Laplace transform - Continuous and Discrete Time Signals and Systems - page 285

Interchanging the order of integration, we get

∞

L x 1 ( t ) ∗ x 2 ( t ) = ∞ x 2 ( t − τ )e − st d t x 1 ( τ ) d τ.

−∞

0 −

By noting that the inner integration ∫ x 2 ( t − τ ) exp( − st )d t = X 2 ( s ) exp( − s τ ),
the above integral simplifies to

∞

x 1 ( τ )e − s τ d τ = X 2 ( s ) X 1 ( s ) ,

L x 1 ( t ) ∗ x 2 ( t ) = X 2 ( s )

−∞

which proves Eq. (6.25). The s-plane convolution property may be proved in a
similar fashion.

Like the CTFT convolution property discussed in Section 5.5.8, the Laplace
time-convolution property provides us with an alternative approach to cal-
culate the output y ( t ) when a CT signal x ( t ) is applied at the input of an
LTIC system with the impulse response h ( t ). In Chapter 3, we proved that
the zero-state output response y ( t ) is obtained by convolving the input signal
x ( t ) with the impulse response h ( t ), i.e. y ( t ) = h ( t ) ∗ x ( t ). Using the time-
convolution property, the Laplace transform Y ( s ) of the resulting output y ( t )is
given by

←→

y ( t ) = x ( t ) ∗ h ( t ) Y ( s ) = X ( s ) H ( s ) ,

where X ( s ) and H ( s ) are the Laplace transforms of the input signal x ( t ) and
the impulse response h ( t ) of the LTIC systems. In other words, the Laplace
transform of the output signal is obtained by multiplying the Laplace transforms
of the input signal and the impulse response. The procedure for calculating the
output y ( t ) of an LTI system in the complex s-domain, therefore, consists of
the following four steps.

(1) Calculate the Laplace transform X ( s ) of the input signal x ( t ). If the input
signal and the impulse response are both causal functions, then the unilateral
Laplace transform is used. If either of the two functions is non-causal, the
bilateral Laplace transform must be used.
(2) Calculate the Laplace transform H ( s ) of the impulse response h ( t ) of the
LTIC system. The Laplace transform H ( s ) is referred to as the Laplace
transfer function of the LTIC system and provides a meaningful insight
into the behavior of the system.
(3) Based on the convolution property, the Laplace transform Y ( s ) of the output
response y ( t ) is given by the product of the Laplace transforms of the input
signal and the impulse response of the LTIC systems. Mathematically, this
implies that Y ( s ) = X ( s ) H ( s ).
(4) Calculate the output response y ( t ) in the time domain by taking the inverse
Laplace transform of Y ( s ) obtained in step (3).