166 C H A P T E R 3: The Laplace Transform
and of transient and steady-state responses. This is a significant reason to study the Laplace
analysis before the Fourier analysis, which deals exclusively with the frequency characterization
of continuous-time signals and systems. Stability and transients are important issues in classic
control theory, thus the importance of the Laplace transform in this area. The frequency character-
ization of signals and the frequency response of systems—provided by the Fourier transform—are
significant in communications.
n One- and two-sided Laplace transforms—Given the prevalence of causal signals (those that are zero
for negative time) and of causal systems (having zero impulse responses for negative time) the
Laplace transform is typically known as “one-sided,” but the“two-sided” transform also exists.
The impression is that these are two different transforms, but in reality it is the Laplace transform
applied to two different types of signals and systems. We will show that by separating the signal
into its causal and its anti-causal components, we only need to apply the one-sided transform.
Care should be exercised, however, when dealing with the inverse transform so as to get the
correct signal.
n Region of convergence and the Fourier transform—Since the Laplace transform requires integration
over an infinite domain, it is necessary to consider if and where this integration converges—or
the “region of convergence” in thes-plane. Now, if such a region includes thejaxis of thes-
plane, then the Laplace transform exists fors=j, and when computed there it coincides with
the Fourier transform of the signal. Thus, the Fourier transform for a large class of functions
can be obtained directly from their Laplace transforms—a good reason to study first the Laplace
transform. In a subtle way, the Laplace transform is also connected with the Fourier series rep-
resentation of periodic continuous-time signals. Such a connection reduces the computational
complexity of the Fourier series by eliminating integration in cases when we can compute the
Laplace transform of a period.
n Eigenfunctions of LTI systems—LTI systems respond to complex exponentials in a very special way:
The output is the exponential with its magnitude and phase changed by the response of the
system at the exponent. This provides the characterization of the system by the Laplace transform,
in the case of exponents of the complex frequencys, and by the Fourier representation when the
exponent isj. The eigenfunction concept is linked to phasors used to compute the steady-state
response in circuits (see Figure 3.1).3.2 The Two-Sided Laplace Transform
Rather than giving the definitions of the Laplace transform and its inverse, let us see how they could
be obtained intuitively. As indicated before, a basic idea in characterizing signals—and their response
when applied to LTI systems—is to consider them a combination of basic signals for which we can
easily obtain a response. In Chapter 2, when considering the time-domain solutions, we represented
the input as an infinite combination of impulses occurring at all possible times and weighted by
the value of the input signal at those times. The reason we did so is because the response due to
an impulse is the impulse response of the LTI system, which is fundamental in our studies. A similar
approach will be followed when attempting to obtain the frequency-domain representation of signals
and their responses when applied to an LTI system. In this case, the basic functions used are com-
plex exponentials or sinusoids that depend on frequency. The concept of eigenfunction is somewhat