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| {{Other uses}}
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| {{refimprove|date=August 2010}}
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| 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 has also been called '''quadratic-phase signal'''.<ref>http://books.google.com.br/books?id=QuIHjnXQqM8C&lpg=PA703&dq=chirp%20quadratic%20phase&pg=PA703#v=onepage&q&f=false</ref> It is commonly used in [[sonar]] and [[radar]], but has other applications, such as in [[spread spectrum]] communications. In spread spectrum usage, [[surface acoustic wave|SAW]] devices such as [[reflective array compressor|RACs]] 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 chirping in analogy to the sound made by some birds, see [[bird vocalization]].
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| == Types of chirp ==
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| === Linear chirp ===
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| [[File:Linear-chirp.svg|thumb|300px|A linear chirp waveform; a sinusoidal wave that increases in frequency linearly over time]]
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| [[File:LinearChirp.jpg|thumb|300px|[[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.]]
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| {{Listen|filename=Linchirp.ogg|title=Linear chirp|description=Sound example for linear chirp (5 repetitions).|format=[[Ogg]]}}
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| In a ''linear'' chirp, the [[instantaneous phase#Instantaneous frequency|instantaneous frequency]] <math>f(t)</math> varies linearly with time:
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| :<math>f(t) = f_0 + k t</math> | |
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| where <math>f_0</math> is the starting frequency (at time <math>t = 0</math>), and <math>k</math> is the rate of frequency increase or [[chirp rate]].
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| :<math>k = \frac{f_1-f_0}{t_1} </math> | |
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| where <math>f_1</math> is the final frequency and <math> f_0 </math> is the starting frequency.
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| The corresponding time-domain function for the [[Phase (waves)|phase]] of any oscillating signal is the integral of the frequency function, since 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>.
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| For the linear chirp, this results in:
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| :<math>\begin{align} | |
| \phi(t) &= \phi_0 + 2\pi\int_0^t f(\tau)\, d\tau\\
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| & = \phi_0 + 2\pi\int_0^t (f_0 + k \tau)\, d\tau\\
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| & = \phi_0 + 2\pi \left(f_0 t + \frac{k}{2} t^2 \right),
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| \end{align}</math>
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| where <math>\phi_0</math> is the initial phase (at time <math>t = 0</math>).
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| The corresponding time-domain function for a [[sinusoidal]] linear chirp is the sine of the phase in radians:
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| :<math>x(t) = \sin\left[\phi_0 + 2\pi \left(f_0 t + \frac{k}{2} t^2 \right) \right]</math> | |
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| In the frequency domain, the instantaneous frequency described by the equation <math>f(t) = f_0 + k t</math> is accompanied by additional frequencies ([[harmonics]]) which exist as a fundamental consequence of [[frequency modulation]].{{citation needed|date=September 2012}} These harmonics are quantifiably described through the use of [[Bessel function]]s. However, with the aid of a [[Time–frequency representation|frequency vs. time profile]] [[spectrogram]] one can readily see that the linear chirp has spectral components at harmonics of the fundamental chirp.
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| {{clear}}
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| === Exponential chirp ===
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| [[File:exponentialchirp.png|thumb|300px|An exponential chirp waveform; a sinusoidal wave that increases in frequency exponentially over time]]
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| [[File:Expchirp.jpg|thumb|300px|[[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 kHz repeating every second. Also visible in this Spectrogram is a frequency fallback to 6 kHz after peaking, likely an artifact of the specific method employed to generate the waveform.]]
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| {{Listen|filename=Expchirp.ogg|title=Exponential chirp|description=Sound example for exponential chirp (5 repetitions).|format=[[Ogg]]}}
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| 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_2 - t_1</math> is kept constant, the frequency ratio <math>f(t_2)/f(t_1)</math> will also be constant.
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| In an ''exponential'' chirp, the frequency of the signal varies [[exponential function|exponentially]] as a function of time:
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| :<math>f(t) = f_0 k^t</math>
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| 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 increase]] in frequency.
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| Unlike the linear chirp, which has a constant chirp rate, an exponential chirp has an exponentially increasing chirp rate.
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| The corresponding time-domain function for the [[Phase (waves)|phase]] of an exponential chirp is the integral of the frequency:
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| :<math>\begin{align}
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| \phi(t)
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| & = \phi_0 + 2\pi \int_0^t f(\tau)\, d\tau \\
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| & = \phi_0 + 2\pi f_0 \int_0^t k^{\tau} d\tau \\
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| & = \phi_0 + 2\pi f_0 \left( \frac{k^t - 1}{\ln(k)} \right)
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| \end{align}</math>
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| Where <math>\phi_0</math> is the initial phase (at <math>t = 0</math>).
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| The corresponding time-domain function for a sinusoidal exponential chirp is the sine of the phase in radians:
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| :<math>x(t) = \sin\left[\phi_0 + 2\pi f_0 \left( \frac{k^t - 1}{\ln(k)} \right)\right]</math>
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| 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^t</math> accompanied by additional [[harmonics]].{{citation needed|date=September 2012}}
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| == Generation of a chirp signal ==
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| A chirp signal can be generated with [[analog circuit]]ry via a [[voltage-controlled oscillator|VCO]], and a linearly or exponentially ramping control [[voltage]]. It can also be generated [[Digital data|digital]]ly 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. It can also be generated by a YIG oscillator.
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| {{clear}}
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| == Relation to an impulse signal ==
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| [[File:Chirp animation.gif|thumb|Chirp and impulse signals and their (selected) spectral components.]]
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| 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>http://setiathome.berkeley.edu/ap_chirp.php</ref> [[Dispersion (optics)|Dispersion]] of a signal propagation medium may result in unintentional conversion of impulse signals into chirps. On the other hand, many practical applications, such as [[Chirped pulse amplification|chirped pulse amplifiers]] or echolocation systems,<ref>http://www.dspguide.com/ch11/6.htm</ref> use chirp signals instead of impulses because of their inherently lower [[Crest factor|PAPR]].
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| == Uses and occurrences ==
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| === Chirp modulation ===
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| Chirp modulation, or linear frequency modulation for digital communication was patented by [[Sidney Darlington]] in 1954 with significant later work performed by Winkler 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.
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| 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 filters for reception of linear chirps are available.
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| [[File:P-type-chirplets-for-image-processing.png|thumb|300px|(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.]]
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| === Chirplet transform ===
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| {{main|Chirplet transform}}
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| Another kind of chirp is the projective chirp, of the form:
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| : <math>g = f\left[\frac{a \cdot x + b}{c \cdot x + 1}\right]</math>,
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| 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>Mann, Steve and Haykin, Simon; The Chirplet Transform: A Generalization of
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| Gabor's Logon Transform; Vision Interface '91.[http://wearcam.org/chirplet/vi91scans/index.htm]</ref>
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| === Key chirp ===
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| 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 [[RST code]] is given an appended letter 'C'.
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| == See also ==
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| * [[Chirplet transform]] - A signal representation based on a family of localized chirp functions, each member of which can usually be expressed as parameterized transformations of each other.
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| * [[Pulse compression]] - A signal processing technique designed to maximize the sensitivity and resolution of radar systems by modifying transmitted pulses to improve their auto-correlation properties. One way of accomplishing this is to chirp the RADAR signal (also known as [[Chirp Radar]]).
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| * [[Chirp Spread Spectrum]] - A part of the wireless telecommunications standard IEEE 802.15.4a CSS (see [http://www.ieee802.org/15/pub/05/15-05-0002-00-004a-nanotron-chirp-spread-spectrum-css-phy-presentation.ppt Chirp Spread Spectrum (CSS) PHY Presentation for IEEE P802.15.4a]).
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| * [[Continuous-wave radar]]
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| * [[SHARAD]]
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| * [[Chirped pulse amplification]]
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| * [[Chirped mirror]]
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| * [[Dispersion (optics)]]
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| {{Commons category|Chirp}}
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| == References ==
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| {{Reflist}}
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| ==External links==
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| {{Wiktionary|Chirp}}
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| *[http://www.audiocheck.net/audiofrequencysignalgenerator_sweep.php Online Chirp Tone Generator] (wav file output)
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| [[Category:Signal processing]]
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| [[Category:Test items]]
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