## About

In electronics, control systems engineering, and statistics frequency domain is a term used to describe the analysis of mathematical functions or signals with respect to frequency, rather than time.

Speaking non-technically, 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.

A given function or signal can be converted between the time and frequency domains with a pair of mathematical operators called a transform. An example is the Fourier transform, which decomposes a function into the sum of a (potentially infinite) number of sine wave frequency components. The 'spectrum' of frequency components is the frequency domain representation of the signal. The inverse Fourier transform converts the frequency domain function back to a time function.

A spectrum analyzer is the tool commonly used to visualize real-world signals in the frequency domain.

## Magnitude and phase

In using the Laplace, Z-, or Fourier transforms, the frequency spectrum is complex, describing the magnitude and phase of a signal, or of the response of a system, as a function of frequency. 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.

The power spectral density is a frequency-domain description that can be applied to a large class of signals that are neither periodic nor square-integrable; to have a power spectral density a signal needs only to be the output of a wide-sense stationary random process.

## Partial frequency-domain example

Due to popular simplifications of the hearing process and titles such as Plomp's "The Ear as a Frequency Analyzer," the inner ear is often thought of as converting time-domain sound waveforms to frequency-domain spectra. The frequency domain is not actually a very accurate or useful model for hearing, but a time/frequency space or time/place space can be a useful description.[citation needed]

## Different frequency domains

Although "the" frequency domain is spoken of in the singular, there are actually several different frequency domains, each defined by a different mathematical transform, which are used to analyze signals. These are the most common transforms used and the fields in which they are used:

- Fourier series - repetitive signals, oscillating systems
- Fourier transform - nonrepetitive signals
- Laplace transform - electronic circuits and control systems
- Z transform - discrete signals, digital signal processing
- wavelet transform - digital image processing, signal compression