The modulation index is the number that says how hard a carrier is being driven by the signal riding on it, and it means two different things depending on which modulation scheme you are using. In amplitude modulation it is a ratio of amplitudes, written m, and it describes the depth of the envelope. In frequency modulation it is a ratio of frequencies, written β, and it describes how far the carrier swings compared with how fast it is being swung. This calculator handles both, and keeps the definitions separate.
Getting the two confused is the most common error in the topic, because the same phrase covers both and the numbers look nothing alike. An AM index of 1.0 is full modulation and going past it is a fault; an FM index of 5.0 is ordinary broadcast practice. Arb Digital built the page with an explicit mode selector rather than a single formula so that the distinction is unavoidable.
What This Modulation Index Calculator Does
In AM mode, give it either the modulating and carrier amplitudes or the maximum and minimum of the envelope as read from an oscilloscope, and it returns the modulation index and the percentage depth. The grid then reports the occupied bandwidth, the total transmitted power at that depth, the power carried in the sidebands, and the fraction of the total that the sidebands represent — the number that determines how much of your transmitter output is doing useful work.
In FM mode, give it the peak deviation and the modulating frequency, and it returns β, the Carson-rule bandwidth, the number of significant sideband pairs and the band edges around the carrier. Deviation is the transmitter's setting; the modulating frequency comes from the audio, so β changes continuously in real programme material even though the deviation limit does not.
How to Use It
- Select AM or FM. The input fields change, because the two schemes need different quantities.
- For AM, choose your input style. Amplitudes if you know the signal levels, envelope maximum and minimum if you are reading a modulated waveform off a scope.
- For FM, enter the peak deviation in kilohertz — the maximum excursion of the carrier from its centre frequency, one way.
- Enter the modulating frequency. For a bandwidth figure, use the highest frequency in the modulating signal rather than a mid-band tone.
- Add carrier power for AM if you want the power split; FM does not need it, because total power is constant.
The Formula: How It's Calculated
For amplitude modulation, the index is the ratio of the modulating amplitude to the carrier amplitude:
m = Am ÷ Ac, or equivalently m = (Vmax − Vmin) ÷ (Vmax + Vmin)
The second form is what you use with a scope, because the envelope peaks and troughs are directly measurable. Total power at depth m is Pt = Pc(1 + m²÷2), with the sidebands carrying Pcm²÷2 split equally between the upper and lower. The occupied bandwidth is twice the highest modulating frequency, and the sidebands sit at fc ± fm.
For frequency modulation, the index is the ratio of peak deviation to modulating frequency:
β = Δf ÷ fm, with bandwidth from Carson's rule BW ≈ 2(Δf + fm)
FM produces an infinite set of sidebands whose amplitudes follow Bessel functions of the first kind, so its bandwidth is formally unbounded. Carson's rule is the standard engineering approximation that captures roughly 98% of the power, and the calculator estimates the number of significant sideband pairs as those falling inside that bandwidth. Background on how these signals are used in practice, including the frequency allocations and the stereo subcarrier arrangement, is set out in the HyperPhysics page on radio broadcast signals, and the licensing framework that governs occupied bandwidth in the United States is administered by the FCC, which publishes the AM station classes and their operating conditions.
Two worked examples you can reproduce here. AM: an envelope reading 12 V at the peaks and 4 V at the troughs gives m = (12 − 4) ÷ (12 + 4) = 0.5, so 50% depth. On a 100 W carrier, total power is 100 × (1 + 0.25÷2) = 112.5 W, of which 12.5 W is sideband — 11.1% of the transmitted power. FM: 75 kHz deviation with a 15 kHz modulating tone gives β = 5.0, and Carson's rule gives 2 × (75 + 15) = 180 kHz of occupied bandwidth.
Why AM Power Efficiency Is Capped at One Third
Look again at Pt = Pc(1 + m²÷2). The carrier term is fixed. Everything the sidebands get comes from the m²÷2 term, and since m cannot exceed 1 without distortion, the most the sidebands can ever carry is half the carrier power — a third of the total. Two-thirds of a fully modulated AM transmitter's output is carrier, which conveys no information at all and exists only so a simple envelope detector can recover the signal.
At 50% depth it is much worse: the sidebands take 11.1% of the transmitted power and the carrier takes 88.9%. This is the arithmetic that motivated single-sideband, which suppresses the carrier and one sideband entirely and puts all the transmitter output into the one component that carries information. It is also why AM broadcast stations are so concerned with maintaining high average modulation depth: the carrier bill is paid whether or not anyone is talking.
Over-Modulation: What Happens Above m = 1
When m exceeds 1, the envelope tries to go negative and cannot. The carrier is cut off during part of each cycle, and the recovered audio is clipped — a distinct, harsh distortion. Worse, that clipping generates harmonics of the modulating signal, which appear as new sidebands spread far outside the intended channel. In broadcast parlance this is splatter, and it is interference to adjacent channels rather than merely poor audio on your own.
This is why AM transmitters use limiters ahead of the modulator and why regulators specify a maximum negative modulation depth rather than leaving it to the operator. The calculator flags any index above 1 in words. Note that operating a transmitter on most radio spectrum requires a licence in nearly every jurisdiction, and occupied bandwidth and spurious emissions are conditions of that licence; the EIRP calculator covers the radiated power side of the same rules.
Narrowband and Wideband FM Are Genuinely Different
FM behaviour splits at roughly β = 0.5. Below that, only the first sideband pair is significant, and the spectrum looks almost exactly like an AM signal — a carrier with one pair of sidebands, occupying about 2fm. This is narrowband FM, and it is what land-mobile, marine and amateur voice radio use, typically with 5 kHz deviation and 2.5 to 3 kHz of audio.
Above β = 1 the picture changes completely. Multiple sideband pairs become significant, the bandwidth grows roughly in proportion to deviation, and the signal starts to earn FM's characteristic noise advantage. Broadcast FM at 75 kHz deviation with 15 kHz audio gives β = 5 and a 180 kHz Carson bandwidth, which is why the channel spacing is 200 kHz. The extra bandwidth buys signal-to-noise improvement above a threshold carrier level and buys nothing below it, which is the classic FM trade: excellent quality when the signal is strong, an abrupt collapse when it is not, rather than the graceful degradation of AM.
One consequence surprises people: at certain values of β the carrier component vanishes entirely. The carrier amplitude follows the zeroth-order Bessel function, whose first zero is near β = 2.405, so a correctly set FM transmitter can show no carrier at all on a spectrum analyser. That is not a fault, and it is used deliberately as a calibration method for setting deviation.
Choosing the Right Modulating Frequency for a Bandwidth Estimate
Both bandwidth formulas take a single modulating frequency, but real signals are not single tones. For an occupied-bandwidth estimate, use the highest frequency present in the modulating signal after filtering — 15 kHz for broadcast FM audio, around 3 kHz for communications voice. Using a mid-band figure will understate the bandwidth, sometimes substantially.
In FM this also means β is not one number for a real signal. Deviation is set by amplitude and modulating frequency by pitch, so a loud bass note gives a very large β while a quiet treble note gives a small one. The single value people quote for a system, such as 5 for broadcast FM, is the deviation ratio: maximum permitted deviation divided by maximum modulating frequency. It is the worst-case index, which is exactly what a bandwidth calculation needs. Converting between the frequency units involved is quick work in the frequency converter, and channel-width arithmetic is covered by the bandwidth and frequency calculator.
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Browse All Free Tools Talk To Arb DigitalCommon Mistakes to Avoid
- Applying the AM definition to FM — AM's index is a ratio of amplitudes bounded at 1, FM's is a ratio of frequencies that routinely reaches 5 or more.
- Using a mid-band tone for a bandwidth estimate — both bandwidth formulas need the highest modulating frequency present, not an average one.
- Assuming FM total power changes with modulation — it does not; modulation only moves power from the carrier into the sidebands.
- Treating Carson's rule as exact — FM's sideband set is formally infinite, and Carson captures roughly 98% of the power rather than all of it.
- Reading a missing FM carrier as a fault — the carrier component genuinely vanishes near an index of 2.405, where the zeroth-order Bessel function crosses zero.
Related Free Tools From Arb Digital
For the radiated-power side of a transmission, use the EIRP calculator, and for how much of it survives the journey, the free space path loss calculator. Convert between hertz, kilohertz and megahertz with the frequency converter, work channel widths with the bandwidth and frequency calculator, and find the physical wavelength with the wavelength calculator. Power and gain ratios in dB are handled by the decibel calculator, and receiver sensitivity by the noise figure calculator. The rest are in the free online tools hub.
Frequently Asked Questions
They measure different things. The AM index is the modulating amplitude divided by the carrier amplitude, so it is a depth of modulation bounded at 1. The FM index is the peak frequency deviation divided by the modulating frequency, a ratio of frequencies with no upper bound, and values of 5 are normal in broadcasting.
Measure the maximum and minimum height of the modulated envelope, then compute the maximum minus the minimum divided by their sum. An envelope reaching 12 V at the peaks and 4 V at the troughs gives an index of 0.5, or 50% depth.
The envelope cannot go negative, so the carrier is cut off for part of each cycle. The recovered audio is clipped, and the clipping generates harmonics that appear as sidebands well outside the intended channel. That splatter interferes with adjacent channels, which is why transmitters use limiters ahead of the modulator.
Because the carrier carries no information and cannot be removed in conventional AM. The sidebands take a share of the power equal to the index squared over two, so even at full modulation they hold only a third of the total. At 50% depth they hold 11.1%, with the rest spent on the carrier.
An engineering approximation for the bandwidth of a frequency-modulated signal: twice the sum of the peak deviation and the highest modulating frequency. FM strictly produces an infinite series of sidebands, and Carson's rule brackets the roughly 98% of the power that matters in practice.
An index below about 0.5, where only the first sideband pair is significant and the occupied bandwidth is close to twice the modulating frequency. Land-mobile, marine and amateur voice radio work in this region, typically with around 5 kHz of deviation.
No. Frequency modulation varies the instantaneous frequency at constant amplitude, so total transmitted power is the same whether the transmitter is modulated or not. What changes is the distribution of that power between the carrier and the sidebands.
This tool is provided for educational and engineering-estimate use. It applies the single-tone AM and FM relations with Carson's rule as a bandwidth approximation, and does not account for pre-emphasis, composite stereo or data subcarriers, filter shaping, or transmitter non-linearity. Operating a radio transmitter requires a licence in most jurisdictions, and occupied bandwidth and spurious emission limits are conditions of that licence rather than matters of preference.
