### 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
and then we might be tempted to write
which would be wrong. The frequency of a wave is actually defined as . Thus, to obtain the frequency given in Equation (53), we need to have a phase of
For consistency with the notation for phase modulation, we define the modulation index to be
with as the frequency swing – how far the frequency is shifted by the modulation – and the modulation frequency – how fast the frequency is shifted. Thus, a sinusoidal frequency modulation can be written as
which is exactly the same expression as Equation (44) for phase modulation. The practical difference is the typical size of the modulation index, with phase modulation having a modulation index of , while for frequency modulation, typical numbers might be . Thus, in the case of frequency modulation, the approximations for small are not valid. The series expansion using Bessel functions, as in Equation (46), can still be performed, however, very many terms of the resulting sum need to be taken into account.