Analog QAM carrier recovery

Here' my problem:

signal:   s(t) = f(t)*COS(wt) + g(t)*SIN(wt)

f(t) and g(t) are both baseband ANALOG signals.

Question:  How do we recover the carrier at the
receiving side? ( I mean recover COS(wt) and SIN(wt) synchrously)

(If f(t) = +- 1 and g(t) = +- 1, digital signals,
 then Costas Loop or squaring loop can do it.)


-- Harry

Harry wrote:
> Here' my problem:
>
> signal:   s(t) = f(t)*COS(wt) + g(t)*SIN(wt)
>
> f(t) and g(t) are both baseband ANALOG signals.
>
> Question:  How do we recover the carrier at the
> receiving side? ( I mean recover COS(wt) and SIN(wt) synchrously)

In general you can't recover the carrier, unless you put some
constraints on f(t) and g(t).  For example, if f(t) and g(t) are both
zero, you're hosed.

> (If f(t) = +- 1 and g(t) = +- 1, digital signals,
>  then Costas Loop or squaring loop can do it.)

That's the ticket.

Regards,
Allan

On 14 Sep 2005 16:20:07 -0700, "Harry" <harry0d88@aol.com> wrote:

>Here' my problem:
>
>signal:   s(t) = f(t)*COS(wt) + g(t)*SIN(wt)
>
>f(t) and g(t) are both baseband ANALOG signals.
>
>Question:  How do we recover the carrier at the
>receiving side? ( I mean recover COS(wt) and SIN(wt) synchrously)
>
>(If f(t) = +- 1 and g(t) = +- 1, digital signals,
> then Costas Loop or squaring loop can do it.)

It appears that what you're describing is QPSK rather than QAM as
you've indicated in the subject.

Either way, a Costas loop with an appropriate phase detector is
typically how this is done.


Eric Jacobsen
Minister of Algorithms, Intel Corp.
My opinions may not be Intel's opinions.
http://www.ericjacobsen.org

Harry wrote:

> Here' my problem:
> 
> signal:   s(t) = f(t)*COS(wt) + g(t)*SIN(wt)
> 
> f(t) and g(t) are both baseband ANALOG signals.
> 
> Question:  How do we recover the carrier at the
> receiving side? ( I mean recover COS(wt) and SIN(wt) synchrously)
> 
> (If f(t) = +- 1 and g(t) = +- 1, digital signals,
>  then Costas Loop or squaring loop can do it.)
> 
> 
> -- Harry
> 
As Alan Herriman wrote, you're out of luck unless f(t) and g(t) have 
some distinguishing characteristics that will let you figure out when 
the signal is being received correctly.

Is this the case, or are f(t) and g(t) totally arbitrary?  What is it 
that you're trying to do?

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Here we assume that f(t) and g(t) are two uncorrelated baseband
signals.
Is this totally hopeless?

s(t) = f(t)*COS(wt + theta) + g(t)*SIN(wt + theta)

Squaring the signal s(t) can get the frequency w (after dividing the
frequency by 2)
but the original phase is unknown.  (Am I right?) 

-- Harry

> Either way, a Costas loop with an appropriate phase detector is
typically how this is done.

Here's the question:  Costas loop works for digital signals.

Does Costas loop also work for ANALOG signals?

Also, let's assume the case that "both f(t) and g(t) are zero" is a
don't care situation..
(no transmission)

-- Harry

Harry wrote:
>>Either way, a Costas loop with an appropriate phase detector is
> 
> typically how this is done.
> 
> Here's the question:  Costas loop works for digital signals.
> 
> Does Costas loop also work for ANALOG signals?
> 
> Also, let's assume the case that "both f(t) and g(t) are zero" is a
> don't care situation..
> (no transmission)

A Costas loop is a digital PLL; there are plenty of analog PLLs. To 
determine the phase of the received carrier relative to what has been 
transmitted requires some kind of reference. Both f(t) and g(t) being 
zero is a true don't-care case. When there's no modulation, who cares 
about the carrier?

Jerry
--
"Once the rockets go up, who cares where they come down?
That's not my department" says Werner von Braun. -- Tom Lehrer
�����������������������������������������������������������������������

Harry wrote:
> Here we assume that f(t) and g(t) are two uncorrelated baseband
> signals.
> Is this totally hopeless?
> 
> s(t) = f(t)*COS(wt + theta) + g(t)*SIN(wt + theta)
> 
> Squaring the signal s(t) can get the frequency w (after dividing the
> frequency by 2)
> but the original phase is unknown.  (Am I right?) 

You are right that it's totally hopeless.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Harry wrote:

> Here we assume that f(t) and g(t) are two uncorrelated baseband
> signals.
> Is this totally hopeless?
> 
> s(t) = f(t)*COS(wt + theta) + g(t)*SIN(wt + theta)
> 
> Squaring the signal s(t) can get the frequency w (after dividing the
> frequency by 2)
> but the original phase is unknown.  (Am I right?) 
> 
> -- Harry
> 
f(t) and g(t) could be totally uncorrelated signals yet still have 
properties that would let you do this -- in the QPSK example they only 
take the values +1 and -1, and they only change at specific times so you 
can extract things.

The word you're looking for is "arbitrary".  If f(t) and g(t) are 
completely arbitrary then s(t) is also arbitrary -- so not only can you 
not know the phase of the carrier but you can't know the frequency, 
either.  In fact, if an unkind Communications God wanted to be perverse 
he or she could just choose an s(t) and set

f(t) = s(t) cos(wt + theta), g(t) = s(t) sin(wt + theta).

So what are your constraints on f(t) and g(t)?  What are you trying to do?

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

> so not only can you not know the phase of the carrier but you can't know the frequency,

"w" is the carrier frequency.  Usually and also in this case, "w" is
assumed to be known (but not the phase) to the receiver.

-- Harry