Frequency domain

Source: Wikipedia, the free encyclopedia.
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

time-domain graph shows how a signal changes over time, whereas 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 the phase of a set of sinusoids
(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 mathematical

electronic 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 joint

instantaneous frequency
response being a key link between the time domain and the frequency domain.

Advantages

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

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).

Magnitude and phase

In using the

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
.

Types

Although "the" frequency domain is spoken of in the singular, there is 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 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.

Discrete frequency domain

A discrete frequency domain is a frequency domain that is discrete rather than continuous. For example, the

discrete-time signals) to functions that have a continuous frequency domain.[2][3]

A

discrete-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 a discrete Fourier transform
.

History of term

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]

See also

References

  1. Wiley
    . p. 72.
  2. ^ C. Britton Rorabaugh (1998). DSP primer. McGraw-Hill Professional. p. 153.
    ISBN 978-0-07-054004-0
    .
  3. ^ Shanbao Tong and Nitish Vyomesh Thakor (2009). Quantitative EEG analysis methods and clinical applications. Artech House. p. 53.
    ISBN 978-1-59693-204-3
    .
  4. doi:10.1137/0101003
  5. ^ 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.

Further reading
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