Revisiting the Basics - Digital Signal Processing: A Practitioner's Approach - page 22

This is the law of scaling. We see that a simple function, which looks apparently
linear, fails to adhere to these two basic rules. Consider f ð x Þ¼ 2x þ 5. We have
f ð 2 Þ¼ 9 and f ð 5 Þ¼ 15 but f ð 5 þ 2 Þ¼ 19 which is not the same as f ð 2 Þþ f ð 5 Þ

since 19 6¼ 24. Let us look at f ð 3 2 Þ¼ f ð 6 Þ¼ 17 and 3 f ð 2 Þ¼ 27. This
function fails to obey both the rules, hence it is not linear. We can modify this
function by a simple transformation g ð x Þ¼ f ð x

;

5

2 Þ and make it linear.

2.1.1 Linear Systems
Consider a single-input single-output (SISO) system that can be excited by a
sequence u k resulting in an output sequence y k . It is depicted in Figure 2.1. The

u k y k

System

Figure 2.1 A single-input single-output system

sequence u k could be finite while y k could be infinite, which is common. For the
system to be linear,

u k ¼) y k ;
u k ¼) y k ;
ð u k þ u k Þ¼) y k ;
ð au k Þ¼) y k :

ð 2 : 3 Þ

Thus, we must have y k ¼ y k þ y k and y k ¼ ay k , where a is an arbitrary constant.
Consider two cascaded linear systems A and B with input u k and output y k .
Linearity demands that exchanging systems will not alter the output for the same
input. This is indicated in Figure 2.2.

y k

u k

System A System B

u k y k

System B System A

Figure 2.2 Commutative property

2.1.2 Sinusoidal Inputs
Under steady state, linear systems display wonderful properties for sinusoidal
inputs. If u k ¼ cos ð 2

fk Þ then the output is also a sinusoidal signal given as

fk þ Þ . This output also can be written as y k ¼ y i k þ y k , where

y k ¼ A cos ð 2

fk Þ and y k ¼ B 1 sin ð 2 y i k ¼ A 1 cos ð 2

fk Þ . Mathematicians represent the sequence