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  of a signal  is:
The energy spectral density is most suitable for transients—that is, pulse-like signals—having a finite total energy. Finite or not, Parsevals theorem (or Plancherels theorem) gives us an alternate expression for the energy of the signal:
where:
is the value of the Fourier transform of  at frequency  (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 can be interpreted as a density function multiplied by an infinitesimally small frequency interval, describing the energy contained in the signal at frequency  in the frequency interval . 
Therefore, the energy spectral density of  is defined as:
The function  and the autocorrelation of  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  represents the potential (in volts) of an electrical pulse propagating along a transmission line of impedance , 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  is equal to , so the total energy is found by integrating  with respect to time over the duration of the pulse. To find the value of the energy spectral density  at frequency , one could insert between the transmission line and the resistor a bandpass filter which passes only a narrow range of frequencies (, say) near the frequency of interest and then measure the total energy  dissipated across the resistor. The value of the energy spectral density at  is then estimated to be . In this example, since the power  has units of V Ω, the energy  has units of V s Ω = J, and hence the estimate  of the energy spectral density has units of J Hz, as required. In many situations, it is common to forget the step of dividing by  so that the energy spectral density instead has units of V Hz.
This definition generalizes in a straightforward manner to a discrete signal with a countably infinite number of values  such as a signal sampled at discrete times :
where  is the discrete-time Fourier transform of   The sampling interval  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. (also see normalized frequency)