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Inmathematics,physics,electronics,control systems engineering, andstatistics, thefrequency domain refers to the analysis ofmathematical functions orsignals with respect tofrequency (and possibly phase), rather than time, as intime series.[1] While atime-domain graph shows how a signal changes over time, a frequency-domain graph shows how the signal is distributed within different frequency bands over a range of frequencies. A complex valued frequency-domain representation consists of both the magnitude and thephase of a set ofsinusoids (or other basis waveforms) at the frequency components of the signal. Although it is common to refer to the magnitude portion (the real valued frequency-domain) as the frequency response of a signal, the phase portion is required to uniquely define the signal.
A given function or signal can be converted between the time and frequency domains with a pair of mathematicaloperators calledtransforms. An example is theFourier transform, which converts a time function into a complex valued sum or integral ofsine 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. Theinverse Fourier transform converts the frequency-domain function back to the time-domain function. Aspectrum analyzer is a tool commonly used to visualizeelectronic signals in the frequency domain.
A frequency-domain representation may describe either a static function or a particular time period of a dynamic function (signal or system). The frequency transform of a dynamic function is performed over a finite time period of that function and assumes the function repeats infinitely outside of that time period. Some specialized signal processing techniques for dynamic functions use transforms that result in a jointtime–frequency domain, with theinstantaneous frequency response 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 bylinear 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 thedifferential equations toalgebraic 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 asbandwidth,frequency response,gain,phase shift,resonant frequencies,time constant,resonance width,damping factor,Q factor,harmonics,spectrum,power spectral density,eigenvalues,poles, andzeros.
An example of a field in which frequency-domain analysis gives a better understanding than time domain ismusic; the theory of operation of musical instruments and themusical 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 theLaplace,Z-, or Fourier transforms, a signal is described by acomplex function of frequency: the component of the signal at any given frequency is given by acomplex number. Themodulus of the number is theamplitude of that component, and theargument is the relative phase of the wave. For example, using theFourier transform, asound 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 afrequency spectrum orspectral density. Aspectrum analyzer is a device that displays the spectrum, while the time-domain signal can be seen on anoscilloscope.
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:
More generally, one can speak of thetransform 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.
Adiscrete frequency domain is a frequency domain that isdiscrete rather thancontinuous. For example, thediscrete Fourier transform maps a function having adiscrete time domain into one having a discrete frequency domain. Thediscrete-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]
Aperiodic signal has energy only at a base frequency and its harmonics; thus it can be analyzed using a discrete frequency domain. Adiscrete-time signal gives rise to a periodic frequency spectrum. In a situation where both these conditions occur, a signal which is discrete and periodic results in a frequency spectrum which is also discrete and periodic; this is the usual context for adiscrete 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] Seetime domain: origin of term for details.[5]
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.
