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