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.[5]

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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.[1]

Power densityvs energydensity

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;[6] that is, the energy E {\displaystyle E} of a signal x ( t ) {\displaystyle x(t)} is: E ≜ ∫ − ∞ ∞ | x ( t ) | 2   d t . {\displaystyle E\triangleq \int _{-\infty }^{\infty }\left|x(t)\right|^{2}\ dt.}

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}

Power densityRF

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 Fourier transform does not formally exist.[nb 1] Regardless, Parseval's theorem tells us that we can re-write the average power as follows. 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}

Using the same notation and methods as used for the power spectral density derivation, we exploit Parseval's theorem and obtain S x y ( f ) = lim T → ∞ 1 T [ x ^ T ∗ ( f ) y ^ T ( f ) ] S y x ( f ) = lim T → ∞ 1 T [ y ^ T ∗ ( f ) x ^ T ( f ) ] {\displaystyle {\begin{aligned}S_{xy}(f)&=\lim _{T\to \infty }{\frac {1}{T}}\left[{\hat {x}}_{T}^{*}(f){\hat {y}}_{T}(f)\right]&S_{yx}(f)&=\lim _{T\to \infty }{\frac {1}{T}}\left[{\hat {y}}_{T}^{*}(f){\hat {x}}_{T}(f)\right]\end{aligned}}} where, again, the contributions of S x x ( f ) {\displaystyle S_{xx}(f)} and S y y ( f ) {\displaystyle S_{yy}(f)} are already understood. Note that S x y ∗ ( f ) = S y x ( f ) {\displaystyle S_{xy}^{*}(f)=S_{yx}(f)} , 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 T → ∞ {\displaystyle T\to \infty } becomes the Fourier transform of a cross-correlation function.[16] S x y ( f ) = ∫ − ∞ ∞ [ lim T → ∞ 1 T ∫ − ∞ ∞ x T ∗ ( t − τ ) y T ( t ) d t ] e − i 2 π f τ d τ = ∫ − ∞ ∞ R x y ( τ ) e − i 2 π f τ d τ S y x ( f ) = ∫ − ∞ ∞ [ lim T → ∞ 1 T ∫ − ∞ ∞ y T ∗ ( t − τ ) x T ( t ) d t ] e − i 2 π f τ d τ = ∫ − ∞ ∞ R y x ( τ ) e − i 2 π f τ d τ , {\displaystyle {\begin{aligned}S_{xy}(f)&=\int _{-\infty }^{\infty }\left[\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }x_{T}^{*}(t-\tau )y_{T}(t)dt\right]e^{-i2\pi f\tau }d\tau =\int _{-\infty }^{\infty }R_{xy}(\tau )e^{-i2\pi f\tau }d\tau \\S_{yx}(f)&=\int _{-\infty }^{\infty }\left[\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }y_{T}^{*}(t-\tau )x_{T}(t)dt\right]e^{-i2\pi f\tau }d\tau =\int _{-\infty }^{\infty }R_{yx}(\tau )e^{-i2\pi f\tau }d\tau ,\end{aligned}}} where R x y ( τ ) {\displaystyle R_{xy}(\tau )} is the cross-correlation of x ( t ) {\displaystyle x(t)} with y ( t ) {\displaystyle y(t)} and R y x ( τ ) {\displaystyle R_{yx}(\tau )} is the cross-correlation of y ( t ) {\displaystyle y(t)} with x ( t ) {\displaystyle x(t)} . In light of this, the PSD is seen to be a special case of the CSD for x ( t ) = y ( t ) {\displaystyle x(t)=y(t)} . If x ( t ) {\displaystyle x(t)} and y ( t ) {\displaystyle y(t)} are real signals (e.g. voltage or current), their Fourier transforms x ^ ( f ) {\displaystyle {\hat {x}}(f)} and y ^ ( f ) {\displaystyle {\hat {y}}(f)} are usually restricted to positive frequencies by convention. Therefore, in typical signal processing, the full CPSD is just one of the CPSDs scaled by a factor of two. CPSD Full = 2 S x y ( f ) = 2 S y x ( f ) {\displaystyle \operatorname {CPSD} _{\text{Full}}=2S_{xy}(f)=2S_{yx}(f)}

From here we see, again assuming the ergodicity of x ( t ) {\displaystyle x(t)} , that the power spectral density can be found as the Fourier transform of the autocorrelation function (Wiener–Khinchin theorem).[11]

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Power densityunit

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.

Power densityof battery

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).

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 units 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)

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.

For discrete signals xn and yn, the relationship between the cross-spectral density and the cross-covariance is S x y ( f ) = ∑ n = − ∞ ∞ R x y ( τ n ) e − i 2 π f τ n Δ τ {\displaystyle S_{xy}(f)=\sum _{n=-\infty }^{\infty }R_{xy}(\tau _{n})e^{-i2\pi f\tau _{n}}\,\Delta \tau }

From here, due to the convolution theorem, we can also view | 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. Taking into account that 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}}} and making, u ( t ) = x T ∗ ( − t ) {\displaystyle u(t)=x_{T}^{*}(-t)} , we have: | x ^ T ( f ) | 2 = [ x ^ T ( f ) ] ∗ ⋅ x ^ T ( f ) = F { x T ∗ ( − t ) } ⋅ F { x T ( t ) } = F { u ( t ) } ⋅ F { x T ( t ) } = F { u ( t ) ∗ x T ( t ) } = ∫ − ∞ ∞ [ ∫ − ∞ ∞ u ( τ − t ) x T ( t ) d t ] e − i 2 π f τ d τ = ∫ − ∞ ∞ [ ∫ − ∞ ∞ x T ∗ ( t − τ ) x T ( t ) d t ] e − i 2 π f τ   d τ , {\displaystyle {\begin{aligned}\left|{\hat {x}}_{T}(f)\right|^{2}&=[{\hat {x}}_{T}(f)]^{*}\cdot {\hat {x}}_{T}(f)\\&={\mathcal {F}}\left\{x_{T}^{*}(-t)\right\}\cdot {\mathcal {F}}\left\{x_{T}(t)\right\}\\&={\mathcal {F}}\left\{u(t)\right\}\cdot {\mathcal {F}}\left\{x_{T}(t)\right\}\\&={\mathcal {F}}\left\{u(t)\mathbin {\mathbf {*} } x_{T}(t)\right\}\\&=\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }u(\tau -t)x_{T}(t)dt\right]e^{-i2\pi f\tau }d\tau \\&=\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }x_{T}^{*}(t-\tau )x_{T}(t)dt\right]e^{-i2\pi f\tau }\ d\tau ,\end{aligned}}} where the convolution theorem has been used when passing from the 3rd to the 4th line.

The power of the signal in a given frequency band [ f 1 , f 2 ] {\displaystyle [f_{1},f_{2}]} , where 0 < f 1 < f 2 {\displaystyle 0

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.

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 SI units of watts per hertz (abbreviated as W/Hz).[2]

Power densitylaser

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.

Just as with the energy spectral density, the definition of the power spectral density can be generalized to discrete time variables x n {\displaystyle x_{n}} . As before, we can consider a window of − N ≤ n ≤ N {\displaystyle -N\leq n\leq N} with the signal sampled at discrete times t n = t 0 + ( n Δ t ) {\displaystyle t_{n}=t_{0}+(n\,\Delta t)} for a total measurement period T = ( 2 N + 1 ) Δ t {\displaystyle T=(2N+1)\,\Delta t} . S x x ( f ) = lim N → ∞ ( Δ t ) 2 T | ∑ n = − N N x n e − i 2 π f n Δ t | 2 {\displaystyle S_{xx}(f)=\lim _{N\to \infty }{\frac {(\Delta t)^{2}}{T}}\left|\sum _{n=-N}^{N}x_{n}e^{-i2\pi fn\,\Delta t}\right|^{2}} Note that a single estimate of the PSD can be obtained through a finite number of samplings. As before, the actual PSD is achieved when N {\displaystyle N} (and thus T {\displaystyle T} ) approaches infinity and the expected value is formally applied. In a real-world application, one would typically average a finite-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 {\displaystyle T} approach infinity.[13]

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.

The energy spectral density is most suitable for transients—that is, pulse-like signals—having a finite total energy. Finite or not, Parseval's theorem (or Plancherel's theorem) gives us an alternate expression for the energy of the signal:[7] ∫ − ∞ ∞ | 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)\triangleq \int _{-\infty }^{\infty }e^{-i2\pi ft}x(t)\ dt} is the value of 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} .

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.

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. Energy spectral density (ESD) would have units of V2 s Hz−1, since energy has units of power multiplied by time (e.g., watt-hour).[3]

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.[1] According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of any sort of signal (including noise) as analyzed in terms of its frequency content, is called its spectrum.

Now, if we divide the time convolution above by the period T {\displaystyle T} and take the limit as T → ∞ {\displaystyle T\rightarrow \infty } , it becomes the autocorrelation function of the non-windowed signal x ( t ) {\displaystyle x(t)} , which is denoted as R x x ( τ ) {\displaystyle R_{xx}(\tau )} , provided that x ( t ) {\displaystyle x(t)} is ergodic, which is true in most, but not all, practical cases.[nb 2] lim T → ∞ 1 T | x ^ T ( f ) | 2 = ∫ − ∞ ∞ [ lim T → ∞ 1 T ∫ − ∞ ∞ x T ∗ ( t − τ ) x T ( t ) d t ] e − i 2 π f τ   d τ = ∫ − ∞ ∞ R x x ( τ ) e − i 2 π f τ d τ {\displaystyle \lim _{T\to \infty }{\frac {1}{T}}\left|{\hat {x}}_{T}(f)\right|^{2}=\int _{-\infty }^{\infty }\left[\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }x_{T}^{*}(t-\tau )x_{T}(t)dt\right]e^{-i2\pi f\tau }\ d\tau =\int _{-\infty }^{\infty }R_{xx}(\tau )e^{-i2\pi f\tau }d\tau }

Power densityformula

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 units of V2 Ω−1, the energy E ( f ) {\displaystyle E(f)} has units of V2 s Ω−1 = J, and hence the estimate E ( f ) / Δ f {\displaystyle E(f)/\Delta f} of the energy spectral density has units of J Hz−1, as required. In many situations, it is common to forget the step of dividing by Z {\displaystyle Z} so that the energy spectral density instead has units of V2 Hz−1.

Any signal that can be represented as a variable 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.

Given two signals x ( t ) {\displaystyle x(t)} and y ( t ) {\displaystyle y(t)} , each of which possess power spectral densities S x x ( f ) {\displaystyle S_{xx}(f)} and S y y ( f ) {\displaystyle S_{yy}(f)} , it is possible to define a cross power spectral density (CPSD) or cross spectral density (CSD). To begin, let us consider the average power of such a combined signal. P = lim T → ∞ 1 T ∫ − ∞ ∞ [ x T ( t ) + y T ( t ) ] ∗ [ x T ( t ) + y T ( t ) ] d t = lim T → ∞ 1 T ∫ − ∞ ∞ | x T ( t ) | 2 + x T ∗ ( t ) y T ( t ) + y T ∗ ( t ) x T ( t ) + | y T ( t ) | 2 d t {\displaystyle {\begin{aligned}P&=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }\left[x_{T}(t)+y_{T}(t)\right]^{*}\left[x_{T}(t)+y_{T}(t)\right]dt\\&=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }|x_{T}(t)|^{2}+x_{T}^{*}(t)y_{T}(t)+y_{T}^{*}(t)x_{T}(t)+|y_{T}(t)|^{2}dt\\\end{aligned}}}

power densityw/m2

In the general case, the units of PSD will be the ratio of units of variance per unit of frequency; so, for example, a series of displacement values (in meters) over time (in seconds) will have PSD in units of meters squared per hertz, m2/Hz. 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.[4]

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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 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 x 2 ( t ) {\displaystyle x^{2}(t)} over the time domain, as dictated by Parseval's theorem.[1]

Power densitysymbol

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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.

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 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 E ( t ) {\displaystyle E(t)} as it fluctuates at an extremely high frequency. Obtaining a spectrum from time series such as these involves the Fourier transform, and generalizations based on Fourier analysis. 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.

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However, for the sake of dealing with the math that follows, it is more convenient to deal with time limits in the signal itself rather than time limits in the bounds of the integral. As such, we have an alternative representation of the average power, 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. 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.} Clearly, in cases where the above expression for P is non-zero, the integral must grow without bound as T grows without bound. That is the reason why we cannot use the energy of the signal, which is that diverging integral, in such cases.

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.

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