Converting I and Q to real signal

"micdahl" <micdahl@u.washington.edu> writes:
> [...]
> So now I am thoroughly confused as to which equation is correct. I
> understand the origin of the (IQ'-QI')/I^2 equation but don't know where
> the (IQ'-QI')/(I^2+Q^2) equation stems from. This is the one everyone
> seems to quote however. If either of you could give a reason for
> championing the latter equation I would greatly appreciate it.  

We wish to determine dtheta(t)/dt, where theta(t) = arctan(Q(t)/I(t)). 

First, let w(t) = Q(t)/I(t). Then theta(t) = arctan(w(t)). 
From the chain rule of the derivative, 

  dtheta(t)/dt = dtheta(w)/dw * dw(t)/dt

Now darctan(w(t))/dw = 1 / (1+w^2). So now we need dw(t)/dt. 

From the quotient rule, 

  dw(t)/dt = d(Q(t)/I(t))/dt 
           = Q'(t)I(t) - Q(t)I'(T) / I^2(t)

I think you can finish it from here.
-- 
%  Randy Yates                  % "I met someone who looks alot like you,
%% Fuquay-Varina, NC            %             she does the things you do, 
%%% 919-577-9882                %                     but she is an IBM."
%%%% <yates@ieee.org>           %        'Yours Truly, 2095', *Time*, ELO   
http://home.earthlink.net/~yatescr

>"micdahl" <micdahl@u.washington.edu> writes:
>> [...]
>> So now I am thoroughly confused as to which equation is correct. I
>> understand the origin of the (IQ'-QI')/I^2 equation but don't know
where
>> the (IQ'-QI')/(I^2+Q^2) equation stems from. This is the one everyone
>> seems to quote however. If either of you could give a reason for
>> championing the latter equation I would greatly appreciate it.  
>
>We wish to determine dtheta(t)/dt, where theta(t) = arctan(Q(t)/I(t)). 
>
>First, let w(t) = Q(t)/I(t). Then theta(t) = arctan(w(t)). 
>From the chain rule of the derivative, 
>
>  dtheta(t)/dt = dtheta(w)/dw * dw(t)/dt
>
>Now darctan(w(t))/dw = 1 / (1+w^2). So now we need dw(t)/dt. 
>
>From the quotient rule, 
>
>  dw(t)/dt = d(Q(t)/I(t))/dt 
>           = Q'(t)I(t) - Q(t)I'(T) / I^2(t)
>
>I think you can finish it from here.
>-- 
>%  Randy Yates                  % "I met someone who looks alot like
you,
>%% Fuquay-Varina, NC            %             she does the things you do,

>%%% 919-577-9882                %                     but she is an
IBM."
>%%%% <yates@ieee.org>           %        'Yours Truly, 2095', *Time*, ELO
  
>http://home.earthlink.net/~yatescr
>


Beautiful!!

So we are not approximating arctan(Q/I) as (Q/I) at all but carrying out
the derivative analytically. I guess I got caught up in the approximation
red herring. Thank you both for your help! 

-Michael


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