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 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 bandwidth, frequency response, gain, phase shift, resonant frequencies, time constant, resonance width, damping factor, Q factor, harmonics, spectrum, power spectral density, eigenvalues, poles, and 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 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 modulus of the number is the amplitude of that component, and the argument 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 signal can be seen on an oscilloscope. ==Types==

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 – periodic signals, oscillating systems. • Fourier transform – aperiodic signals, transients. • Laplace transform – electronic circuits and control systems. • Z transform – discrete-time signals, digital signal processing. • Wavelet transform — image analysis, data compression. More generally, one can speak of the '''''' 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 Fourier transform maps a function having a discrete time domain into one having a discrete frequency domain. The discrete-time Fourier transform, on the other hand, maps functions with discrete time (discrete-time signals) to functions that have a continuous frequency domain. A periodic signal has energy only at a base frequency and its harmonics; thus it can be analyzed using a discrete frequency domain. 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. See time domain: origin of term for details. ==See also==