Signals and Systems - Electrical Engineering

3.3 The One-Sided Laplace Transform 179

=

−ejθ

s−j 0

e−σt−j(−^0 )t|∞t= 0 =

ejθ

s−j 0

ROC: σ > 0

According to Euler’s identity

cos( 0 t+θ)=

ej(^0 t+θ)+e−j(^0 t+θ)

2

by the linearity of the integral and using the above result, we get that

L[cos( 0 t+θ)u(t)]=0.5L[ej(^0 t+θ)u(t)]+0.5L[e−j(^0 t+θ)u(t)]

=0.5

ejθ(s+j 0 )+e−jθ(s−j 0 )

s^2 +^20

=

scos(θ)− 0 sin(θ)

s^2 +^20

and a region of convergence{(σ,):σ >0,−∞<  <∞}.

Now if we letθ=0,−π/2 in the above equation we have the following Laplace transforms:

L[cos( 0 t)u(t)]=

s

s^2 +^20

L[sin( 0 t)u(t)]=

 0

s^2 +^20

as cos( 0 t−π/ 2 )=sin( 0 t). The ROC of the above Laplace transforms is{(σ,):σ >0,−∞<

 <∞}, or the open right-hands-plane (i.e., not including thejaxis). See Figure 3.6 for the

pole-zero plots and the corresponding signals forθ=0,θ=π/4, and 0 =2. n

nExample 3.4

Use MATLAB symbolic computation to find the Laplace transform of a real exponential,x(t)=

e−tu(t), and ofx(t)modulated by a cosine ory(t)=e−tcos( 10 t)u(t). Plot the signals and the poles

and zeros of their Laplace transforms.

Solution

The following script is used. The MATLAB functionlaplaceis used for the computation of the

Laplace transform and the functionezplotallows us to do the plotting. For the plotting of the poles

and zeros we use our functionsplane. When you run the script you obtain the Laplace transforms

X(s)=

1

s+ 1

Y(s)=

s+ 1

s^2 + 2 s+ 101