SISO

linear systems

superposition principle

linear time-invariant

LTI

transfer function

Chapter 1

Systems, Sampling and

Quantization

1.1

Continuous-Time Systems

Sound is usually considered as a mono-dimensional signal (i.e., a function of

time) representing the air pressure in the ear canal. For the purpose of this

book, a Single-Input Single-Output (SISO) System is defined as any algorithm

or device that takes a signal in input and produces a signal in output. Most of

our discussion will regard linear systems, that can be defined as those systems

for which the superposition principle holds:

Superposition Principle : if y

1

and y

2

are the responses to the input se-

quences x

1

and x

2

, respectively, then the input ax

1

+ bx

2

produces the

response ay

1

+ by

2

.

The superposition principle allows us to study the behavior of a linear sys-

tem starting from test signals such as impulses or sinusoids, and obtaining the

responses to complicated signals by weighted sums of the basic responses.

A linear system is said to be linear time-invariant (LTI), if a time shift in

the input results in the same time shift in the output or, in other words, if it

does not change its behavior in time.

Any continuous-time LTI system can be described by a differential equation.

The Laplace transform, defined in appendix A.8.1 is a mathematical tool that is

used to analyze continuous-time LTI systems, since it allows to transform com-

plicated differential equations into ratios of polynomials of a complex variable

s. Such ratio of polynomials is called the transfer function of the LTI system.
Example

1. Consider the LTI system having as input and output the

functions of time (i.e., the signals) x(t) and y(t), respectively, and described by

the differential equation

dy

dt

- s

0

y = x .

(1)

This equation, transformed into the Laplace domain according to the rules of

appendix A.8.1, becomes
sY

L

(s) - s

0

Y

L

(s) = X

L

(s) .

(2)

1