Fundamentals of System Analysis and Synthesis - Digital Signal Processing in Power System Protection and Control - page 46

In all matters concerning digital filters or digital signal processing in general,
the systems which are linear, time invariant and causal are considered in this topic.
The system is called linear if for any pair of equations between input and output
values of the form:

y 1 ð n Þ¼ F f x 1 ð n Þg y 2 ð n Þ¼ F f x 2 ð n Þg ð 4 : 28a Þ

the following relationship holds:

y ð n Þ¼ F f a 1 x 1 ð n Þþ a 2 x 2 ð n Þg ¼ a 1 F f x 1 ð n Þg þ a 2 F f x 2 ð n Þg ¼ a 1 y 1 ð n Þþ a 2 y 2 ð n Þ
ð 4 : 28b Þ

It is easy to check that the system described by the equation
y ð n Þ¼ 0 : 5 ½ x ð n Þþ x ð n 2 Þ is linear but those described by the equations y ð n Þ¼
x 2 ð n Þ or y ð n Þ¼ x ð n j j are not.
The system is called time invariant when the shift of input signals by certain
number of samples results in the same shift of output signals, i.e. when for y 1 ð n Þ¼
F f x 1 ð n Þg and x 2 ð n Þ¼ x 1 ð n þ m Þ one gets:

y 2 ð n Þ¼ F f x 2 ð n Þg ¼ F f x 1 ð n þ m Þg ¼ y 1 ð n þ m Þ: ð 4 : 29 Þ

One can easily check that the systems described by equations y ð n Þ¼ x ð n Þþ
x ð n 2 Þ or y ð n Þ¼ x 2 ð n Þ are time invariant, however, the one given by equation
y ð n Þ¼ nx ð n Þ is not.
A system is called causal if for zero input signal for all instants before given
reference instant, say n = 0, the corresponding output signal for those instants
remains also zero. In other words in a causal system a non-zero output signal may
appear not earlier but after a cause, which means that if x(n) = 0 for n \ 0 then
also y(n) = 0 for n \ 0.

Discrete Convolution
It is easy to notice that, according to discrete signal definition as well as simple and
inverse Z transform equations, any discrete signal can be written in the form:

x ð n Þ¼ X

1

x ð k Þ d ð n k Þ; ð 4 : 30 Þ

k ¼ 0

where

d ð n Þ¼ 1

for n ¼ 0

:

0 for other n

Then the output of any linear system can be expressed as:

( ) ¼ X

y ð n Þ¼ F f x ð n Þg ¼ F X

1 1

x ð k Þ d ð n k Þ x ð k Þ F f d ð n k Þg: ð 4 : 31 Þ

k ¼ 0 k ¼ 0