Steve intended to post this to the list but it came directly to me instead.
As he doesnt have a copy of it I am posting it for him.
Bruce
Hello,
(as requested..)
Here is the Euler derivation. For simplicity I will use "B" for the
first part then expand it. I made a boo-boo in my earlier email.
sin(B) = (e^iB-e^-iB)/2i
sin(B) * sin(B) = (e^iB-e^-iB) * (e^iB-e^-iB) / (2i*2i)
= (e^2*iB + e^(-2)*iB - 2 * e^(iB-iB)) / (-4)
= (e^2*iB + e^(-2)*iB - 2) / (-4)
= (1/2) - 0.5*(e^2*iB + e^(-2)*iB)/2
= (1/2) - 0.5*cos(2B)
Now if B means omega*t + sin(omega2*t) where omega is a phase velocity for
the oscillator and omega2 is the pure phase modulation at some frequency.
We get: (1/2) - 0.5 * cos((2*omega *t) + (2*sin(omega2*t))) which is twice the
original frequency and twice the original amplitude of pure phase modulation
but at the same omega2 frequency.
And of course 20 log 2 gives us 6 dB.
-steve
_______________________________________________
time-nuts mailing list -- [email protected]
To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts
and follow the instructions there.
