Signals and Systems - Electrical Engineering

3.3 The One-Sided Laplace Transform 187

which are located on thejaxis. The farther away from the origin of thejaxis the poles are, the
higher the frequency 0 , and the closer the poles are to the origin, the lower the frequency. Thus,
thejaxis corresponds to the frequency axis. Furthermore, notice that to generate the real-valued
signalg(t)we need two complex conjugate poles, one at+j 0 and the other at−j 0. Although
frequency, as measured by frequency meters, is a positive value, “negative” frequencies are needed to
represent “real” signals (if the poles are not complex conjugate pairs, the inverse Laplace transform is
complex—rather than real valued).

The conclusion is that the Laplace transform of a sinusoid has a pair of poles on thejaxis. For these poles

to correspond to a real-valued signal they should be complex conjugate pairs, requiring negative as well as

positive values of the frequency. Furthermore, when these poles are moved away from the origin of thej

axis, the frequency increases, and the frequency decreases whenever the poles are moved toward the origin.

Finally, consider the case of a signald(t)=Ae−αtcos( 0 t)u(t)or a causal sinusoid multiplied (or
modulated) bye−αt. According to Euler’s identity,

d(t)=A

[

e(−α+j^0 )t

2

u(t)+

e(−α−j^0 )t

2

u(t)

]

and as such we can again use linearity to get

D(s)=

A(s+α)

(s+α)^2 +^20

(3.10)

Notice the connection between Equations (3.9) and (3.10). GivenG(s), thenD(s)=G(s+α), with
G(s)corresponding tog(t)=Acos( 0 t)andD(s)tod(t)=g(t)e−αt. Multiplying a functiong(t)by an
exponentiale−αt, withαreal or imaginary, shifts the transform toG(s+α)—that is, it is acomplex
frequency-shiftproperty. The poles ofD(s)have as the real part the damping factor−αand as the
imaginary part the frequencies± 0. The real part of the pole indicates decay (ifα >0) or growth
(ifα <0) in the signal, while the imaginary part indicates the frequency of the cosine in the signal.
Again, the poles will be complex conjugate pairs since the signald(t)is real valued.

The conclusion is that the location of the poles (and to some degree the zeros), as indicated in the previous two

cases, determines the characteristics of the signal. Signals are characterized by their damping and frequency

and as such can be described by the poles of its Laplace transform.

If we were to add the different signals considered above, then the Laplace transform of the resulting
signal would be the sum of the Laplace transform of each of the signals and the poles would be
the aggregation of the poles from each. This observation will be important when finding the inverse
Laplace transform, then we would like to do the opposite: To isolate poles or pairs of poles (when
they are complex conjugate) and associate with each a general form of the signal with parameters that
are found by using the zeros and the other poles of the transform. Figure 3.10 provides an example
illustrating the importance of the location of the poles, and the significance of theσandjaxes.