In signal processing, the power spectrum S x x ( f ) {\displaystyle S_{xx}(f)} of a continuous time signal x ( t ) {\displaystyle x(t)} describes the distribution of power into frequency components f {\displaystyle f} composing that signal. Fourier analysis shows that any physical signal can be decomposed into a distribution of frequencies over a continuous range, where some of the power may be concentrated at discrete frequencies. The statistical average of the energy or power of any type of signal (including noise) as analyzed in terms of its frequency content, is called its spectral density. 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 (PSD, 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 PSD then refers to the spectral power distribution that would be found, 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 x 2 ( t ) {\displaystyle x^{2}(t)} over the time domain, as dictated by Parseval's theorem. The spectrum of a physical process x ( t ) {\displaystyle x(t)} often contains essential information about the nature of x {\displaystyle x} . For instance, the pitch and timbre of a musical instrument can be determined from a spectral analysis. The color of a light source is determined by the spectrum of the electromagnetic wave's electric field E ( t ) {\displaystyle E(t)} as it oscillates at an extremely high frequency. Obtaining a spectrum from time series data such as these involves the Fourier transform, and generalizations based on Fourier analysis. In many cases the time domain is not directly captured 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). 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.
Units
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 density of the signal as a function of frequency. Power spectral density is commonly expressed in the SI unit watt per hertz (W/Hz). When a signal is defined in terms of only a voltage varying in time, for instance, there is no specific power associated with a given voltage. 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 the unit V2⋅Hz−1 for the PSD. Energy spectral density (ESD) would have the unit V2⋅s⋅Hz−1, since energy is power multiplied by time (e.g., watt-hour). In the general case, the unit of PSD will be the ratio of unit of variance per unit of frequency; so, for example, a series of displacement values (in meters) over time (in seconds) will have PSD with the unit m2/Hz. In the analysis of random vibrations, the unit g02⋅Hz−1 may be used for the PSD of acceleration, where g0 denotes standard gravity. 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.
One-sided vs. two-sided A PSD can be either a one-sided function of only positive frequencies or a two-sided function of both positive and negative frequencies but with only half the amplitude. Noise PSDs are generally one-sided in engineering and two-sided in physics.
Definition
Energy spectral density
In signal processing, the energy of a signal x ( t ) {\displaystyle x(t)} is given by
E ≜ ∫ − ∞ ∞ | x ( t ) | 2 d t . {\displaystyle E\triangleq \int _{-\infty }^{\infty }\left|x(t)\right|^{2}\ dt.}
Assuming the total energy is finite (i.e. x ( t ) {\displaystyle x(t)} is a square-integrable function) allows applying Parseval's theorem (or Plancherel's theorem). That is,
∫ − ∞ ∞ | x ( t ) | 2 d t = ∫ − ∞ ∞ | x ^ ( f ) | 2 d f , {\displaystyle \int _{-\infty }^{\infty }|x(t)|^{2}\,dt=\int _{-\infty }^{\infty }\left|{\hat {x}}(f)\right|^{2}\,df,}
where
x ^ ( f ) = ∫ − ∞ ∞ e − i 2 π f t x ( t ) d t , {\displaystyle {\hat {x}}(f)=\int _{-\infty }^{\infty }e^{-i2\pi ft}x(t)\ dt,}
is the Fourier transform of x ( t ) {\displaystyle x(t)} at frequency f {\displaystyle f} (in Hz). The theorem also holds true in the discrete-time cases. Since the integral on the left-hand side is the energy of the signal, the value of | x ^ ( f ) | 2 d f {\displaystyle \left|{\hat {x}}(f)\right|^{2}df} can be interpreted as a density function multiplied by an infinitesimally small frequency interval, describing the energy contained in the signal at frequency f {\displaystyle f} in the frequency interval f + d f {\displaystyle f+df} . Therefore, the energy spectral density of x ( t ) {\displaystyle x(t)} is defined as
The function S ¯ x x ( f ) {\displaystyle {\bar {S}}_{xx}(f)} and the autocorrelation of x ( t ) {\displaystyle x(t)} form a Fourier transform pair, a result also known as the Wiener–Khinchin theorem (see also Periodogram). As a physical example of how one might measure the energy spectral density of a signal, suppose V ( t ) {\displaystyle V(t)} represents the potential (in volts) of an electrical pulse propagating along a transmission line of impedance Z {\displaystyle Z} , and suppose the line is terminated with a matched resistor (so that all of the pulse energy is delivered to the resistor and none is reflected back). By Ohm's law, the power delivered to the resistor at time t {\displaystyle t} is equal to V ( t ) 2 / Z {\displaystyle V(t)^{2}/Z} , so the total energy is found by integrating V ( t ) 2 / Z {\displaystyle V(t)^{2}/Z} with respect to time over the duration of the pulse. To find the value of the energy spectral density S ¯ x x ( f ) {\displaystyle {\bar {S}}_{xx}(f)} at frequency f {\displaystyle f} , one could insert between the transmission line and the resistor a bandpass filter which passes only a narrow range of frequencies ( Δ f {\displaystyle \Delta f} , say) near the frequency of interest and then measure the total energy E ( f ) {\displaystyle E(f)} dissipated across the resistor. The value of the energy spectral density at f {\displaystyle f} is then estimated to be E ( f ) / Δ f {\displaystyle E(f)/\Delta f} . In this example, since the power V ( t ) 2 / Z {\displaystyle V(t)^{2}/Z} has the unit V2⋅Ω−1, the energy E ( f ) {\displaystyle E(f)} has the unit V2⋅s⋅Ω−1 = J, and hence the estimate E ( f ) / Δ f {\displaystyle E(f)/\Delta f} of the energy spectral density has the unit J⋅Hz−1. In many situations, it is common to omit the step of dividing by Z {\displaystyle Z} so that the energy spectral density instead has the unit V2⋅s·Hz−1. This definition generalizes in a straightforward manner to a discrete signal with a countably infinite number of values x n {\displaystyle x_{n}} such as a signal sampled at discrete times t n = t 0 + ( n Δ t ) {\displaystyle t_{n}=t_{0}+(n\,\Delta t)} :
S ¯ x x ( f ) = lim N → ∞ ( Δ t ) 2 | ∑ n = − N N x n e − i 2 π f n Δ t | 2 ⏟ | x ^ d ( f ) | 2 , {\displaystyle {\bar {S}}_{xx}(f)=\lim _{N\to \infty }(\Delta t)^{2}\underbrace {\left|\sum _{n=-N}^{N}x_{n}e^{-i2\pi fn\,\Delta t}\right|^{2}} _{\left|{\hat {x}}_{d}(f)\right|^{2}},}
where x ^ d ( f ) {\displaystyle {\hat {x}}_{d}(f)} is the discrete-time Fourier transform of x n . {\displaystyle x_{n}.} The sampling interval Δ t {\displaystyle \Delta t} is needed to keep the correct physical unit and to ensure that we recover the continuous case in the limit Δ t → 0 {\displaystyle \Delta t\to 0} . But in the mathematical sciences the interval is often set to 1, which simplifies the results at the expense of generality. (Also see Normalized frequency (unit))
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, 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. 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 x ( t ) {\displaystyle x(t)} (or over another independent variable), and using an analogy with electrical signals (among other physical processes), it is customary to refer to it as the power spectrum even when there is no physical power involved. If one were to create a physical voltage source which followed x ( t ) {\displaystyle x(t)} and applied it to the terminals of a one ohm resistor, then indeed the instantaneous power dissipated in that resistor would be given by x 2 ( t ) {\displaystyle x^{2}(t)} watts. The average power P {\displaystyle P} of a signal x ( t ) {\displaystyle x(t)} over all time is therefore given by the following time average, where the period T {\displaystyle T} is centered about some arbitrary time t = t 0 {\displaystyle t=t_{0}} :
P = lim T → ∞ 1 T ∫ t 0 − T / 2 t 0 + T / 2 | x ( t ) | 2 d t {\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{t_{0}-T/2}^{t_{0}+T/2}\left|x(t)\right|^{2}\,dt}
Whenever it is more convenient to deal with time limits in the signal itself rather than time limits in the bounds of the integral, the average power can also be written as
P = lim T → ∞ 1 T ∫ − ∞ ∞ | x T ( t ) | 2 d t , {\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }\left|x_{T}(t)\right|^{2}\,dt,}
where x T ( t ) = x ( t ) w T ( t ) {\displaystyle x_{T}(t)=x(t)w_{T}(t)} and w T ( t ) {\displaystyle w_{T}(t)} is unity within the arbitrary period and zero elsewhere. When P {\displaystyle P} is non-zero, the integral must grow to infinity at least as fast as T {\displaystyle T} does. That is the reason why we cannot use the energy of the signal, which is that diverging integral. In analyzing the frequency content of the signal x ( t ) {\displaystyle x(t)} , one might like to compute the ordinary Fourier transform x ^ ( f ) {\displaystyle {\hat {x}}(f)} ; however, for many signals of interest the ordinary Fourier transform does not formally exist. However, under suitable conditions, certain generalizations of the Fourier transform (e.g. the Fourier–Stieltjes transform) still adhere to Parseval's theorem. As such,
P = lim T → ∞ 1 T ∫ − ∞ ∞ | x ^ T ( f ) | 2 d f , {\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }|{\hat {x}}_{T}(f)|^{2}\,df,}
where the integrand defines the power spectral density:
The convolution theorem then allows regarding | x ^ T ( f ) | 2 {\displaystyle |{\hat {x}}_{T}(f)|^{2}} as the Fourier transform of the time convolution of x T ∗ ( − t ) {\displaystyle x_{T}^{*}(-t)} and x T ( t ) {\displaystyle x_{T}(t)} , where * represents the complex conjugate. In order to prove the claim below Eq.2, we will find an expression for [ x ^ T ( f ) ] ∗ {\displaystyle [{\hat {x}}_{T}(f)]^{*}} that will be useful for the purpose. In fact, we will demonstrate that [ x ^ T ( f ) ] ∗ = F { x T ∗ ( − t ) } {\displaystyle [{\hat {x}}_{T}(f)]^{*}={\mathcal {F}}\left\{x_{T}^{*}(-t)\right\}} . Start by noting that
F { x T ∗ ( − t ) } = ∫ − ∞ ∞ x T ∗ ( − t ) e − i 2 π f t d t {\displaystyle {\begin{aligned}{\mathcal {F}}\left\{x_{T}^{*}(-t)\right\}&=\int _{-\infty }^{\infty }x_{T}^{*}(-t)e^{-i2\pi ft}dt\end{aligned}}}
and let z = − t {\displaystyle z=-t} , so that z → − ∞ {\displaystyle z\rightarrow -\infty } when t → ∞ {\displaystyle t\rightarrow \infty } and vice versa. So
∫ − ∞ ∞ x T ∗ ( − t ) e − i 2 π f t d t = ∫ ∞ − ∞ x T ∗ ( z ) e i 2 π f z ( − d z ) = ∫ − ∞ ∞ x T ∗ ( z ) e i 2 π f z d z = ∫ − ∞ ∞ x T ∗ ( t ) e i 2 π f t d t {\displaystyle {\begin{aligned}\int _{-\infty }^{\infty }x_{T}^{*}(-t)e^{-i2\pi ft}dt&=\int _{\infty }^{-\infty }x_{T}^{*}(z)e^{i2\pi fz}\left(-dz\right)\\&=\int _{-\infty }^{\infty }x_{T}^{*}(z)e^{i2\pi fz}dz\\&=\int _{-\infty }^{\infty }x_{T}^{*}(t)e^{i2\pi ft}dt\end{aligned}}}
where, in the last line, use has been made of z {\displaystyle z} and t {\displaystyle t} being dummy variables. So, we have
F { x T ∗ ( − t ) } = ∫ − ∞ ∞ x T ∗ ( − t ) e − i 2 π f t d t = ∫ − ∞ ∞ x T ∗ ( t ) e i 2 π f t d t = ∫ − ∞ ∞ x T ∗ ( t ) [ e − i 2 π f t ] ∗ d t = [ ∫ − ∞ ∞ x T ( t ) e − i 2 π f t d t ] ∗ = [ F { x T ( t ) } ] ∗ = [ x ^ T ( f ) ] ∗ {\displaystyle {\begin{aligned}{\mathcal {F}}\left\{x_{T}^{*}(-t)\right\}&=\int _{-\infty }^{\infty }x_{T}^{*}(-t)e^{-i2\pi ft}dt\\&=\int _{-\infty }^{\infty }x_{T}^{*}(t)e^{i2\pi ft}dt\\&=\int _{-\infty }^{\infty }x_{T}^{*}(t)[e^{-i2\pi ft}]^{*}dt\\&=\left[\int _{-\infty }^{\infty }x_{T}(t)e^{-i2\pi ft}dt\right]^{*}\\&=\left[{\mathcal {F}}\left\{x_{T}(t)\right\}\right]^{*}\\&=\left[{\hat {x}}_{T}(f)\right]^{*}\end{aligned}}}
q.e.d. Now, let's demonstrate the claim below eq.2 by using the demonstrated identity. In addition, we will make the substitution u ( t ) = x T ∗ ( − t ) {\displaystyle u(t)=x_{T}^{*}(-t)} . In this way, we have:
| x ^ T ( f ) | 2 = [ x ^ T ( f ) ] ∗ ⋅ x ^ T ( f ) = F { x T ∗ ( − t ) } ⋅ F { x T ( t ) } = F { u ( t ) } ⋅