Re: A Stupid FM Demod Question?

Randy Yates wrote:
"robert bristow-johnson" <rbj@xxxxxxxxxxxxxxxxxxxx> writes:

Randy Yates wrote:
Vladimir Vassilevsky <antispam_bogus@xxxxxxxxxxx> writes:
Randy Yates wrote:

If you want to extract the frequency from a complex signal that
contains a single, time-varying frequency component, then the
straightforward method would be to determine the phase of the
input signal and then run that phase into a differentiator.
The straightforward way would be not to use the phase
representation. Work with the complex derivatives directly:

IdQ/dt - QdI/dt
---------------
I^2 + Q^2
I guess the key advantage is that you don't have to implement an
arctan function?

Would you use an FIR differentiator to get Q' and I', making sure
they're properly time-aligned with the I and Q values?
actually Randy, you *do* have to implement an arctan() function for
small angles, but you *don't* have to compute a derivative explicitly
from these discrete samples (using some big FIR).

omega[n] = arg{ I[n] + j*Q[n] } - arg{ I[n-1] + j*Q[n-1] }

= arctan( Q[n]/I[n] ) - arctan( Q[n-1]/I[n-1] )

= arctan( (Q[n]*I[n-1] - I[n]*Q[n-1]) / (I[n]*I[n-1]
+Q[n]*Q[n-1]) )

How did you get from the second line to the third?

the measure omega[n] is simple that of how much the phase of I + j*Q
has advanced from the phase of the previous sample. isn't that what
frequency is? how much (in a given period of time) the phase angle
advances?

I always thought that frequency is the derivative of phase, f = dtheta/dt.
That's the gist of my question. Do I need to perform a full derivative
or is theta2 - theta1 perfect or good enough?

Seems like, going back to first semester calculus, theta2-theta1
is the average slope in the period t1 to t2, and that the true
instantaneous slope would be dtheta/dt.

Not only does [theta(n) - theta(n-1)] represent a secant through the continuous frequency curve passing through theta(n-1) and theta(n), it has a half sample delay. It is a pretty darn good match to the tangent at the midway point if the frequency slope is slow enough not to generate significant sidebands. (Yeah: that's a tautology that hinges on "significant".) I suspect you can sleep easy.

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

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