FM total bandwidth is approximately twice the sum of frequency deviation and modulating frequency

The approximate bandwidth of an FM signal is BW ≈ 2(d + m), where d is the peak frequency deviation and m is the modulating frequency. Since I = d/m, this can be rewritten as BW ≈ 2m(I + 1). This rule (Carson’s rule, from radio engineering) gives an upper bound on the spectral extent. In audio FM synthesis, it means that raising either the modulation index or the modulating frequency will widen the spectrum. Doubling the modulator frequency doubles the bandwidth even at the same index. This formula is approximate because Bessel functions decay gradually rather than abruptly, but it captures the design-level relationship a synthesist needs to predict spectral range.

Examples

c=440Hz, m=440Hz, I=2: BW≈2×(880+440)=2640Hz — sidebands extend roughly ±1320Hz around the carrier. At I=5: BW≈2×(2200+440)=5280Hz.

Assessment

A patch has c=200Hz, m=100Hz. What index is needed to reach sidebands at 1000Hz? Verify using BW≈2(d+m) and I=d/m.