On 12/05/2010 11:49 AM, MRR wrote:
 > Hello Randy,
 >
 > Then, could you provide a reference/s by which complex sampling is defined?

I couldn't find a good one on my bookshelf. It may be one of those things
that are so basic folks don't think about defining it precisely.

My best shot would be this:

   Complex Sequence: A sequence of samples x[n] where x[n] \in C and
   n \in Z.

Note that "\in" is the set "element of" operator, and "C" is
the set of complex numbers and "Z" is the set of integers.

 > I am interested in understanding well the relationship between a REAL
 > signal (the signal which flights over the air), complex sampling and
 > Nyquist theorem.

 > I am interested in practical cases, since I seem to understand that complex
 > sampling is carried out for a real signal that initially meets the even
 > parity in the frequency domain (its module has a simmetry respect positive
 > and negative frequencies). For example, many receivers implement I/Q
 > demodulation obtaining a baseband representation of a passband signal
 > (e.g., QAM demodulation).

Since there are very few (if any, but that could probably be debated)
real-world complex signals, complex signals are derived from real
signals as you have stated, usually (at least in signal processing)
either by Hilbert transform or by quadrature downconversion.
-- 
Randy Yates                      % "My Shangri-la has gone away, fading like
Digital Signal Labs              %  the Beatles on 'Hey Jude'"
mailto://yates@ieee.org          %
http://www.digitalsignallabs.com % 'Shangri-La', *A New World Record*, ELO

Hello Randy,

Then, could you provide a reference/s by which complex sampling is defined?

I am interested in understanding well the relationship between a REAL
signal (the signal which flights over the air), complex sampling and
Nyquist theorem.
I am interested in practical cases, since I seem to understand that complex
sampling is carried out for a real signal that initially meets the even
parity in the frequency domain (its module has a simmetry respect positive
and negative frequencies). For example, many receivers implement I/Q
demodulation obtaining a baseband representation of a passband signal
(e.g., QAM demodulation).

Cheers,

M
>On 12/05/2010 07:54 AM, MRR wrote:
>> Thanks a lot for this debate.
>> I continued this issue in wikipedia and the article "sampling" have
been
>> modified. One of the editors has added "complex sampling", which may be
>> useful for understanding how the demodulation with complex baseband
>> representation is carried out.
>>
>> Here you are the link:
>> http://en.wikipedia.org/wiki/Sampling_%28signal_processing%29
>>
>> Cheers,
>>
>> M
>
>The definition of complex sampling in that article is
>not correct. The real and imaginary components of a
>complex signal are not required to be related. Complex
>sampling is not necessarily of Hilbert transform pairs.
>
>--Randy
>
>

On 12/05/2010 07:54 AM, MRR wrote:
> Thanks a lot for this debate.
> I continued this issue in wikipedia and the article "sampling" have been
> modified. One of the editors has added "complex sampling", which may be
> useful for understanding how the demodulation with complex baseband
> representation is carried out.
>
> Here you are the link:
> http://en.wikipedia.org/wiki/Sampling_%28signal_processing%29
>
> Cheers,
>
> M

The definition of complex sampling in that article is
not correct. The real and imaginary components of a
complex signal are not required to be related. Complex
sampling is not necessarily of Hilbert transform pairs.

--Randy

Thanks a lot for this debate. 
I continued this issue in wikipedia and the article "sampling" have been
modified. One of the editors has added "complex sampling", which may be
useful for understanding how the demodulation with complex baseband
representation is carried out.

Here you are the link:
http://en.wikipedia.org/wiki/Sampling_%28signal_processing%29

Cheers,

M

Randy Yates <yates@ieee.org> writes:

> Randy Yates <yates@ieee.org> writes:
>> [...]
>>     g(w) = 0, w_l < |w| < w_h
>
> Correction:
>
>      G(w) = 0, w_l < |w| < w_h

Correction to the correct: (!) (thanks Dilip!)


      G(w) = 0, w <= w_l and w >= w_h

-- 
Randy Yates                      % "Remember the good old 1980's, when 
Digital Signal Labs              %  things were so uncomplicated?"
mailto://yates@ieee.org          % 'Ticket To The Moon' 
http://www.digitalsignallabs.com % *Time*, Electric Light Orchestra

Randy Yates <yates@ieee.org> writes:
> [...]
>     g(w) = 0, w_l < |w| < w_h

Correction:

     G(w) = 0, w_l < |w| < w_h
-- 
Randy Yates                      % "I met someone who looks alot like you,
Digital Signal Labs              %             she does the things you do, 
mailto://yates@ieee.org          %                     but she is an IBM."
http://www.digitalsignallabs.com %        'Yours Truly, 2095', *Time*, ELO

Randy Yates <yates@ieee.org> writes:

> Hi,
>
> Just about any signal you deal with in reality is going to be a real
> signal (no pun intended!). So yes, any signal you transmit to or
> receive from an antenna is going to be real.
>
> In my opinion, the easiest way to see what's going on with real and
> complex I/Q signals in digital communication systems is as follows:
>
>   1. Understand the fact that a signal g(t) is real in the time domain
>   if and only if it has a Hermitian-symmetric frequency domain function
>   G(w). Hermitian-symmetric means this:
>
>     Re(G(-w)) = Re(G(w)) and Im(G(-w)) = -Im(G(w))
>
>   2. Realize that most digital communication signals received from an
>   antenna or transmitted to an antenna are bandpass, real signals, i.e.
>
>     g(t) is real, and
>
>     g(w) = 0, w_l < |w| < w_h
>
> (By the way, "w" means "omega," or 2 x pi x f, where f is frequency.)
>
> So what we usually do in a digital communication system is to take the
> real, bandpass signal to or from the antenna, which (because it's real)
> is necessarily symmetric in frequency, i.e., it has two bands of energy,
> one in positive frequency and one in negative frequency, and translate
> ONE of those bands (either the positive or negative - doesn't matter
> since they both carry the same information) down to "baseband" (i.e.,
> DC). So, unless the bandpass signal was Hermitian-symmetric about Fc
> (the carrier frequency), the translated signal will NOT be Hermitian-
> symmetric and thus will necessarily be complex. But it's the "same"
> signal in the sense that, due to the symmetry that was present in its
> real form, no information was lost in the translation.
>
> I hope this is clear and helps you understand what's going on in these
> types of system "translations."

PS: The way you "recover" a complex, baseband signal from a received
real, bandpass signal g(t) is to "mix" g(t) with a complex exponential
at the carrier frequency, y(t) = g(t) * e(t) (this "translates" one of
the bandpass bands down to DC), and then lowpass filter the result.
-- 
Randy Yates                      % "She's sweet on Wagner-I think she'd die for Beethoven.
Digital Signal Labs              %  She love the way Puccini lays down a tune, and
mailto://yates@ieee.org          %  Verdi's always creepin' from her room." 
http://www.digitalsignallabs.com % "Rockaria", *A New World Record*, ELO

Hi,

Just about any signal you deal with in reality is going to be a real
signal (no pun intended!). So yes, any signal you transmit to or
receive from an antenna is going to be real.

In my opinion, the easiest way to see what's going on with real and
complex I/Q signals in digital communication systems is as follows:

  1. Understand the fact that a signal g(t) is real in the time domain
  if and only if it has a Hermitian-symmetric frequency domain function
  G(w). Hermitian-symmetric means this:

    Re(G(-w)) = Re(G(w)) and Im(G(-w)) = -Im(G(w))

  2. Realize that most digital communication signals received from an
  antenna or transmitted to an antenna are bandpass, real signals, i.e.

    g(t) is real, and

    g(w) = 0, w_l < |w| < w_h

(By the way, "w" means "omega," or 2 x pi x f, where f is frequency.)

So what we usually do in a digital communication system is to take the
real, bandpass signal to or from the antenna, which (because it's real)
is necessarily symmetric in frequency, i.e., it has two bands of energy,
one in positive frequency and one in negative frequency, and translate
ONE of those bands (either the positive or negative - doesn't matter
since they both carry the same information) down to "baseband" (i.e.,
DC). So, unless the bandpass signal was Hermitian-symmetric about Fc
(the carrier frequency), the translated signal will NOT be Hermitian-
symmetric and thus will necessarily be complex. But it's the "same"
signal in the sense that, due to the symmetry that was present in its
real form, no information was lost in the translation.

I hope this is clear and helps you understand what's going on in these
types of system "translations."
-- 
Randy Yates                      % "Rollin' and riding and slippin' and
Digital Signal Labs              %  sliding, it's magic."
mailto://yates@ieee.org          %  
http://www.digitalsignallabs.com % 'Living' Thing', *A New World Record*, ELO

Thanks for the answers, but the last one has nothing to do with I-Q
modulation I think (apart from two sine-cosine signals are sent from the
transmitter).

Thanks ,

-M

>You're modulating a radio frequency carrier, that's the reason why you're
>able to transmit two real-valued signals and separate them at the
>receiver.
>
>Think of your radio frequency carrier as a continuous sine wave: You can
>make it bigger or smaller (1st parameter), and you can shift it forwards
or
>backwards in time (2nd parameter). The receiver *knows how the
unmodulated
>carrier wave should look at any point in time*, and can tell the
magnitude
>*and* the time shift simultaneously.
>
>For a "normal" real-valued signal such as from a microphone, you can't
have
>both, because the receiver couldn't distinguish between amplitude scaling
>and time shift. 
>
>