Complex baseband signal. Inphase-Quadrature data

Hello everybody,

I am beginning on this communication stuff and I have some doubts I´d like
to clarify. One of them is regarding I/Q structure for transmitting and
receiving a signal. I´d like to someone to tell me if I am right on the
following:

- Two signals are generated in the transmitter with an specificic
modulation (BPSK, QPSK...), and multiplied by a cosine and a sine
(Inphase-Quadrature componentes, both at a carrier frequency w. Then this
REAL signal is generated (*let´s suppose amplitude is 1): 

x(t) = Xi(phi)*coswt-Xq(phi)*sinwt. *phi is time dependent

Thus, it is the signal that "flies" out the antenna (after amplifying and
filtering process).

- This signal can be mathematically treated (we are on the transmitter) on
the complex plane by the SIGNAL ENVELOPE, which is: 

x´(phi) = cos(phi)+j*sin(phi)= Xi+jXq. 
Then, the equivalent transsmitted signal is:

x(t) = Re{x'(t)*e^(jwt)}


So, in complex baseband we can say (I omit ADC and DAC steps) :

TRANSMITTER                                           RECEIVER

signal shifted 
with the carrier at w  ----> REAL SIGNAL FLIES -----> I-Q DATA?      


These are my question:

1- Am I right in the procedure described before? Specially that only a REAL
signal can "fly".

2- On the receiver part, how can I recover the complex data from the real
signal (the signal which flies), and are they supposed to be in baseband?


Thank you,

M

"MRR" <mario.ruzruiz@n_o_s_p_a_m.gmail.com> wrote in message 
news:4vKdnRsshe-PNkvRnZ2dnUVZ_oCdnZ2d@giganews.com...
>
> 1- Am I right in the procedure described before? Specially that only a 
> REAL
> signal can "fly".

Yes, you are correct, at all times you only have real signals.

Two real signals that are in quadrature can be REPRESENTED by complex
arithmetic where the real and imaginary axes are at right angles. This 
representation
is used to simplify the mathematical treatment.

If you wish, you can do it with sines, cosines and PI/2 differences, but 
it's
a lot easier with complex arithmetic!


> 2- On the receiver part, how can I recover the complex data from the real
> signal (the signal which flies), and are they supposed to be in baseband?

The two signals in quadrature are also ORTHOGONAL and so can be
separately demodulated by two inserted carriers that are 90 degrees apart in
phase.

Do you remember your derivation of the coefficients of the Fourier Series? 
That
also relied on the orthogoanlity of sine and cosine.

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. 

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. 
>
>

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

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   

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:

> 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

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

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