Frequency Modulation
Frequency sways, amplitude holds
Raise the frequency deviation Δf. The carrier crowds at the message peak and thins at the trough — yet the top and bottom envelopes stay perfectly flat the whole time.
The message rides the frequency
AM swung the amplitude; FM leaves the amplitude alone and instead pushes and pulls how many times per second the carrier oscillates — its instantaneous frequency — with the message. That frequency is f(t) = fc + kf m(t): the higher the message, the faster it oscillates. Because the amplitude never changes, the envelope is a flat straight line.
The index β sets the width
The most the frequency strays is the deviation Δf. Divided by the message frequency fm, β = Δf/fm is the modulation index. When β is below 1 the frequency wiggles only slightly — narrowband FM; when β is above 1 the frequency is swung hard — wideband FM. Phase is the integral of frequency, so the signal reads s(t) = Ac cos(2π fc t + β sin(ωm t)).
Carson’s rule and noise immunity
FM’s bandwidth is Carson’s rule, B = 2(Δf + fm): the larger the deviation, the more band it uses. In return, since the information lives only in the frequency, a receiver can clip the amplitude to a fixed value (a limiter) and throw away whatever noise rode the amplitude. Spending more bandwidth to buy noise immunity is FM’s central bargain.
Back to the first screen
As you raised Δf the carrier crowded harder at the message peak and thinned more at the trough, yet the top and bottom envelopes stayed flat to the end. With deviation near zero the carrier was unchanged — effectively no modulation; with deviation large enough it became wideband FM and turned gold. The one thing to hear in FM is this: the message lives in the instantaneous frequency, not the amplitude, and that is why the amplitude’s noise can be clipped away.
In the next unit
AM and FM finish analog modulation. From here we turn the message into numbers. First, sample the continuous signal at regular intervals, round each value to steps (quantize), and write it as bits — PCM. The starting point is the sampling theorem: sample often enough and the original signal can be reconstructed exactly.