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Power Density Spectrum ○◂｜Definition｜1st｜20260414124631-00-⌔

Spectral density - Wikipedia#Power_spectral_density

Power spectral density

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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 $t=t_0$:

$P=\underset{T\to \infty }{\lim }\frac{1}{T}{\int }_{t_0-T/2}^{t_0+T/2}{\left|x(t)\right|}^{2}dt$

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

$P=\underset{T\to \infty }{\lim }\frac{1}{T}{\int }_{-\infty }^{\infty }{\left|x_T(t)\right|}^{2}dt,$

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.¹ However, under suitable conditions, certain generalizations of the Fourier transform (e.g. the Fourier–Stieltjes transform) still adhere to Parseval’s theorem. As such,

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

where the integrand defines the power spectral density:²³

The convolution theorem then allows regarding $|{\hat{x}}_T(f){|}^2$ as the Fourier transform of the time convolution of $x_T^{\ast }(-t)$ and $x_T(t)$, where ﹡ represents the complex conjugate.

In order to prove the claim below Eq.2, we will find an expression for $[{\hat{x}}_T(f){]}^{\ast }$ that will be useful for the purpose. In fact, we will demonstrate that $[{\hat{x}}_T(f){]}^{\ast }=\mathcal{F}\left\{x_T^{\ast }(-t)\right\}$. Start by noting that

$\begin{array}{rl}\mathcal{F}\left\{x_T^{\ast }(-t)\right\}& ={\int }_{-\infty }^{\infty }{x}_T^{\ast }(-t)e^{-i2\pi ft}dt\end{array}$

and let $z=-t$, so that $z\to -\infty$ when $t\to \infty$ and vice versa. So

$\begin{array}{rl}{\int }_{-\infty }^{\infty }{x}_T^{\ast }(-t)e^{-i2\pi ft}dt& ={\int }_{\infty }^{-\infty }{x}_T^{\ast }(z)e^{i2\pi fz}\left(-dz\right)\\ & ={\int }_{-\infty }^{\infty }{x}_T^{\ast }(z)e^{i2\pi fz}dz\\ & ={\int }_{-\infty }^{\infty }{x}_T^{\ast }(t)e^{i2\pi ft}dt\end{array}$

where, in the last line, use has been made of $z$ and $t$ being dummy variables. So, we have

$\begin{array}{rl}\mathcal{F}\left\{x_T^{\ast }(-t)\right\}& ={\int }_{-\infty }^{\infty }{x}_T^{\ast }(-t)e^{-i2\pi ft}dt\\ & ={\int }_{-\infty }^{\infty }{x}_T^{\ast }(t)e^{i2\pi ft}dt\\ & ={\int }_{-\infty }^{\infty }{x}_T^{\ast }(t)[e^{-i2\pi ft}{]}^{\ast }dt\\ & ={\left[{\int }_{-\infty }^{\infty }{x}_T(t)e^{-i2\pi ft}dt\right]}^{\ast }\\ & ={\left[\mathcal{F}\left\{x_T(t)\right\}\right]}^{\ast }\\ & ={\left[{\hat{x}}_T(f)\right]}^{\ast }\end{array}$

q.e.d.

Now, let’s demonstrate the claim below eq.2 by using the demonstrated identity. In addition, we will make the substitution $u(t)=x_T^{\ast }(-t)$. In this way, we have:

$\begin{array}{rl}{\left|{\hat{x}}_T(f)\right|}^2& =[{\hat{x}}_T(f){]}^{\ast }\cdot {\hat{x}}_T(f)\\ & =\mathcal{F}\left\{x_T^{\ast }(-t)\right\}\cdot \mathcal{F}\left\{x_T(t)\right\}\\ & =\mathcal{F}\left\{u(t)\right\}\cdot \mathcal{F}\left\{x_T(t)\right\}\\ & =\mathcal{F}\left\{u(t)\ast x_T(t)\right\}\\ & ={\int }_{-\infty }^{\infty }\left[{\int }_{-\infty }^{\infty }u(\tau -t)x_T(t)dt\right]e^{-i2\pi f\tau }d\tau \\ & ={\int }_{-\infty }^{\infty }\left[{\int }_{-\infty }^{\infty }{x}_T^{\ast }(t-\tau )x_T(t)dt\right]e^{-i2\pi f\tau }d\tau ,\end{array}$

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 $T\to \infty$, 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.⁴

$\underset{T\to \infty }{\lim }\frac{1}{T}{\left|{\hat{x}}_T(f)\right|}^2={\int }_{-\infty }^{\infty }\left[\underset{T\to \infty }{\lim }\frac{1}{T}{\int }_{-\infty }^{\infty }{x}_T^{\ast }(t-\tau )x_T(t)dt\right]e^{-i2\pi f\tau }d\tau ={\int }_{-\infty }^{\infty }{R}_{xx}(\tau )e^{-i2\pi f\tau }d\tau$

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.⁵

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.⁶

The power of the signal in a given frequency band $[f_1,f_2]$, where $0<f_1<f_2$, 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):

$P_{\text{band-limited}}=2{\int }_{f_1}^{f_2}{S}_{xx}(f)df$

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 $T=(2N+1)\Delta t$.

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

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 $N$ (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.⁷

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.

Printed 2026-06-28.

(echo:: @ ᯤ)

$⟨\hat{x}(\omega ){\hat{x}}^{\ast }({\omega }^{\prime })⟩=2\pi f(\omega )\delta (\omega -{\omega }^{\prime }),$

where $\delta (\omega -{\omega }^{\prime })$ is the Dirac delta function. Such formal statements may sometimes be useful to guide the intuition, but should always be used with utmost care.

Footnotes

1. Some authors, e.g., (Risken & Frank 1996, p. 30) still use the non-normalized Fourier transform in a formal way to formulate a definition of the power spectral density ↩

2. Oppenheim & Verghese 2016, pp. 422–423. ↩

3. Miller & Childers 2012, pp. 429–431. ↩

4. The Wiener–Khinchin theorem makes sense of this formula for any wide-sense stationary process under weaker hypotheses: $R_{xx}$ 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 $R_{xx}$ 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). ↩

5. Miller & Childers 2012, p. 433. ↩

6. Dennis Ward Ricker (2003). Echo Signal Processing. Springer. ISBN 978-1-4020-7395-3. ↩

7. Brown & Hwang 1997. ↩

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