Appendices to Part I

Abstract

A linear system or filter is a linear, homogeneous function of time which transforms an input signal (u_n)_{n ∈ ℤ} to an output signal (y_n)_{n ∈ ℤ} thus it may be represented by the convolution

$$ {y_n} = \sum\limits_{{k \in \mathbb{Z}}} {{s_k}{u_{{n - k}}}} $$

(6.1.1)

(6.1.1) where S ≔ (s_k)_{k ∈ ℤ} is the impulse response of the system. To avoid convergence problems, we shall assume in this paragraph that the entries (u_n) are almost always zero, which means that they are zero except for a finite number of instants n.