Signals and Systems - Electrical Engineering

CHAPTER 3 The Laplace Transform............................................................................

What we know is not much.

What we do not know is immense.

Pierre-Simon marquis de Laplace (1749–1827)

French mathematician and astronomer

3.1 Introduction

The material in this chapter is very significant for the analysis of continuous-time signals and systems.
The main issues discussed are:

n Frequency domain analysis of continuous-time signals and systems—We begin the frequency domain
analysis of continuous-time signals and systems using transforms. The Laplace transform, the
most general of these transforms, will be followed by the Fourier transform. Both provide com-
plementary representations of a signal to its own in the time domain, and an algebraic character-
ization of systems. The Laplace transform depends on a complex variables=σ+j, composed
of dampingσand frequency, while the Fourier transform considers only frequency.
n Damping and frequency characterization of continuous-time signals—The growth or decay of a signal—
damping—as well as its repetitive nature—frequency—in the time domain are characterized in
the Laplace domain by the location of the roots of the numerator and denominator, or zeros and
poles, of the Laplace transform of the signal.
n Transfer function characterization of continuous-time LTI systems—The Laplace transform provides a
significant algebraic characterization of continuous-time systems: The ratio of the Laplace trans-
form of the output to that of the input—or the transfer function of the system. It unifies the
convolution integral and the differential equations system representations. The concept of trans-
fer function is not only useful in analysis but also in design, as we will see later. The location
of the poles and the zeros of the transfer function relates to the dynamic characteristics of the
system.
n Stability, and transient and steady-state responses—Certain characteristics of continuous-time sys-
tems can only be verified or understood via the Laplace transform. Such is the case of stability,

Signals and Systems Using MATLAB®. DOI: 10.1016/B978-0-12-374716-7.00006-5
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