[time-nuts] 20logN was Re: phase noise questions (long)

[Bruce Griffiths]

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