Editing Chirp

{{Short description|Frequency swept signal}}
{{Other uses}}
{{more citations needed|date=August 2010}}
[[File:Linear-chirp.svg|thumb|upright=1.3|A linear chirp waveform; a sinusoidal wave that increases in frequency linearly over time]]

A '''chirp''' is a [[signal (information theory)|signal]] in which the [[frequency]] increases (''up-chirp'') or decreases (''down-chirp'') with time. In some sources, the term ''chirp'' is used interchangeably with '''sweep signal'''.<ref>Weisstein, Eric W. "Sweep Signal". From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SweepSignal.html</ref> It is commonly applied to [[sonar]], [[radar]], and [[laser]] systems, and to other applications, such as in [[Spread spectrum|spread-spectrum]] communications (see [[chirp spread spectrum]]). This signal type is biologically inspired and occurs as a phenomenon due to dispersion (a non-linear dependence between frequency and the propagation speed of the wave components). It is usually compensated for by using a matched filter, which can be part of the propagation channel. Depending on the specific performance measure, however, there are better techniques both for radar and communication. Since it was used in radar and space, it has been adopted also for communication standards. For automotive radar applications, it is usually called linear frequency modulated waveform (LFMW).<ref>{{cite journal |last1=Lee |first1=Tae-Yun |last2=Jeon |first2=Se-Yeon |last3=Han |first3=Junghwan |last4=Skvortsov |first4=Vladimir |last5=Nikitin |first5=Konstantin |last6=Ka |first6=Min-Ho |title=A Simplified Technique for Distance and Velocity Measurements of Multiple Moving Objects Using a Linear Frequency Modulated Signal |journal=IEEE Sensors Journal |date=August 2016 |volume=16 |issue=15 |pages=5912–5920 |doi=10.1109/JSEN.2016.2563458 |bibcode=2016ISenJ..16.5912L }}</ref>

In spread-spectrum usage, [[surface acoustic wave]] (SAW) devices are often used to generate and demodulate the chirped signals. In [[optics]], [[ultrashort pulse|ultrashort]] [[laser]] pulses also exhibit chirp, which, in optical transmission systems, interacts with the [[dispersion (optics)|dispersion]] properties of the materials, increasing or decreasing total pulse dispersion as the signal propagates. The name is a reference to the chirping sound made by birds; see [[bird vocalization]].

== Definitions ==
The basic definitions here translate as the common physics quantities location (phase), speed (angular velocity), acceleration (chirpyness).
If a [[waveform]] is defined as:
<math display="block">x(t) = \sin\left(\phi(t)\right)</math>

then the [[instantaneous angular frequency]], ''ω'', is defined as the phase rate as given by the first derivative of phase,
with the instantaneous ordinary frequency, ''f'', being its normalized version:
<math display="block">
\omega(t) = \frac{d\phi(t)}{dt}, \,
f(t) = \frac{\omega(t)}{2\pi}
</math>

Finally, the '''instantaneous angular chirpyness''' (symbol ''γ'') is defined to be the second derivative of instantaneous phase or the first derivative of instantaneous angular frequency, 
<math display="block">
\gamma(t) = \frac{d^2\phi(t)}{dt^2} = \frac{d\omega(t)}{dt}
</math>
Angular chirpyness has units of radians per square second (rad/s<sup>2</sup>); thus, it is analogous to ''[[angular acceleration]]''.

The '''instantaneous ordinary chirpyness''' (symbol ''c'') is a normalized version, defined as the rate of change of the instantaneous frequency:<ref name=Mann/>
<math display="block">
c(t) = \frac{\gamma(t)}{2\pi} = \frac{df(t)}{dt}
</math>
Ordinary chirpyness has units of square reciprocal seconds (s<sup>−2</sup>); thus, it is analogous to ''[[rotational acceleration]]''.

== Types ==

=== 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}}

=== 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}}

=== 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>

== Generation ==

A chirp signal can be generated with [[analog circuit]]ry via a [[voltage-controlled oscillator]] (VCO), and a linearly or exponentially ramping control [[voltage]].{{fact|date=February 2025}} It can also be generated [[Digital data|digitally]] by a [[digital signal processor]] (DSP) and [[digital-to-analog converter]] (DAC), using a [[direct digital synthesizer]] (DDS) and by varying the step in the numerically controlled oscillator.<ref>{{cite book |doi=10.1109/ICTC.2014.6983343 |chapter=Implementation of DDS chirp signal generator on FPGA |title=2014 International Conference on Information and Communication Technology Convergence (ICTC) |date=2014 |last1=Yang |first1=Heein |last2=Ryu |first2=Sang-Burm |last3=Lee |first3=Hyun-Chul |last4=Lee |first4=Sang-Gyu |last5=Yong |first5=Sang-Soon |last6=Kim |first6=Jae-Hyun |pages=956–959 |isbn=978-1-4799-6786-5 }}</ref> It can also be generated by a [[YIG oscillator]].{{clarify|date=April 2015}}
{{clear}}

== Relation to an impulse signal ==

[[File:Chirp animation.gif|thumb|Chirp and impulse signals and their (selected) [[spectral component]]s. On the bottom given four [[monochromatic]] components, sine waves of different frequency. The red line in the waves give the relative [[phase shift]] to the other sine waves, originating from the chirp characteristic. The animation removes the phase shift step by step (like with [[matched filter]]ing), resulting in a [[Sinc function|sinc pulse]] when no relative phase shift is left.]]
A chirp signal shares the same spectral content with an [[Dirac delta function|impulse signal]]. However, unlike in the impulse signal, spectral components of the chirp signal have different phases,<ref name="berkeley">{{cite web | url=http://setiathome.berkeley.edu/ap_chirp.php|title=Chirped pulses|publisher=setiathome.berkeley.edu|access-date=2014-12-03}}</ref><ref>{{cite book |last1=Easton Jr |first1=Roger L. |title=Fourier Methods in Imaging |date=2010 |publisher=John Wiley & Sons |isbn=978-1-119-99186-1 |page=700 }}</ref><ref name="dspguide">{{cite web|url=http://www.dspguide.com/ch11/6.htm|title=Chirp Signals | publisher=dspguide.com | access-date=2014-12-03}}</ref><ref name=arxiv>{{cite arXiv | eprint=1907.04186 | last1=Nikitin | first1=Alexei V. | last2=Davidchack | first2=Ruslan L. | title=Bandwidth is Not Enough: "Hidden" Outlier Noise and Its Mitigation | year=2019 | class=eess.SP }}</ref> i.e., their power spectra are alike but the [[phase spectrum|phase spectra]] are distinct. [[Dispersion (optics)|Dispersion]] of a signal propagation medium may result in unintentional conversion of impulse signals into chirps ([[Whistler_(radio)|Whistler]]). On the other hand, many practical applications, such as [[Chirped pulse amplification|chirped pulse amplifiers]] or echolocation systems,<ref name="dspguide"/> use chirp signals instead of impulses because of their inherently lower [[Crest factor|peak-to-average power ratio]] (PAPR).<ref name=arxiv/>

== Uses and occurrences ==
=== Chirp modulation ===
Chirp modulation, or linear frequency modulation for digital communication, was patented by [[Sidney Darlington]] in 1954 with significant later work performed by Winkler{{Who|date=February 2025}} in 1962. This type of modulation employs sinusoidal waveforms whose instantaneous frequency increases or decreases linearly over time. These waveforms are commonly referred to as linear chirps or simply chirps.

Hence the rate at which their frequency changes is called the ''chirp rate''. In binary chirp modulation, binary data is transmitted by mapping the bits into chirps of opposite chirp rates. For instance, over one bit period "1" is assigned a chirp with positive rate ''a'' and "0" a chirp with negative rate −''a''. Chirps have been heavily used in [[radar]] applications and as a result advanced sources for transmission and [[matched filter]]s for reception of linear chirps are available.

[[File:P-type-chirplets-for-image-processing.png|thumb|upright=1.3|(a) In image processing, direct periodicity seldom occurs, but, rather, periodicity-in-perspective is encountered. (b) Repeating structures like the alternating dark space inside the windows, and light space of the white concrete, "chirp" (increase in frequency) towards the right. (c) Thus the best fit chirp for image processing is often a projective chirp.]]

=== Chirplet transform ===
{{main article|Chirplet transform}}

Another kind of chirp is the projective chirp, of the form:
<math display="block">g = f\left[\frac{a \cdot x + b}{c \cdot x + 1}\right],</math>
having the three parameters ''a'' (scale), ''b'' (translation), and ''c'' (chirpiness). The projective chirp is ideally suited to [[image processing]], and forms the basis for the projective [[chirplet transform]].<ref name="Mann">Mann, Steve and Haykin, Simon;  The Chirplet Transform: A generalization of Gabor's Logon Transform; Vision Interface '91.[http://wearcam.org/chirplet/vi91scans/index.htm]</ref>

=== Key chirp ===
A change in frequency of [[Morse code]] from the desired frequency, due to poor stability in the [[Radio frequency|RF]] [[oscillator]], is known as '''chirp''',<ref>The Beginner's Handbook of Amateur Radio By Clay Laster</ref> and in the [[R-S-T system]] is given an appended letter 'C'.
{{clear}}

== See also ==
* [[Chirp spectrum]] - Analysis of the frequency spectrum of chirp signals
* [[Chirp compression]] - Further information on compression techniques
* [[Chirp spread spectrum]] - A part of the wireless telecommunications standard IEEE 802.15.4a CSS
* [[Chirped mirror]]
* [[Chirped pulse amplification]]
* [[Chirplet transform]] - A signal representation based on a family of localized chirp functions.
* [[Continuous-wave radar]]
* [[Dispersion (optics)]]
* [[Pulse compression]] 
* {{slink|Radio propagation#Measuring HF propagation}}

== References ==
{{reflist}}

==External links==
{{Commons category|Chirp}}
{{Wiktionary|chirp}}
* [http://www.audiocheck.net/audiofrequencysignalgenerator_sweep.php Online Chirp Tone Generator] (WAV file output)
* [https://fishfinderbrand.com/what-is-chirp-sonar-on-fish-finder/ CHIRP Sonar on FishFinder]
* [https://allbestfishfinder.com/what-is-chirp-on-a-fish-finder/ CHIRP Sonar on FishFinder]

[[Category:Signal processing]]
[[Category:Test items]]