Signals and Systems - Electrical Engineering

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

whereσmaxis the maximum of the real parts of the poles ofX(s). Since in this section we only

consider causal signals, the region of convergence will be assumed and will not be shown with the

Laplace transform.

The most common inverse Laplace method is the so-calledpartial fraction expansion, which consists in

expanding the given function insinto a sum of components of which the inverse Laplace transforms

can be found in a table of Laplace transform pairs. Assume the signal we wish to find has a rational

Laplace transform—that is,

X(s)=

N(s)

D(s)

(3.18)

whereN(s)andD(s)are polynomials inswith real-valued coefficients. In order for the partial fraction

expansion to be possible, it is required thatX(s)beproper rational, which means that the degree of

the numerator polynomialN(s)is less than that of the denominator polynomialD(s). IfX(s)is not

proper, then we need to do long division until we obtain a proper rational function—that is,

X(s)=g 0 +g 1 s+···+gmsm+

B(s)

D(s)

(3.19)

where the degree ofB(s)is now less than that ofD(s)—so that we can perform partial expansion for

B(s)/D(s). The inverse ofX(s)is then given by

x(t)=g 0 δ(t)+g 1

dδ(t)

dt

+···+gm

dmδ(t)

dtm

+L−^1

[

B(s)

D(s)

]

(3.20)

The presence ofδ(t)and its derivatives (called doublets, triplets, etc.) are very rare in actual signals,

and as such the typical rational function has a numerator polynomial that is of lower degree than the

denominator polynomial.

Remarks

n Things to remember before performing the inversion are:

n The poles of X(s)provide the basic characteristics of the signal x(t).

n If N(s)and D(s)are polynomials in s with real coefficients, then the zeros and poles of X(s)are real

and/or complex conjugate pairs, and can be simple or multiple.

n In the inverse, u(t)should be included since the result of the inverse is causal—the function u(t)is an

integral part of the inverse.

n The basic idea of the partial expansion is to decompose proper rational functions into a sum of rational

components of which the inverse transform can be found directly in tables. Table 3.1 displays common

one-sided Laplace transform pairs, while Table 3.2 provides properties of the one-sided Laplace transform.

We will consider now how to obtain a partial fraction expansion when the poles are real, simple and

multiple, and in complex conjugate pairs, simple and multiple.