Spectral density

The statistical average of any sort of signal (including noise) as analyzed in terms of its frequency content, is called its spectrum.

For instance, the pitch and timbre of a musical instrument are immediately determined from a spectral analysis.

The color of a light source is determined by the spectrum of the electromagnetic wave's electric field

In many cases the time domain is not specifically employed in practice, such as when a dispersive prism is used to obtain a spectrum of light in a spectrograph, or when a sound is perceived through its effect on the auditory receptors of the inner ear, each of which is sensitive to a particular frequency.

However this article concentrates on situations in which the time series is known (at least in a statistical sense) or directly measured (such as by a microphone sampled by a computer).

Power spectral density is commonly expressed in SI units of watts per hertz (abbreviated as W/Hz).

[2] When a signal is defined in terms only of a voltage, for instance, there is no unique power associated with the stated amplitude.

In the analysis of random vibrations, units of g2 Hz−1 are frequently used for the PSD of acceleration, where g denotes the g-force.

In the following discussion the meaning of x(t) will remain unspecified, but the independent variable will be assumed to be that of time.

represents the potential (in volts) of an electrical pulse propagating along a transmission line of impedance

, one could insert between the transmission line and the resistor a bandpass filter which passes only a narrow range of frequencies (

This definition generalizes in a straightforward manner to a discrete signal with a countably infinite number of values

is needed to keep the correct physical units and to ensure that we recover the continuous case in the limit

But in the mathematical sciences the interval is often set to 1, which simplifies the results at the expense of generality.

For continuous signals over all time, one must rather define the power spectral density (PSD) which exists for stationary processes; this describes how the power of a signal or time series is distributed over frequency, as in the simple example given previously.

, the power spectral density can be found once more as the Fourier transform of the autocorrelation function (Wiener–Khinchin theorem).

Just using a single such time series, the estimated power spectrum will be very "noisy"; however this can be alleviated if it is possible to evaluate the expected value (in the above equation) using a large (or infinite) number of short-term spectra corresponding to statistical ensembles of realizations of

Note that a single estimate of the PSD can be obtained through a finite number of samplings.

This periodogram converges to the true PSD as the number of estimates as well as the averaging time interval

Using the same notation and methods as used for the power spectral density derivation, we exploit Parseval's theorem and obtain

, so the full contribution to the cross power is, generally, from twice the real part of either individual CPSD.

Just as before, from here we recast these products as the Fourier transform of a time convolution, which when divided by the period and taken to the limit

Therefore, in typical signal processing, the full CPSD is just one of the CPSDs scaled by a factor of two.

Depending on what is known about the signal, estimation techniques can involve parametric or non-parametric approaches, and may be based on time-domain or frequency-domain analysis.

For example, a common parametric technique involves fitting the observations to an autoregressive model.

In some cases the frequency spectrum may include a distinct peak corresponding to a sine wave component.

The concept and use of the power spectrum of a signal is fundamental in electrical engineering, especially in electronic communication systems, including radio communications, radars, and related systems, plus passive remote sensing technology.

Electronic instruments called spectrum analyzers are used to observe and measure the power spectra of signals.

The spectrum analyzer measures the magnitude of the short-time Fourier transform (STFT) of an input signal.

If the signal being analyzed can be considered a stationary process, the STFT is a good smoothed estimate of its power spectral density.