# Nyquist rate for sampling complex-valued data?

Started by November 11, 2004
```Stephan M. Bernsee wrote:

> On 2004-11-12 06:58:05 +0100, Jerry Avins <jya@ieee.org> said:
>
>> Tim Wescott wrote:
>>
>>> Jerry Avins wrote:
>>>
>>>> The signal, its derivative _and_ its integral? Three samples?
>>>> <raised eyebrow>
>>>>
>>>> Jerry
>>>
>>>
>>> Well, if that troubles you, how about the signal, it's first derivative,
>>> it's second derivative, it's third, ad infinitum -- then you can make a
>>> Taylor's series expansion and get the whole signal back all from samples
>>> taken at one point in time.[snip]
>>
>>
>> Certainly not per cycle. What are you illustrating?
>>
>> Jerry
>
>
> You were asking why he would take three samples (f, df, F) instead of
> two (re, im).
>
> You were referring to consecutive samples in time (the sample rate can
> be lowered if you put the required information into the individual
> measurements) while Tim says you need to sample only one point in time
> if you have the signal value and its 1st, 2nd, 3rd, Nth derivative at
> that point.
>
> In a way this is the extreme case of what complex-valued sampling does.
> I believe you were still thinking about using half the sampling rate -
> while Tim's example would require only 1/Nth of the sampling rate.

Thanks. In the end, if a signal is bandlimited to f_max and sampled for
a time T_total, there is a definite number of sines and cosines -- call
the aggregate N -- that will exactly reproduce it. Any N independent
measurements will in theory provide enough information to perform the
reproduction. The need for independence limits how closely in time
measurements of the same type of quantity (value, derivative order) can
be made, which leads us to uniform sampling for the duration of the
signal. The higher-order quantities work on paper and sometimes in
retrospect, but not in practice in real time. Try it and see! :-)

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;
```
```Jerry Avins wrote:
> Tim Wescott wrote:
>
>
>>Jerry Avins wrote:
>>
>>
>>>The signal, its derivative _and_ its integral? Three samples?
>>><raised eyebrow>
>>>
>>>Jerry
>>
>>
>>Well, if that troubles you, how about the signal, it's first derivative,
>>it's second derivative, it's third, ad infinitum -- then you can make a
>>Taylor's series expansion and get the whole signal back all from samples
>>taken at one point in time.
>>
>>Now keep in mind that I'm speaking of doings in mathmagic land, where
>>there is no such thing as noise to corrupt one's measurements, and it
>>would take me some real work to figure out just what would work here in
>>the real world.
>
>
> Certainly not per cycle. What are you illustrating?
>
> Jerry

Just that the Nyquist theorem itself only requires that you have
independent samples coming in at an adequate average rate -- the
Taylor's series is an extreme example where you have an infinite number
of independent samples all at one point, which (in theory) allows you to
reconstruct a signal over all time.

There is, of course, a bandwidth limitation -- the Taylor's series only
works for signals that are continuous in all their derivatives, which is
a way of saying that the signal's "bandwidthishness" needs to converge
faster than the series.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com
```
```Jerry Avins wrote:

> Stephan M. Bernsee wrote:
>
>
>>On 2004-11-12 06:58:05 +0100, Jerry Avins <jya@ieee.org> said:
>>
>>
>>>Tim Wescott wrote:
>>>
>>>
>>>>Jerry Avins wrote:
>>>>
>>>>
>>>>>The signal, its derivative _and_ its integral? Three samples?
>>>>><raised eyebrow>
>>>>>
>>>>>Jerry
>>>>
>>>>
>>>>Well, if that troubles you, how about the signal, it's first derivative,
>>>>it's second derivative, it's third, ad infinitum -- then you can make a
>>>>Taylor's series expansion and get the whole signal back all from samples
>>>>taken at one point in time.[snip]
>>>
>>>
>>>Certainly not per cycle. What are you illustrating?
>>>
>>>Jerry
>>
>>
>>You were asking why he would take three samples (f, df, F) instead of
>>two (re, im).
>>
>>You were referring to consecutive samples in time (the sample rate can
>>be lowered if you put the required information into the individual
>>measurements) while Tim says you need to sample only one point in time
>>if you have the signal value and its 1st, 2nd, 3rd, Nth derivative at
>>that point.
>>
>>In a way this is the extreme case of what complex-valued sampling does.
>>I believe you were still thinking about using half the sampling rate -
>>while Tim's example would require only 1/Nth of the sampling rate.
>
>
> Thanks. In the end, if a signal is bandlimited to f_max and sampled for
> a time T_total, there is a definite number of sines and cosines -- call
> the aggregate N -- that will exactly reproduce it. Any N independent
> measurements will in theory provide enough information to perform the
> reproduction. The need for independence limits how closely in time
> measurements of the same type of quantity (value, derivative order) can
> be made, which leads us to uniform sampling for the duration of the
> signal. The higher-order quantities work on paper and sometimes in
> retrospect, but not in practice in real time. Try it and see! :-)
>
> Jerry

Yes, I was just illustrating a point.  In the real world you'll have
measurement noise and non-ideal bandwidth limitations that'll muck up
your theoretical ideal.  You could, in theory, use more measurements
than two per sample, but if you asked me to do it I'd insist on doing
some careful background work to insure that I meet whatever performance

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com
```
```Jeroen Boschma wrote:

>
> Tim Wescott wrote:
>

>>Or sample the signal and it's derivative, or the signal, its derivative
>>and its integral, etc.  Nyquist only demands that you take unique
>>samples and that the average rate be more than twice the lowest frequency.
>
>                                                            ^^^^^^
>
> Should be highest frequency, I guess. This is sort of a generalized requirement. For bandlimited
> signals which are centered around a non-zero center frequency, the required sampling rate is *at
> least* more than the bandwidth of the signal. In those cases, the sampling also shifts the signal to
> a lower frequency (around zero if the sampling frequency is properly chosen) and the average
> sampling rate only has to be twice the highest frequency of the resulting signal. So if you want to
> make a DSP-receiver for the FM radio band (around 100 MHz), you'll only need to sample the radio
> signals with a few hundred kHz.
>

Sorry, typo.  You need to sample at a rate higher than twice the
bandwidth of the signal, and in cases where you're sampling a signal
riding on a carrier that you actually get unique samples.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com
```
```Tim Wescott <tim@wescottnospamdesign.com> wrote in message
>
> Or sample the signal and it's derivative, or the signal, its derivative
> and its integral, etc.  Nyquist only demands that you take unique
> samples and that the average rate be more than twice the lowest frequency.
>
> Of course, defining "unique" may take a little bit of math...

2X the lowest frequency? That would mean if I add a constant to any
signal then it can be always be characterized by a single sample. My
life just got a lot simpler ;)

Dirk
```