Digital Signal Processing: A Practitioner's Approach

2

Revisiting the Basics

In this chapter we revisit the basics, covering linear systems, some important relevant concepts in number systems, basics of optimisation [1–1], 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. As an abstraction, the behaviour of the system when two input conditions occur simultaneously is the same as the sum of the outputs if the two occur separately. Mathematically, for example, consider a function f(x) whose values at x₁ and x₂ are f(x₁) and f(x₂), then

This is the law of superposition. The scaling property demands a proportionality ...