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.^{ [1] } Put simply, a time-domain graph shows how a signal changes over time, whereas a frequency-domain graph shows how much of the signal lies within each given frequency band over a range of frequencies. A frequency-domain representation can also include information on the phase shift that must be applied to each sinusoid in order to be able to recombine the frequency components to recover the original time signal.

- Advantages
- Magnitude and phase
- Types
- Discrete frequency domain
- History of term
- See also
- References
- Further reading

A given function or signal can be converted between the time and frequency domains with a pair of mathematical operators called transforms. An example is the Fourier transform, which converts a time function into a complex valued sum or integral of sine waves of different frequencies, with amplitudes and phases, each of which represents a frequency component. The "spectrum" of frequency components is the frequency-domain representation of the signal. The inverse Fourier transform converts the frequency-domain function back to the time-domain function. A spectrum analyzer is a tool commonly used to visualize electronic signals in the frequency domain.

Some specialized signal processing techniques use transforms that result in a joint time–frequency domain, with the instantaneous frequency being a key link between the time domain and the frequency domain.

One of the main reasons for using a frequency-domain representation of a problem is to simplify the mathematical analysis. For mathematical systems governed by linear differential equations, a very important class of systems with many real-world applications, converting the description of the system from the time domain to a frequency domain converts the differential equations to algebraic equations, which are much easier to solve.

In addition, looking at a system from the point of view of frequency can often give an intuitive understanding of the qualitative behavior of the system, and a revealing scientific nomenclature has grown up to describe it, characterizing the behavior of physical systems to time varying inputs using terms such as bandwidth, frequency response, gain, phase shift, resonant frequencies, time constant, resonance width, damping factor, Q factor, harmonics, spectrum, power spectral density, eigenvalues, poles, and zeros.

An example of a field in which frequency-domain analysis gives a better understanding than time domain is music; the theory of operation of musical instruments and the musical notation used to record and discuss pieces of music is implicitly based on the breaking down of complex sounds into their separate component frequencies (musical notes).

In using the Laplace, Z-, or Fourier transforms, a signal is described by a complex function of frequency: the component of the signal at any given frequency is given by a complex number. The modulus of the number is the amplitude of that component, and the argument is the relative phase of the wave. For example, using the Fourier transform, a sound wave, such as human speech, can be broken down into its component tones of different frequencies, each represented by a sine wave of a different amplitude and phase. The response of a system, as a function of frequency, can also be described by a complex function. In many applications, phase information is not important. By discarding the phase information, it is possible to simplify the information in a frequency-domain representation to generate a frequency spectrum or spectral density. A spectrum analyzer is a device that displays the spectrum, while the time-domain signal can be seen on an oscilloscope.

Although "*the*" frequency domain is spoken of in the singular, there are a number of different mathematical transforms which are used to analyze time-domain functions and are referred to as "frequency domain" methods. These are the most common transforms, and the fields in which they are used:

- Fourier series – periodic signals, oscillating systems.
- Fourier transform – aperiodic signals, transients.
- Laplace transform – electronic circuits and control systems.
- Z transform – discrete-time signals, digital signal processing.
- Wavelet transform — image analysis, data compression.

More generally, one can speak of the **transform domain** with respect to any transform. The above transforms can be interpreted as capturing some form of frequency, and hence the transform domain is referred to as a frequency domain.

A **discrete frequency domain** is a frequency domain that is discrete rather than continuous. For example, the discrete Fourier transform maps a function having a discrete time domain into one having a discrete frequency domain. The discrete-time Fourier transform, on the other hand, maps functions with discrete time (discrete-time signals) to functions that have a continuous frequency domain.^{ [2] }^{ [3] }

The Fourier transform of a periodic signal has energy only at a base frequency and its harmonics. Another way of saying this is that a periodic signal can be analyzed using a discrete frequency domain. Dually, a discrete-time signal gives rise to a periodic frequency spectrum. Combining these two, if we start with a time signal which is both discrete and periodic, we get a frequency spectrum which is also both discrete and periodic. This is the usual context for a discrete Fourier transform.

The use of the terms "frequency domain" and "time domain" arose in communication engineering in the 1950s and early 1960s, with "frequency domain" appearing in 1953.^{ [4] } See time domain: origin of term for details.^{ [5] }

- Bandwidth
- Blackman–Tukey transformation
- Fourier analysis for computing periodicity in evenly spaced data
- Least-squares spectral analysis for computing periodicity in unevenly spaced data
- Short-time Fourier transform
- Time–frequency representation
- Time–frequency analysis
- Wavelet
- Wavelet transform – digital image processing, signal compression

**Digital signal processing** (**DSP**) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are a sequence of numbers that represent samples of a continuous variable in a domain such as time, space, or frequency. In digital electronics, a digital signal is represented as a pulse train, which is typically generated by the switching of a transistor.

In mathematics, **Fourier analysis** is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer.

**Harmonic analysis** is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms. In the past two centuries, it has become a vast subject with applications in areas as diverse as number theory, representation theory, signal processing, quantum mechanics, tidal analysis and neuroscience.

In Fourier analysis, the **cepstrum** is the result of computing the inverse Fourier transform (IFT) of the logarithm of the estimated signal spectrum. The method is a tool for investigating periodic structures in frequency spectra. The *power cepstrum* has applications in the analysis of human speech.

A **wavelet** is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the number and direction of its pulses. Wavelets are imbued with specific properties that make them useful for signal processing.

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 representation.

The power spectrum of a time series describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of a certain signal or sort of signal as analyzed in terms of its frequency content, is called its spectrum.

A **spectrogram** is a visual representation of the spectrum of frequencies of a signal as it varies with time. When applied to an audio signal, spectrograms are sometimes called **sonographs**, **voiceprints**, or **voicegrams**. When the data are represented in a 3D plot they may be called *waterfall displays*.

In mathematics, **deconvolution** is the operation inverse to convolution. Both operations are used in signal processing and image processing. For example, it may be possible to recover the original signal after a filter (convolution) by using a deconvolution method with a certain degree of accuracy. Due to the measurement error of the recorded signal or image, it can be demonstrated that the worse the SNR, the worse the reversing of a filter will be; hence, inverting a filter is not always a good solution as the error amplifies. Deconvolution offers a solution to this problem.

In signal processing and electronics, the **frequency response** of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of systems, such as audio and control systems, where they simplify mathematical analysis by converting governing differential equations into algebraic equations. In an audio system, it may be used to minimize audible distortion by designing components so that the overall response is as flat (uniform) as possible across the system's bandwidth. In control systems, such as a vehicle's cruise control, it may be used to assess system stability, often through the use of Bode plots. Systems with a specific frequency response can be designed using analog and digital filters.

A **sine wave**, **sinusoidal wave**, or just **sinusoid** is a mathematical curve defined in terms of the *sine* trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in mathematics, as well as in physics, engineering, signal processing and many other fields.

The Fourier transform of a function of time, s(t), is a complex-valued function of frequency, S(f), often referred to as a frequency spectrum. Any linear time-invariant operation on s(t) produces a new spectrum of the form H(f)•S(f), which changes the relative magnitudes and/or angles (phase) of the non-zero values of S(f). Any other type of operation creates new frequency components that may be referred to as **spectral leakage** in the broadest sense. Sampling, for instance, produces leakage, which we call *aliases* of the original spectral component. For Fourier transform purposes, sampling is modeled as a product between s(t) and a Dirac comb function. The spectrum of a product is the convolution between S(f) and another function, which inevitably creates the new frequency components. But the term 'leakage' usually refers to the effect of *windowing*, which is the product of s(t) with a different kind of function, the window function. Window functions happen to have finite duration, but that is not necessary to create leakage. Multiplication by a time-variant function is sufficient.

A **time–frequency representation** (**TFR**) is a view of a signal represented over both time and frequency. Time–frequency analysis means analysis into the time–frequency domain provided by a TFR. This is achieved by using a formulation often called "Time–Frequency Distribution", abbreviated as TFD.

In signal processing, **time–frequency analysis** comprises those techniques that study a signal in both the time and frequency domains *simultaneously,* using various time–frequency representations. Rather than viewing a 1-dimensional signal and some transform, time–frequency analysis studies a two-dimensional signal – a function whose domain is the two-dimensional real plane, obtained from the signal via a time–frequency transform.

In mathematics and signal processing, an **analytic signal** is a complex-valued function that has no negative frequency components. The real and imaginary parts of an analytic signal are real-valued functions related to each other by the Hilbert transform.

In mathematics, a **wavelet series** is a representation of a square-integrable function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an **orthonormal wavelet** and of the **integral wavelet transform**.

**Geophysical survey** is the systematic collection of geophysical data for spatial studies. Detection and analysis of the geophysical signals forms the core of Geophysical signal processing. The magnetic and gravitational fields emanating from the Earth's interior hold essential information concerning seismic activities and the internal structure. Hence, detection and analysis of the electric and Magnetic fields is very crucial. As the Electromagnetic and gravitational waves are multi-dimensional signals, all the 1-D transformation techniques can be extended for the analysis of these signals as well. Hence this article also discusses multi-dimensional signal processing techniques.

In statistical signal processing, the goal of **spectral density estimation** (**SDE**) or simply **spectral estimation** is to estimate the spectral density of a signal from a sequence of time samples of the signal. Intuitively speaking, the spectral density characterizes the frequency content of the signal. One purpose of estimating the spectral density is to detect any periodicities in the data, by observing peaks at the frequencies corresponding to these periodicities.

A **bin-centres** test signal is one which has been constructed such that it has frequency components at FFT bin-centre frequencies. This allows analysis without an FFT window function on a synchronous measurement system.

- ↑ Broughton, S. A.; Bryan, K. (2008).
*Discrete Fourier Analysis and Wavelets: Applications to Signal and Image Processing*. New York: Wiley. p. 72. - ↑ C. Britton Rorabaugh (1998).
*DSP primer*. McGraw-Hill Professional. p. 153. ISBN 978-0-07-054004-0. - ↑ Shanbao Tong and Nitish Vyomesh Thakor (2009).
*Quantitative EEG analysis methods and clinical applications*. Artech House. p. 53. ISBN 978-1-59693-204-3. - ↑ Zadeh, L. A. (1953), "Theory of Filtering",
*Journal of the Society for Industrial and Applied Mathematics*,**1**: 35–51, doi:10.1137/0101003 - ↑ Earliest Known Uses of Some of the Words of Mathematics (T), Jeff Miller, March 25, 2009

Goldshleger, N., Shamir, O., Basson, U., Zaady, E. (2019). Frequency Domain Electromagnetic Method (FDEM) as tool to study contamination at the sub-soil layer. Geoscience 9 (9), 382.

- Boashash, B. (Sep 1988). "Note on the Use of the Wigner Distribution for Time Frequency Signal Analysis" (PDF).
*IEEE Transactions on Acoustics, Speech, and Signal Processing*.**36**(9): 1518–1521. doi:10.1109/29.90380.. - Boashash, B. (April 1992). "Estimating and Interpreting the Instantaneous Frequency of a Signal-Part I: Fundamentals".
*Proceedings of the IEEE*.**80**(4): 519–538. doi:10.1109/5.135376..

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