PM component of a noisy signal in Oscillators
I am reading the oscillator chapter in RF Microelectronics textbook by Razavi. I have one question that is bothering me for a long time that I am not able to completely understand. In the chapter on oscillator when he is talking about Conversion of Additive Noise to Phase noise, he considers
x(t)=A cos(wot)+a cos(wo+Δw)t
Where the second term is the additive noise component.
He says that after passing the above signal through a limiter (so that amplitude noise component gets suppressed), we get the following FM (or PM) signal at the output of the limiter.
xlim(t) = (A/2) cos wot - (a/2) cos(wo+Δw)t + (a/2) cos(wo-Δw)t
I am able to understand the second and third terms of the above equation, but I don't understand how we are specifically getting (A/2) (in (A/2) coswot) . I mean why is it half of A? This has been bothering me a lot
Thanks in advance
It seems a typo to me.
If, for sake of simplicity, we call x=wo*y and y=Δw*t, then
x(t) = A*cos(x) + a*cos(x+y)
developing the last term we get:
x(t) = A*cos(x) + a*cos(x)*cos(y) - a*sin(x)*sin(y)
developing again the last term:
x(t) = A*cos(x) + a*cos(x)*cos(y) - a/2*cos(x-y) + a/2*cos(x+y)
notice that I've the last to sign reversed with respect to your equation.
However I can write the last equation as:
x(t) = [A + a*cos(y)]*cos(x) - a/2*cos(x-y) + a/2*cos(x+y)
That means the AM is given by the term A + a*cos(y) that has to be limited. In case of a passive (frequency selective) hard limiter the limiting value will be given by A-a, since -1<cos(y)<1, then
xlim(t) = [A - a]*cos(x) - a/2*cos(x-y) + a/2*cos(x+y)
However I'm not very sure about this my derivation