This preview shows pages 1–3. Sign up to view the full content.
255
Chapter 9
The Laplace Transform
Chapter Outline
9.1 DEFINITION OF THE LAPLACE TRANSFORM.
..................................
256
9.1.1 Definitions.
...................................................................................................................................
256
9.1.2 Existence of the Laplace Transform .
...............................................................................
258
9.1.3 The Impulse Function.
............................................................................................................
260
9.1.4 Region of Convergence.
.........................................................................................................
260
9.2 PROPERTIES OF THE LAPLACE TRANSFORM.
................................
262
9.3 PARTIAL FRACTION EXPANSION.
.....................................................
268
9.3.1 Definition.
.....................................................................................................................................
268
9.3.2 Partial Fraction Inversion.
.....................................................................................................
271
9.3.3 MATLAB Experiments.
..........................................................................................................
274
9.4 LAPLACE TRANSFORM SOLUTION TO DIFFERENTIAL
EQUATIONS .
.......................................................................................
274
9.4.1 Solving Differential Equations.
...........................................................................................
274
9.4.2 Implications of the Pole Locations.
..................................................................................
276
9.5 RELATIONSHIP TO FOURIER TRANSFORMS.
...................................
279
9.6 CHAPTER SUMMARY.
........................................................................
282
9.7 HOMEWORK FOR CHAPTER 9.
..........................................................
286
A signal is a function that represents the time variation of a physical variable.
To
this point we have introduced two quite distinct representations of a signal.
The first
class of representations of a signal, developed in Chapter 5, is the class of functions
that depend on time.
This type of representation is quite obvious.
The second class
of signal representations, derived in Chapter 7, is based on the Fourier transform of
the signal representations in the first class (when the Fourier transform exists).
These signal representations depend on a real frequency variable
ϖ
.
In this chapter
we will define a third class of signal representations that depend on a complex
variable
s
.
This class of signal representations is the Laplace transform of the first
class of signal representations (when the Laplace transform exists).
This third class
of signal representations, like the second class of signal representations, doesn’t
represent the signals directly in the time domain.
There are many advantages and
insights to this type of signal representation, which we will develop in the coming
chapters.
We can make similar comments about system representations, which we
will discuss in detail in Chapters 10  15.
In this chapter we introduce the Laplace transform as a mathematical tool for
analyzing signals and systems.
After the definition of the Laplace transform the
properties of the transform are developed.
Finally, these properties are applied to the
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document256
Chapter 9
The Laplace Transform
solution of a differential equation.
The last section of this chapter discusses the
relationship between the definitions of the Laplace and Fourier transforms.
In
keeping with the philosophy of the text, the primary focus of this chapter is on the
mathematical details of the Laplace transform.
In the following chapters we will
apply this transform to the analysis of signals and systems.
Computer Usage
:
The Laplace
transform has also appeared as a useful
representation for signals and systems for computer packages.
Hence, knowledge of
the transform is required for effective use of many computer packages.
Summary of Sections
Section 9.1:
We define the Laplace transform.
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '09
 MEEHAN

Click to edit the document details