Signals and Systems - Electrical Engineering

174 C H A P T E R 3: The Laplace Transform

the Laplace transform to test for convergence, we lets=σ+jand the term|ej|=1. Thus, all

regions of convergence will contain−∞<  <∞.

If{σi}are the real parts of the poles ofF(s)=L[f(t)], the region of convergence corresponding to

different types of signals or impulse responses is determined from its poles as follows:

n For a causalf(t),f(t)=0 fort<0, the region of convergence of its Laplace transformF(s)is a

plane to the right of the poles,

Rc={(σ,):σ >max{σi},−∞<  <∞}

n For an anti-causalf(t),f(t)=0 fort>0, the region of convergence of its Laplace transformF(s)

is a plane to the left of the poles,

Rac={(σ,):σ <min{σi},−∞<  <∞}

n For a noncausalf(t)(i.e.,f(t)defined for−∞<t<∞), the region of convergence of its Laplace

transformF(s)is the intersection of the regions of convergence corresponding to the causal

component,Rc, andRaccorresponding to the anti-causal component:

Rc

⋂

Rac

See Figure 3.5 for an example illustrating how the ROCs connect with the poles and the type of signal.

Special case:The Laplace transform of a functionf(t)of finite supportt 1 ≤t≤t 2 , has the wholes-plane as

ROC.

FIGURE 3.5

ROC for (a) causal signal with poles with

σmax= 0 ; (b) causal signal with poles with

σmax< 0 ; (c) anti-causal signal with poles with

σmin> 0 ; (d) noncausal signal where ROC is

bounded by poles (poles on the left-hand

s-plane give causal component and poles on

the right-hands-plane give the anti-causal

component of the signal). The ROCs do not

contain poles, but they can contain zeros.

(a)

jΩ

o o

×

×

× σ

(b)

jΩ

o

×

×

σ

(c)

jΩ

o

×

o

o σ

(d)

jΩ

o

o

×

×

×

× σ