Energy spectral density

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;^[5] that is, the energy ${\displaystyle E}$ of a signal ${\displaystyle x(t)}$ is:

${\displaystyle E\triangleq \int _{-\infty }^{\infty }\left|x(t)\right|^{2}\ dt.}$

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^[6] (or Plancherel's theorem) gives us an alternate expression for the energy of the signal:

${\displaystyle \int _{-\infty }^{\infty }|x(t)|^{2}\,dt=\int _{-\infty }^{\infty }\left|{\hat {x}}(f)\right|^{2}\ df,}$

where:

${\displaystyle {\hat {x}}(f)\triangleq \int _{-\infty }^{\infty }e^{-i2\pi ft}x(t)\ dt}$

is the value of the Fourier transform of ${\displaystyle x(t)}$ at frequency ${\displaystyle f}$ (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${\displaystyle \left|{\hat {x}}(f)\right|^{2}df}$ can be interpreted as a density function multiplied by an infinitesimally small frequency interval, describing the energy contained in the signal at frequency ${\displaystyle f}$ in the frequency interval ${\displaystyle f+df}$.

Therefore, the energy spectral density of ${\displaystyle x(t)}$ is defined as:^[6]

${\displaystyle {\bar {S}}_{xx}(f)\triangleq \left|{\hat {x}}(f)\right|^{2}}$

(Eq.1)

The function ${\displaystyle {\bar {S}}_{xx}(f)}$ and the autocorrelation of ${\displaystyle x(t)}$ 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 ${\displaystyle V(t)}$ represents the potential (in volts) of an electrical pulse propagating along a transmission line of impedance ${\displaystyle Z}$, 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 ${\displaystyle t}$ is equal to ${\displaystyle V(t)^{2}/Z}$, so the total energy is found by integrating ${\displaystyle V(t)^{2}/Z}$ with respect to time over the duration of the pulse. To find the value of the energy spectral density ${\displaystyle {\bar {S}}_{xx}(f)}$ at frequency ${\displaystyle f}$, one could insert between the transmission line and the resistor a bandpass filter which passes only a narrow range of frequencies (${\displaystyle \Delta f}$, say) near the frequency of interest and then measure the total energy ${\displaystyle E(f)}$ dissipated across the resistor. The value of the energy spectral density at ${\displaystyle f}$ is then estimated to be ${\displaystyle E(f)/\Delta f}$. In this example, since the power ${\displaystyle V(t)^{2}/Z}$ has units of V² Ω⁻¹, the energy ${\displaystyle E(f)}$ has units of V² s Ω⁻¹ = J, and hence the estimate ${\displaystyle E(f)/\Delta f}$ 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 ${\displaystyle Z}$ so that the energy spectral density instead has units of V² Hz⁻¹.

This definition generalizes in a straightforward manner to a discrete signal with a countably infinite number of values ${\displaystyle x_{n}}$ such as a signal sampled at discrete times ${\displaystyle t_{n}=t_{0}+(n\,\Delta t)}$:

${\displaystyle {\bar {S}}_{xx}(f)=\lim _{N\to \infty }(\Delta t)^{2}\underbrace {\left|\sum _{n=-N}^{N}x_{n}e^{-i2\pi fn\,\Delta t}\right|^{2}} _{\left|{\hat {x}}_{d}(f)\right|^{2}},}$

where ${\displaystyle {\hat {x}}_{d}(f)}$ is the discrete-time Fourier transform of ${\displaystyle x_{n}.}$  The sampling interval ${\displaystyle \Delta t}$ is needed to keep the correct physical units and to ensure that we recover the continuous case in the limit ${\displaystyle \Delta t\to 0.}$  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)

Power spectral density

The power spectrum of the measured cosmic microwave background radiation 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 ${\displaystyle x(t)}$ (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 ${\displaystyle x(t)}$ and applied it to the terminals of a one ohm resistor, then indeed the instantaneous power dissipated in that resistor would be given by ${\displaystyle x(t)^{2}}$ watts.

The average power ${\displaystyle P}$ of a signal ${\displaystyle x(t)}$ over all time is therefore given by the following time average, where the period ${\displaystyle T}$ is centered about some arbitrary time ${\displaystyle t=t_{0}}$:

${\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{t_{0}-T/2}^{t_{0}+T/2}|x(t)|^{2}\,dt}$

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 ${\displaystyle x_{T}(t)=x(t)w_{T}(t)}$ and ${\displaystyle w_{T}(t)}$ is unity within the arbitrary period and zero elsewhere.

${\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }|x_{T}(t)|^{2}\,dt}$

Clearly in cases where the above expression for P is non-zero (even as T grows without bound) the integral itself must also grow without bound. That is the reason that we cannot use the energy spectral density itself, which is that diverging integral, in such cases.

In analyzing the frequency content of the signal ${\displaystyle x(t)}$, one might like to compute the ordinary Fourier transform ${\displaystyle {\hat {x}}(f)}$; however, for many signals of interest the Fourier transform does not formally exist.^{[N 1]} Regardless, Parseval's theorem tells us that we can re-write the average power as follows.

${\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }|{\hat {x}}_{T}(f)|^{2}\,df}$

Then the power spectral density is simply defined as the integrand above.^[8][9]

${\displaystyle S_{xx}(f)=\lim _{T\to \infty }{\frac {1}{T}}|{\hat {x}}_{T}(f)|^{2}\,}$

(Eq.2)

From here, we can also view ${\displaystyle |{\hat {x}}_{T}(f)|^{2}}$ as the Fourier transform of the time convolution of ${\displaystyle x_{T}^{*}(-t)}$ and ${\displaystyle x_{T}(t)}$, where * represents the complex conjugate.

${\displaystyle \left|{\hat {x}}_{T}(f)\right|^{2}={\mathcal {F}}\left\{x_{T}^{*}(-t)\mathbin {\mathbf {*} } x_{T}(t)\right\}=\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }x_{T}^{*}(\tau -t)x_{T}(t)dt\right]e^{-i2\pi f\tau }\ d\tau }$

Now, if we divide the time convolution above by the period ${\displaystyle T}$ and take the limit as ${\displaystyle T\rightarrow \infty }$, it becomes the autocorrelation function of the non-windowed signal ${\displaystyle x(t)}$, which is denoted as ${\displaystyle R_{xx}(\tau )}$, provided that ${\displaystyle x(t)}$ is ergodic, which is true in most, but not all, practical cases.^[10].

${\displaystyle \lim _{T\to \infty }{\frac {1}{T}}\left|{\hat {x}}_{T}(f)\right|^{2}=\int _{-\infty }^{\infty }\left[\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }x_{T}^{*}(\tau -t)x_{T}(t)dt\right]e^{-i2\pi f\tau }\ d\tau =\int _{-\infty }^{\infty }R_{xx}(\tau )e^{-i2\pi f\tau }d\tau }$

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

${\displaystyle S_{xx}(f)=\int _{-\infty }^{\infty }R_{xx}(\tau )e^{-i2\pi f\tau }\,d\tau ={\hat {R}}_{xx}(f)}$

(Eq.3)

Many authors use this equality to actually define the power spectral density.^[11]

The power of the signal in a given frequency band ${\displaystyle [f_{1},f_{2}]}$, where ${\displaystyle 0<f_{1}<f_{2}}$, can be calculated by integrating over frequency. Since ${\displaystyle S_{xx}(-f)=S_{xx}(f)}$, 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):

${\displaystyle P_{\textsf {bandlimited}}=2\int _{f_{1}}^{f_{2}}S_{xx}(f)\,df}$

More generally, similar techniques may be used to estimate a time-varying spectral density. In this case the time interval ${\displaystyle T}$ is finite rather than approaching infinity. This results in decreased spectral coverage and resolution since frequencies of less than ${\displaystyle 1/T}$ are not sampled, and results at frequencies which are not an integer multiple of ${\displaystyle 1/T}$ 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 ${\displaystyle x(t)}$ 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 ${\displaystyle x_{n}}$. As before, we can consider a window of ${\displaystyle -N\leq n\leq N}$ with the signal sampled at discrete times ${\displaystyle t_{n}=t_{0}+(n\,\Delta t)}$ for a total measurement period ${\displaystyle T=(2N+1)\,\Delta t}$.

${\displaystyle S_{xx}(f)=\lim _{N\to \infty }{\frac {(\Delta t)^{2}}{T}}\left|\sum _{n=-N}^{N}x_{n}e^{-i2\pi fn\,\Delta t}\right|^{2}}$

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 ${\displaystyle N}$ (and thus ${\displaystyle T}$) 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 ${\displaystyle T}$ approach infinity (Brown & Hwang).^[12]

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.

Cross power spectral density

Given two signals ${\displaystyle x(t)}$ and ${\displaystyle y(t)}$, each of which possess power spectral densities ${\displaystyle S_{xx}(f)}$ and ${\displaystyle S_{yy}(f)}$, 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.

${\displaystyle {\begin{aligned}P&=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }\left[x_{T}(t)+y_{T}(t)\right]^{*}\left[x_{T}(t)+y_{T}(t)\right]dt\\&=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }|x_{T}(t)|^{2}+x_{T}^{*}(t)y_{T}(t)+y_{T}^{*}(t)x_{T}(t)+|y_{T}(t)|^{2}dt\\\end{aligned}}}$

Using the same notation and methods as used for the power spectral density derivation, we exploit Parseval's theorem and obtain

${\displaystyle {\begin{aligned}S_{xy}(f)&=\lim _{T\to \infty }{\frac {1}{T}}\left[{\hat {x}}_{T}^{*}(f){\hat {y}}_{T}(f)\right]&S_{yx}(f)&=\lim _{T\to \infty }{\frac {1}{T}}\left[{\hat {y}}_{T}^{*}(f){\hat {x}}_{T}(f)\right]\end{aligned}}}$

where, again, the contributions of ${\displaystyle S_{xx}(f)}$ and ${\displaystyle S_{yy}(f)}$ are already understood. Note that ${\displaystyle S_{xy}^{*}(f)=S_{yx}(f)}$, 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 ${\displaystyle T\to \infty }$ becomes the Fourier transform of a cross-correlation function.^[15]

${\displaystyle {\begin{aligned}S_{xy}(f)&=\int _{-\infty }^{\infty }\left[\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }x_{T}^{*}(t-\tau )y_{T}(t)dt\right]e^{-i2\pi f\tau }d\tau =\int _{-\infty }^{\infty }R_{xy}(\tau )e^{-i2\pi f\tau }d\tau \\S_{yx}(f)&=\int _{-\infty }^{\infty }\left[\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }y_{T}^{*}(t-\tau )x_{T}(t)dt\right]e^{-i2\pi f\tau }d\tau =\int _{-\infty }^{\infty }R_{yx}(\tau )e^{-i2\pi f\tau }d\tau \end{aligned}}}$

where ${\displaystyle R_{xy}(\tau )}$ is the cross-correlation of ${\displaystyle x(t)}$ with ${\displaystyle y(t)}$ and ${\displaystyle R_{yx}(\tau )}$ is the cross-correlation of ${\displaystyle y(t)}$ with ${\displaystyle x(t)}$. In light of this, the PSD is seen to be a special case of the CSD for ${\displaystyle x(t)=y(t)}$. If ${\displaystyle x(t)}$ and ${\displaystyle y(t)}$ are real signals (e.g. voltage or current), their Fourier transforms ${\displaystyle {\hat {x}}(f)}$ and ${\displaystyle {\hat {y}}(f)}$ 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.

${\displaystyle \operatorname {CPSD} _{\text{Full}}=2S_{xy}(f)=2S_{yx}(f)}$

For discrete signals x_n and y_n, the relationship between the cross-spectral density and the cross-covariance is

${\displaystyle S_{xy}(f)=\sum _{n=-\infty }^{\infty }R_{xy}(\tau _{n})e^{-i2\pi f\tau _{n}}\,\Delta \tau }$

Electrical engineering

Spectrogram of an FM radio signal with frequency on the horizontal axis and time increasing upwards on the vertical axis.

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.

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.

Cosmology

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.

Climate Science

Power spectral-analysis have been used to examine the spatial structures for climate research.^[21] These results suggests atmospheric turbulence link climate change to more local regional volatility in weather conditions.^[22]