Spectral density

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The spectral density of a fluorescent light as a function of optical wavelength shows peaks at atomic transitions, indicated by the numbered arrows.
The voice waveform over time (left) has a broad audio power spectrum (right).

In statistical signal processing, statistics, and physics, the spectrum of a time-series or signal is a positive real function of a frequency variable associated with a stationary stochastic process, or a deterministic function of time, which has dimensions of power per hertz (Hz), or energy per hertz. Intuitively, the spectrum decomposes the content of a stochastic process into different frequencies present in that process, and helps identify periodicities. More specific terms which are used are the power spectrum, spectral density, power spectral density, or energy spectral density.

Explanation[edit]

In physics, the signal is usually a wave, such as an electromagnetic wave, random vibration, or an acoustic wave. The spectral density of the wave, when multiplied by an appropriate factor, will give the power carried by the wave, per unit frequency, known as the power spectral density (PSD) of the signal. Power spectral density is commonly expressed in watts per hertz (W/Hz).[1]

For voltage signals, it is customary to use units of V2 Hz−1 for the PSD and V2 s Hz−1 for the ESD (energy spectral density).[2] Often it is convenient to work with an amplitude spectral density (ASD), which is the square root of the PSD; the ASD of a voltage signal has units of V Hz−1/2.[3] For random vibration analysis, units of g2 Hz−1 are sometimes used for the PSD of acceleration. Here g denotes the g-force.[4]

Although it is not necessary to assign physical dimensions to the signal or its argument, in the following discussion the terms used will assume that the signal varies in time.

Preliminary conventions on notations for time series[edit]

The phrase time series has been defined as "... a collection of observations made sequentially in time."[5] But it is also used to refer to a stochastic process that functions as the underlying theoretical model for the process that generated the data (and thus includes consideration of all the other possible sequences of data that could have been observed, but were not). Furthermore, the 'time' can be either continuous or discrete. There are, therefore, four different but closely related definitions and formulas for the power spectrum of a time series.

If  X_n (discrete time) or  X_t (continuous time) is a stochastic process, we will refer to a possible time series of data coming from it as a sample or path or signal of the stochastic process. To avoid confusion, we will reserve the word process for a stochastic process, and use one of the words signal, or sample, to refer to a time series of data.

For any random variable X, standard notations of angle brackets or \operatorname{E} will be used for ensemble average, also known as statistical expectation, and \operatorname{Var} for the theoretical variance.

Motivating example[edit]

Suppose x_n, from n=0 to N-1 is a time series (discrete time) with zero mean. Suppose that it is a sum of a finite number of periodic components (all frequencies are positive):


x_n =  \sum_k [a_k \cos (2\pi \nu_k n) + b_k \sin (2\pi \nu_k n)].

The variance of x_n is, for a zero-mean function as above, given by \frac 1N \sum_{n=0}^{N-1} x_n^2. If these data were samples taken from an electrical signal, this would be its average power (power is energy per unit time, so it is analogous to variance if energy is analogous to the amplitude squared).

Now, for simplicity, suppose the signal extends infinitely in time, so we pass to the limit as N\rightarrow \infty. If the average power is bounded, which is almost always the case in reality, then the following limit exists and is the variance of the data.


\lim _ {N\rightarrow \infty} \frac 1N \sum_{n=0}^{N-1} x_n^2.

Again, for simplicity, we will pass to continuous time, and assume that the signal extends infinitely in time in both directions. Then these two formulas become

x(t) =  \sum_k [a_k \cos (2\pi \nu_k t) + b_k \sin (2\pi \nu_k t)]

and


\lim _ {T\rightarrow\infty} \frac 1{2T} \int_{-T}^T x(t)^2 dt.

But obviously the root mean square of either \cos or \sin is 1/\sqrt{2}, so the variance of a_k \cos (2\pi \nu_k t) is a_k^2 / 2 and that of b_k \sin (2\pi \nu_k t) is  b_k^2 / 2. Hence, the power of x(t) which comes from the component with frequency \nu_k is  (a_k^2 + b_k^2) / 2. All these contributions add up to the power of x(t).

Then the power as a function of frequency is obviously (a_k^2 + b_k^2) / 2, and its statistical cumulative distribution function F(\nu) will be

F(\nu) = \sum _ {k : \nu_k < \nu} \frac 12 (a_k^2 + b_k^2).

F is a step function, monotonically non-decreasing. Its jumps occur at the frequencies of the periodic components of x, and the value of each jump is the power or variance of that component.

The variance is the covariance of the data with itself. If we now consider the same data but with a lag of \tau, we can take the covariance of x(t) with x(t+\tau), and define this to be the autocorrelation function c of the signal (or data) x:


c (\tau) = \lim _ {T\rightarrow\infty} \frac 1{2T} \int_{-T}^T x(t) x(t+\tau) dt.

When it exists, it is an even function of \tau. If the average power is bounded, then c exists everywhere, is finite, and is bounded by c(0), which is the power or variance of the data.

It is elementary to show that c can be decomposed into periodic components with the same periods as x:


c(\tau) = 
 \sum_k \frac 12 (a_k^2 + b_k^2)\cos (2\pi \nu_k \tau).

This is in fact the spectral decomposition of c over the different frequencies, and is obviously related to the distribution of power of x over the frequencies: the amplitude of a frequency component of c is its contribution to the power of the signal.

Definition[edit]

Energy spectral density[edit]

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 x(t) is[6]

 \int\limits_{-\infty}^\infty |x(t)|^2\, dt.

The energy spectral density is most suitable for transients—that is, pulse-like signals—having a finite total energy. In this case, Parseval's theorem gives us an alternate expression for the energy of the signal in terms of its Fourier transform, \hat{x}(\omega):[6]

\int\limits_{-\infty}^\infty |x(t)|^2\, dt = \frac{1}{2\pi}\int\limits_{-\infty}^\infty |\hat{x}(\omega)|^2\, d\omega.

Here \omega is the angular frequency. Since the integral on the right-hand side is the energy of the signal, the integrand |\hat{x}(\omega)|^2 can be interpreted as a density function describing the energy per unit frequency contained in the signal at frequency \omega. In light of this, the energy spectral density of a signal x(t) is defined as[6][N 1]

 S_{xx}(\omega) = |\hat{x}(\omega)|^2 = \left|\int\limits_{-\infty}^\infty x(t)e^{-i\omega t}\,{d}t\right|^2.

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 S_{xx}(\omega) at frequency \omega, one could insert between the transmission line and the resistor a bandpass filter which passes only a narrow range of frequencies (\Delta \omega, say) near the frequency of interest and then measure the total energy E(\omega) dissipated across the resistor. The value of the energy spectral density at \omega is then estimated to be E(\omega)/\Delta\omega. In this example, since the power V(t)^2/Z has units of V2 Ω−1, the energy E(\omega) has units of V2 s Ω−1 = J, and hence the estimate E(\omega)/\Delta\omega of the energy spectral density has units of J Hz−1, as required. In many situations, it is common to forgo the step of dividing by Z so that the energy spectral density instead has units of V2 s Hz−1.

This definition generalizes in a straightforward manner to a discrete signal with an infinite number of values x_n such as a signal sampled at discrete times x_n=x(n\,\Delta t):

S_{xx}(\omega) = (\Delta t)^2\left|\sum_{n=-\infty}^\infty x_n e^{-i\omega n}\right|^2= (\Delta t)^2 \hat x_d(\omega)\hat x_d^*(\omega),

where \hat x_d(\omega) is the discrete Fourier transform of x_n . The sampling interval \Delta t is needed to keep the correct physical units and to ensure that we recover the continuous case in the limit \Delta t\rightarrow 0; however, in the mathematical sciences, the interval is often set to 1.

Power spectral density[edit]

The above definition of energy spectral density is most suitable for transients, i.e., pulse-like signals, for which the Fourier transforms of the signals exist. For continued signals that describe, for example, stationary physical processes, it makes more sense to define a power spectral density (PSD), which describes how the power of a signal or time series is distributed over the different frequencies, as in the simple example given previously. Here, power can be the actual physical power, or more often, for convenience with abstract signals, can be defined as the squared value of the signal. The total power P of a signal x(t) is the following time average:

 P = \lim_{T\rightarrow \infty} \frac 1 {2T} \int_{-T}^T x(t)^2\,dt.

The power of a signal may be finite even if the energy is infinite. For example, a 10-volt power supply connected to a 1 kΩ resistor delivers (10 V)2 / (1 kΩ) = 0.1 W of power at any given time; however, if the supply is allowed to operate for an infinite amount of time, it will deliver an infinite amount of energy (0.1 J each second for an infinite number of seconds).

In analyzing the frequency content of the signal x(t), one might like to compute the ordinary Fourier transform \hat{x}(\omega); however, for many signals of interest this Fourier transform does not exist.[N 2] Because of this, it is advantageous to work with a truncated Fourier transform  \hat{x}_T(\omega), where the signal is integrated only over a finite interval [0, T]:

 \hat{x}_T(\omega) = \frac{1}{\sqrt{T}} \int_0^T x(t) e^{-i\omega t}\, dt.

Then the power spectral density can be defined as[8][9]

 S_{xx}(\omega) = \lim_{T \rightarrow \infty} \mathbf{E} \left[ | \hat{x}_T(\omega) | ^ 2 \right].

Here E denotes the expected value; explicitly, we have[9]

 \mathbf{E} \left[ | \hat{x}_T(\omega) |^2 \right] = \mathbf{E} \left[ \frac{1}{T} \int\limits_0^T x^*(t) e^{i\omega t}\, dt \int\limits_0^T x(t') e^{-i\omega t'}\, dt' \right] = \frac{1}{T} \int\limits_0^T \int\limits_0^T \mathbf{E}\left[x^*(t) x(t')\right] e^{i\omega (t-t')}\, dt\, dt'.

Using such formal reasoning, one may already guess that for a stationary random process, the power spectral density f(\omega) and the autocorrelation function of this signal  \gamma(\tau)=\langle X(t) X(t+\tau)\rangle should be a Fourier transform pair. Provided that  \gamma(\tau) is absolutely integrable, which is not always true, then

S_{xx}(\omega)=\int_{-\infty}^\infty \,\gamma(\tau)\,e^{-i\omega\tau}\,d \tau=\hat \gamma(\omega).

The Wiener–Khinchin theorem makes sense of this formula for any wide-sense stationary process under weaker hypotheses:  \gamma does not need to be absolutely integrable, it only needs to exist. But the integral can no longer be interpreted as usual. The formula also makes sense if interpreted as involving distributions (in the sense of Laurent Schwartz, not in the sense of a statistical Cumulative distribution function) instead of functions. If  \gamma is continuous, Bochner's theorem can be used to prove that its Fourier transform exists as a positive measure, whose distribution function is F (but not necessarily as a function and not necessarily possessing a probability density).

Many authors use this equality to actually define the power spectral density.[10]

The power of the signal in a given frequency band [\omega_1,\omega_2] can be calculated by integrating over positive and negative frequencies,


\int_{\omega_1}^{\omega_2}\,S_{xx}(\omega)+S_{xx}(-\omega) \,d \omega = F(\omega_2) - F(-\omega_2)

where F is the integrated spectrum whose derivative is S_{xx}.

More generally, similar techniques may be used to estimate a time-varying spectral density.[citation needed]

The definition of the power spectral density generalizes in a straightforward manner to finite time-series x_n with 1\le n\le N, such as a signal sampled at discrete times x_n=x(n\Delta t) for a total measurement period T=N \Delta t.

S_{xx}(\omega)=\frac{(\Delta t)^2}{T}\left|\sum_{n=1}^N x_n e^{-i\omega n}\right|^2.

In a real-world application, one would typically average this single-measurement PSD over several repetitions of the measurement to obtain a more accurate estimate of the theoretical PSD of the physical process underlying the individual measurements. This computed PSD is sometimes called periodogram. One can prove that this periodogram converges to the true PSD when the averaging time interval T goes to infinity (Brown & Hwang[11]) to approach the Power Spectral Density (PSD).

If two signals both possess power spectral densities, then a cross-spectral density can be calculated by using their cross-correlation function.

Properties of the power spectral density[edit]

Some properties of the PSD include:[12]

  • The spectrum of a real valued process is an even function of frequency: S_{xx}(-\omega) = S_{xx}(\omega).
  • If the process is continuous and purely indeterministic, the autocovariance function can be reconstructed by using the Inverse Fourier transform
  • it describes the distribution of the variance over frequency. In particular,
    \text{Var}(X_n) = \gamma_0 = 2 \int_0^{\infty} S_{xx}(\omega) d\omega.
  • It is a linear function of the autocovariance function in the sense that if \gamma is decomposed into two functions \gamma(\tau) = \alpha_1 \gamma_1(\tau) + \alpha_2 \gamma_2(\tau), then
    f = \alpha_1 S_{xx,1} + \alpha_2 S_{xx,2}.

The integrated spectrum or power spectral distribution F(\omega) is defined as[13]

F(\omega)=  \int _{-\infty}^\omega S_{xx}(\omega')\, d\omega'.

Cross-spectral density[edit]

Given two signals x(t) and y(t), each of which possess power spectral densities S_{xx}(\omega) and S_{yy}(\omega), it is possible to define a cross-spectral density (CSD) given by

S_{xy}(\omega) = \lim_{T\rightarrow\infty} \mathbf{E}\left\{\left[F_x^T(\omega)\right]^*F_y^T(\omega)\right\}.

The cross-spectral density (or 'cross power spectrum') is thus the Fourier transform of the cross-correlation function.

S_{xy}(\omega) = \int_{-\infty}^{\infty} R_{xy}(t) e^{-j \omega t} dt = \int_{-\infty}^{\infty} \left[ \int_{-\infty}^{\infty} x(\tau) \cdot y(\tau+t) d\tau \right] \, e^{-j \omega t} dt,

where R_{xy}(t) is the cross-correlation of x(t) and y(t).

By an extension of the Wiener–Khinchin theorem, the Fourier transform of the cross-spectral density S_{xy}(\omega) is the cross-covariance function.[14] In light of this, the PSD is seen to be a special case of the CSD for x(t) = y(t).

For discrete signals xn and yn, the relationship between the cross-spectral density and the cross-covariance is


S_{xy}(\omega)=\frac{1}{2\pi}\sum_{n=-\infty}^\infty R_{xy}(n)e^{-j\omega n}

Estimation[edit]

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.

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.

Properties[edit]

  • The spectral density of f(t) and the autocorrelation of f(t) form a Fourier transform pair (for PSD versus ESD, different definitions of autocorrelation function are used).
  • One of the results of Fourier analysis is Parseval's theorem which states that the area under the energy spectral density curve is equal to the area under the square of the magnitude of the signal, the total energy:
\int_{-\infty}^\infty \left| f(t) \right|^2\, dt = \int_{-\infty}^\infty ESD(\omega)\, d\omega.
The above theorem holds true in the discrete cases as well. A similar result holds for power: the area under the power spectral density curve is equal to the total signal power, which is  R(0) , the autocorrelation function at zero lag. This is also (up to a constant which depends on the normalization factors chosen in the definitions employed) the variance of the data comprising the signal.

Related concepts[edit]

  • Most spectrum graphs really display only the power spectral density. Sometimes[when?] the complete frequency spectrum is graphed in two parts, amplitude versus frequency and phase versus frequency (which contains the rest of the information from the frequency spectrum). The original function f(t) cannot be recovered from the amplitude spectral density part alone — the temporal information is lost. See spectral phase annd phase noise.
  • The spectral centroid of a signal is the midpoint of its spectral density function, i.e. the frequency that divides the distribution into two equal parts.
  • The spectral edge frequency of a signal is an extension of the previous concept to any proportion instead of two equal parts.
  • Spectral density is a function of frequency, not a function of time. However, the spectral density of small windows of a longer signal may be calculated, and plotted versus time associated with the window. Such a graph is called a spectrogram. This is the basis of a number of spectral analysis techniques such as the short-time Fourier transform and wavelets.
  • In radiometry and colorimetry (or color science more generally), the spectral power distribution (SPD) of a light source is a measure of the power carried by each frequency or color in a light source. The light spectrum is usually measured at points (often 31) along the visible spectrum, in wavelength space instead of frequency space, which makes it not strictly a spectral density. Some spectrophotometers can measure increments as fine as one to two nanometers. Values are used to calculate other specifications and then plotted to demonstrate the spectral attributes of the source. This can be a helpful tool in analyzing the color characteristics of a particular source.

Applications[edit]

Electrical engineering[edit]

Spectrogram of an FM radio signal with frequency on the horizontal axis and time increrasing 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. Much effort has been expended and millions of dollars spent on developing and producing electronic instruments called "spectrum analyzers" for aiding electrical engineers and technicians in observing and measuring the power spectra of signals. The cost of a spectrum analyzer varies depending on its frequency range, its bandwidth, and its accuracy. The higher the frequency range (S-band, C-band, X-band, Ku-band, K-band, Ka-band, etc.), the more difficult the components are to make, assemble, and test and the more expensive the spectrum analyzer is. Also, the wider the bandwidth that a spectrum analyzer possesses, the more costly that it is, and the capability for more accurate measurements increases costs as well.

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. These devices work in low frequencies and with small bandwidths.

Coherence[edit]

See Coherence (signal processing) for use of the cross-spectral density.

See also[edit]

Notes[edit]

  1. ^ Here the Fourier transform is defined so that the forward transform (from x(t) to \hat{x}(\omega)) carries no factor of 1/2π; this factor instead appears on the inverse transform.
  2. ^ Some authors (e.g. Risken[7] ) still use the non-normalized Fourier transform in a formal way to formulate a definition of the power spectral density
     \langle \hat x(\omega) \hat x^\ast(\omega') \rangle = 2\pi\,f(\omega)\,\delta(\omega-\omega'),
    where  \delta(\omega-\omega') is the Dirac delta function. Such formal statements may sometimes be useful to guide the intuition, but should always be used with utmost care.

References[edit]

  1. ^ Gérard Maral (2003). VSAT Networks. John Wiley and Sons. ISBN 0-470-86684-5. 
  2. ^ Michael Peter Norton and Denis G. Karczub (2003). Fundamentals of Noise and Vibration Analysis for Engineers. Cambridge University Press. ISBN 0-521-49913-5. 
  3. ^ Michael Cerna and Audrey F. Harvey (2000). "The Fundamentals of FFT-Based Signal Analysis and Measurement". 
  4. ^ Alessandro Birolini (2007). Reliability Engineering. Springer. p. 83. ISBN 978-3-540-49388-4. 
  5. ^ C. Chatfield (1989). The Analysis of Time Series—An Introduction (fourth ed.). Chapman and Hall, London. p. 1. ISBN 0-412-31820-2. 
  6. ^ a b c Oppenheim; Verghese. Signals, Systems, and Inference. pp. 32–4. 
  7. ^ Hannes Risken (1996). The Fokker–Planck Equation: Methods of Solution and Applications (2nd ed.). Springer. p. 30. ISBN 9783540615309. 
  8. ^ Fred Rieke, William Bialek, and David Warland (1999). Spikes: Exploring the Neural Code (Computational Neuroscience). MIT Press. ISBN 978-0262681087. 
  9. ^ a b Scott Millers and Donald Childers (2012). Probability and random processes. Academic Press. pp. 370–5. 
  10. ^ Dennis Ward Ricker (2003). Echo Signal Processing. Springer. ISBN 1-4020-7395-X. 
  11. ^ Robert Grover Brown & Patrick Y.C. Hwang (1997). Introduction to Random Signals and Applied Kalman Filtering. John Wiley & Sons. ISBN 0-471-12839-2. 
  12. ^ Storch, H. Von; F. W Zwiers (2001). Statistical analysis in climate research. Cambridge University Press. ISBN 0-521-01230-9. 
  13. ^ An Introduction to the Theory of Random Signals and Noise, Wilbur B. Davenport and Willian L. Root, IEEE Press, New York, 1987, ISBN 0-87942-235-1
  14. ^ William D Penny (2009). "Signal Processing Course, chapter 7". 

External links[edit]