Laplace transform - Continuous and Discrete Time Signals and Systems - page 271

The above argument is not true because the CTFT of the unit step function
contains a discontinuity at ω = 0 due to the presence of the impulse function
δ ( ω ). Therefore, the CTFT violates the condition for the existence of CTFT. In
such cases, the CTFT is not derived from its definition but is expressed using
the impulse function, which is not a mathematical function in the strict sense.
It is therefore natural to expect Eq. (6.10) to be invalid. Likewise, the ROC for
the Laplace transform of the sine wave, cosine wave, squared cosine wave, and
squared sine wave do not contain the j ω -axis, and Eq. (6.10) is also not valid
in these cases.

6.2.2 Region of convergence

As a side note to our discussion, we observe that the Laplace transform is
guaranteed to exist at all points within the ROC. For example, consider the
causal sine wave h ( t ) = sin(4 t ) u ( t ). We are interested in calculating the values
of the Laplace transform at two points, s 1 = 2 + j3 and s 2 = j3 in the complex
s-plane. Since s 1 lies within the ROC, Re { s > 0, the value of the Laplace
transform at s 1 is given by

4
(2 + j3) 2 + 4 2

4
4 + j12 − 9 + 16

= 4

11 + j12 ,
which, as expected, is a finite value. The point s 2 = j3 lies outside the ROC.
However, the Laplace transform is not necessarily infinite at s 2 . In fact, the
Laplace transform of the causal sine wave h ( t ) = sin(4 t ) u ( t ) is finite for all
values of s on the imaginary axis except at s

H (2 + j3) = =

= j4. The value of the Laplace

transform at s 2 is given by

4
( j3) 2 + 4 2

=− 4

5 .
Since the Laplace transform is not defined at two points ( s = j4) on the
imaginary axis, the entire imaginary axis is excluded from the ROC. In short, if
a point lies on the boundary of the ROC, it is possible that the Laplace transform
exists, though the point may not be included in the ROC.

H ( j3) =

6.2.3 Spectra for the Laplace transform

In Chapter 5, the magnitude and phase spectra of the CTFT provided mean-
ingful insights into the frequency properties of the reference function. Except
for one difference, the magnitude and phase spectra of the Laplace transform
(collectively referred to as the Laplace spectra) can be plotted in a similar way.
Since the Laplace variable s is a complex variable, the Laplace spectra are
plotted with respect to a 2D complex plane with Re { s =σ and Im { s =ω
being the two independent axes. For the magnitude spectrum, the magnitude
of the Laplace transform is plotted along the z -axis within the ROC defined
in the 2D complex plane. Likewise, for the phase spectrum, the phase of the