View source for Spectral window

''of an estimator of the spectral density''

A function of an angular frequency <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086550/s0865501.png" /> defining a weight function used in the non-parametric estimation of the [[Spectral density|spectral density]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086550/s0865502.png" /> of a [[Stationary stochastic process|stationary stochastic process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086550/s0865503.png" /> by smoothing the [[Periodogram|periodogram]] constructed from the observed data of the process. As an estimator of the value of the spectral density at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086550/s0865504.png" /> one usually takes the integral with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086550/s0865505.png" /> of the product of the periodogram at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086550/s0865506.png" /> and an expression of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086550/s0865507.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086550/s0865508.png" /> is a real number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086550/s0865509.png" /> is a fixed function of the frequency which takes its greatest value at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086550/s08655010.png" /> and is such that its integral over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086550/s08655011.png" /> is equal to one. This function is usually called a spectral window generator, while the term  "spectral window"  is used for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086550/s08655012.png" />. The width of the spectral window is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086550/s08655013.png" />, and depends on the size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086550/s08655014.png" /> of the sample (that is, on the length of the observed realization of the process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086550/s08655015.png" />) and tends to zero as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086550/s08655016.png" /> (but more slowly than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086550/s08655017.png" />). The Fourier transform of the spectral window (and in the case of discrete time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086550/s08655018.png" />, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086550/s08655019.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086550/s08655020.png" /> 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|Spectral density, estimator of the]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.B. Blackman,   J.W. Tukey,   "The measurement of power spectra: From the point of view of communications engineering" , Dover, reprint  (1959)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.M. Jenkins,   D.G. Watts,   "Spectral analysis and its applications" , '''1–2''' , Holden-Day  (1968)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  D.R. Brillinger,   "Time series. Data analysis and theory" , Holt, Rinehart &amp; Winston  (1975)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.B. Priestley,   "Spectral analysis and time series" , '''1–2''' , Acad. Press  (1981)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.M. Yaglom,   "Correlation theory of stationary and related random functions" , '''1–2''' , Springer  (1987)  (Translated from Russian)</TD></TR></table>