PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Fourier and Laplace 351

poles =

-1.0000 + 1.0000i

-1.0000 - 1.0000i

-1.0000

**************************************************

The ROC lie in the region given by: real part greater than

real _ s =

-1.0000

**************************************************

Region of convergence

Pole−zero map

1.5

0.5

0

−0.5

− 1

−1.5

− 3 −2.5 − 2 −1.5 − 1 −0.5 0 0.5 1 1.5 2

1

Real axis

Imaginary axis

FIGURE 4.8

Pole/zero plot of R.4.95.

R.4.96 Recall that the linear time invariant (LTI) system transfer function H(s) is a rational

function, given as the ratio of two polynomials in s. These rational functions can be

expressed in terms of a partial fraction expansion, a format that can be used in the

evaluation of the ILT, by using the transformation Table 4.2.

Recall from Chapter 7 of the book entitled Practical MATLAB® Basics for Engi-

neers, that the MATLAB function [r, p, k] = residue(num, den) returns the coeffi cients

(residues) r, the poles p, and the stand-alone term k of the partial fraction expansion

given by the ratio of the num(Y) and den(X) polynomials expressed as vectors con-

sisting of its coeffi cients arranged in descending powers of s.

Recall also that the poles can be distinct real, repeated, and complex, where com-

plex poles always occur as conjugate pairs. In the evaluation process of the LT of a

real function of t, distinct real poles are rather easy to deal with (see Chapter 7 of

the book entitled Practical MATLAB® Basics for Engineers).

When repeated poles are present in the form (s + a)n, the partial fraction expan-

sion must include the following terms: b 1 /(s + a), b 2 /(s + a)^2 , ..., bn/(s + a)n.

R.4.97 For example, let

Hs

ssss

ssss

()





573530

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422

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