3.4 Inverse Laplace Transform 197
If the input of an LTI system is the causal signalx(t)and the impulse response of the system ish(t),
then the outputy(t)can be written as
y(t)=
∫∞
0
x(τ)h(t−τ)dτ t≥ 0
and zero otherwise. Its Laplace transform is
Y(s)=L
∫∞
0
x(τ)h(t−τ)dτ
=
∫∞
0
∫∞
0
x(τ)h(t−τ)dτ
e−stdt
=
∫∞
0
x(τ)
∫∞
0
h(t−τ)e−s(t−τ)dt
e−sτdτ=X(s)H(s)
where the internal integral is shown to beH(s)=L[h(t)] (change variable toν=t−τ) using the
causality ofh(t). The remaining integral is the Laplace transform ofx(t).
The system function or transfer functionH(s)=L[h(t)], the Laplace transform of the impulse responseh(t)of
an LTI system, can be expressed as the ratio
H(s)=
L[y(t)]
L[x(t)]
=
L[output]
L[input]
(3.17)
This function is calledtransfer functionbecause it transfers the Laplace transform of the input to the output.
Just as with the Laplace transform of signals,H(s)characterizes an LTI system by means of its poles and
zeros. Thus, it becomes a very important tool in the analysis and synthesis of systems.
3.4 INVERSE LAPLACE TRANSFORM
Inverting the Laplace transform consists in finding a function (either a signal or an impulse response
of a system) that has the given transform with the given region of convergence. We will consider three
cases:
n Inverse of one-sided Laplace transforms giving causal functions.
n Inverse of Laplace transforms with exponentials.
n Inverse of two-sided Laplace transforms giving anti-causal or noncausal functions.
The given functionX(s)we wish to invert can be the Laplace transform of a signal or a transfer
function—that is, the Laplace transform of an impulse response.
3.4.1 Inverse of One-Sided Laplace Transforms
When we consider a causal functionx(t), the region of convergence ofX(s)is of the form
{(σ,):σ > σmax,−∞< <∞}
