Signals and Systems - Electrical Engineering

3.4 Inverse Laplace Transform 203

Remarks

n An equivalent partial fraction expansion consists in expressing the numerator N(s)of X(s), for some
constants a and b, as N(s)=a+b(s+α), a first-order polynomial, so that

X(s)=

a+b(s+α)

(s+α)^2 +^20

=

a

 0

 0

(s+α)^2 +^20

+b

s+α

(s+α)^2 +^20

so that the inverse is a sum of a sine and a cosine multiplied by a decaying exponential. The inverse

Laplace transform is

x(t)=

[

a

 0

e−αtsin( 0 t)+be−αtcos( 0 t)

]

u(t)

which can be simplified, using the sum of phasors corresponding to sine and cosine, to

x(t)=

√

a^2

^20

+b^2 e−αtcos

(

 0 t−tan−^1

(

a

 0 b

))

u(t)

n Whenα= 0 the above indicates that the inverse Laplace transform of

X(s)=

a+bs

s^2 +^20

is

x(t)=

√

a^2

^20

+b^2 cos

(

 0 t−tan−^1

(

a

 0 b

))

u(t)

which is transform of a cosine with a phase shift not commonly found in tables.
n When the frequency 0 = 0 , we get that the inverse Laplace transform of

X(s)=

a+b(s+α)

(s+α)^2

=

a

(s+α)^2

+

b

s+α

(corresponds to a double pole at−α) is

x(t)= lim

 0 → 0

[

a

 0

e−αtsin( 0 t)+be−αtcos( 0 t)

]

u(t)

=[ate−αt+be−αt]u(t)

where the first limit is found by L’Hˆopital’s rule. Notice that when computing the partial fraction expansion

of the double pole s=−αthe expansion is composed of two terms, one with denominator(s+α)^2 and

the other with denominator s+αof which the sum gives a first-order numerator and a second-order

denominator to satisfy the proper rational condition.