Editing Chirp (section)

=== Hyperbolic ===
Hyperbolic chirps are used in radar applications, as they show maximum matched filter response after being distorted by the Doppler effect.<ref>{{cite journal |last1=Yang |first1=J. |last2=Sarkar |first2=T. K. |title=Doppler-invariant property of hyperbolic frequency modulated waveforms |journal=Microwave and Optical Technology Letters |date=June 2006 |volume=48 |issue=6 |pages=1174–1179 |doi=10.1002/mop.21573 }}</ref>

In a hyperbolic chirp, the frequency of the signal varies hyperbolically as a function of time:
<math display="block">f(t) = \frac{f_0 f_1 T}{(f_0-f_1)t+f_1T}</math>

The corresponding time-domain function for the phase of a hyperbolic chirp is the integral of the frequency:
<math display="block">\begin{align}
  \phi(t)
    &= \phi_0 + 2\pi     \int_0^t f(\tau)\, d\tau \\
    &= \phi_0 + 2\pi \frac{-f_0 f_1 T}{f_1-f_0} \ln\left(1-\frac{f_1-f_0}{f_1T}t\right)
\end{align}</math>
where <math>\phi_0</math> is the initial phase (at <math>t = 0</math>).

The corresponding time-domain function for a sinusoidal hyperbolic chirp is the sine of the phase in radians:
<math display="block">x(t) = \sin\left[ \phi_0 + 2\pi \frac{-f_0 f_1 T}{f_1-f_0} \ln\left(1-\frac{f_1-f_0}{f_1T}t\right)\right]</math>