xiv Preface
Our approach of presenting first the Laplace transform and then the Fourier series and Fourier transform is
justified by several reasons. For one, students coming into a signals and systems course have been familiarized
with the Laplace transform in their previous circuits or differential equations courses, and will continue using
it in control courses. So expertise in this topic is important and the learned material will stay with them longer.
Another is that a common difficulty students have in applying the Fourier series and the Fourier transform is
connected with the required integration. The Laplace transform can be used not only to sidestep the integration
but to provide a more comprehensive understanding of the frequency representation. By asking students to
consider the two-sided Laplace transform and the significance of its region of convergence, they will appreciate
better the Fourier representation as a special case of Laplace’s in many cases. More importantly, these transforms
can be seen as a continuum rather than as different transforms. It also makes theoretical sense to deal with
the Laplace representation of systems first to justify the existence of the steady-state solution considered in the
Fourier representations, which would not exist unless stability of the system is guaranteed, and stability can only
be tested using the Laplace transform. The paradigm of interest is the connection of transient and steady-state
responses that must be understood by students before they can understand the connections between Fourier and
Laplace analyses.
Chapter 6 presents applications of the Laplace and the Fourier transforms to control, communications, and fil-
tering. The intent of the chapter is to motivate interest in these areas. The chapter illustrates the significance of
the concepts of transfer function, response of systems, and stability in control, and of modulation in communi-
cations. An introduction to analog filtering is provided. Analytic as well as MATLAB examples illustrate different
applications to control, communications, and filter design.
Using the sampling theory as a bridge, the third part of the book covers the theory and illustrates the application
of discrete-time signals and systems. Chapter 7 presents the theory of sampling: the conditions under which the
signal does not lose information in the sampling process and the recovery of the analog signal from the sampled
signal. Once the basic concepts are given, the analog-to-digital and digital-to-analog converters are considered
to provide a practical understanding of the conversion of analog-to-digital and digital-to-analog signals.
Discrete-time signals and systems are discussed in Chapter 8, while Chapter 9 introduces the Z-transform.
Although the treatment of discrete-time signals and systems in Chapter 8 mirrors that of continuous-time sig-
nals and systems, special emphasis is given in this chapter to issues that are different in the two domains. Issues
such as the discrete nature of the time, the periodicity of the discrete frequency, the possible lack of periodicity
of discrete sinusoids, etc. are considered. Chapter 9 provides the basic theory of the Z-transform and how it
relates to the Laplace transform. The material in this chapter bears similarity to the one on the Laplace trans-
form in terms of operational solution of difference equations, transfer function, and the significance of poles and
zeros.
Chapter 10 presents the Fourier analysis of discrete signals and systems. Given the accumulated experience of
the students with continuous-time signals and systems, we build the discrete-time Fourier transform (DTFT) on
the Z-transform and consider special cases where the Z-transform cannot be used. The discrete Fourier transform
(DFT) is obtained from the Fourier series of discrete-time signals and sampling in frequency. The DFT will be
of great significance in digital signal processing. The computation of the DFT of periodic and aperiodic discrete-
time signals using the fast Fourier transform (FFT) is illustrated. The FFT is an efficient algorithm for computing
the DFT, and some of the basics of this algorithm are discussed in Chapter 12.
Chapter 11 introduces students to discrete filtering, thus extending the analog filtering in Chapter 6. In this
chapter we show how to use the theory of analog filters to design recursive discrete low-pass filters. Frequency
transformations are then presented to show how to obtain different types of filters from low-pass prototype
filters. The design of finite-impulse filters using the window method is considered next. Finally, the implementa-
tion of recursive and nonrecursive filters is shown using some basic techniques. By using MATLAB for the design
of recursive and nonrecursive discrete filters, it is expected that students will be motivated to pursue on their
own the use of more sophisticated filter designs.