In applied mathematics and physics, the spectral density is a general concept applied to a signal which may have any physical dimensions or none at all. Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains. ...   Physics (from the Greek, φυσικός (physikos), natural, and φύσις (physis), nature) is the science of the natural world, dealing with the fundamental constituents of the universe, the forces they exert on one another, and the results produced by these forces. ...

Explanation

In physics, the signal is usually a wave, such as an electromagnetic wave, or an acoustic wave. The spectral density of the wave, when multiplied by an appropriate factor, will give the power carried by the wave, per unit frequency. This is then known as the power spectral density (PSD) or spectral power distribution (SPD) of the signal. The units of spectral power density are commonly expressed in watts per hertz (W/Hz) or watts per nanometer (W/nm) (for a measurement versus wavelength instead of frequency).   Physics (from the Greek, φυσικός (physikos), natural, and φύσις (physis), nature) is the science of the natural world, dealing with the fundamental constituents of the universe, the forces they exert on one another, and the results produced by these forces. ... Electromagnetic radiation is a propagating wave in space with electric and magnetic components. ... This article is about compression waves. ... In physics, power (symbol: P) is the amount of work done per unit of time. ... The watt (symbol: W) is the SI derived unit of power. ... The hertz (symbol: Hz) is the SI unit of frequency. ...

Although it is not necessary to assign physical dimensions to the signal or its argument, in the following discussion the terms used will assume that the signal varies in time.

Definition

The energy spectral density describes how the energy (or variance) of a signal or a time series is distributed with frequency. If f(t) is a signal, the spectral density Φ(ω) of the signal is the square of the magnitude of the continuous Fourier transform of the signal. In statistics and signal processing, a time series is a sequence of data points, measured typically at successive times, spaced apart at uniform time intervals. ... In mathematics, the continuous Fourier transform is a certain linear operator that maps functions to other functions. ...

where ω is the angular frequency (2π times the cyclic frequency) and F(ω) is the continuous Fourier transform of f(t). If the signal is discrete with components f_n, we may approximate f(t) by:

where δ(x) is the Dirac delta function and the sum over n may be over a finite or infinite number of elements. If the number is infinite we have: The Dirac delta function, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere. ...

where F(ω) is the discrete-time Fourier transform of f_n. If the number is finite (=N) we may define ω = 2πm / N and: A discrete-time Fourier transform (or DTFT) is a Fourier transform of a function of an integer (discrete) time variable n with an unbounded domain. ...

where F_m is the discrete Fourier transform of f_n. As is always the case, the multiplicative factor of 1 / 2π is not absolute, but rather depends on the particular normalizing constants used in the definition of the various Fourier transforms. In mathematics, the discrete Fourier transform (DFT), sometimes called the finite Fourier transform, is a Fourier transform widely employed in signal processing and related fields to analyze the frequencies contained in a sampled signal, solve partial differential equations, and to perform other operations such as convolutions. ...

The above definitions of energy spectral density require that the Fourier transforms of the signals exist, that is, that the signals are square-integrable or square-summable. An often more useful alternative is the power spectral density (PSD), which describes how the power of a signal or time series is distributed with frequency. In physics, power (symbol: P) is the amount of work done per unit of time. ...

Since a signal with nonzero average power is not square integrable, the Fourier transforms do not exist in this case. Fortunately, the Wiener-Khinchin theorem provides a simple alternative. The PSD is the Fourier transform of the autocorrelation function of the signal if the signal can be treated as a stationary random process. The Wiener-Kinchen theorem states that the power spectral density of X(t) is the Fourier transorm of <Rx(t)>. ... Autocorrelation is a mathematical tool used frequently in signal processing for analysing functions or series of values, such as time domain signals. ...

The power spectral density of a signal exists if and only if the signal is a wide-sense stationary process. If the signal is not stationary then the same methods used to calculate the spectral density can still be used, but the result cannot be called the spectral density. In the mathematical sciences, a stationary process (or strict(ly) stationary process) is a stochastic process in which the probability density function of some random variable X does not change over time or position. ...

Properties

• The spectral density of f(t) and the autocorrelation of f(t) form a Fourier transform pair.

• The spectral density is usually estimated using Fourier transform techniques, but other techniques such as Welch's method and the maximum entropy method can also be used.

• One of the results of Fourier analysis is Parseval's theorem which states that the area under the energy spectral density curve is equal to the area under the square of the magnitude of the signal, the total energy:

The above theorem holds true in the discrete cases as well. A similar result holds for the total power in a power spectral density being equal to the corresponding mean total signal power, which is the autocorrelation function at zero lag.

Autocorrelation is a mathematical tool used frequently in signal processing for analysing functions or series of values, such as time domain signals. ... In physics and applied mathematics, Welchs method, named after P.D. Welch, is used for estimating power spectra. ... The principle of maximum entropy is a method for analyzing the available information in order to determine a unique epistemic probability distribution. ... In mathematics, Parsevals theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. ...

Related concepts

• Most "frequency" graphs really display only the spectral density. Sometimes the complete frequency spectrum is graphed in 2 parts, "amplitude" versus frequency (which is the spectral density) and "phase" versus frequency (which contains the rest of the information from the frequency spectrum). The signal f(t) can be recovered from complete frequency spectrum. Note that the signal f(t) cannot be recovered from the spectral density part alone -- the "temporal information" is lost.

• The spectral centroid of a signal is the midpoint of its spectral density function, i.e. the frequency that divides the distribution into two equal parts.

• Spectral density is a function of frequency, not a function of time. However, the spectral density of small "windows" of a longer signal may be calculated, and plotted versus time associated with the window. Such a graph is called a spectrogram. This is the basis of a number of spectral analysis techniques such as the short-time Fourier transform and wavelets.

Waves with the same phase Waves with different phases The phase of a wave relates the position of a feature, typically a peak or a trough of the waveform, to that same feature in another part of the waveform (or, which amounts to the same, on a second waveform). ... Familiar concepts associated with a frequency are colors, musical notes, radio/TV channels, and even the regular rotation of the earth. ... A spectrogram of violin playing with linear frequency on the vertical axis and time on the horizontal axis. ... The short-time Fourier transform (STFT), or short-term Fourier transform, is a Fourier-related transform used to determine the sinusoidal frequency and phase content of a signal as it changes over time. ... All wavelet transforms consider a function (taken to be a function of time) in terms of oscillations which are localised in both time and frequency. ...

Applications

Electronics engineering

The concept and use of the power spectrum of a signal is fundamental in electronic engineering, especially in electronic communication systems (e.g. radio & microwave communications, radars, and related systems). Much effort had been made and millions of dollars spent on developing and producing electronic instruments called "Spectrum Analyzers" for aiding electronics engineers, technologists, and technicians in observing and measuring the power spectrum of electronic signals. The cost of a spectrum analyzer varies according to its bandwidth and its accuracy. The top quality instruments cost over $100,000.

Colorimetry

Main article: Colorimetry

The power spectral density of a light source is a measure of the power carried by each frequency or "color" in a light source. The light spectrum is usually measured at points (often 31) along the visible spectrum, in wavelength space instead of frequency space, which makes it not strictly a spectral density. Some spectrophotometers can measure increments as fine as 1 or 2 nanometers. Values are used to calculate other specifications and then plotted to demonstrate the spectral attributes of the source. This can be a helpful tool in analyzing the color characteristics of a particular source. incandescent and fluorescent Spectral Power Distributions File links The following pages link to this file: Color temperature Spectral density Spectrophotometry Spectroradiometer Categories: GFDL images ... Colorimetry is the science that describes colors in numbers, or provides a physical color match using a variety of measurement instruments. ... Prism splitting light Light is electromagnetic radiation with a wavelength that is visible to the eye, or in a more general sense, any electromagnetic radiation in the range from infrared to ultraviolet. ... The visible spectrum (or sometimes optical spectrum) is the portion of the electromagnetic spectrum that is visible to the human eye. ... A Spectrophotometer In physics, spectrophotometry is the quantitative study of electromagnetic spectra. ... A nanometre (American spelling: nanometer) is 1. ... Color is an important part of the visual arts. ...

See also

• colours of noise
• Spectral leakage
• Window function
• Frequency domain
• Frequency spectrum
• Bispectrum

Many of these definitions assume a signal with components at all frequencies, with a spectral density per unit of bandwidth proportional to 1/fβ. For instance, white noise is flat, with β = 0, while brown has β = 2. ... When taking the discrete Fourier transform (DFT) of a signal (e. ... In signal processing, a window function (or apodization function) is a function that is zero-valued outside of some chosen interval. ... Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ... Familiar concepts associated with a frequency are colors, musical notes, radio/TV channels, and even the regular rotation of the earth. ... The bispectrum is a statistic used to search for nonlinear interactions. ...

External links

• Power Spectral Density Function - from Wolfram Research's "Time Series Documentation"