# Time domain

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.

Time domain refers to the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the case of continuous time, or at various separate instants in the case of discrete time. An oscilloscope is a tool commonly used to visualize real-world signals in the time domain. A time-domain graph shows how a signal changes with time, whereas a frequency-domain graph shows how much of the signal lies within each given frequency band over a range of frequencies.

Though most precisely referring to time in physics, the term time domain may occasionally informally refer to position in space when dealing with spatial frequencies, as a substitute for the more precise term spatial domain.

## Origin of term

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.[1] When an analysis uses the second or one of its multiples as a unit of measurement, then it is in the time domain. When analysis concerns the reciprocal units such as Hertz, then it is in the frequency domain.

## References

1. ^ Lee, Y. W.; Cheatham, T. P., Jr.; Wiesner, J. B. (1950). "Application of Correlation Analysis to the Detection of Periodic Signals in Noise". Proceedings of the IRE. 38 (10): 1165–1171. doi:10.1109/JRPROC.1950.233423.

Fourier transform time and frequency domains (small).gif
A visualization of the relationship between the time domain and the frequency domain of a function, based on its Fourier transform.

The Fourier transform takes an input function f (in red) in the "time domain" and converts it into a new function f-hat (in blue) in the "frequency domain".

In other words, the original function can be thought of as being "amplitude given time", and the Fourier transform of the function is "amplitude given frequency".

Shown here, a simple 6-component approximation of the square wave is decomposed (exactly, for simplicity) into 6 sine waves. These component frequencies show as very sharp peaks in the frequency domain of the function, shown as the blue graph.
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Author/Creator: David Vignoni (original icon); Flamurai (SVG convertion); bayo (color), Licence: GPL
Square root of x formula. Symbol of mathematics.