# Spectral window

*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).

#### References

[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: http://www.encyclopediaofmath.org/index.php?title=Spectral_window&oldid=13835