Periodogram

The periodogram is an estimator for the spectral power density of a signal . We are looking for a function that specifies the distribution of the power (or energy ) of the signal over the angular frequency . The term was coined by Arthur Schuster in 1898. The method is used in signal processing , electrical engineering, physics and econometrics . Spectrum analyzers are an important example . ${\displaystyle P\left(\omega \right)}$ ${\displaystyle \omega }$

In the mathematical sense, the periodogram is an inconsistent estimator, see also spectral density estimation .

Context and conventions

As a rule, only samples of the signal are given at discrete points in time with the sampling duration , and the estimation is limited to samples, e.g. B. . ${\displaystyle f\left(t\right)}$ ${\displaystyle t_{n}=n\cdot T}$ ${\displaystyle T}$ ${\displaystyle N}$ ${\displaystyle 0\leq n<N}$

An essential step in the process is a discrete Fourier transform . The limitation of the Fourier transformation to intervals of length can be achieved by multiplying the signal with a window function . ${\displaystyle N\cdot T}$ ${\displaystyle w\left(t\right)}$

In the simplest case, the width , i.e. the length of the sample values, is a rectangular function . ${\displaystyle w\left(t\right)}$ ${\displaystyle N\cdot T}$ ${\displaystyle N}$

In order to reduce artifacts in the spectrum (due to the infinitely steep slope at the edges of the rectangular window), windows with slower changes and their own names are usually used, e.g. B. the Parzen window or the "Welch window". Then there is talk of a modified periodogram.

The notation is used for the discrete Fourier transform of the signal . Here are angular frequencies with permissible. ${\displaystyle f\left(t\right)w\left(t\right)}$ ${\displaystyle F^{\left(w\right)}\left(\omega _{m}\right)=\sum \limits _{n}f\left(t_{n}\right)w\left(t_{n}\right)e^{i\omega _{m}t_{n}}}$ ${\displaystyle \omega _{m}=m\cdot \Delta \Omega =m\cdot {\frac {2\pi }{N\cdot T}}}$ ${\displaystyle 0\leq m<N}$

definition

The periodogram is defined according to

${\displaystyle P^{\left(w\right)}\left(\omega \right)={\frac {\left|F^{\left(w\right)}\left(\omega \right)\right|^{2}}{\sum _{n=0}^{N-1}w\left(t_{n}\right)^{2}}}.}$

In accordance with the sampling theorem , the periodogram is -periodic. One therefore restricts oneself to an interval ( Brillouin zone ) or . ${\displaystyle 2\pi /T}$ ${\displaystyle 0\leq \omega \leq 2\pi /T}$ ${\displaystyle -\pi /T\leq \omega \leq \pi /T}$

There are various conventions regarding the normalization factor. An important parameter here is the mean square amplitude (mean power) of the signal. The normalization is chosen so that the mean value of corresponds as closely as possible with . ${\displaystyle \left\langle A^{2}\right\rangle ={\frac {1}{N}}\sum _{n}\left|f\left(t_{n}\right)\right|^{2}}$ ${\displaystyle P^{\left(w\right)}\left(\omega \right)}$ ${\displaystyle \left\langle A^{2}\right\rangle }$

If the amplitude of the signal is digitized and has a maximum value , the periodogram can also be normalized relative to the maximum ( full scale ). The maximum is reached for monochromatic signals , which is the full-scale periodogram ${\displaystyle A}$ ${\displaystyle f=Ae^{-i\omega t}}$

${\displaystyle P_{FS}^{\left(w\right)}\left(\omega \right)={\frac {\left|F^{\left(w\right)}\left(\omega \right)\right|^{2}}{\left(\sum _{n=0}^{N-1}w\left(t_{n}\right)\right)^{2}}}.}$

Examples

White noise

It is a white noise with variance , . The ensemble mean of the square of the Fourier transforms is then ${\displaystyle f\left(t\right)}$ ${\displaystyle \left\langle A^{2}\right\rangle }$ ${\displaystyle \left\langle f\left(t_{m}\right)f^{*}\left(t_{n}\right)\right\rangle =\delta _{m,n}\left\langle A^{2}\right\rangle }$

${\displaystyle \left\langle \left|F^{\left(w\right)}\left(\omega \right)\right|^{2}\right\rangle =\left\langle A^{2}\right\rangle \sum \limits _{m,n}e^{i\omega \left(t_{m}-t_{n}\right)}\delta _{m,n}w\left(t_{m}\right)w\left(t_{n}\right)=\left\langle A^{2}\right\rangle \sum _{n}w\left(t_{n}\right)^{2}.}$

The periodogram has the mean value , regardless of the window length. All frequencies give the same energy contribution. ${\displaystyle \left\langle A^{2}\right\rangle }$

Constant signal

General statements can be made for the average frequency of . Starting point is ${\displaystyle \left|F^{\left(w\right)}\left(\omega \right)\right|^{2}}$

${\displaystyle \sum _{\omega }\left|F^{\left(w\right)}\left(\omega \right)\right|^{2}=\sum _{\omega }\sum \limits _{t,t'}e^{i\omega \left(t-t'\right)}f\left(t\right)w\left(t\right)f^{*}\left(t'\right)w\left(t'\right)=N\sum _{t}\left|f\left(t\right)\right|^{2}w^{2}\left(t\right).}$

For constant signal will ${\displaystyle f\left(t\right)=A}$

${\displaystyle \sum _{\omega }\left|F^{\left(w\right)}\left(\omega \right)\right|^{2}=N\left\langle A^{2}\right\rangle \sum _{t}w^{2}\left(t\right).}$

The mean value of the periodogram is also (regardless of ) . With a constant signal, the periodogram provides a peak at frequency . As it increases , this peak becomes higher and narrower. ${\displaystyle N}$ ${\displaystyle \left\langle A^{2}\right\rangle }$ ${\displaystyle 0}$ ${\displaystyle N}$

Rectangle window

In the case of a rectangular window , the Parseval equation applies . Division by gives the mean value of the periodogram . This value is independent provided that this applies to the mean amplitude square. ${\displaystyle w=1}$ ${\displaystyle \sum _{\omega }\left|F\left(\omega \right)\right|^{2}=N^{2}\left\langle A^{2}\right\rangle }$ ${\displaystyle N^{2}}$ ${\displaystyle \sum _{\omega }P\left(\omega \right)/N=\left\langle A^{2}\right\rangle }$ ${\displaystyle N}$ ${\displaystyle \left\langle A^{2}\right\rangle }$

Limitations and improvements

The number of values ​​in the periodogram grows with the window length , but the values ​​are not more precise. In the case of white noise with amplitude , the variance of the periodogram values remains of the order of magnitude with increasing window length . This can be remedied by averaging neighboring values ​​or averaging over several periodograms. ${\displaystyle N}$ ${\displaystyle A}$ ${\displaystyle A^{4}}$

Continuous signal

For a signal defined on the time continuum , the Fourier transform is the product of the signal and the window function ${\displaystyle f(t)}$

${\displaystyle F^{\left(w\right)}\left(\omega \right)=\int \limits _{-\infty }^{\infty }f\left(t\right)w\left(t\right)e^{i\omega t}dt.}$

The periodogram is

${\displaystyle P^{\left(w\right)}\left(\omega \right)={\frac {\left|F^{\left(w\right)}\left(\omega \right)\right|^{2}}{T\int _{-\infty }^{\infty }w^{2}\left(t\right)dt}}.}$

As with the sampled signal, the standard deviation of the periodogram values remains of the same order of magnitude as the values ​​themselves with increasing time series length in the worst case. ${\displaystyle T}$

Individual evidence

1. ↑ Schuster, Arthur: On the investigation of hidden periodicities with application to a supposed 26 day period of meteorological phenomena , Terrestrial Magnetism and Atmospheric Electricity , 3, pp. 13–41, 1898
2. ↑ ^{a b} William H. Press, Saul A. Teukolsky, William T. Vetter Ling, Brian P. Flannery, Michael Metcalf: Numerical Recipes in C . Cambridge University Press , 1992, ISBN 0-521-43108-5 .
3. ^ Monson H. Hayes: Statistical Digital Signal Processing an Modeling . John Wiley & sons, inc., 1996, ISBN 978-0-471-59431-4 .