FM synthesis modulates a carrier's instantaneous frequency using a modulator oscillator, requiring phase integration

In FM synthesis, a carrier oscillator’s instantaneous frequency is continuously varied by a modulator oscillator. The simplest setup is sine-carrier plus sine-modulator. The correct implementation requires integrating the time-varying frequency to obtain phase, then evaluating sin(phase), because angular frequency equals the derivative of phase. A common error is plugging f(t) directly into sin(2π·f(t)·t), which is mathematically wrong and produces an incorrect signal. The carrier frequency sets the center pitch; the modulator frequency and amplitude determine the spectral content.

Examples

f(t) = fC + AM·cos(2π·fM·t). Output: s(t) = AC·sin(2π∫f(τ)dτ). In Pd or Max, connect modulator output to frequency inlet of carrier, then sum with the carrier frequency.

Assessment

Write out the correct FM equation from memory and explain why f(t) must be integrated rather than used directly as sin’s argument.