Frequency domain

In 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.

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 converts the time function into a sum of sine waves of different frequencies, 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 a time function. A spectrum analyzer is the tool commonly used to visualize real-world 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.

Magnitude and phase

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 magnitude of the number is the amplitude of that component, and the angle 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 frequency can be seen on an oscilloscope.

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.

Different frequency domains

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 – repetitive signals, oscillating systems
• Fourier transform – nonrepetitive signals, transients
• Laplace transform – electronic circuits and control systems
• Z transform – discrete 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.

Discrete frequency domain

The Fourier transform of a periodic signal only has energy 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 both periodic and discrete. 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.^[2] See time domain: origin of term for details.^[3]

See also

• Bandwidth
• Time domain
• Short-time Fourier transform
• Time–frequency representation
• Time–frequency analysis
• Wavelet
• Wavelet transform – digital image processing, signal compression

References

1. Broughton, S.A.; Bryan, K. (2008). Discrete Fourier Analysis and Wavelets: Applications to Signal and Image Processing. New York: Wiley. p. 72.
2. Zadeh, L. A. (1953), "Theory of Filtering", Journal of the Society for Industrial and Applied Mathematics 1: 35–51, doi:10.1137/0101003
3. Earliest Known Uses of Some of the Words of Mathematics (T), Jeff Miller, March 25, 2009

Further reading

• Boashash, B. (Sep 1988). "Note on the Use of the Wigner Distribution for Time Frequency Signal Analysis". 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..