320 Practical MATLAB® Applications for Engineers
controls, heat and wave propagation, electronic power supplies, and signal generation, as
well as nonengineering areas such as biology, physiology, economics, and music.
The illogical concept that a continuous varying function such as a sinusoidal could be
used to approximate functions with square corners and discontinuities was fi rst received
by the best minds of the time, such as the members of the French Academy of Science, with
skepticism. Probably this concept will also be received with skepticism by the reader, but
Fourier was right.
The Fourier transform (FT) is an extension of the FS for the case of nonperiodic func-
tions as well as the series; it represents the original function in a format that is, in many
instances, easier to analyze and understand. The FT is a compact, clever, and symmetrical
relation between a function and its Fourier complex expansion.
The transform of a function is referred to as its spectrum, and physically constitutes
a model that is as good as, or better than the function itself. For example, an electrical,
optical, or acoustical wave can be observed and studied using a spectrum analyzer in fre-
quency, as well as an oscilloscope in time.
The Laplace transform (LT), a cousin of the FT, is introduced later in this chapter as a tool
to solve linear, ordinary differential equations (DEs) (with initial conditions).
The main idea is to convert the DE such as the loop or node equations of an electric net-
work from the time to the frequency domain (referred as the s-domain). This conversion
transforms the DE into an algebraic equation. The algebraic frequency domain equation
provides, in many situations, information and insight into a system, not evident in the
time domain, such as stability and its pole/zero constellation, and is, in general, an easier
system model to deal with.
Once a solution of the algebraic equation is found in the frequency domain (in terms
of s), an inverse transform is required to convert the solution from s (frequency) into t (t i me
domain).
For this purpose, a conversion table (Table 4.2) can be used, or if the symbolic MATLAB®
toolbox is available, the commands laplace and ilaplace can be used for the evaluation of
the direct and inverse transformations. The forerunner of the LT method was the opera-
tional calculus, created by Oliver Heaviside (1850–1925), which was used to solve tran-
sients in a circuit described by a set of DEs; a method not fully understood and accepted
by the leading scientists and mathematicians of his time. Heaviside was a gifted practical
engineer who had the physical insight to pick the correct solution from a number of
possible solutions—a heuristic approach that lacked the mathematical rigor. Years after
Heaviside’s publications, his method was rigorously verifi ed by men such as Bromwich,
Giorgi, Carson, and others.
The verifi cation came basically from Laplace’s work in 1807. The Laplace method of solv-
ing DEs is particularly useful in circuit analysis since initial conditions are automatically
incorporated into the equation, as sources, in the fi rst step rather than the last.
The FTs as well as the LTs are sophisticated mathematical techniques that engineers
and scientists have developed over the past 100 years, which are extensively employed
in research and development. Many achievements, discoveries, major contributions, and
applications in modern technology are based on the concepts developed by Fourier and
Laplace in a variety of areas such as analysis and fi ltering of data, image and music pro-
cessing, image and music enhancement, sound effects, system analysis, controls, and many
other applications.
The Fourier and Laplace methods introduced over the past century a new point of view
of system analysis and synthesis, new terminology and new concepts that evolved and
over time became a part of the specialized vocabulary used by engineers and scientists to
communicate with one another.