Spectral window

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of an estimator of the spectral density

A function of an angular frequency defining a weight function used in the non-parametric estimation of the spectral density of a stationary stochastic process by smoothing the periodogram constructed from the observed data of the process. As an estimator of the value of the spectral density at a point one usually takes the integral with respect to of the product of the periodogram at and an expression of the form . Here is a real number and is a fixed function of the frequency which takes its greatest value at and is such that its integral over is equal to one. This function is usually called a spectral window generator, while the term "spectral window" is used for the function . The width of the spectral window is , and depends on the size of the sample (that is, on the length of the observed realization of the process ) and tends to zero as (but more slowly than ). The Fourier transform of the spectral window (and in the case of discrete time , when , the set of its Fourier coefficients) is called the lag window of an estimator of the spectral density. It defines a weight function of a discrete or continuous argument (depending on whether is discrete or continuous), by which one must multiply the sample auto-correlations evaluated from the given sample to make the Fourier transform of the resulting product coincide with the desired estimator of the spectral density (cf. Spectral density, estimator of the).


[1] R.B. Blackman, J.W. Tukey, "The measurement of power spectra: From the point of view of communications engineering" , Dover, reprint (1959)
[2] G.M. Jenkins, D.G. Watts, "Spectral analysis and its applications" , 1–2 , Holden-Day (1968)
[3] D.R. Brillinger, "Time series. Data analysis and theory" , Holt, Rinehart & Winston (1975)
[4] M.B. Priestley, "Spectral analysis and time series" , 1–2 , Acad. Press (1981)
[5] A.M. Yaglom, "Correlation theory of stationary and related random functions" , 1–2 , Springer (1987) (Translated from Russian)
How to Cite This Entry:
Spectral window. A.M. Yaglom (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098