Laplace transform - Continuous and Discrete Time Signals and Systems - page 306

Solution
The Laplace transform of the input signal x ( t )isgivenby

L sin(3 t ) u ( t ) = 3

s 2 + 9 .
The Laplace transfer function of the RC series circuit is given by

X ( s ) =

= 1

1 + sCR .
Substituting the value of the product CR = 0 . 5 yields

H ( s ) = H ( ω ) j ω= s

1

= 2

s + 2 .
The Laplace transform Y ( s ) of the output signal is given by

H ( s ) =

1 + 0 . 5 s

6
( s + 2) ( s 2 + 9)

6
13( s + 2)

6 s − 12
13( s 2 + 9)

Y ( s ) = H ( s ) X ( s ) = ≡ −

or

3
( s 2 + 9) .
Taking the inverse transform leads to the following expression for the overall
output in the time domain:

6
13( s + 2)

6
13

s
( s 2 + 9)

4
13

Y ( s ) = − +

6
13 e

6
13 cos(3 t ) +

4
13 sin(3 t )

− 2 t −

y ( t ) =

6
13 e

√

− 2 t ◦

u ( t ) = +

13 sin(3 t − 56

) u ( t ) .

The steady state value of the output is computed by applying the limit t →∞

to the overall output:

√

◦

y ss ( t ) =

t ←∞ y ( t ) =

lim sin(3 t − 56 ) u ( t ) .

13

In Chapters 5 and 6, we presented two frequency-domain approaches to analyze
CT signals and systems. The CTFT-based approach introduced in Chapter 5 uses
the real frequency ω , whereas the Laplace-transform-based approach uses the
complex frequency σ . Both approaches have advantages. Depending upon the
application under consideration, the appropriate transform is selected.
Comparing Example 6.22 with Example 5.26, the Laplace transform appears
to be a more convenient tool for the transient analysis. For the steady state
analysis, the Laplace transform does not seem to offer any advantage over
the CTFT. The transient analysis is very important for applications in control
systems, including process control and guided missiles. In signal processing
applications, such as audio, image, and video processing, the transients are
generally ignored. In such applications, the CTFT is sufficient to analyze the
steady state response. This is precisely why most signal processing literature
uses the CTFT, while the control systems literature uses the Laplace transform.
Important applications of the Laplace transforms such as analysis of the spring