4-32
Consider the “simple” case of sinusoidal phase modulation at frequency fm.  Then, f(t) = fo(t)sin(2pfmt), and V(t) = Vocos[2pfct + f(t)] = Vocos[2pfct + f0(t)sin(pfmt)], where fo(t)= peak phase excursion, and fc=carrier frequency.  Cosine of a sine function suggests a Bessel function expansion of V(t) into its components at various frequencies via the identities:
After some messy algebra, SV(f) and Sf(f) are as shown on the next page.  Then,          
Sf(f) to SSB Power Ratio Relationship
   According to IEEE Standard 1139-1999, the phase noise, L(f), is defined as

L(f)   ˝S(f)

which is equal to the SSB power ratio only when <f2(t)> = the integral of Sf(f) from f = 0 to f = , is less than about 0.1 rad2.  S(f) can always be measured unambiguously, whereas the SSB power ratio (the pre-1988 definition of phase noise) diverges close to the carrier.


   The above analysis and the graph on the next page were provided by Raymond L. Filler, U.S. Army LABCOM, 1989.