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

TABLE I.

The transformation to apply to each carrier tone in order to apply linear modulation. Each carrier tone is replaced with a base tone and two sidebands of linearly offset frequency.

Linear modulation sidebands
	Frequency	Amplitude^a	Phase
Base	f_n	A_n	$ϕ_{n}$
Upper band	$f_{n} + ω$	$- \frac{1}{2} m A_{n}$	$ϕ_{n} - \frac{π}{2} + Φ_{n}$
Lower band	$f_{n} - ω$	$\frac{1}{2} m A_{n}$	$ϕ_{n} - \frac{π}{2} + Φ_{n}$

^{^a}

Whereas it is improper to express amplitude as a negative value, it does greatly simplify the comparison with the exponential sidebands in this format. To recover the proper amplitudes and phase shifts, remove the factor of −1 appearing in an amplitude term and add a phase shift of π.