Signals and Systems - Electrical Engineering

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

u = sym(’Heaviside(t)’)

U=laplace(u)

% Delta function

d = sym(’Dirac(t)’)

D = laplace(d)

giving

u = Heaviside(t)

U = 1/s

d = Dirac(t)si

D = 1

whereUandDstand for the Laplace transforms ofuandd. The naming ofu(t)andδ(t)as Heaviside

and Dirac functions is used in MATLAB.^3

The pulsep(t)=u(t)−u(t− 1 )is a finite support signal and so its ROC is the wholes-plane. Its

Laplace transform is

P(s)=L[u(t+ 1 )−u(t− 1 )]=

∫^1

− 1

e−stdt=

−e−st

s

|^1 t=− 1 =

1

s

[es−e−s]=

es

s

[1−e−^2 s]

which as shown before has an infinite number of zeros, and the one at the origin cancels the pole,

so that

P(s)=

∏∞

k=−∞,k6= 0

(s−jπk)

n

nExample 3.3

Let us find and use the Laplace transform ofej(^0 t+θ)u(t)to obtain the Laplace transform ofx(t)=

cos( 0 t+θ)u(t). Consider the special cases forθ=0 andθ=−π/2. Determine the ROCs. Use

MATLAB to plot the signals and the corresponding poles/zeros when 0 =2 andθ=0 andπ/4.

Solution

The Laplace transform of the complex causal signalej(^0 t+θ)u(t)is found to be

L[ej(^0 t+θ)u(t)]=

∫∞

0

ej(^0 t+θ)e−stdt=ejθ

∫∞

0

e−(s−j^0 )tdt

(^3) Oliver Heaviside (1850–1925) was an English electrical engineer who adapted the Laplace transform to the solution of differential
equations (the so-called operational calculus), while Paul Dirac (1902–1984) was also an English electrical engineer, better known for
his work in physics.