3.3 Frequency modulation

For small modulation, indices, phase modulation and frequency modulation can be understood as different descriptions of the same effect [29]. Following the same spirit as above we would assume a modulated frequency to be given by
′ ω = ω0 + m cos(Ωt ), (53 )
and then we might be tempted to write
( ) E = E exp i(ω + m ′cos(Ωt))t , (54 ) 0 0
which would be wrong. The frequency of a wave is actually defined as ω ∕(2π) = f = d φ∕dt. Thus, to obtain the frequency given in Equation (53View Equation), we need to have a phase of
m-′ ω0t + Ω sin (Ωt). (55 )
For consistency with the notation for phase modulation, we define the modulation index to be
′ m = m-- = Δ-ω-, (56 ) Ω Ω
with Δ ω as the frequency swing – how far the frequency is shifted by the modulation – and Ω the modulation frequency – how fast the frequency is shifted. Thus, a sinusoidal frequency modulation can be written as
( ( Δ ω )) E = E0 exp(iφ) = E0 exp i ω0t + ----cos (Ωt ) , (57 ) Ω
which is exactly the same expression as Equation (44View Equation) for phase modulation. The practical difference is the typical size of the modulation index, with phase modulation having a modulation index of m < 10, while for frequency modulation, typical numbers might be 4 m > 10. Thus, in the case of frequency modulation, the approximations for small m are not valid. The series expansion using Bessel functions, as in Equation (46View Equation), can still be performed, however, very many terms of the resulting sum need to be taken into account.
  Go to previous page Go up Go to next page