Chirp

Template:Short description {{#invoke:other uses|otheruses}} Template:More citations needed

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

A chirp is a 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 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>Template:Cite journal</ref>

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

DefinitionsEdit

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/s²); 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⁻²); thus, it is analogous to rotational acceleration.

TypesEdit

LinearEdit

File:LinearChirpMatlab.png

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.

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 Template:Nowrap

The corresponding time-domain function for the 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">Template:Cite book</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>

ExponentialEdit

File:Exponentialchirp.png

An exponential chirp waveform; a sinusoidal wave that increases in frequency exponentially over time

File:Expchirp.jpg

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.

In a geometric chirp, also called an exponential chirp, the frequency of the signal varies with a 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>Template:Citation</ref><ref>Template:Cite journal</ref>

In an exponential chirp, the frequency of the signal varies 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 change in frequency.

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 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.Template:Citation needed

HyperbolicEdit

Hyperbolic chirps are used in radar applications, as they show maximum matched filter response after being distorted by the Doppler effect.<ref>Template:Cite journal</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>

GenerationEdit

A chirp signal can be generated with analog circuitry via a voltage-controlled oscillator (VCO), and a linearly or exponentially ramping control voltage.Template:Fact It can also be generated 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>Template:Cite book</ref> It can also be generated by a YIG oscillator.Template:Clarify

Relation to an impulse signalEdit

File:Chirp animation.gif

Chirp and impulse signals and their (selected) spectral components. 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 filtering), resulting in a sinc pulse when no relative phase shift is left.

A chirp signal shares the same spectral content with an impulse signal. However, unlike in the impulse signal, spectral components of the chirp signal have different phases,<ref name="berkeley">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>Template:Cite book</ref><ref name="dspguide">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref name=arxiv>Template:Cite arXiv</ref> i.e., their power spectra are alike but the phase 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 as chirped pulse amplifiers or echolocation systems,<ref name="dspguide"/> use chirp signals instead of impulses because of their inherently lower peak-to-average power ratio (PAPR).<ref name=arxiv/>

Uses and occurrencesEdit

Chirp modulationEdit

Chirp modulation, or linear frequency modulation for digital communication, was patented by Sidney Darlington in 1954 with significant later work performed by WinklerTemplate: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 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.

File:P-type-chirplets-for-image-processing.png

(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 transformEdit

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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.[1]</ref>

Key chirpEdit

A change in frequency of Morse code from the desired frequency, due to poor stability in the 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'.

ReferencesEdit

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External linksEdit

Template:Sister project Template:Sister project

Online Chirp Tone Generator (WAV file output)
CHIRP Sonar on FishFinder
CHIRP Sonar on FishFinder