Signals and Systems - Electrical Engineering

3.3 The One-Sided Laplace Transform 191

which is a first-order linear differential equation with constant coefficients, zero initial condition,

and a causal input so that it is a linear time-invariant system, as discussed before.

Lettingvs(t)=δ(t)and computing the Laplace transform of the above equation (using the linearity

and the derivative properties of the transform and remembering the initial condition is zero), we

obtain the following equation in thes-domain:

L[δ(t)]=L

[

L

di(t)

dt

+Ri(t)

]

1 =sLI(s)+RI(s)

whereI(s)is the Laplace transform ofi(t). Solving forI(s)we have that

I(s)=

1 /L

s+R/L

which as we have seen is the Laplace transform of

i(t)=

1

L

e−(R/L)tu(t)

Notice thati( 0 −)=0 and that the response has the form of a decaying exponential trying to follow

the input signal, a delta function. n

nExample 3.9

In this example we consider the duality between the time and the Laplace domains. The differentia-

tion property indicates that computing the derivative of a function in the time domain corresponds

to multiplying bysthe Laplace transform of the function (assuming initial conditions are zero).

We will illustrate in this example the dual of this—that is, when we differentiate a function in the

s-domain its effect in the time domain is to multiply by−t. Consider the connection betweenδ(t),

u(t), andr(t)(i.e., the unit impulse, the unit step, and the ramp, respectively), and relate it to the

indicated duality. Explain how this property connects with the existence of multiple poles, real and

complex, in general.

Solution

The relation between the signalsδ(t),u(t), andr(t)is seen from

L[r(t)]=

1

s^2

L

[

u(t)=

dr(t)

dt

]

=s

1

s^2

=

1

s

L

[

δ(t)=

du(t)

dt

]

=s

1

s

= 1

which also shows that a double pole at the origin, 1/s^2 , corresponds to a ramp functionr(t)=tu(t).