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CM · Modulation basics

Frequency Modulation

FM loads the message onto the carrier's instantaneous frequency. The amplitude stays put while the spacing between carrier crests tightens and loosens, so the information lives in the frequency, not in the amplitude. This unit shows why that makes FM resistant to noise.

Frequency sways, amplitude holds

The faint gold curve on top is the message, and the blue trace below is the FM carrier. Raise the deviation with the slider: under the message peak the carrier crests crowd together, and under the trough they spread apart. All the while the dashed gold lines above and below the waveform, that is, the amplitude, do not budge.

Frequency deviation Δfβ = 0.2
The message frequency fm is held fixed and only the deviation Δf grows, so the value shown on the right is β = Δf/fm. As β grows, FM passes from narrowband to wideband.
Instantaneous frequency at this deviation
f(t) = fc + Δf cos(ωm t)
Barely modulated

The message rides the frequency

Where AM made the amplitude swell, FM leaves the amplitude alone and instead pushes and pulls how many times per 1 second the carrier oscillates, its instantaneous frequency, in step with the message. The instantaneous frequency is written f(t) = fc + kf m(t): the larger the message value, the faster the carrier oscillates. Since the amplitude never changes, the envelope is a flat straight line, like the dashed gold lines in the plot.

The index β sets the width

The largest amount by which the instantaneous frequency strays from the carrier frequency is called the deviation Δf. Dividing it by the message frequency fm gives β = Δf/fm, the modulation index of FM. When β is less than 1 the frequency only wobbles a little, which is narrowband FM; when it is greater than 1 the frequency swings widely, which is wideband FM. Phase is the time integral of frequency, so the signal is written s(t) = Ac cos(2π fc t + β sin(ωm t)).

Observef(t) = fc + Δf cos(ωm t)
The message pushes and pulls the instantaneous frequency.
Chooseβ = Δf / ?
The modulation index is the deviation divided by the message frequency.

Carson's rule and noise immunity

Carson's rule estimates the FM bandwidth as B = 2(Δf + fm). A larger deviation therefore occupies more spectrum. In return, because the information lives only in the frequency, the receiver can clip the amplitude to a fixed level with a limiter and throw away the noise riding on the amplitude in one stroke. Giving up bandwidth to buy noise immunity is the central bargain of FM.

Fill inB = 2(Δf ? fm)
Carson's rule adds the deviation and the message frequency, then doubles.
On your ownB ≈ ?
In wideband FM the deviation dominates, giving B ≈ 2Δf.

Back to the first screen

As you raised the deviation, the carrier crowded tighter under the message peak and spread wider under the trough, yet the amplitude drawn as dashed gold lines stayed flat to the end. At the start, with the deviation near 0, the carrier was essentially unmodulated and showed red; once β grew large enough for wideband FM, the meter turned gold. What to remember from FM is that the message sits in the instantaneous frequency, not the amplitude, so noise on the amplitude can simply be clipped away.

Frequency modulation (FM) loads the message onto the carrier's instantaneous frequency, making f(t) = fc + kf m(t). The amplitude is constant, and the information lives in the frequency. With modulation index β = Δf/fm less than 1 the scheme is narrowband, and larger means wideband; the signal is s(t) = Ac cos(2π fc t + β sin(ωm t)). The bandwidth follows Carson's rule, B = 2(Δf + fm), and because the amplitude is constant, amplitude noise can be clipped off, which makes FM robust to noise.

In the next unit

AM and FM complete analog modulation. From the next unit on, the message becomes numbers. First comes PCM, which samples a continuous signal at regular intervals, chops the values into steps, and writes them down as bits. The starting point is the sampling theorem: sample often enough and the original signal can be rebuilt exactly.