= Instantaneous frequency
{title2=$f_i(t)=\phi'(t)/(2\pi)$}
For a <signal> with a specified differentiable, continuously unwrapped <wave phase> $\phi(t)$, its instantaneous frequency is
$$
f_i(t)=\frac{1}{2\pi}\frac{d\phi}{dt}.
$$
Unwrapping means following successive turns instead of resetting the <wave phase> at $2\pi$. Differentiating a reset phase would introduce artificial jumps. An arbitrary real <signal> does not by itself specify a unique amplitude-phase representation; this definition refers to the phase chosen in the model.
For the single-message-<frequency> <FM> model,
$$
\phi(t)=2\pi f_ct+\beta\sin(2\pi f_mt),\qquad
f_i(t)=f_c+\beta f_m\cos(2\pi f_mt).
$$
The proof is the <chain rule>: the <derivative> of the <sine> contributes $2\pi f_m$, which cancels the $2\pi$ in the definition. In the $20\,\mathrm{Hz}$ example, $f_i(0)=25$, $f_i(1/4)=20$, and $f_i(1/2)=15\,\mathrm{Hz}$. Although the <frequency> offset vanishes at $t=1/4$, the <wave phase> offset is then $5$ <radians>; the <wave phase> offset and its rate of change need not vanish together.
Between times $t_1,t_2$, the accumulated number of cycles is $(\phi(t_2)-\phi(t_1))/(2\pi)=\int_{t_1}^{t_2}f_i(t)\,dt$. Thus a crossing-to-crossing measurement yields an average <frequency> over that interval. Its reciprocal time is only an approximation to the <instantaneous frequency> at one point when $f_i$ varies slowly over a cycle.
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