. 2021 Mar 2;149(3):1434–1443. doi: 10.1121/10.0003604

TABLE II.

The transformation to apply to each carrier tone in order to apply exponential modulation. Each carrier tone is replaced with a base tone and its associated sidebands. The first sidebands correspond to k = 1. The even-numbered sidebands use the appropriate Even k row, the odd-numbered sidebands use the respective Odd k row. The amplitude terms fall off exponentially for successive sidebands. For 20-dB peak-to-valley modulation (m = 10), by the fifth set of sidebands (k = 5), each subsequent amplitude term is more than 1 order of magnitude smaller than the prior.

Exponential modulation sidebands
	Frequency	Band	Amplitude^a,^b,^c	Phase
Base	f_n	—	$I_{0} (M') A_{n}$	$ϕ_{n}$
Upper bands	$f_{n} + k ω$	(Even k)	${(- 1)}^{k / 2} I_{k} (M') A_{n}$	$ϕ_{n} + k Φ_{n}$
		(Odd k)	${(- 1)}^{(k + 1) / 2} I_{k} (M') A_{n}$	$ϕ_{n} - \frac{π}{2} + k Φ_{n}$
Lower bands	$f_{n} - k ω$	(Even k)	${(- 1)}^{k / 2} I_{k} (M') A_{n}$	$ϕ_{n} - k Φ_{n}$
		(Odd k)	${(- 1)}^{(k - 1) / 2} I_{k} (M') A_{n}$	$ϕ_{n} - \frac{π}{2} - k Φ_{n}$

^{^a}

The substitution $M' \equiv (m / 20) ln (10)$ is made for readability.

^{^b}

$I_{ν} (z)$ is the modified Bessel function of the first kind of order ν and argument z.

^³

See the footnote regarding negative amplitudes in Table I.