The expressions for x(t) and y(t) are found by using properties of the Fourier
transform: First, x(t) is found by utilizing the differentiation property of the Fourier
transform and a well-known Fourier transform pair.
x(t) = F = e-Ru(t) (3-38)
-v C R- +jc) d7
The term u(t) is the unit-step function and is defined as
1 t>0
u(t) = (3-39)
0 t<0
The time-derivative can be computed by using the product rule of differentiation
and the fact that the derivative of the unit-step function is the Dirac delta function, 8(t),
which is defined as
CO t=0
s(t) = (3-40)
0 t O
Therefore, the expression for x(t) is
x(t) =- e Cu(t) + 6(t) (3-41)
The quantity Eor,,,(o) is an arbitrary function of co, therefore the expression for
y(t) is
y(t) = OAplateEort) (3-42)
The quantity Enorm (t) is simply the inverse Fourier transform of Enorm (o).
Substituting the expressions for x(t) and y(t) into Equation 3-35 yields
vt(t) = (-Re RFu(t) +8(t)) *OApateEnorm(t) (3-43)
V,,=1i+~) KL I ~la