Spectral window

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).

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)