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

Reply by Tim Wescott●November 12, 20042004-11-12

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

Reply by Tim Wescott●November 12, 20042004-11-12

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
goals you were asking for.
--
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Reply by Tim Wescott●November 12, 20042004-11-12

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

Reply by Jerry Avins●November 12, 20042004-11-12

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.
�����������������������������������������������������������������������

>"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.

Reply by Randy Yates●November 12, 20042004-11-12

"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
http://home.earthlink.net/~yatescr

Reply by Jeroen Boschma●November 12, 20042004-11-12

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

Reply by Stephan M. Bernsee●November 12, 20042004-11-12

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

Reply by Jerry Avins●November 12, 20042004-11-12

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.
�����������������������������������������������������������������������