FM Equation and Control Parameters

• As previously noted, the instantaneous frequency change applied to the carrier oscillator frequency input is given by

  $$f(t) = f_c + \Delta f \sin(2 \pi f_m t).$$ (1)

• To write the time-varying output of the FM algorithm, we need to integrate the instantaneous frequency as
  

  $$y(t) = A \sin(2 \pi \int_0^t f(t) dt)$$

  $$= A \sin(2 \pi f_c t - \Delta f/f_m \cos(2 \pi f_m t)$$

  $$= A \sin(2 \pi f_c t - I \cos(2 \pi f_m t),$$ (2)

  
  where $I = \Delta f / f_m$ is referred to as the “modulation index”.

• Increasing the modulation index ($I$) generally increases the effective bandwidth of the FM spectrum.

• The Matlab script FM1.m demonstrates the use of both Eq. (2) and Eq. (1) to implement FM synthesis, with an aural and visual demonstration of the modulator's affect on the resulting FM spectrum.

• The Matlab script FM2.m demonstrates the difference between using the standard FM algorithm, Eq. (2), to calculate an FM synthesis result versus phase increment approaches (including an instantaneous frequency as in Eq. (1)).

• Another parameter useful for analyzing the spectrum of simple FM is called the harmonicity ratio, $H = f_m / f_c$.

• For $H = 1$, the carrier and modulator frequencies are equal and the resulting spectrum is harmonic, with the fundamental frequency equal to the carrier frequency ( $f_c, f_c \pm f_m, f_c \pm 2 f_m, \ldots = f_c, 0, 2 f_c, -f_c, 3 f_c, \ldots$).

• For $H = 1/2$, the carrier frequency is double the modulator frequency and the resulting spectrum is harmonic, though with fundamental frequency equal to the modulator frequency.

• For $H = 2$, the spectrum is harmonic with odd integer multiples of the fundamental $f_c$.

• In general, if $H$ has the form $N_1 / N_2$, where $N_1$ and $N_2$ are positive integers, the resulting spectrum will be harmonic.

• If $H$ is not of this form, the spectrum will be inharmonic.