Frequency Modulation Explained: Deviation, Index and Carson's Rule
In frequency modulation (FM) the carrier amplitude stays constant and the instantaneous frequency moves up and down with the message. Because information is not stored in amplitude, noise spikes and amplitude fading that ruin AM reception have far less effect. That is why FM broadcast between 88 and 108 MHz sounds clean, and why walkie-talkies and many telemetry links use FM or its digital cousin FSK.
This tutorial defines frequency deviation and modulation index, shows where the sidebands come from, and uses Carson's rule to size the bandwidth of real FM systems.
Frequency Deviation and Modulation Index
For a single-tone message Am cos(2 pi fm t), the modulator shifts the carrier frequency by an amount proportional to the message voltage. The constant of proportionality is the frequency sensitivity kf, in hertz per volt.
Instantaneous frequency: f(t) = fc + kf Am cos(2 pi fm t) Peak frequency deviation: delta_f = kf x Am Modulation index: beta = delta_f / fm FM signal: s(t) = Ac cos(2 pi fc t + beta sin(2 pi fm t))
Note two differences from AM. First, the FM index can be much larger than 1; there is no over-modulation in the AM sense. Second, beta depends on both the message amplitude (through delta_f) and the message frequency. A loud low note produces a large beta; the same loudness at a higher pitch produces a smaller one.
Worked Example: Finding Deviation and Index
A modulator has kf = 5 kHz/V. A 2 V, 2.5 kHz tone is applied.
delta_f = 5 kHz/V x 2 V = 10 kHz beta = 10 kHz / 2.5 kHz = 4 The carrier swings between fc - 10 kHz and fc + 10 kHz, 2500 times per second.
The FM Spectrum and Bessel Functions
Unlike AM, a single tone does not produce just two sidebands. Expanding the FM equation gives an infinite set of sidebands spaced fm apart, with amplitudes given by Bessel functions of the first kind, Jn(beta):
s(t) = Ac x sum over n of Jn(beta) cos(2 pi (fc + n fm) t) Carrier amplitude: Ac J0(beta) n-th sideband pair: Ac Jn(beta) at fc +/- n fm
| beta | J0 (carrier) | J1 | J2 | J3 |
|---|---|---|---|---|
| 0.25 | 0.98 | 0.12 | 0.01 | - |
| 1.0 | 0.77 | 0.44 | 0.11 | 0.02 |
| 2.405 | 0.00 | 0.52 | 0.43 | 0.20 |
Two useful facts follow. The total power stays constant at Ac^2 / 2R whatever the modulation, because modulation only redistributes power between carrier and sidebands. And at beta = 2.405 the carrier component vanishes completely. Technicians use this carrier null on a spectrum analyser to calibrate deviation: set fm, raise the audio level until the carrier disappears, and the deviation is 2.405 times fm.
Carson's Rule for Bandwidth
Since the sidebands theoretically extend forever, engineers need a practical bandwidth. Carson's rule captures about 98 percent of the signal power:
BW = 2 (delta_f + fm) = 2 fm (beta + 1) Narrowband FM (beta much less than 1): BW is about 2 fm (like AM) Wideband FM (beta much greater than 1): BW is about 2 delta_f
Worked Example 1: Commercial FM Broadcast
Maximum deviation delta_f = 75 kHz Highest audio frequency fm = 15 kHz beta = 75 / 15 = 5 BW = 2 (75 + 15) = 180 kHz Channel spacing allotted: 200 kHz (leaves a guard band)
Worked Example 2: Two-Way Narrowband Radio
delta_f = 5 kHz, voice limited to fm = 3 kHz beta = 5 / 3 = 1.67 BW = 2 (5 + 3) = 16 kHz -> fits a 25 kHz channel
Worked Example 3: Effect of Changing the Tone
Keep delta_f = 75 kHz but drop the tone to 1 kHz. Then beta = 75 and BW = 2(75 + 1) = 152 kHz. The bandwidth barely changes, because in wideband FM it is set mostly by deviation, not by the message frequency.
FM Compared with AM
| Property | AM (DSB-FC) | FM |
|---|---|---|
| Information carried in | Amplitude | Frequency |
| Bandwidth | 2 fm | 2 (delta_f + fm), usually much wider |
| Transmitted power | Changes with modulation | Constant |
| Noise performance | Poor; noise adds directly to envelope | Better; limiter removes amplitude noise, improvement grows with beta |
| Transmitter amplifier | Must be linear | Can be efficient non-linear (class C) |
| Capture effect | No | Yes; stronger station suppresses weaker one on the same channel |
FM buys its noise immunity with bandwidth. Above a threshold carrier-to-noise ratio, the output SNR of wideband FM improves roughly with the square of beta. Below that threshold, reception collapses suddenly into clicks and hiss, which you may have heard when driving away from an FM station.
Pre-emphasis and De-emphasis
Noise at the output of an FM discriminator rises with audio frequency. To compensate, the transmitter boosts high audio frequencies with a high-pass RC network (pre-emphasis) and the receiver applies the matching low-pass network (de-emphasis). The time constant is 50 microseconds in India and Europe and 75 microseconds in North America.
Corner frequency f = 1 / (2 pi tau) tau = 50 us -> f = 1 / (2 pi x 50e-6) = 3.18 kHz tau = 75 us -> f = 2.12 kHz
Common Mistakes
- Using deviation alone as the bandwidth. The occupied bandwidth is roughly twice the deviation plus twice fm.
- Assuming beta must stay below 1. That limit applies to AM, not FM.
- Forgetting that a receiver with the wrong de-emphasis sounds dull or harsh.
- Confusing FM with PM. In phase modulation the phase, not frequency, is proportional to the message; the two differ by an integration of the message.
Summary
FM varies carrier frequency, keeps power constant and spreads energy into many Bessel sidebands spaced fm apart. The modulation index is beta = delta_f / fm, and Carson's rule BW = 2(delta_f + fm) gives a reliable bandwidth estimate: 180 kHz for broadcast FM and 16 kHz for a typical narrowband voice radio. FM trades bandwidth for noise immunity, a trade-off you will meet again in digital modulation.