Signals and Systems - Electrical Engineering

3.2 The Two-Sided Laplace Transform 171

giving as the system function for the channel,

H(s)=α 0 e−st^0 +···+αNe−stN

Notice that the time shifts in the input–output equation became exponentials in the Laplace

domain, a property we will see later. n

Let us consider the different types of functions (either continuous-time signals or the impulse
responses of continuous-time systems) we might be interested in calculating Laplace transforms of.

n Finite support functions:the functionf(t)in this case is

f(t)= 0 fort6∈finite segmentt 1 ≤t≤t 2

for any finite, positive or negativet 1 andt 2 , and so thatt 1 <t 2. We will see that the Laplace trans-

form of these finite support signals is of particular interest in the computation of the coefficients

of the Fourier series of periodic signals.

n Infinite support functions:In this case,f(t)is defined over an infinite support (e.g.,t 1 <t<t 2 where
eithert 1 ort 2 are infinite, or both are infinite as long ast 1 <t 2 ).

A finite, or infinite, support functionf(t)is called (see examples in Figure 3.3):

nCasualiff(t)= 0 t<0,

nAnti-causaliff(t)= 0 t≥0,

nNon causalif a combination of the above.

In each of these cases we need to consider the region in thes-plane where the transform exists or its
region of convergence (ROC). This is obtained by looking at the convergence of the transform.

FIGURE 3.3
Examples of different types of signals:
(a) noncausal finite support signalx 1 (t), (b) causal
finite support signalx 2 (t), (c) noncausal infinite
support signalx 3 (t), and (d) causal infinite
support signalx 4 (t).

(a)

x 1 (t)

t

(b)

x 2 (t)

t

(d)

x 4 (t)

t

(c)

x 3 (t)

t