Editing Spectral density (section)

=== Energy spectral density{{Anchor|Energy}} ===
{{distinguish-redirect|Energy spectral density|Energy spectrum}}

In [[signal processing]], the [[Energy (signal processing)|energy]] of a signal <math>x(t)</math> is given by
<math display="block"> E \triangleq \int_{-\infty}^\infty \left|x(t)\right|^2\ dt.</math>
Assuming the total energy is finite (i.e. <math>x(t)</math> is a [[square-integrable function]]) allows applying [[Parseval's theorem]] (or [[Plancherel's theorem]]).{{sfn | Oppenheim | Verghese | 2016 | p=60}} That is,
<math display="block">\int_{-\infty}^\infty |x(t)|^2\, dt = \int_{-\infty}^\infty \left|\hat{x}(f)\right|^2\, df,</math>
where
<math display="block">\hat{x}(f) = \int_{-\infty}^\infty e^{-i 2\pi ft}x(t) \ dt,</math>
is the [[Fourier transform]] of <math>x(t)</math> at [[frequency]] <math>f</math> (in [[Hz]]).{{sfn | Stein | 2000 | pp=108,115}} 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<math>\left| \hat{x}(f) \right|^2 df</math> can be interpreted as a [[density function]] multiplied by an infinitesimally small frequency interval, describing the energy contained in the signal at frequency <math>f</math> in the frequency interval <math>f + df</math>. 

Therefore, the '''energy spectral density''' of <math>x(t)</math> is defined as:{{sfn | Oppenheim | Verghese | 2016 | p=14}}

{{Equation box 1
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The function <math>\bar{S}_{xx}(f)</math> and the [[autocorrelation]] of <math>x(t)</math> form a Fourier transform pair, a result also known as the [[Wiener–Khinchin theorem]] (see also [[Periodogram#Definition|Periodogram]]).

As a physical example of how one might measure the energy spectral density of a signal, suppose <math>V(t)</math> represents the [[electric potential|potential]] (in [[volt]]s) of an electrical pulse propagating along a [[transmission line]] of [[Electrical impedance|impedance]] <math>Z</math>, and suppose the line is terminated with a [[impedance matching|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 <math>t</math> is equal to <math>V(t)^2/Z</math>, so the total energy is found by integrating <math>V(t)^2/Z</math> with respect to time over the duration of the pulse. To find the value of the energy spectral density <math>\bar{S}_{xx}(f)</math> at frequency <math>f</math>, one could insert between the transmission line and the resistor a [[bandpass filter]] which passes only a narrow range of frequencies (<math>\Delta f</math>, say) near the frequency of interest and then measure the total energy <math>E(f)</math> dissipated across the resistor. The value of the energy spectral density at <math>f</math> is then estimated to be <math>E(f)/\Delta f</math>. In this example, since the power <math>V(t)^2/Z</math> has units of V<sup>2</sup> Ω<sup>−1</sup>, the energy <math>E(f)</math> has units of V<sup>2</sup>&nbsp;s&nbsp;Ω<sup>−1</sup>&nbsp;= [[Joule|J]], and hence the estimate <math>E(f)/\Delta  f</math> of the energy spectral density has units of J&nbsp;Hz<sup>−1</sup>, as required. In many situations, it is common to forget the step of dividing by <math>Z</math> so that the energy spectral density instead has units of V<sup>2</sup>&nbsp;Hz<sup>−1</sup>.

This definition generalizes in a straightforward manner to a discrete signal with a [[countably infinite]] number of values <math>x_n</math> such as a signal sampled at discrete times <math>t_n=t_0 + (n\,\Delta t)</math>:
<math display="block">\bar{S}_{xx}(f) = \lim_{N\to \infty} (\Delta t)^2 \underbrace{\left|\sum_{n=-N}^N x_n e^{-i 2\pi f n \, \Delta t}\right|^2}_{\left|\hat x_d(f)\right|^2},</math>
where <math>\hat x_d(f)</math> is the [[discrete-time Fourier transform]] of <math>x_n.</math>&nbsp; The sampling interval <math>\Delta t</math> is needed to keep the correct physical units and to ensure that we recover the continuous case in the limit <math>\Delta t\to 0.</math>&nbsp; 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)|normalized frequency]])