Tutorials - Digital Signal Processing: An Introduction with MATLAB and Applications - page 256

Similarly we get: b = 6 and c =-9/2.
From Tables we find L 1

of Y(s) as follows:

y ð t Þ¼ ae t þ be 2t þ ce 3t ¼ 1

2 e t þ 6e 2t 9 2 e 3t :

MATLAB: The above question can be solved on MATLAB r as follows:

A ¼½ 21312 ;B ¼½ 16116 ; ½ R ; P ; K ¼ residue ð A ; B Þ

where R gives the coefficients c i , P gives the poles p i , and K is empty if degree( A )
\ degree( B ). Note that B can be obtained using B = conv([1 1],[1 5 6])
{multiplication of two polynomials}.

Tutorial 22

Q: A linear electrical system is described by the second-order differential

equation:

y 00 ð t Þþ 5y 0 ð t Þþ 6y ð t Þ¼ x ð t Þ

ð 1 Þ

with initial conditions:

y ð 0 Þ¼ 2; y 0 ð 0 Þ¼ 1 :

If the input voltage is given by x(t) = e -t u(t), find the output y(t).

Solution: From L Tables we have:

y 0 ð t Þ L sY ð s Þ y ð 0 Þ¼ sY ð s Þ 2
y 00 ð t Þ L s ½ sY ð s Þ y ð 0 Þ y 0 ð 0 Þ¼ s 2 Y ð s Þ 2s 1
x ð t Þ¼ e t u ð t Þ L X ð s Þ¼ 1
s þ 1

Taking L of both sides of Eq. 1, we get:

½ s 2 Y ð s Þ 2s 1 þ 5 ½ sY ð s Þ 2 þ 6Y ð s Þ¼ 1
s þ 1

s þ 1 ¼ 2s 2 þ 13s þ 12

Y ð s Þ½ s 2 þ 5s þ 6 ¼ 2s þ 11 þ 1

s þ 1
) Y ð s Þ¼ 2s 2 þ 13s þ 12
ð s þ 1 Þð s 2 þ 5s þ 6 Þ
) y ð t Þ¼ 1

2 e t þ 6e 2t 9 2 e 3t ð From Tutorial 21 Þ