Energy spectral density In
signal processing, the
energy of a signal x(t) is given by E \triangleq \int_{-\infty}^\infty \left|x(t)\right|^2\ dt. Assuming the total energy is finite (i.e. x(t) is a
square-integrable function) allows applying
Parseval's theorem (or
Plancherel's theorem). That is, \int_{-\infty}^\infty |x(t)|^2\, dt = \int_{-\infty}^\infty \left|\hat{x}(f)\right|^2\, df, where \hat{x}(f) = \int_{-\infty}^\infty e^{-i 2\pi ft}x(t) \ dt, is the
Fourier transform of x(t) at
frequency 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\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 in the frequency interval f + df. Therefore, the
energy spectral density of x(t) is defined as The function \bar{S}_{xx}(f) and the
autocorrelation of 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) represents the
potential (in
volts) of an electrical pulse propagating along a
transmission line of
impedance 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 is equal to V(t)^2/Z, so the total energy is found by integrating V(t)^2/Z with respect to time over the duration of the pulse. To find the value of the energy spectral density \bar{S}_{xx}(f) at frequency f, one could insert between the transmission line and the resistor a
bandpass filter which passes only a narrow range of frequencies (\Delta f, say) near the frequency of interest and then measure the total energy E(f) dissipated across the resistor. The value of the energy spectral density at f is then estimated to be E(f)/\Delta f. In this example, since the power V(t)^2/Z has the unit V2⋅Ω−1, the energy E(f) has the unit V2⋅s⋅Ω−1 =
J, and hence the estimate 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 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 such as a signal sampled at discrete times t_n=t_0 + (n\,\Delta t): \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}, where \hat x_d(f) is the
discrete-time Fourier transform of x_n. The sampling interval \Delta t is needed to keep the correct physical unit and to ensure that we recover the continuous case in the limit \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 temperature anisotropy in terms of the angular scale. The solid line is a theoretical model, for comparison. 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) (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) 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)
watts. The average power P of a signal x(t) over all time is therefore given by the following time average, where the period T is centered about some arbitrary time t=t_{0}: 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\to \infty} \frac 1 {T} \int_{-\infty}^{\infty} \left|x_{T}(t)\right|^2\,dt, where x_{T}(t) = x(t)w_{T}(t) and w_{T}(t) is unity within the arbitrary period and zero elsewhere. When P is non-zero, the integral must grow to infinity at least as fast as 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), one might like to compute the ordinary Fourier transform \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\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 |\hat{x}_{T}(f)|^2 as the
Fourier transform of the time
convolution of x_{T}^*(-t) and x_{T}(t), where * represents the complex conjugate. In order to prove the claim below Eq.2, we will find an expression for [ \hat{x}_{T}(f) ] ^* that will be useful for the purpose. In fact, we will demonstrate that [ \hat{x}_{T}(f) ] ^* = \mathcal{F}\left\{ x_{T}^* ( - t ) \right\}. Start by noting that \begin{align} \mathcal{F}\left\{ x_{T}^* ( - t ) \right\} &= \int ^\infty _{ - \infty} x_{T}^* ( - t ) e^{-i 2\pi f t } dt \end{align} and let z = -t , so that z \rightarrow - \infty when t \rightarrow \infty and vice versa. So \begin{align} \int ^\infty _{ - \infty} x_{T}^* ( - t ) e^{-i 2\pi f t } dt &= \int ^{ - \infty} _\infty x_{T}^* ( z ) e^{i 2\pi f z } \left( -dz \right) \\ &= \int ^\infty _{ - \infty} x_{T}^* ( z ) e^{i 2\pi f z } dz \\ &= \int ^\infty _{ - \infty} x_{T}^* ( t ) e^{i 2\pi f t } dt \end{align} where, in the last line, use has been made of z and t being dummy variables. So, we have \begin{align} \mathcal{F}\left\{ x_{T}^* ( - t ) \right\} &= \int ^\infty _{ - \infty} x_{T}^* ( - t ) e^{-i 2\pi f t } dt \\ &= \int ^\infty _{ - \infty} x_{T}^* ( t ) e^{ i 2\pi f t } dt \\ &= \int ^\infty _{ - \infty} x_{T}^* ( t ) [ e^{ - i 2\pi f t } ]^* dt \\ &= \left[\int ^\infty _{ - \infty} x_{T} ( t ) e^{ - i 2\pi f t } dt \right]^* \\ &= \left[\mathcal{F} \left\{ x_{T} ( t )\right\}\right] ^* \\ &= \left[\hat{x}_T(f) \right] ^* \end{align} 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). In this way, we have: \begin{align} \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^{-i 2\pi f\tau} d\tau \\ &= \int_{-\infty}^\infty \left[\int_{-\infty}^\infty x_{T}^*(t - \tau)x_{T}(t) dt \right]e^{-i 2\pi f\tau} \ d\tau, \end{align} where the convolution theorem has been used when passing from the 3rd to the 4th line. Now, if we divide the time convolution above by the period T and take the limit as T \rightarrow \infty, it becomes the
autocorrelation function of the non-windowed signal x(t), which is denoted as R_{xx}(\tau), provided that x(t) is
ergodic, which is true in most, but not all, practical cases. \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^{-i 2\pi f\tau} \ d\tau = \int_{-\infty}^\infty R_{xx}(\tau)e^{-i 2\pi f\tau} d\tau Assuming the ergodicity of x(t), the power spectral density can be found once more as the Fourier transform of the
autocorrelation function R_{xx}, a property known as the
Wiener–Khinchin theorem. Many authors use this relationship to define the power spectral density in terms of the autocorrelation function instead of the Fourier transform of the signal as we have done. The power of the signal in a given frequency band [f_1, f_2], where 0, can be calculated by integrating over frequency. Since S_{xx}(-f) = S_{xx}(f), an equal amount of power can be attributed to positive and negative frequency bands, which accounts for the factor of 2 in the following form (such trivial factors depend on the conventions used): P_\textsf{band-limited} = 2 \int_{f_1}^{f_2} S_{xx}(f) \, df More generally, similar techniques may be used to estimate a time-varying spectral density. In this case the time interval T is finite rather than 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. Just as with the energy spectral density, the definition of the power spectral density can be generalized to
discrete time variables x_n. As before, we can consider a window of -N\le n\le N with the signal sampled at discrete times t_n = t_0 + (n\,\Delta t) for a total measurement period T = (2N + 1) \,\Delta t. S_{xx}(f) = \lim_{N\to \infty}\frac{(\Delta t)^2}{T}\left|\sum_{n=-N}^N x_n e^{-i 2\pi f n \,\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 (and thus 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 approach infinity. 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: {{bulleted list P = \operatorname{E}(x^2) = \int_{-\infty}^{\infty}\! S_{xx}(f) \, df F(f) = 2 \int _0^f S_{xx}(f')\, df'. Note that the previous expression for total power (signal variance) is a special case where . }}
Cross power spectral density Given two signals x(t) and y(t), each of which possess power spectral densities S_{xx}(f) and 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. \begin{align} 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{align} Using the same notation and methods as used for the power spectral density derivation, we exploit Parseval's theorem and obtain \begin{align} 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{align} where, again, the contributions of S_{xx}(f) and S_{yy}(f) are already understood. Note that 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\to\infty becomes the Fourier transform of a
cross-correlation function. \begin{align} 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^{-i 2 \pi f \tau} d\tau= \int_{-\infty}^{\infty} R_{xy}(\tau) e^{-i 2 \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^{-i 2 \pi f \tau} d\tau= \int_{-\infty}^{\infty} R_{yx}(\tau) e^{-i 2 \pi f \tau} d\tau, \end{align} where R_{xy}(\tau) is the
cross-correlation of x(t) with y(t) and R_{yx}(\tau) is the cross-correlation of y(t) with x(t). In light of this, the PSD is seen to be a special case of the CSD for x(t) = y(t). If x(t) and y(t) are real signals (e.g. voltage or current), their Fourier transforms \hat{x}(f) and \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. \operatorname{CPSD}_\text{Full} = 2S_{xy}(f) = 2 S_{yx}(f) For discrete signals and , the relationship between the cross-spectral density and the cross-covariance is S_{xy}(f) = \sum_{n=-\infty}^\infty R_{xy}(\tau_n)e^{-i 2 \pi f \tau_n}\,\Delta\tau == Estimation ==