"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
Converting I and Q to real signal
Started by ●May 6, 2007
Reply by ●May 7, 20072007-05-07
Reply by ●May 7, 20072007-05-07
>"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 knowwhere>> 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 likeyou,>%% Fuquay-Varina, NC % she does the things you do,>%%% 919-577-9882 % but she is anIBM.">%%%% <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 _____________________________________ Do you know a company who employs DSP engineers? Is it already listed at http://dsprelated.com/employers.php ?