Editing Chirp (section)

=== Linear ===
[[File:LinearChirpMatlab.png|thumb|upright=1.5|[[Spectrogram]] of a linear chirp. The spectrogram plot demonstrates the linear rate of change in frequency as a function of time, in this case from 0 to 7 kHz, repeating every 2.3 seconds. The intensity of the plot is proportional to the energy content in the signal at the indicated frequency and time.]]
{{Listen|filename=Linchirp.ogg|title=Linear chirp|description=Sound example for linear chirp (five repetitions)|format=[[Ogg]]}}

In a '''linear-frequency chirp''' or simply '''linear chirp''', the instantaneous frequency <math>f(t)</math> varies exactly linearly with time:
<math display="block">f(t) = c t + f_0,</math>
where <math>f_0</math> is the starting frequency (at time <math>t = 0</math>) and <math>c</math> is the chirp rate, assumed constant:
<math display="block">c = \frac{f_1 - f_0}{T} = \frac{\Delta f}{\Delta t}.</math>

Here, <math>f_1</math> is the final frequency and <math> T </math> is the time it takes to sweep from <math> f_0 </math> to {{nowrap|<math>f_1</math>.}}

The corresponding time-domain function for the [[Phase (waves)|phase]] of any oscillating signal is the integral of the frequency function, as one expects the phase to grow like <math>\phi(t + \Delta t) \simeq \phi(t) + 2\pi f(t)\,\Delta t</math>, i.e., that the derivative of the phase is the angular frequency <math>\phi'(t) = 2\pi\,f(t)</math>.

For the linear chirp, this results in:
<math display="block">\begin{align}
  \phi(t) &= \phi_0 + 2\pi\int_0^t f(\tau)\, d\tau\\
          &= \phi_0 + 2\pi\int_0^t \left(c \tau+f_0\right)\, d\tau\\
          &= \phi_0 + 2\pi         \left(\frac{c}{2} t^2+f_0 t\right),
\end{align}</math>

where <math>\phi_0</math> is the initial phase (at time <math>t = 0</math>).  Thus this is also called a '''quadratic-phase signal'''.<ref name="google">{{cite book|title=Fourier Methods in Imaging|author=Easton, R.L.| date=2010| publisher=Wiley| isbn=9781119991861|url=https://books.google.com/books?id=QuIHjnXQqM8C|page=703|access-date=2014-12-03}}</ref>

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