Chirp

For other uses, see Chirp (disambiguation).

A chirp is a signal in which the frequency increases ('up-chirp') or decreases ('down-chirp') with time. In the fields of communications, Signal processing, and in Electrical engineering more generally a signal is any time-varying or spatial-varying quantity Frequency is a measure of the number of occurrences of a repeating event per unit Time. It is commonly used in sonar and radar, but has other applications, such as in spread spectrum communications. Sonar (which started as an Acronym for sound navigation and ranging) is a technique that uses Sound propagation (usually underwater to navigate Radar is a system that uses electromagnetic waves to identify the range altitude direction or speed of both moving and fixed objects such as Aircraft, ships Spread-spectrum techniques are methods by which Energy generated in a particular bandwidth is deliberately spread in the Frequency domain, resulting In spread spectrum usage, SAW devices such as RACs are often used to generate and demodulate the chirped signals. A surface acoustic wave ( SAW) is an Acoustic wave traveling along the surface of a material having some elasticity, with an Amplitude that In optics, ultrashort laser pulses also exhibit chirp due to the dispersion of the materials they propagate through. In Optics, an ultrashort pulse of light is an Electromagnetic pulse whose time duration is on the order of the femtosecond (10^{-15} second A laser is a device that emits Light ( Electromagnetic radiation) through a process called Stimulated emission.

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

In a linear chirp, the instantaneous frequency f(t ) varies linearly with time:

f (t) = f 0 + k t

where f₀ is the starting frequency (at time t = 0), and k is the rate of frequency increase or chirp rate. In Signal processing, the instantaneous phase (or "local phase" or simply "phase" of a complex-valued function x(t\ is the real-valued Chirp rate is the instantaneous rate of change of the Frequency of a waveform A corresponding time-domain function for a sinusoidal chirp is:

$x(t) = \sin(2 \pi \int_0^t f(t')\, dt') = \sin\left(2\pi (f_0 + \frac{k}{2} t) t \right)$

Linear chirp

Sound example for linear chirp (5 repetitions).

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

In a geometric chirp, the frequency of the signal varies with a geometric relationship over time. In Mathematics, a geometric progression, also known as a geometric sequence, is a Sequence of Numbers where each term after the first is found In other words, if two points in the waveform are chosen, t₁ and t₂, and the time interval between them t₂ − t₁ is kept constant, the frequency ratio f(t₂)/f(t₁) will also be constant.

In an exponential chirp, the frequency of the signal varies exponentially as a function of time:

f (t) = f 0 k t

where f₀ is the frequency at t=0, and k is the rate of exponential increase in frequency. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) Exponential growth (including Exponential decay) occurs when the growth rate of a mathematical function is proportional to the function's current value A corresponding sinusoidal chirp waveform would be defined by:

$x(t) = \sin(2 \pi f_0 \int_0^t k^{t'} dt') = \sin\left(\frac{2\pi f_0}{\ln(k)} ( k^t - 1)\right)$

Exponential chirp

Sound example for exponential chirp (5 repetitions).

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Although somewhat harder to generate, the geometric type does not suffer from reduction in correlation gain if Doppler-shifted by a moving target. In Probability theory and Statistics, correlation, (often measured as a correlation coefficient) indicates the strength and direction of a linear The Doppler effect (or Doppler shift) named after Christian Doppler, is the change in Frequency and Wavelength of a Wave for This is because the Doppler shift actually scales the frequencies of a wave by a multiplier (shown below as the constant c).

f (t) Doppler = c f (t) Original

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. Analogue electronics (or analog in American English) are those electronic systems with a continuously Variable signal A voltage-controlled oscillator or VCO is an Electronic oscillator designed to be controlled in Oscillation Frequency by a Voltage Electrical tension (or voltage after its SI unit, the Volt) is the difference of electrical potential between two points of an electrical It can also be generated digitally by a DSP and DAC, perhaps by varying the phase angle coefficient in the sinusoid generating function. A digital system uses discrete (discontinuous values usually but not always Symbolized Numerically (hence called "digital" to represent information for A digital signal processor ( DSP or DSP micro) is a specialized Microprocessor designed specifically for Digital signal processing, generally In Electronics, a digital-to-analog converter ( DAC or D-to-A) is a device for converting a digital (usually binary code to an Analog signal

Chirp modulation

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. Sidney Darlington ( July 18, 1906 - Exeter New Hampshire, October 31, 1997) was an electrical engineer inventor of a Transistor 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[1].

(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) to towards the right. (c) Thus the best fit chirp for image processing is often a projective chirp.

Chirplet transform

Main article: Chirplet transform

Another kind of chirp is the projective chirp, of the form $g = f[(a \cdot x + b)/(c \cdot x + 1)]$ , having the three parameters a (scale), b (translation), and c (chirpiness). In Signal processing, the chirplet transform is an Inner product of an input signal with a family of analysis primitives called chirplets. The projective chirp is ideally suited to image processing, and forms the basis for the projective chirplet transform. Image processing is any form of Signal processing for which the input is an image such as photographs or frames of video the output of image processing can be either an image In Signal processing, the chirplet transform is an Inner product of an input signal with a family of analysis primitives called chirplets.

Dictionary

-noun

A short, sharp or high note or noise, as of a bird or insect.

-verb

(intransitive) To make a shop, sharp, cheerful, as of small birds or crickets.
(intransitive) to speak in a high-pitched staccato
(slang) (intransitive) to insult

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Chirp modulation

Chirplet transform

See also

Dictionary

chirp

-noun

-verb