Modulation Calculator (AM index & depth, FM index & Carson bandwidth)

Understanding the modulation calculator — concept and principle

This calculator works out the key figures of amplitude modulation (AM) and frequency modulation (FM). For AM it gives the modulation index and depth from the envelope peaks and troughs; for FM the modulation index from the frequency deviation. It also gives the occupied bandwidth, the share of power in the sidebands and, for FM, the remaining carrier amplitude. It is used in radio, telecommunications and electronics courses.

With β = 5 almost all the power moves out of the carrier into sidebands spread over about 180 kHz.

How it is calculated

AM index m = (V_max − V_min) ÷ (V_max + V_min); depth = m × 100%

AM bandwidth = 2 f_m; total power = P_c (1 + m²/2)

FM index β = Δf ÷ f_m

Carson's rule bandwidth B ≈ 2 (Δf + f_m)

FM carrier amplitude = J₀(β); share of power in the sidebands = 1 − J₀(β)²

f_m is the highest modulating frequency, Δf the peak frequency deviation, J₀ the Bessel function of order zero.

The principle

AM varies the carrier's amplitude with the message, creating two sidebands at the carrier frequency ± f_m; the carrier itself carries most of the power and no information. FM varies the frequency instead. Its spectrum contains many sidebands spaced f_m apart, with amplitudes given by Bessel functions of β. Because the total power of an FM signal is constant, power taken from the carrier goes into the sidebands. Carson's rule estimates the bandwidth that holds about 98% of the power.

Worked example

Default inputs: FM, f_m = 15 kHz, Δf = 75 kHz.

β = 75 ÷ 15 = 5.
Carson bandwidth = 2 × (75 + 15) = 180 kHz.
Carrier amplitude J₀(5) = −0.1776.
Power in the sidebands = 1 − 0.1776² = 96.8%.
Depth and total-to-carrier power apply to AM only, so they show —.

Each point is the calculator re-run with only “Highest modulating (audio) frequency (kHz)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Highest modulating (audio) frequency (kHz)	Modulation index
3.75	20
7.5	10
11.2	6.667
15	5
18.8	4
22.5	3.333
30	2.5

How to read the result

These are the classic figures of FM broadcasting: 75 kHz deviation with audio up to 15 kHz. A larger β spreads the signal wider but makes it more resistant to noise. In AM, even at 100% modulation only one third of the total power is in the sidebands, which is why AM is less power-efficient than FM. The FM carrier disappears entirely when β ≈ 2.405, a trick used to calibrate deviation.

Common mistakes

Using the average audio frequency instead of the highest one.
Treating Carson's rule as an exact bandwidth; it is an estimate.
Over-modulating AM above 100%, which distorts the envelope and splatters into neighbouring channels.

Keep in mind: Allowed deviation and bandwidth are set by radio regulations for each service. Check the rules that apply before transmitting.

Frequently asked questions

Why is FM broadcast bandwidth about 200 kHz?

Carson's rule with 75 kHz deviation and 15 kHz audio gives 180 kHz, plus guard space.

What does a negative J₀ mean?

The carrier component is inverted in phase; its power is J₀².

What is 100% AM modulation?

The envelope just reaches zero at its troughs.

Is narrowband FM different?

Yes. With β well below 1, the bandwidth is close to 2 f_m, like AM.

Sources: J. R. Carson (1922), Carson's bandwidth rule; standard communication-theory results for AM power and the Bessel spectrum of FM

Checked 9 Nov 2026. Standard communication-theory results: AM index from the envelope, AM power P_t = P_c(1 + m²/2), FM index β = Δf/f_m, Carson’s bandwidth rule B ≈ 2(Δf + f_m) (J. R. Carson, 1922) and the Bessel-function spectrum of sinusoidal FM. 75 kHz deviation: FM broadcast channel standard.

Your numbers