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

[steve the knife]

Hi,

I did the same thing using the Euler relation for sin(w1+sin(w2t)) and straight
multiplication and collecting terms gave 1-0.5*cos(2*w1+2*sin(w2t)). I wanted
to actually contribute but was too slow :(

-steve


Quoting Bruce Griffiths <bruce.griffiths at xtra.co.nz>:

> Christophe Huygens wrote:
> > Hi John, Steve, et al,
> >
> > While I am not a phase noise buff at all, in talking to many on this
> > subject
> > I feel that this is not well understood. When I ask where the 6db/Hz for
> > doubling or 20logN in general comes from, I very often get an
> > unsatisfying
> > answer and I have seen strange notes on this mailing list on this
> > subject as
> > well...
> >
> > For me to understand what happens in a simplified way 2 things are key:
> >
> > 1. Phase noise is subject to FM theory - you can think of the carrier
> > being FM modulated with a very low modulation index, with
> > modulation frequency the offset from the carrier. This is easy
> > enough to accept for most. The noise phasor sits on top of the carrier.
> > This give amplitude noise, that can be limited away, and well...
> > phase noise. The actual modulation index in our case is always
> > very small I guess, except when you looking real close to the
> > carrier, but then still - if the oscillator is good, the deviation will
> > still be small hence low modulation index theory still applies..
> >
> > 2. What happens with an FM signal when applied to an ideal doubler -
> > this is a bit of a trickier. Say I have a narrowband (low modulation
> > index) signal of 200Hz, modulated by 20Hz.
> >
> > a. The spectrum is:
> > sideband 1 (180) - carrier (200) - sideband 2 (220).
> > AFTER the doubler the spectrum is:
> > sideband 1 (380) - carrier (400) - sideband 2 (420).
> >
> > I have a hard time to find an intuitive explanation for this,
> > but it only takes 20 lines of octave/matlab code to verify...
> > I am getting too old (or I m too young) to get into the Bessel
> > functions myself.
> >
> > So no need to multiply the offset also by N as sometimes seen.
> >
> > b. The amplitude of the sidebands does grow with respect to
> > the carrier (all this for small modulation indexes) by about
> > 6 db.  Also easy to show in a a simulation.
> >
> > The duality of multiplication in the time domain and convolution
> > in the frequency domain also explains this I think, like it
> > can explain a.
> >
> > Maybe somebody on the list can step in and give a clear and
> > concise explanation for the above.
> >
> >
> >
> > Christophe
> Christophe
> 
> 1) There's no need to resort to using Matlab, simple trigonometric
> identities will suffice.
> 
> For a multiplier type frequency doubler with small modulation index FM:
> fc(t) = (1- 0.5*beta*beta)*coswct - 0.5*beta[cos ((wm-wc)t) - cos
> ((wc+wm)t)]
> 
> fc(t)*fc(t) = (1- 0.5*beta*beta)*coswct*(1- 0.5*beta*beta)*coswct - (1-
> 0.5*beta*beta)*coswct*0.5*beta[cos ((wm-wc)t) - cos ((wc+wm)t)]  +
> 0.5*beta[cos ((wm-wc)t) - cos ((wc+wm)t)] *0.5*beta[cos ((wm - wc)t) -
> cos ((wc + wm)t)]
> 
> fc(t)*fc(t) = (1- 0.5*beta*beta)*(1- 0.5*beta*beta)*[0.5*(1 + cos(2wct)]
> + (1- 0.5*beta*beta)*0.5*beta*[0.5*[cos((wm-2wc)t) + cos((2wc+wm)t)]]
>  + 0.25*beta*beta*[0.5*(1 + cos(2*(wm-wc)t)) + 0.5*(1 + cos(2*(wm +
> wc)t)) + (cos ((2wc)t) + cos ((2wm)t)]
> 
> fc(t) = (0.5 - 0.25*beta^2  + 0.125*beta^4) +
> (0.5 - 0.25*beta^2 + 0.125*beta^4) cos(2wct) +
> (0.25*beta - 0.125*beta^3 )* cos((2wc-wm)t) +
> (0.25*beta - 0.125*beta^3 )* cos((2wc+wm)t) +
> (0.125*beta^2)*cos(2*(wm-wc)t) +
> (0.125*beta^2)*cos(2*(wm + wc)t)
> 
> When beta <<1, the AC components can be approximated by
> 
> 0.5*cos((2wc)t) +
> 0.25*beta*cos((2wc - wm)t) +
> 0.25*beta*cos((2wc + wm)t)
> 
> Thus the ratio of the sideband components to the carrier has increased
> by 6 dB at the output of the frequency doubler.
> 
> It is perhaps even simpler to use the complex exponential representation
> of the signal (this is especially true for multiplication by N) as
> exponentiation is an easier operation (by hand at least) than complex
> trigonometric identities.
> 
> fc(t) = Re[exp(jwct)*(1+ 0.5*beta*{exp(jwmt) - exp(-jwmt)})]
> 
> fc(t)^N = Re[exp(jNwct)*(1+0.5*beta*{exp(jwmt) - exp(-jwmt)})^N)
> 
> From which the largest components are
> 
> Re[exp(jNwct)*(1 + 0.5*beta*N*{exp(jwmt) - exp(-jwmt)}]
> or
> Re[exp(jNwct) + 0.5*beta*N*exp(jNwct + jwmt) + 0.5*beta*N*exp(jNwct -jwmt)]
> or
> cos (Nwct) + 0.5*m*N cos((Nwc+wm)t) + 0.5*m*N*cos((Nwc +wm)t)
> 
> 2) Bessel functions are only necessary to calculate the amplitudes of
> the various components, their functional form is unaltered.
> 
> 3) The even harmonic FM sidelobes are in effect amplitude modulation
> components of the signal.
> 
> 4) The phase sensitive detector used in phase noise measurements is
> insensitive to AM.
> 
> Bruce
> 
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