Editing Chirp (section)

=== Exponential ===

[[File:exponentialchirp.png|thumb|upright=1.3|An exponential chirp waveform; a sinusoidal wave that increases in frequency exponentially over time]]
[[File:Expchirp.jpg|thumb|upright=1.3|[[Spectrogram]] of an exponential chirp. The exponential rate of change of frequency is shown as a function of time, in this case from nearly 0 up to 8&nbsp;kHz repeating every second. Also visible in this spectrogram is a frequency fallback to 6&nbsp;kHz after peaking, likely an artifact of the specific method employed to generate the waveform.]]
{{Listen|filename=Expchirp.ogg|title=Exponential chirp|description=Sound example for exponential chirp (five repetitions)|format=[[Ogg]]}}

In a '''geometric chirp''', also called  an '''exponential chirp''', the frequency of the signal varies with a [[geometric progression|geometric]] relationship over time. In other words, if two points in the waveform are chosen, <math>t_1</math> and <math>t_2</math>, and the time interval between them <math>T = t_2 - t_1</math> is kept constant, the frequency ratio <math>f\left(t_2\right)/f\left(t_1\right)</math> will also be constant.<ref>{{Citation |last=Li |first=X. |title=Time and Frequency Analysis Methods on GW Signals |date=2022-11-15 |url=https://github.com/xli2522/GW-SignalGen |access-date=2023-02-10}}</ref><ref>{{Cite journal |journal= IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control|year=2008 |pmc=2652352 |last1=Mamou |first1=J. |last2=Ketterling |first2=J. A. |last3=Silverman |first3=R. H. |title=Chirp-coded excitation imaging with a high-frequency ultrasound annular array |volume=55 |issue=2 |pages=508–513 |doi=10.1109/TUFFC.2008.670 |pmid=18334358 }}</ref>

In an exponential chirp, the frequency of the signal varies [[exponential function|exponentially]] as a function of time:
<math display="block">f(t) = f_0 k^\frac{t}{T}</math>
where <math>f_0</math> is the starting frequency (at <math>t = 0</math>), and <math>k</math> is the rate of [[exponential growth|exponential change]] in frequency.

<math display="block">k = \frac{f_1}{f_0}</math>

Where <math>f_1</math> is the ending frequency of the chirp (at  <math>t = T</math>).

Unlike the linear chirp, which has a constant chirpyness, an exponential chirp has an exponentially increasing frequency rate.

The corresponding time-domain function for the [[Phase (waves)|phase]] of an exponential 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 f_0 \int_0^t k^\frac{\tau}{T} d\tau \\
    &= \phi_0 + 2\pi f_0 \left(\frac{T k^\frac{t}{T}}{\ln(k)}\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 exponential chirp is the sine of the phase in radians:
<math display="block">x(t) = \sin\left[\phi_0 + 2\pi f_0 \left(\frac{T k^\frac{t}{T}}{\ln(k)}\right) \right]</math>

As was the case for the Linear Chirp, the instantaneous frequency of the Exponential Chirp consists of the fundamental frequency <math>f(t) = f_0 k^\frac{t}{T}</math> accompanied by additional [[harmonics]].{{citation needed|date=September 2012}}