PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Fourier and Laplace 367

(c) Example (#3), the frequency function F(s) is:

8 s + 30

--------

s^2 + 25

its inverse Laplace transform, f(t) is:

8 cos(5 t) + 6 sin(5 t)

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

(d) Example(#4), the frequency function Y(s) is:

4 s + 10

-----------------

s (s + 1) (s + 2)

its inverse Laplace transform, y(t) is given by:

5 - 6 exp(-t) + exp(-2 t)

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

Note that the results obtained fully agree with the results obtained in R.4.98,

with a signifi cant reduction of mathematical insight and labor.

R.4.119 Let fT(t) be a periodic function (with a period T ≠ 0). Then its LT is given by

£ ft

e

T()
 TsFs()

1

1  

where

Fs e f tdtst

T

()
 T()

0

∫

Observe that this frequency–time relation pair was presented in R.4.82 (#h), using a

slightly different notation. It is important to understand that the LT can be equally

useful in the analysis of periodic, as well as nonperiodic functions, as was the case

of the FT.

R.4.120 A way to evaluate the ILT of a given transfer function H(s) = num/den is by evaluat-

ing its impulse response h(t). Recall that the MATLAB function impulse (num,den,t)

returns the system impulse response, where num and den are the numerator and

denominator polynomials of H(s) entered as row vectors arranged in descending

powers of s, and t defi nes the time range of interest.

R.4.121 For example,

Let Fs

s

ss

s

s

()

()

()





325

(^232)
num
den
then the program that returns the expression for f(t) (f(t) → F(s)), and the plot f(t)
versus t, evaluated by numerical and symbolic techniques, as well as the plot F(s)
versus s, over the ranges: 0 ≤ t ≤ 10, and 0 ≤ s ≤ 10, is given as follows by the
script fi le F f.
MATLAB Solution
% Script file: F f
num = [0 3 25];
den = [1 3 2];
t = 0:.1:10;
f = impulse (num, den, t);
subplot(3,1,1)