= Frequency modulation
{wiki}
= FM
{c}
{synonym}
Frequency modulation carries a message by changing the <instantaneous frequency> of a <carrier wave>. In the ideal model the <wave amplitude> stays constant: information changes the rate at which the <wave phase> advances. Write a real electrical <signal> and its chosen complex representation as
$$
s(t)=A\cos\phi(t),\qquad z(t)=Ae^{\mathrm i\phi(t)},\qquad A>0.
$$
The <wave envelope> in this representation has <modulus> $|z(t)|=A$ throughout, although $s(t)$ continually passes through zero. A smaller instantaneous value of $s(t)$ therefore does not mean a smaller <wave amplitude>. The continuously unwrapped <wave phase> $\phi$ is measured in <radians>; its <derivative> is an <angular frequency>, while division by $2\pi$ gives <frequency> in cycles per second, or <hertz>.
For a message $m(t)$ expressed as an <electric potential difference>, an ideal <voltage-controlled oscillator> has sensitivity $K_f$ in <hertz> per volt and obeys
$$
f_i(t)=f_c+K_fm(t),\qquad
\phi(t)=\phi(0)+2\pi f_ct+2\pi K_f\int_0^t m(\tau)\,d\tau.
$$
The <integral> is essential: putting the message directly into the <wave phase> gives <phase modulation>, whose <instantaneous frequency> instead depends on the message's <derivative>. <Amplitude modulation> changes the multiplying <wave amplitude>. These are different ways to encode the same kind of message <signal>.
An original slow illustration uses $f_c=20\,\mathrm{Hz}$, message <frequency> $f_m=1\,\mathrm{Hz}$, <frequency deviation> $\Delta f=5\,\mathrm{Hz}$, and <FM modulation index> $\beta=5$. With zero initial <wave phase>,
$$
\phi(t)=2\pi\cdot20t+5\sin(2\pi t),\qquad
s(t)=A\cos\bigl(2\pi\cdot20t+5\sin(2\pi t)\bigr),\qquad
f_i(t)=20+5\cos(2\pi t).
$$
Here $t$ is in seconds. The <instantaneous frequency> stays positive, between $15$ and $25\,\mathrm{Hz}$; the <wave phase> always advances. The oscillations bunch up where $f_i$ is large and spread out where it is small, while the <wave envelope> remains at $A$.
\Image[/frequency-modulation-waveform-and-instantaneous-frequency.png]
{title=Constant amplitude and changing cycle spacing in a frequency-modulated carrier}
{description=Original illustration with $f_c=20\,\mathrm{Hz}$, $f_m=1\,\mathrm{Hz}$, $\Delta f=5\,\mathrm{Hz}$ and $\beta=5$. The upper bounds are $s/A=\pm1$, not measured data; the lower curve is the exact derivative of the specified wave phase.}
This separation is useful in a receiver: changing a positive multiplying <wave amplitude> does not change the zero crossings of $\cos\phi$, so their timing still carries the message. <Noise> that moves those crossings, a vanishing <wave amplitude>, and <frequency response> distortion can still damage it. The surprising beauty of <FM> is that a constant <wave envelope> can conceal a rich <frequency spectrum>: the components interfere to change timing instead of the envelope's height.
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