Spectral density

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In applied mathematics and physics, the spectral density, power spectral density, or energy spectral density is a general concept applied to a signal which may have physical dimensions such as power per Hz, or energy per Hz, or none at all.

1 Explanation
2 Definition
- 2.1 Energy spectral density
- 2.2 Power spectral density
3 Properties
4 Related concepts
5 Applications
- 5.1 Electronics engineering
- 5.2 Colorimetry
6 See also
7 External links

[edit] Explanation

In physics, the signal is usually a wave, such as an electromagnetic wave, random vibration, 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 power spectral density are commonly expressed in watts per hertz (W/Hz) or watts per nanometer (W/nm) (for a measurement versus wavelength instead 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.

[edit] Definition

[edit] Energy spectral density

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 finite-energy (square integrable) signal, the spectral density $Φ(ω)$ of the signal is the square of the magnitude of the continuous Fourier transform of the signal.

$\Phi(\omega)=\left|\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(t)e^{-i\omega t}\,dt\right|^2 = \frac{F(\omega)F^*(\omega)}{2\pi}$

where $ω$ is the angular frequency ( $2π$ times the cycle frequency) and $F (ω)$ is the continuous Fourier transform of $f (t)$ .

If the signal is discrete with components $f n$ , over an infinite number of elements, we still have an energy spectral density:

$\Phi(\omega)=\left|\frac{1}{\sqrt{2\pi}}\sum_{n=-\infty}^\infty f_n e^{-i\omega n}\right|^2=\frac{F(\omega)F^*(\omega)}{2\pi}$

where $F (ω)$ is the discrete-time Fourier transform of $f n$ . If the number of defined values is finite, the sequence can be treated as periodic, using a discrete Fourier transform to make a discrete spectrum, or it can be extended with zeros and a spectral density can be computed as in the infinite-sequence case.

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.

[edit] Power spectral density

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.

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 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 autocorrelation function must be a function of two variables, so no PSD exists, but similar techniques may be used to estimate a time-varying spectral density.

[edit] Properties

The spectral density of $f (t)$ and the autocorrelation of $f (t)$ form a Fourier transform pair (for PSD versus ESD, different definitions of autocorrelation function are used).

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:

$\int_{-\infty}^\infty \left| f(t) \right|^2\, dt = \int_{-\infty}^\infty \Phi(\omega)\, d\omega.$

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.

[edit] 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.

[edit] Applications

[edit] 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 has 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.

The spectrum analyzer measures essentially the magnitude of the short-time Fourier transform (STFT) of an input signal. If the signal being analyzed is stationary, the STFT is a good smoothed estimate of its power spectral density.

[edit] Colorimetry

Main article: Colorimetry

The spectrum 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.