458 Practical MATLAB® Applications for Engineers
Some historians traced the basis of the algorithm used in this research paper to the great
nineteenth century German mathematician, Carl Friedrich Gauss. In any case, this paper
and the subsequent research and publications led to a set of techniques known collectively
as FFT, or radix 2 dissemination, in which the computation effi ciency was highly improved
from 4N^2 + 2N(2N − 1) to 2Nlog 2 (N) + 3Nlog 2 (N) operations in the evaluations of the DFT
using the fft, where N denotes the length of the discrete sequence.
The other discrete transform, such as the ZT, was developed by the discrete control
research group at Columbia University, integrated by professors and graduate students, in
the early 1950s and gained acceptance in the 1960s. The ZT is equivalent to the LT for the
analog system model, in the sense that a difference equation can be transformed into an
algebraic equation.
The discrete signals and systems are analogous to the continuous case in its analysis,
synthesis, and representation in the traditional two domains—time and frequency.
Recall that in the continuous case, differential equations were used to describe a net-
work behavior in the time domain, and an algebraic equivalent relation (equation) was
used to describe the system in the frequency domain (Fourier and Laplace in Chapter 4 of
this book). Discrete systems analysis synthesis follows the same general approach.
The concepts of poles, zeros, transfer function, convergence, impulse response, step
response; the convolution process in time and frequency, stability, superposition, power,
and energy considerations; and other system concepts and techniques developed for the
case of continuous systems can be extended with minor modifi cations into the analysis and
synthesis approach used in discrete-time systems. Principles, concepts, properties, and
general analytical tools and techniques of the different transforms and their applications
will be stated, tested, and explored in this chapter by using the power of MATLAB®.
5.2 Objectives
After completing this chapter, the reader should be able to
Express a discrete system in terms of an input and output sequences by means of
a sampling rule
Know the defi ning characteristics of discrete systems such as linear, time invari-
ance, causal, stable, passive, SISO, and bounded input–bounded output (BIBO)
Know the discrete-system elements such as unit delay, unit shift, and multiplier
Transform an input sequence f(n) into an output sequence g(n) by means of a
difference equation
Derive the discrete-system transfer function from a given difference equation
Determine the conditions for system stability in both domains—time and frequency
Use the ZT as the tool to describe the behavior of a discrete-time system or signal• • • • • • •
FIGURE 5.1
System block diagram.
f(n) g(n)Input OutputDiscrete-time system [H]