Frequency domain

The Fourier transform converts the function's time-domain representation, shown in red, to the function's frequency-domain representation, shown in blue. The component frequencies, spread across the frequency spectrum, are represented as peaks in the frequency domain.

In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time.

- Frequency domain

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Fourier transform

Mathematical transform that decomposes functions depending on space or time into functions depending on spatial frequency or temporal frequency.

An example application of the Fourier transform is determining the constituent pitches in a musical waveform. This image is the result of applying a Constant-Q transform (a Fourier-related transform) to the waveform of a C major piano chord. The first three peaks on the left correspond to the frequencies of the fundamental frequency of the chord (C, E, G). The remaining smaller peaks are higher-frequency overtones of the fundamental pitches. A pitch detection algorithm could use the relative intensity of these peaks to infer which notes the pianist pressed.
Animation showing the Fourier Transform of a time shifted signal. [Top] the original signal (yellow), is continuously time shifted (blue). [Bottom] The resultant Fourier Transform of the time shifted signal. Note how the higher frequency components revolve in complex plane faster than the lower frequency components.
The rectangular function is Lebesgue integrable.
The sinc function, which is the Fourier transform of the rectangular function, is bounded and continuous, but not Lebesgue integrable.
Some problems, such as certain differential equations, become easier to solve when the Fourier transform is applied. In that case the solution to the original problem is recovered using the inverse Fourier transform.
Original function showing oscillation 3 Hz.
Real and imaginary parts of integrand for Fourier transform at 3 Hz
Real and imaginary parts of integrand for Fourier transform at 5 Hz
Magnitude of Fourier transform, with 3 and 5 Hz labeled.

The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time.

Control engineering

Control engineering or control systems engineering

Control systems play a critical role in space flight
Control of fractionating columns is one of the more challenging applications

A system can be mechanical, electrical, fluid, chemical, financial or biological, and its mathematical modelling, analysis and controller design uses control theory in one or many of the time, frequency and complex-s domains, depending on the nature of the design problem.

Time domain

Time domain refers to the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time.

The Fourier transform relates the function in the time domain, shown in red, to the function in the frequency domain, shown in blue. The component frequencies, spread across the frequency spectrum, are represented as peaks in the frequency domain.

The use of the contrasting terms time domain and frequency domain developed in U.S. communication engineering in the late 1940s, with the terms appearing together without definition by 1950.

Frequency response

Quantitative measure of the magnitude and phase of the output as a function of input frequency.

Magnitude response of a low pass filter with 6 dB per octave or 20 dB per decade roll-off

The frequency response characterizes systems in the frequency domain, just as the impulse response characterizes systems in the time domain.

Fourier series

Sum that represents a periodic function as a sum of sine and cosine waves.

Fig 1. The top graph shows a non-periodic function s(x) in blue defined only over the red interval from 0 to P. The function can be analyzed over this interval to produce the Fourier series in the bottom graph. The Fourier series is always a periodic function, even if original function s(x) wasn't.
Heat distribution in a metal plate, using Fourier's method
The atomic orbitals of chemistry are partially described by spherical harmonics, which can be used to produce Fourier series on the sphere.
A square wave (represented as the blue dot) is approximated by its sixth partial sum (represented as the purple dot), formed by summing the first six terms (represented as arrows) of the square wave's Fourier series. Each arrow starts at the vertical sum of all the arrows to its left (i.e. the previous partial sum).
The first four partial sums of the Fourier series for a square wave. As more harmonics are added, the partial sums converge to (become more and more like) the square wave.
Function <math>s_6(x)</math> (in red) is a Fourier series sum of 6 harmonically related sine waves (in blue). Its Fourier transform <math>S(f)</math> is a frequency-domain representation that reveals the amplitudes of the summed sine waves.
link={{filepath:Fourier_series_square_wave_circle_animation.svg}}|Four partial sums (Fourier series) of lengths 1, 2, 3, and 4 terms, showing how the approximation to a square wave improves as the number of terms increases [{{filepath:Fourier_series_square_wave_circle_animation.svg}} (animation)]
link={{filepath:Fourier_series_sawtooth_wave_circles_animation.svg}}|Four partial sums (Fourier series) of lengths 1, 2, 3, and 4 terms, showing how the approximation to a sawtooth wave improves as the number of terms increases [{{filepath:Fourier_series_sawtooth_wave_circles_animation.svg}} (animation)]
Example of convergence to a somewhat arbitrary function. Note the development of the "ringing" (Gibbs phenomenon) at the transitions to/from the vertical sections.

This transform thus can generate frequency domain representations of non-periodic functions as well as periodic functions, allowing a waveform to be converted between its time domain representation and its frequency domain representation.

Digital signal processing

Use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations.

An example of the 2D discrete wavelet transform that is used in JPEG2000. The original image is high-pass filtered, yielding the three large images, each describing local changes in brightness (details) in the original image. It is then low-pass filtered and downscaled, yielding an approximation image; this image is high-pass filtered to produce the three smaller detail images, and low-pass filtered to produce the final approximation image in the upper-left.

Nonlinear signal processing is closely related to nonlinear system identification and can be implemented in the time, frequency, and spatio-temporal domains.

Discrete Fourier transform

Complex-valued function of frequency.

Relationship between the (continuous) Fourier transform and the discrete Fourier transform. Left column: A continuous function (top) and its Fourier transform (bottom). Center-left column: Periodic summation of the original function (top). Fourier transform (bottom) is zero except at discrete points. The inverse transform is a sum of sinusoids called Fourier series.  Center-right column: Original function is discretized (multiplied by a Dirac comb) (top).  Its Fourier transform (bottom) is a periodic summation (DTFT) of the original transform. Right column: The DFT (bottom) computes discrete samples of the continuous DTFT.  The inverse DFT (top) is a periodic summation of the original samples.  The FFT algorithm computes one cycle of the DFT and its inverse is one cycle of the DFT inverse.
Depiction of a Fourier transform (upper left) and its periodic summation (DTFT) in the lower left corner. The spectral sequences at (a) upper right and (b) lower right are respectively computed from (a) one cycle of the periodic summation of s(t) and (b) one cycle of the periodic summation of the s(nT) sequence. The respective formulas are (a) the Fourier series integral and (b) the DFT summation .  Its similarities to the original transform, S(f), and its relative computational ease are often the motivation for computing a DFT sequence.
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Discrete transforms embedded in time & space.

The DFT is therefore said to be a frequency domain representation of the original input sequence.

Z-transform

ROC shown in blue, the unit circle as a dotted grey circle and the circle &#124;z&#124; = 0.5 is shown as a dashed black circle
ROC shown in blue, the unit circle as a dotted grey circle and the circle &#124;z&#124; = 0.5 is shown as a dashed black circle
ROC shown as a blue ring 0.5 < &#124;z&#124; < 0.75

In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation.

Wavelet transform

Representation of a square-integrable function by a certain orthonormal series generated by a wavelet.

An example of the 2D discrete wavelet transform that is used in JPEG2000.
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But smooth, periodic signals are better compressed using other methods, particularly traditional harmonic analysis in the frequency domain with Fourier-related transforms.

Data compression

Process of encoding information using fewer bits than the original representation.

Comparison of spectrograms of audio in an uncompressed format and several lossy formats. The lossy spectrograms show bandlimiting of higher frequencies, a common technique associated with lossy audio compression.
Solidyne 922: The world's first commercial audio bit compression sound card for PC, 1990
Processing stages of a typical video encoder

To determine what information in an audio signal is perceptually irrelevant, most lossy compression algorithms use transforms such as the modified discrete cosine transform (MDCT) to convert time domain sampled waveforms into a transform domain, typically the frequency domain.