Spectral window

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of an estimator of the spectral density

A function of an angular frequency $ \lambda $ defining a weight function used in the non-parametric estimation of the spectral density $ f ( \lambda ) $ of a stationary stochastic process $ X ( t) $ 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 $ \lambda _ {0} $ one usually takes the integral with respect to $ d \lambda $ of the product of the periodogram at $ \lambda $ and an expression of the form $ B _ {N} A ( B _ {N} ( \lambda - \lambda _ {0} ) ) $. Here $ B _ {N} $ is a real number and $ A ( \lambda ) $ is a fixed function of the frequency which takes its greatest value at $ \lambda = 0 $ and is such that its integral over $ \lambda $ is equal to one. This function is usually called a spectral window generator, while the term "spectral window" is used for the function $ B _ {N} A ( B _ {N} \lambda ) $. The width of the spectral window is $ B _ {N} ^ {-} 1 $, and depends on the size $ N $ of the sample (that is, on the length of the observed realization of the process $ X ( t) $) and tends to zero as $ N \rightarrow \infty $( but more slowly than $ N ^ {-} 1 $). The Fourier transform of the spectral window (and in the case of discrete time $ t $, when $ - \pi \leq \lambda < \pi $, 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 $ t $ 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. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.M. Yaglom (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article