Signals and Systems - Electrical Engineering

202 C H A P T E R 3: The Laplace Transform

Simple Complex Conjugate Poles

The partial fraction expansion of a proper rational function

X(s)=

N(s)

(s+α)^2 +^20

=

N(s)

(s+α−j 0 )(s+α+j 0 )

(3.24)

with complex conjugate poles{s1,2=−α±j 0 }is given by

X(s)=

A

s+α−j 0

+

A∗

s+α+j 0

where

A=X(s)(s+α−j 0 )|s=−α+j 0 =|A|ejθ

so that the inverse is the function

x(t)= 2 |A|e−αtcos( 0 t+θ)u(t) (3.25)

Because the numerator and the denominator polynomials ofX(s)have real coefficients, the zeros

and poles whenever complex appear as complex conjugate pairs. One could thus think of the case

of a pair of complex conjugate poles as similar to the case of two simple real poles presented above.

Notice that the numeratorN(s)must be a first-order polynomial forX(s)to be proper rational. The

poles ofX(s),s1,2=−α±j 0 , indicate that the signalx(t)will have an exponentiale−αt, given that

the real part of the poles is−α, multiplied by a sinusoid of frequency 0 , given that the imaginary

parts of the poles are± 0. We have the expansion

X(s)=

A

s+α−j 0

+

A∗

s+α+j 0

where the expansion coefficients are complex conjugate of each other. From the pole information,

the general form of the inverse is

x(t)=Ke−αtcos( 0 t+8)u(t)

for some constantsKand 8. As before, we can findAas

A=X(s)(s+α−j 0 )|s=−α+j 0 =|A|ejθ

and thatX(s)(s+α+j 0 )|s=−α−j 0 =A∗can be easily verified. Then the inverse transform is given by

x(t)=

[

Ae−(α−j^0 )t+A∗e−(α+j^0 )t

]

u(t)

=|A|e−αt(ej(^0 t+θ)+e−j(^0 t+θ))u(t)

= 2 |A|e−αtcos( 0 t+θ)u(t).