CHAPTER 10 Fourier Analysis of Discrete-Time Signals and Systems...........................
Diligence is the mother of good luck.
Benjamin Franklin (1706–1790)
Printer, inventor, scientist, and diplomat
I am a great believer in luck,
and I find the harder I work,
the more I have of it.
President Thomas Jefferson (1743–1826)
Main author of the U.S. Declaration of Independence10.1 Introduction
In this chapter we will consider the Fourier representation of discrete-time signals and systems. Simi-
lar to the connection between the Laplace and the Fourier transforms of continuous-time signals
and systems, if the region of convergence of the Z-transform of a signal or of the transfer function
of a discrete system includes the unit circle, then the discrete-time Fourier transform (DTFT) of the
signal or the frequency response of the system is easily found. Duality in time and frequency is
used whenever signals and systems do not satisfy this condition. We can thus obtain the Fourier
representation of most discrete-time signals and systems.
Two computational disadvantages of the DTFT are that the direct DTFT is a function of a contin-
uously varying frequency, and the inverse DTFT requires integration. These disadvantages can be
removed by sampling in frequency the DTFT, resulting in the so-called discrete Fourier transform
(DFT) (notice the difference in the naming of these two related frequency representations). An
interesting connection determines their computational feasibility: adiscrete–timesignal has aperi-
odic continuous-frequencytransform—the DTFT—while aperiodic discrete-timesignal has aperiodic and
discrete-frequencytransform—the DFT. As we will discuss in this chapter, any periodic or aperiodic sig-
nal can be represented by the DFT, a computationally feasible transformation where both time and
frequency are discrete and no integration is required, and that can be implemented very efficiently by
the Fast Fourier Transform (FFT) algorithm.
Signals and Systems Using MATLAB®. DOI: 10.1016/B978-0-12-374716-7.00014-4
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