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Chirp

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Frequency swept signal

For other uses, seeChirp (disambiguation).

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A linear chirp waveform; a sinusoidal wave that increases in frequency linearly over time

Achirp is asignal in which thefrequency increases (up-chirp) or decreases (down-chirp) with time. In some sources, the termchirp is used interchangeably withsweep signal.^[1] It is commonly applied tosonar,radar, andlaser systems, and to other applications, such as inspread-spectrum communications (seechirp 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).^[2]

In spread-spectrum usage,surface acoustic wave (SAW) devices are often used to generate and demodulate the chirped signals. Inoptics,ultrashort laser pulses also exhibit chirp, which, in optical transmission systems, interacts with thedispersion 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; seebird vocalization.

Definitions

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The basic definitions here translate as the common physics quantities location (phase), speed (angular velocity), acceleration (chirpyness).If awaveform is defined as: $x(t)=\sin \left(\phi (t)\right)$

then theinstantaneous 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: $\omega (t)={\frac {d\phi (t)}{dt}},\,f(t)={\frac {\omega (t)}{2\pi }}$

Finally, theinstantaneous angular chirpyness (symbolγ) is defined to be the second derivative of instantaneous phase or the first derivative of instantaneous angular frequency, $\gamma (t)={\frac {d^{2}\phi (t)}{dt^{2}}}={\frac {d\omega (t)}{dt}}$ Angular chirpyness has units of radians per square second (rad/s²); thus, it is analogous toangular acceleration.

Theinstantaneous ordinary chirpyness (symbolc) is a normalized version, defined as the rate of change of the instantaneous frequency:^[3] $c(t)={\frac {\gamma (t)}{2\pi }}={\frac {df(t)}{dt}}$ Ordinary chirpyness has units of square reciprocal seconds (s⁻²); thus, it is analogous torotational acceleration.

Types

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Linear

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Linear chirp

Sound example for linear chirp (five repetitions)

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In alinear-frequency chirp or simplylinear chirp, the instantaneous frequency $f(t)$ varies exactly linearly with time: $f(t)=ct+f_{0},$ where $f_{0}$ is the starting frequency (at time $t=0$ ) and $c {\displaystyle c}$ is the chirp rate, assumed constant: $c={\frac {f_{1}-f_{0}}{T}}={\frac {\Delta f}{\Delta t}}.$

Here, $f_{1}$ is the final frequency and $T {\displaystyle T}$ is the time it takes to sweep from $f_{0}$ to $f_{1}$ .

The corresponding time-domain function for thephase of any oscillating signal is the integral of the frequency function, as one expects the phase to grow like $\phi (t+\Delta t)\simeq \phi (t)+2\pi f(t)\,\Delta t$ , i.e., that the derivative of the phase is the angular frequency $\phi '(t)=2\pi \,f(t)$ .

For the linear chirp, this results in: ${\begin{aligned}\phi (t)&=\phi _{0}+2\pi \int _{0}^{t}f(\tau )\,d\tau \\&=\phi _{0}+2\pi \int _{0}^{t}\left(c\tau +f_{0}\right)\,d\tau \\&=\phi _{0}+2\pi \left({\frac {c}{2}}t^{2}+f_{0}t\right),\end{aligned}}$

where $\phi _{0}$ is the initial phase (at time $t=0$ ). Thus this is also called aquadratic-phase signal.^[4]

The corresponding time-domain function for asinusoidal linear chirp is the sine of the phase in radians: $x(t)=\sin \left[\phi _{0}+2\pi \left({\frac {c}{2}}t^{2}+f_{0}t\right)\right]$

Exponential

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Exponential chirp

Sound example for exponential chirp (five repetitions)

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In ageometric chirp, also called anexponential chirp, the frequency of the signal varies with ageometric relationship over time. In other words, if two points in the waveform are chosen, $t_{1}$ and $t_{2}$ , and the time interval between them $T=t_{2}-t_{1}$ is kept constant, the frequency ratio $f\left(t_{2}\right)/f\left(t_{1}\right)$ will also be constant.^[5]^[6]

In an exponential chirp, the frequency of the signal variesexponentially as a function of time: $f(t)=f_{0}k^{\frac {t}{T}}$ where $f_{0}$ is the starting frequency (at $t=0$ ), and $k {\displaystyle k}$ is the rate ofexponential change in frequency.

$k={\frac {f_{1}}{f_{0}}}$

Where $f_{1}$ is the ending frequency of the chirp (at $t=T$ ).

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

The corresponding time-domain function for thephase of an exponential chirp is the integral of the frequency: ${\begin{aligned}\phi (t)&=\phi _{0}+2\pi \int _{0}^{t}f(\tau )\,d\tau \\&=\phi _{0}+2\pi f_{0}\int _{0}^{t}k^{\frac {\tau }{T}}d\tau \\&=\phi _{0}+2\pi f_{0}\left({\frac {T\left(k^{\frac {t}{T}}-1\right)}{\ln(k)}}\right)\end{aligned}}$ where $\phi _{0}$ is the initial phase (at $t=0$ ).

The corresponding time-domain function for a sinusoidal exponential chirp is the sine of the phase in radians: $x(t)=\sin \left[\phi _{0}+2\pi f_{0}\left({\frac {T\left(k^{\frac {t}{T}}-1\right)}{\ln(k)}}\right)\right]$

As was the case for the Linear Chirp, the instantaneous frequency of the Exponential Chirp consists of the fundamental frequency $f(t)=f_{0}k^{\frac {t}{T}}$ accompanied by additionalharmonics.^{[citation needed]}

Hyperbolic

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Hyperbolic chirps are used in radar applications, as they show maximum matched filter response after being distorted by the Doppler effect.^[7]

In a hyperbolic chirp, the frequency of the signal varies hyperbolically as a function of time: $f(t)={\frac {f_{0}f_{1}T}{(f_{0}-f_{1})t+f_{1}T}}$

The corresponding time-domain function for the phase of a hyperbolic chirp is the integral of the frequency: ${\begin{aligned}\phi (t)&=\phi _{0}+2\pi \int _{0}^{t}f(\tau )\,d\tau \\&=\phi _{0}+2\pi {\frac {-f_{0}f_{1}T}{f_{1}-f_{0}}}\ln \left(1-{\frac {f_{1}-f_{0}}{f_{1}T}}t\right)\end{aligned}}$ where $\phi _{0}$ is the initial phase (at $t=0$ ).

The corresponding time-domain function for a sinusoidal hyperbolic chirp is the sine of the phase in radians: $x(t)=\sin \left[\phi _{0}+2\pi {\frac {-f_{0}f_{1}T}{f_{1}-f_{0}}}\ln \left(1-{\frac {f_{1}-f_{0}}{f_{1}T}}t\right)\right]$

Generation

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A chirp signal can be generated withanalog circuitry via avoltage-controlled oscillator (VCO), and a linearly or exponentially ramping controlvoltage.^{[citation needed]} It can also be generateddigitally by adigital signal processor (DSP) anddigital-to-analog converter (DAC), using adirect digital synthesizer (DDS) and by varying the step in the numerically controlled oscillator.^[8] It can also be generated by aYIG oscillator.^{[clarification needed]}

Relation to an impulse signal

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Chirp and impulse signals and their (selected)spectral components. On the bottom given fourmonochromatic components, sine waves of different frequency. The red line in the waves give the relativephase shift to the other sine waves, originating from the chirp characteristic. The animation removes the phase shift step by step (like withmatched filtering), resulting in asinc pulse when no relative phase shift is left.

A chirp signal shares the same spectral content with animpulse signal. However, unlike in the impulse signal, spectral components of the chirp signal have different phases,^[9]^[10]^[11]^[12] i.e., their power spectra are alike but thephase spectra are distinct.Dispersion of a signal propagation medium may result in unintentional conversion of impulse signals into chirps (whistler). On the other hand, many practical applications, such aschirped pulse amplifiers or echolocation systems,^[11] use chirp signals instead of impulses because of their inherently lowerpeak-to-average power ratio (PAPR).^[12]

Uses and occurrences

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Chirp modulation

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Chirp modulation, or linear frequency modulation for digital communication, was patented bySidney Darlington in 1954 with significant later work performed by Winkler^[who?] 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 thechirp 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 ratea and "0" a chirp with negative rate −a. Chirps have been heavily used inradar applications and as a result advanced sources for transmission andmatched filters for reception of linear chirps are available.

(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

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Main article:Chirplet transform

Another kind of chirp is the projective chirp, of the form: $g=f\left[{\frac {a\cdot x+b}{c\cdot x+1}}\right],$ having the three parametersa (scale),b (translation), andc (chirpiness). The projective chirp is ideally suited toimage processing, and forms the basis for the projectivechirplet transform.^[3]

Key chirp

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A change in frequency ofMorse code from the desired frequency, due to poor stability in theRF oscillator, is known aschirp,^[13] and in theR-S-T system is given an appended letter 'C'.

References

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^Weisstein, Eric W. "Sweep Signal". From MathWorld--A Wolfram Web Resource.http://mathworld.wolfram.com/SweepSignal.html
^Lee, Tae-Yun; Jeon, Se-Yeon; Han, Junghwan; Skvortsov, Vladimir; Nikitin, Konstantin; Ka, Min-Ho (August 2016). "A Simplified Technique for Distance and Velocity Measurements of Multiple Moving Objects Using a Linear Frequency Modulated Signal".IEEE Sensors Journal.16 (15):5912–5920.Bibcode:2016ISenJ..16.5912L.doi:10.1109/JSEN.2016.2563458.
^^a ^bMann, Steve and Haykin, Simon; The Chirplet Transform: A generalization of Gabor's Logon Transform; Vision Interface '91.[1]
^Easton, R.L. (2010).Fourier Methods in Imaging. Wiley. p. 703.ISBN 9781119991861. Retrieved2014-12-03.
^Li, X. (2022-11-15),Time and Frequency Analysis Methods on GW Signals, retrieved2023-02-10
^Mamou, J.; Ketterling, J. A.; Silverman, R. H. (2008)."Chirp-coded excitation imaging with a high-frequency ultrasound annular array".IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control.55 (2):508–513.Bibcode:2008ITUFF..55..508M.doi:10.1109/TUFFC.2008.670.PMC 2652352.PMID 18334358.
^Yang, J.; Sarkar, T. K. (June 2006). "Doppler-invariant property of hyperbolic frequency modulated waveforms".Microwave and Optical Technology Letters.48 (6):1174–1179.doi:10.1002/mop.21573.
^Yang, Heein; Ryu, Sang-Burm; Lee, Hyun-Chul; Lee, Sang-Gyu; Yong, Sang-Soon; Kim, Jae-Hyun (2014). "Implementation of DDS chirp signal generator on FPGA".2014 International Conference on Information and Communication Technology Convergence (ICTC). pp. 956–959.doi:10.1109/ICTC.2014.6983343.ISBN 978-1-4799-6786-5.
^"Chirped pulses". setiathome.berkeley.edu. Retrieved2014-12-03.
^Easton Jr, Roger L. (2010).Fourier Methods in Imaging. John Wiley & Sons. p. 700.ISBN 978-1-119-99186-1.
^^a ^b"Chirp Signals". dspguide.com. Retrieved2014-12-03.
^^a ^bNikitin, Alexei V.; Davidchack, Ruslan L. (2019). "Bandwidth is Not Enough: "Hidden" Outlier Noise and Its Mitigation".arXiv:1907.04186 [eess.SP].
^The Beginner's Handbook of Amateur Radio By Clay Laster