Chirp

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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 some sources, the term chirp is used interchangeably with sweep signal.[1] It has also been called quadratic-phase signal.[2] 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, 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 chirping in analogy to the sound made by some birds, see bird vocalization.

Types of chirp[edit]

Linear chirp[edit]

A linear chirp waveform; a sinusoidal wave that increases in frequency linearly over time
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.
Sound example for linear chirp (5 repetitions).

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

f(t) = f_0 + k t

where f_0 is the starting frequency (at time t = 0), and k is the rate of frequency increase or chirp rate.

k = \frac{f_1-f_0}{t_1}

where f_1 is the final frequency and  f_0 is the starting frequency.

The corresponding time-domain function for the phase of any oscillating signal is the integral of the frequency function, since 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{align}
\phi(t) &= \phi_0 + 2\pi\int_0^t f(\tau)\, d\tau\\
& = \phi_0 + 2\pi\int_0^t (f_0 + k \tau)\, d\tau\\
& = \phi_0 + 2\pi \left(f_0 t + \frac{k}{2} t^2 \right),
\end{align}

where \phi_0 is the initial phase (at time t = 0).

The corresponding time-domain function for a sinusoidal linear chirp is the sine of the phase in radians:

x(t) = \sin\left[\phi_0 + 2\pi \left(f_0 t + \frac{k}{2} t^2 \right) \right]

In the frequency domain, the instantaneous frequency described by the equation f(t) = f_0 + k t is accompanied by additional frequencies (harmonics) which exist as a fundamental consequence of frequency modulation.[citation needed] These harmonics are quantifiably described through the use of Bessel functions. However, with the aid of a frequency vs. time profile spectrogram one can readily see that the linear chirp has spectral components at harmonics of the fundamental chirp.

Exponential chirp[edit]

An exponential chirp waveform; a sinusoidal wave that increases in frequency exponentially over time
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.
Sound example for exponential chirp (5 repetitions).

Problems playing this file? See media help.

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, t_1 and t_2, and the time interval between them t_2 - t_1 is kept constant, the frequency ratio f(t_2)/f(t_1) 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 f_0 is the starting frequency (at t = 0), and k is the rate of exponential increase in frequency. Unlike the linear chirp, which has a constant chirp rate, an exponential chirp has an exponentially increasing chirp rate.

The corresponding time-domain function for the phase of an exponential chirp is the integral of the frequency:

\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^{\tau} d\tau \\
& = \phi_0 + 2\pi f_0 \left( \frac{k^t - 1}{\ln(k)} \right)
\end{align}

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{k^t - 1}{\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^t accompanied by additional harmonics.[citation needed]

Generation of a chirp signal[edit]

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, using a Direct digital synthesizer (DDS) and by varying the step in the numerically controlled oscillator. It can also be generated by a YIG oscillator.

Relation to an impulse signal[edit]

Chirp and impulse signals and their (selected) spectral components.

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,[3][4][5] 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. On the other hand, many practical applications, such as chirped pulse amplifiers or echolocation systems,[5] use chirp signals instead of impulses because of their inherently lower PAPR.

Uses and occurrences[edit]

Chirp modulation[edit]

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. 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.

(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[edit]

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 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.[6]

Key chirp[edit]

A change in frequency of Morse code from the desired frequency, due to poor stability in the RF Oscillator is known as chirp,[7] and in the RST code is given an appended letter 'C'.

Chirp rate [edit]

Chirp rate is the instantaneous rate of change of the frequency of a waveform. If a waveform is defined as:

x(t)=\sin\left(\phi(t)\right)

then the instantaneous frequency is defined to be:

f(t)=\frac{1}{2\pi} \frac{d\phi(t)}{dt}

and the chirp rate is defined to be:

c(t)=\frac{1}{2\pi} \frac{d^2\phi(t)}{dt^2}

See also[edit]

References[edit]

  1. ^ Weisstein, Eric W. "Sweep Signal." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SweepSignal.html
  2. ^ Easton, R.L. (2010). Fourier Methods in Imaging. Wiley. p. 703. ISBN 9781119991861. Retrieved 2014-12-03. 
  3. ^ "Chirped pulses". setiathome.berkeley.edu. Retrieved 2014-12-03. 
  4. ^ Easton, R.L. (2010). Fourier Methods in Imaging. Wiley. p. 700. ISBN 9781119991861. Retrieved 2014-12-03. 
  5. ^ a b "Chirp Signals". dspguide.com. Retrieved 2014-12-03. 
  6. ^ Mann, Steve and Haykin, Simon; The Chirplet Transform: A Generalization of Gabor's Logon Transform; Vision Interface '91.[1]
  7. ^ The Beginner's Handbook of Amateur Radio By Clay Laster

External links[edit]