Revisiting the Basics - Digital Signal Processing: A Practitioner's Approach - page 21

2

Revisiting the Basics

In this chapter we revisit the basics, covering linear systems, some important
relevant concepts in number systems, basics of optimisation [1-3], fundamentals of
random variables, and general engineering background with illustrations. Algebra is
the best language by which an engineer can communicate with precision and accuracy,
hence we have used it extensively where normal communication is difficult.
A good understanding of linear system [4] is the basic foundation of digital signal
processing. In reality only very few systems are linear and most of the systems we
encounter in real life are non-linear. Human beings are non-linear and their
behaviour cannot be predicted accurately based on past actions. In spite of this,
it is essential to thoroughly understand linear systems, with an intention to
approximate a complex system by a piecewise linear model. Sometimes we
could also have short-duration linear systems, in a temporal sense.

2.1 Linearity
The underlying principles of linear systems are superposition and scaling.Asan
abstraction, the behaviour of the system when two input conditions occur simulta-
neously is the same as the sum of the outputs if the two occur separately.
Mathematically, for example, consider a function f ð x Þ whose values at x 1 and x 2
are f ð x 1 Þ and f ð x 2 Þ , then

f ð x 1 þ x 2 Þ¼ f ð x 1 Þþ f ð x 2 Þ:

ð 2 :

1 Þ

This is the law of superposition. The scaling property demands a proportionality
feature from the system. If the function value at ax is f ð ax Þ , then this property
demands

af ð x Þ¼ f ð ax Þ: ð 2 : 2 Þ