Spectral Analysis of Time-Limited Observations of Infinitely Long Processes

Abstract

Due to experimental constraints, geophysical observations take place within a finite time interval. In this chapter, therefore, we will investigate the effects of such limitations on the analysis of deterministic nonperiodic processes of infinite length. The limitation to a finite time interval T can be simulated by multiplication of an infinitely long signal s(t) by a window or weighting function that is equal to 1 within the interval T and zero outside this interval. This rectangular function, also called the boxcar function, is defined as follows:

$$ w\left( t \right) = \left\{ {\begin{array}{*{20}{c}} 1 & {for} \\ 0 & {for} \\ \end{array} } \right.\begin{array}{*{20}{c}} {\left| t \right|} \\ {\left| t \right|} \\ \end{array} \begin{array}{*{20}{c}} \leqslant \\ > \\ \end{array} \begin{array}{*{20}{c}} {\frac{T}{2}} \\ {\frac{T}{2}} \\ \end{array} . $$

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