Signals and Systems - Electrical Engineering

3.2 The Two-Sided Laplace Transform 175

Indeed, the integral defining the Laplace transform is bounded for any value ofσ6=0. IfA=
max(|f(t)|), then

|F(s)|≤

∫t^2

t 1

|f(t)||e−st|dt≤A

∫t^2

t 1

e−σtdt=A

e−σt^1 −e−σt^2

σ

<∞ σ6= 0

The Laplace transform of a

n Finite support function (i.e.,f(t)= 0 fort<t 1 andt>t 2 , fort 1 <t 2 ) is

L[f(t)]=L[f(t)[u(t−t 1 )−u(t−t 2 )]] wholes-plane

n Causal function (i.e.,f(t)= 0 fort< 0 ) is

L[f(t)u(t)] Rc={(σ,):σ >max{σi},−∞<  <∞}

n Anti-causal function (i.e.,f(t)= 0 fort> 0 ) is

L[f(t)u(−t)] Rac={(σ,):σ <min{σi},−∞<  <∞}

n Noncausal function (i.e.,f(t)=fac(t)+fc(t)=f(t)u(−t)+f(t)u(t)) is

L[f(t)]=L[fac(−t)u(t)](−s)+L[fc(t)u(t)] Rc

⋂

Rac

Although redundant, a causal functionf(t)(i.e.,f(t)=0 fort<0) is denoted asf(t)u(t). Its Laplace
transform is thus

L[f(t)u(t)]=

∫∞

−∞

f(t)u(t)e−stdt=

∫∞

0

f(t)e−stdt

which is called theone-sided Laplace transform. Likewise, iff(t)is anti-causal (i.e.,f(t)=0 fort>0),
we will denote it asf(t)u(−t)and its Laplace transform is given by

L[f(t)u(−t)]=

∫^0

−∞

f(t)u(−t)e−stdt=

∫∞

0

f(−t′)u(t′)est

′

dt′

or the one-sided Laplace transform of the causal signalf(−t)u(t), withschanged into−s.

A noncausal signalf(t)is defined for all values oft(i.e., for−∞<t<∞). Such a signal has a causal
componentfc(t), which is obtained by multiplyingf(t)by the unit-step function,fc(t)=f(t)u(t), and
an anti-causal componentfac(t), which is obtained by multiplyingf(t)byu(−t), so that

f(t)=fac(t)+fc(t)

=f(t)u(−t)+f(t)u(t) (3.6)