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The power spectrum
Contents
- Explanation
- Energy spectral density
- Power spectral density
- Properties of the power spectral density
- Cross spectral density
- Estimation
- Properties
- Related concepts
- Electrical engineering
- Cosmology
- References
When the energy of the signal is concentrated around a finite time interval, especially if its total energy is finite, one may compute the energy spectral density. More commonly used is the power spectral density (or simply power spectrum), which applies to signals existing over all time, or over a time period large enough (especially in relation to the duration of a measurement) that it could as well have been over an infinite time interval. The power spectral density (PSD) then refers to the spectral energy distribution that would be found per unit time, since the total energy of such a signal over all time would generally be infinite. Summation or integration of the spectral components yields the total power (for a physical process) or variance (in a statistical process), identical to what would be obtained by integrating
The spectrum of a physical process
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). The power spectrum is important in statistical signal processing and in the statistical study of stochastic processes, as well as in many other branches of physics and engineering. Typically the process is a function of time but one can similarly discuss data in the spatial domain being decomposed in terms of spatial frequency.
Explanation
Any signal that can be represented as an amplitude that varies in time has a corresponding frequency spectrum. This includes familiar entities such as visible light (perceived as color), musical notes (perceived as pitch), radio/TV (specified by their frequency, or sometimes wavelength) and even the regular rotation of the earth. When these signals are viewed in the form of a frequency spectrum, certain aspects of the received signals or the underlying processes producing them are revealed. In some cases the frequency spectrum may include a distinct peak corresponding to a sine wave component. And additionally there may be peaks corresponding to harmonics of a fundamental peak, indicating a periodic signal which is not simply sinusoidal. Or a continuous spectrum may show narrow frequency intervals which are strongly enhanced corresponding to resonances, or frequency intervals containing almost zero power as would be produced by a notch filter.
In physics, the signal might be a wave, such as an electromagnetic wave, an acoustic wave, or the vibration of a mechanism. The power spectral density (PSD) of the signal describes the power present in the signal as a function of frequency, per unit frequency. Power spectral density is commonly expressed in watts per hertz (W/Hz).
When a signal is defined in terms only of a voltage, for instance, there is no unique power associated with the stated amplitude. In this case "power" is simply reckoned in terms of the square of the signal, as this would always be proportional to the actual power delivered by that signal into a given impedance. So one might use units of V2 Hz−1 for the PSD and V2 s Hz−1 for the ESD (energy spectral density) even though no actual "power" or "energy" is specified.
Sometimes one encounters an amplitude spectral density (ASD), which is the square root of the PSD; the ASD of a voltage signal has units of V Hz−1/2. This is useful when the shape of the spectrum is rather constant, since variations in the ASD will then be proportional to variations in the signal's voltage level itself. But it is mathematically preferred to use the PSD, since only in that case is the area under the curve meaningful in terms of actual power over all frequency or over a specified bandwidth.
For random vibration analysis, units of g2 Hz−1 are frequently used for the PSD of acceleration. Here g denotes the g-force.
Mathematically it is not necessary to assign physical dimensions to the signal or to the independent variable. In the following discussion the meaning of x(t) will remain unspecified, but the independent variable will be assumed to be that of time.
Energy spectral density
Energy spectral density describes how the energy of a signal or a time series is distributed with frequency. Here, the term energy is used in the generalized sense of signal processing; that is, the energy
The energy spectral density is most suitable for transients—that is, pulse-like signals—having a finite total energy. In this case, Parseval's theorem gives us an alternate expression for the energy of the signal:
where
is the Fourier transform transform of the signal and
As a physical example of how one might measure the energy spectral density of a signal, suppose
This definition generalizes in a straightforward manner to a discrete signal with an infinite number of values
where
Power spectral density
The above definition of energy spectral density is suitable for transients (pulse-like signals) whose energy is concentrated around one time window; then the Fourier transforms of the signals generally exist. For continuous signals over all time, such as stationary processes, one must rather define the power spectral density (PSD); this describes how power of a signal or time series is distributed over frequency, as in the simple example given previously. Here, power can be the actual physical power, or more often, for convenience with abstract signals, is simply identified with the squared value of the signal. For example, statisticians study the variance of a function over time
The average power P of a signal
Note that a stationary process, for instance, may have a finite power but an infinite energy. After all, energy is the integral of power, and the stationary signal continues over an infinite time. That is the reason that we cannot use the energy spectral density as defined above in such cases.
In analyzing the frequency content of the signal
Then the power spectral density can be defined as
Here E denotes the expected value; explicitly, we have
In the latter form (for a stationary random process), one can make the change of variables
or more generally as
in the case that X(t) is complex-valued. Provided that
Many authors use this equality to actually define the power spectral density.
The power of the signal in a given frequency band
More generally, similar techniques may be used to estimate a time-varying spectral density. In this case the truncated Fourier transform defined above over the finite time interval (0, T) is not evaluated in the limit of T approaching infinity. This results in decreased spectral coverage and resolution since frequencies of less than 1/T are not sampled, and results at frequencies which are not an integer multiple of 1/T are not independent. 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 x(t) evaluated over the specified time window.
This definition of the power spectral density can be generalized to discrete time variables
As before, the actual PSD is achieved when N (and thus T) approach infinity and the expected value is formally applied. In a real-world application, one would typically average this single-measurement PSD over many trials to obtain a more accurate estimate of the theoretical PSD of the physical process underlying the individual measurements. This computed PSD is sometimes called a periodogram. This periodogram converges to the true PSD as the number of estimates as well as the averaging time interval T approach infinity (Brown & Hwang).
If two signals both possess power spectral densities, then the #Cross-spectral density can similarly be calculated; as the PSD is related to the autocorrelation, so is the cross-spectral density related to the cross-correlation.
Properties of the power spectral density
Some properties of the PSD include:
The integrated spectrum or power spectral distribution
Cross-spectral density
Given two signals
The cross-spectral density (or 'cross power spectrum') is thus the Fourier transform of the cross-correlation function.
where
By an extension of the Wiener–Khinchin theorem, the Fourier transform of the cross-spectral density
For discrete signals xn and yn, the relationship between the cross-spectral density and the cross-covariance is
Estimation
The goal of spectral density estimation is to estimate the spectral density of a random signal from a sequence of time samples. 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. A common non-parametric technique is the periodogram.
The spectral density is usually estimated using Fourier transform methods (such as the Welch method), but other techniques such as the maximum entropy method can also be used.
Properties
Related concepts
Electrical engineering
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.
Cosmology
Primordial fluctuations, density variations in the early universe, are quantified by a power spectrum which gives the power of the variations as a function of spatial scale.