# Spectral window

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).