Nyquist rate for sampling complex-valued data?

Started by November 11, 2004
```I (vaguely) heard that sampling complex-valued data does not abide by the
Nyquist rate criteria, i.e., the sampling rate fs can go lower than Nyquist
rate and it still can avoid aliasing and reconstruct perfectly...

Is that true? Any theory behind it?

Thanks a lot

```
```kiki wrote:

> I (vaguely) heard that sampling complex-valued data does not abide by the
> Nyquist rate criteria, i.e., the sampling rate fs can go lower than Nyquist
> rate and it still can avoid aliasing and reconstruct perfectly...
>
> Is that true? Any theory behind it?
>
> Thanks a lot

Complex samples consist of a a real part and an imaginary part; another
way to look at that is as two samples. If you do look at it that way,
each complex sample counting for two real ones, then the sample rates
are the same: two real samples or one complex sample in the time it
takes for one cycle if the highest frequency in the signal.

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;
```
```Jerry Avins wrote:

> kiki wrote:
>
>
>>I (vaguely) heard that sampling complex-valued data does not abide by the
>>Nyquist rate criteria, i.e., the sampling rate fs can go lower than Nyquist
>>rate and it still can avoid aliasing and reconstruct perfectly...
>>
>>Is that true? Any theory behind it?
>>
>>Thanks a lot
>
>
> Complex samples consist of a a real part and an imaginary part; another
> way to look at that is as two samples. If you do look at it that way,
> each complex sample counting for two real ones, then the sample rates
> are the same: two real samples or one complex sample in the time it
> takes for one cycle if the highest frequency in the signal.
>
> Jerry

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...

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com
```
```Tim Wescott wrote:
> Jerry Avins wrote:
>
>> kiki wrote:
>>
>>
>>> I (vaguely) heard that sampling complex-valued data does not abide by
>>> the Nyquist rate criteria, i.e., the sampling rate fs can go lower
>>> than Nyquist rate and it still can avoid aliasing and reconstruct
>>> perfectly...
>>>
>>> Is that true? Any theory behind it?
>>>
>>> Thanks a lot
>>
>>
>>
>> Complex samples consist of a a real part and an imaginary part; another
>> way to look at that is as two samples. If you do look at it that way,
>> each complex sample counting for two real ones, then the sample rates
>> are the same: two real samples or one complex sample in the time it
>> takes for one cycle if the highest frequency in the signal.
>>
>> Jerry
>
>
> 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...

The signal, its derivative _and_ its integral? Three samples?
<raised eyebrow>

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;
```
```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.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com
```
```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
--
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;
```
```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.
--
Stephan M. Bernsee
http://www.dspdimension.com

```
```
Tim Wescott wrote:
>
> Jerry Avins wrote:
>
> > kiki wrote:
> >
> >
> >>I (vaguely) heard that sampling complex-valued data does not abide by the
> >>Nyquist rate criteria, i.e., the sampling rate fs can go lower than Nyquist
> >>rate and it still can avoid aliasing and reconstruct perfectly...
> >>
> >>Is that true? Any theory behind it?
> >>
> >>Thanks a lot
> >
> >
> > Complex samples consist of a a real part and an imaginary part; another
> > way to look at that is as two samples. If you do look at it that way,
> > each complex sample counting for two real ones, then the sample rates
> > are the same: two real samples or one complex sample in the time it
> > takes for one cycle if the highest frequency in the signal.
> >
> > Jerry
>
> 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.

>
> Of course, defining "unique" may take a little bit of math...
>
> --
>
> Tim Wescott
> Wescott Design Services
> http://www.wescottdesign.com
```
```"kiki" <lunaliu3@yahoo.com> writes:

> I (vaguely) heard that sampling complex-valued data does not abide by the
> Nyquist rate criteria, i.e., the sampling rate fs can go lower than Nyquist
> rate and it still can avoid aliasing and reconstruct perfectly...
>
> Is that true?

Yes.  Real sampling at Fs samples/second provides a "usable" bandwidth
of Fs/2 Hz while complex sampling provides a usable bandwidth of Fs Hz
at the same sample rate.

> Any theory behind it?

Yes. Use two pieces of knowledge: a) the Fourier transform property
that H(f) = H*(-f) (this is known as "Hermitian symmetry") for a real
signal h(t), H(f) = F[h(t)], and b) the fact that sampling can be
viewed in the frequency domain as replicating the band from -Fs/2 to
+Fs/2 every Fs Hz.

More simply, a real signal has bandwidth from 0 to Fs/2 available, while
a complex signal has bandwidth from -Fs/2 to +Fs/2 available.

As I recall, Richard Lyon's book "Understanding Digital Signal Processing"
(2nd ed.) discusses this phenomenom at great length.
--
%  Randy Yates                  % "So now it's getting late,
%% Fuquay-Varina, NC            %    and those who hesitate
%%% 919-577-9882                %    got no one..."
%%%% <yates@ieee.org>           % 'Waterfall', *Face The Music*, ELO
```
```You might check out the references given at the bottom of

http://www.ece.eps.hw.ac.uk/Research/oceans/projects/sonar/beamform/

which point to many papers in complex (quadrature) sampling.

In article <pt2juyeq.fsf@ieee.org>, Randy Yates <yates@ieee.org> wrote:
>"kiki" <lunaliu3@yahoo.com> writes:
>
>> I (vaguely) heard that sampling complex-valued data does not abide by the
>> Nyquist rate criteria, i.e., the sampling rate fs can go lower than Nyquist
>> rate and it still can avoid aliasing and reconstruct perfectly...
>>
>> Is that true?
>
>Yes.  Real sampling at Fs samples/second provides a "usable" bandwidth
>of Fs/2 Hz while complex sampling provides a usable bandwidth of Fs Hz
>at the same sample rate.
>
>> Any theory behind it?
>
>Yes. Use two pieces of knowledge: a) the Fourier transform property
>that H(f) = H*(-f) (this is known as "Hermitian symmetry") for a real
>signal h(t), H(f) = F[h(t)], and b) the fact that sampling can be
>viewed in the frequency domain as replicating the band from -Fs/2 to
>+Fs/2 every Fs Hz.
>
>More simply, a real signal has bandwidth from 0 to Fs/2 available, while
>a complex signal has bandwidth from -Fs/2 to +Fs/2 available.
>
>As I recall, Richard Lyon's book "Understanding Digital Signal Processing"
>(2nd ed.) discusses this phenomenom at great length.
```