# Chirp

For the characteristic sounds of birds, see bird song. For that of crickets, see stridulation.

A chirp is a signal in which the frequency increases ('up-chirp') or decreases ('down-chirp') with time. It is commonly used in sonar and radar, but has other applications, such as in spread spectrum communications. In spread spectrum usage, SAW devices such as RACs are often used to generate and demodulate the chirped signals. In optics, ultrashort laser pulses also exhibit chirp due to the dispersion of the materials they propagate through.

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Linearchirp.png
a sinusoidal wave that increases in frequency linearly over time

A linear chirp waveform

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Exponentialchirp.png
a sinusoidal wave that increases in frequency exponentially over time

An exponential chirp waveform

In a linear chirp, the frequency varies linearly with time:

[itex]f(t) = f_0 + k t[itex]

where f0 is the starting frequency (at time t = 0), and k is the rate of frequency increase. A corresponding time domain function for a sinusoidal chirp is:

[itex]x(t) = \sin(2 \pi f(t) t) = \sin(2\pi (f_0 + k t) t)[itex]

In a geometric or 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, t1 and t2, and the time interval between them t2t1 is kept constant, the frequency ratio f(t2)/f(t1) will also be constant. The frequency varies exponentially as a function of time:

[itex]f(t) = f_0 k^t[itex]

In this case, f0 is the frequency at t=0, and k is the rate of exponential increase in frequency. A corresponding sinusoidal chirp waveform would be defined by:

[itex]x(t) = \sin(2 \pi f(t) t) = \sin(2\pi f_0 k^t t)[itex]

Although somewhat harder to generate, the geometric type does not suffer from reduction in correlation gain if Doppler shifted by a moving target. This is because the Doppler shift actually scales the frequencies of a wave by a multiplier (shown below as the constant c).

[itex]f(t)_{\mathrm{Doppler}} = c f(t)_{\mathrm{Original}}[itex]

From the equations above, it can be seen that this actually changes the rate of frequency increase of a linear chirp (kt multiplied by a constant) so that the correlation of the original function with the reflected function is low.

Because of the geometric relationship, the Doppler shifted geometric chirp will effectively start at a different frequency (f0 multiplied by a constant), but follow the same pattern of exponential frequency increase, so the end of the original wave, for instance, will still overlap perfectly with the beginning of the reflected wave, and the magnitude of the correlation will be high for that section of the wave.

A chirp signal can be generated with analog circuitry via a VCO, and a linearly or exponentially ramping control voltage. It can also be generated digitally by a DSP and DAC, perhaps by varying the phase angle coefficient in the sinusoid generating function.

## Chirplet transform

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P-type-chirplets-for-image-processing.png
(a) In image processing, we seldom have direct periodicity, but, rather, we encounter periodicity-in-perspective. (b) Repeating structures like the alternating dark space inside the windows, and light space of the white cement, "chirp" (increase in frequency) to towards the right. (c) Thus the best fit chirp for image processing is often a projective chirp.

Another kind of chirp is the projective chirp, of the form g = f((a·x + b)/(c·x + 1)), 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.

• 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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