Statistical Variables

Abstract

Consider a series expansion of a signal F(θ):

$$ F(\theta ) = \sum\limits_{j = 0}^\infty {a(j)f(j,\theta )} $$

There are three basic operations that can be distinguished with the help of this expansion: filtering, shifting and signal design. A filtered signal F_f (θ) is obtained by multiplying a(j) with an attenuation function K(j) and by time shifting f(j,θ) by θ(j):

$$ {F_f}(\theta ) = \sum\limits_{j = 0}^\infty {K(j)a(j)f[j,\theta - \theta (j)]} $$