possibly occur between the
samples."

>When there is ambiguity about terms like this it's best to understand
>the underlying theory; then you have a much better chance of figuring
>out what form the author is using.

Agreed.

-- Greg
```
```krish_dsp wrote:

> What is the dirrerence between nyquist frequency, nyquist rate and nyquist
> bandwidth? I am sure that they dont point to the same concept. I know what
> is a nyquist frequency. But I am not able to understand the concept of the
> other two.

Rate is usually the change in something per unit time, which in
this case should be the same as frequency.

The Nyquist bandwidth includes cases where the spectrum
doesn't include zero frequency.  Even so, it is a little
strange.  Even for a non-baseband signal, sampling has
a rate or frequency but not a bandwidth.

-- glen

```
```Greg Berchin wrote:

> There are a few statements in that with which I disagree:

> "The Nyquist theorem states that if you have a signal that is
> perfectly band limited to a bandwidth of f0 then you can collect all
> the information there is in that signal by sampling, as long as your
> sample rate is 2f0 or more."

> Two items here:
> First, if you are using the colloquial definition of "band limited" to
> mean "a baseband signal having finite amplitude between DC and f0,
> inclusive, and zero amplitude at higher frequencies", then the actual
> bandwidth of that signal is 2f0.  This generalizes to the case that
> you later describe in section 3.4.

Including negative frequencies does give a convenient factor
of two, but it is a little strange in the case of non-baseband
signals.

If you have an AM DSB (double side band) signal then the bandwidth
is twice the modulating frequency, and you can argue that the upper
and lower sidebands are the positive and negative frequencies of
the modulating signal.  But again this is a fake factor of two,
as your source has redundancy in the sidebands.

If you have a non-baseband signal that isn't AM DSB then you
still need a factor of two.

> Second, if the signal is perfectly band limited to f0 (bandwidth of
> 2f0), then you cannot collect all the information if you sample AT
> 2f0; you must sample incrementally greater than 2f0.

If the spectrum has finite amplitude at zero and f0 then you
won't be able to tell the difference.  It has to be infinite
(delta function) at f0 to cause problems.  Also, this only
applies to either infinite time or periodic signals.

> "By ignoring anything that goes on between samples the sampling
> process throws away information about the original signal."

> NOT if the Nyquist Criterion is satisfied!  There is NO loss of
> information.

> "Interpolation is the process where the value of the continuous-time
> signals between samples is constructed from previous values of the
> discrete-time signal."

> I'm being pedantic, but in the ideal case (using an ideal
> reconstruction filter), the value between samples is constructed from
> ALL values of the discrete-time signal.  In the more general case,
> where delay is allowed, Interpolation may also include future values
> of the discrete-time signal.

The question, then, is what kind of interpolation you do.
Ideally you need an infinite degree interpolation polynomial.
It you don't need to be exact (all real cases) then a finite
polynomial will get you close enough.

> As for the OP's questions:  This was discussed here
> and I don't know if the ultimate answer was ever found.  Two schools
> of thought emerged in that discussion.  The first argues that the
> definitions should be based upon the characteristics of the sampling,
> while the second argues that the definitions should be based upon the
> characteristics of the signal being sampled.

> In the sampling-based point of view, the "Nyquist Frequency" is "half
> the sampling frequency".  In the signal-based point of view, the
> "Nyquist Frequency" is "the highest frequency of significant amplitude
> contained within the baseband signal to be sampled", so that the
> "Nyquist Rate" (the minimum frequency above which the signal must be
> sampled) is "twice the Nyquist Frequency".

So the Nyquist bandwidth is half the sampling bandwidth?
That doesn't sound right.

> Both of these can easily be generalized to the baseband sampling
> described in section 3.4 of Tim Wescott's reference, in which case the
> "Nyquist Bandwidth" represents either the overall bandwidth that a
> bandpass signal must not equal or exceed for the selected sampling
> frequency, or the minimum sampling frequency above which the selected
> bandpass signal must be sampled, respectively.

-- glen

```
```glen herrmannsfeldt wrote:

...

> So the Nyquist bandwidth is half the sampling bandwidth?
> That doesn't sound right.

I think I understand sampling rate. What is sampling bandwidth?

Jerry
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```
```>What is the dirrerence between nyquist frequency, nyquist rate and
nyquist
>bandwidth? I am sure that they dont point to the same concept. I know
what
>is a nyquist frequency. But I am not able to understand the concept of
the
>other two.

My understanding:
Nyquist (sampling) frequency = Nyquist (sampling) rate = 2 * (Nyquist)
bandwidth

The first two are well-defined.  I deduced the last one from the Wikipedia
entry for raised-cosine filter
(http://en.wikipedia.org/wiki/Raised-cosine_filter).  However it is not
standard usage, as far as I know.  Many rightfully call it just bandwidth.

Hope this helps,

Emre
```