Instantaneous frequency ($f_i(t)={\varphi }^{\prime }(t)/(2\pi )$)

For a signal with a specified differentiable, continuously unwrapped wave phase $\varphi (t)$, its instantaneous frequency isUnwrapping 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,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 $(\varphi (t_2)-\varphi (t_1))/(2\pi )={\int }_{t_1}^{t_2}{f}_i(t)dt$. 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.