Signals and Systems - Electrical Engineering

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

(c) The inverse Laplace transform of

X(s)=

3 s− 4

s(s+ 1 )(s+ 2 )

should give a response of the form

x(t)=[Ae−t+B+Ce−^2 t]u(t)

Find the values ofA,B, andC. Use the MATLAB functionilaplaceto get the inverse.

3.9. Steady state and transient

Consider the following cases where we want to determine either the steady state, transient, or both.

(a) Without computing the inverse of the Laplace transform

X(s)=

1

s(s^2 + 2 s+ 10 )

corresponding to a causal signalx(t), determine its steady-state solution. What is the value ofx( 0 )?

Show how to obtain this value without computing the inverse Laplace transform.

(b) The Laplace transform of the output of an LTI system is

Y(s)=

1

s((s+ 2 )^2 + 1 )

What would be the steady-state responseyss(t)?

(c) The Laplace transform of the output of an LTI system is

Y(s)=

e−s

s((s− 2 )^2 + 1 )

How would you determine if there is a steady state or not? Explain.

(d) The Laplace transform of the output of an LTI system is

Y(s)=

s+ 1

s((s+ 1 )^2 + 1 )

Determine the steady-state and the transient responses corresponding toY(s).

3.10. Inverse Laplace transformation—MATLAB

Consider the following inverse Laplace problems using MATLAB for causal signalx(t):

(a) Use MATLAB to compute the inverse Laplace transform of

X(s)=

s^2 + 2 s+ 1

s(s+ 1 )(s^2 + 10 s+ 50 )

and determine the value ofx(t)in the steady state. How would you be able to obtain this value without

computing the inverse? Explain

(b) Find the poles and zeros of

X(s)=

( 1 −se−s)

s(s+ 2 )

Find the inverse Laplace transformx(t)(use MATLAB to verify your result).