Time-Domain Analysis and z Transform - Introduction to Digital Signal Processing and Filter Design - page 93

where

x = 0 . 1000 0 . 2100 − 0 . 1340 − 0 . 0514

r =

0000 . 0876 0 . 0257

From these output data, we get

0 . 1 + 0 . 25 z − 1

= 0 . 1 + 0 . 21 z − 1 − 0 . 134 z − 2 − 0 . 0514 z − 3 X(z) =

1 + 0 . 4 z − 1 + 0 . 5 z − 2

0 . 086 z − 4 0 . 0257 z − 5

−

+ .

1 + 0 . 4 z − 1 + 0 . 5 z − 2

Therefore we get x 0 = 0 . 1 ,x 1 = 0 . 21 ,x 2 =− 0 . 134 ,x 3 =− 0 . 0514, which agrees
with the result obtained from long division, by hand calculation. Note that the
vector b hastobeaugmentedby( n

− 1) zeros in the above program above, as
pointed out by Ifeachor and Jervis [6].
Students may find it useful to know the following additional MATLAB func-
tions in their analysis of discrete-time systems, in addition to those used in the
examples above presented. Given a vector of zeros, the coefficients of the poly-
nomial having these zeros is obtained by the function poly . A complex number
entered as a zero must be accompanied by its conjugate so that the coefficients
become real. Given the coefficients of the polynomial in a row vector, its zeros
are found from the function roots . The poles and zeros of a rational function
F(z) are plotted in the z plane by the function zplane . Two other functions
that may be interesting to the students are tfdata and tf . Typing the com-
mands help poly, help roots , help zplane , help tfdata ,and help tf
will display the details for using these commands. A list of all MATLAB func-
tions available in the Signal Processing Toolbox is displayed when the command
help signal is typed in the command window and is given in the MATLAB
primer in Chapter 9. Typing Type functionname displays the MATLAB code
as well as the help manual for the function where functionname is the name of
the function. Using the help command, students become familiar with and pro-
ficient in the use of MATLAB functions that are available for conducting many
tasks in the analysis and design of discrete-time systems. It is only by trying as
many functions in MATLAB and the Signal Processing Toolbox as possible that
one becomes familiar with and proficient in their use, and the topics by Ingle and
Proakis [9] and Mitra [10] are highly recommended for this purpose, in addition
to the functions we have included in this textbook.

2.10 SUMMARY

In this chapter, we have described several ways of modeling linear shift-invariant
discrete-time systems, highlighting that we should learn how to obtain the one