Signals and Systems - Electrical Engineering

3.2 The Two-Sided Laplace Transform 173

100

50

0

− 2 0

− 2

2

1

− 1

− 3

0

Damping

Frequency

− 50

− 100

FIGURE 3.4
Three-dimensional plot of the logarithm of the magnitude ofF(s)= 2 (s^2 + 1 )/(s^2 + 2 s+ 5 )as a function of
dampingσand frequency. The poles shoot up, while the zeros shoot down. In the logarithmic scale both
poles and zeros will have infinite value: WhenF(s)= 0 (zero) its logarithm is−∞, while whenF(s)→∞(pole)
the logarithm is∞.

for an integerk=0,±1,±2,.... Thus, the zeros aresk=jπk,k=0,±1,±2,.... Now, whenk=0,
the zero at 0 cancels the pole at zero; therefore,P(s)has only zeros, an infinite number of them,
{jπk,k=±1,±2,...}.

Poles and ROC
The ROC consists of the values ofσsuch that

∣ ∣ ∣ ∣ ∣ ∣

∫∞

−∞

x(t)e−stdt

∣ ∣ ∣ ∣ ∣ ∣

≤

∫∞

−∞

|x(t)||e−(σ+j)t|dt=

∫∞

−∞

|x(t)|e−σtdt<∞ (3.5)

This is equivalent to choosing values ofσfor whichx(t)e−σtis absolutely integrable.

Two general comments that apply to all types of signals when finding ROCs are:

n No poles are included in the ROC, which means that for the ROC to be that region where the
Laplace transform is defined, the transform cannot become infinite at any point in it. So poles
should not be present in the ROC.
n The ROC is a plane parallel to thejaxis, which means that it is the dampingσthat defines the
ROC, not frequency. This is because when we compute the absolute value of the integrand in