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Chirp
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== 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 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.]] {{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=508β513 |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=1174β1179 |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>
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