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At high FM index or low c/m ratio, the modulated carrier's instantaneous frequency can go negative

Concept L3Craft B · Sound design & synthesis

The FM equation produces an instantaneous frequency that is the sum of the carrier ramp and the modulator sinusoid. When the modulation index I is large enough, the sinusoidal contribution (amplitude I) can exceed the carrier ramp slope, causing the net instantaneous frequency to become negative at certain points in the cycle. This is not a malfunction but a real mathematical condition that produces additional spectral richness via the reflected-frequency mechanism described earlier. An oscillator implementation must handle this by correctly managing phase angles that decrease over time (not just increase), requiring a small but critical modification to the standard table-lookup oscillator code. Ignoring this condition produces incorrect (simplified) output spectra.

Examples

c=100Hz, m=100Hz, I=4: the peak modulation swing is ±400Hz, which overwhelms the 100Hz carrier frequency, causing negative instantaneous frequencies at the modulation peaks.

Assessment

At what minimum value of I, for c=m, will the instantaneous frequency of the modulated carrier first go negative? Explain your reasoning.

Prerequisites — understand these first

Don't confuse with

Sources

“The oscillator, u.g. 3, in Fig. 9, must be able to produce a wave that results from taking the sine of an angle that decreases as well as increases with time.”
corpus · the-synthesis-of-complex-audio-spectra-by-means-of-frequency · chunk 4

Taught in modules