Signals and Systems - Electrical Engineering

348 CHAPTER 5: Frequency Analysis: The Fourier Transform

n If X( 0 )(i.e., the dc value of X()) is zero, then the operator 1 /(j)corresponds to integration in time

of x(t), just like 1 /s in the Laplace domain. For X( 0 )to be zero the integral of the signal from−∞to∞

must be zero.

nExample 5.21

Suppose a system is represented by a second-order differential equation with constant coefficients:

2 y(t)+ 3

dy(t)

dt

+

d^2 y(t)

dt^2

=x(t)

and that the initial conditions are zero. Letx(t)=δ(t). Findy(t).

Solution

Computing the Fourier transform of this equation, we get

[2+ 3 j+(j)^2 ]Y()=X()

ReplacingX()=1 and solving forY(), we have

Y()=

1

2 + 3 j+(j)^2

=

1

(j+ 1 )(j+ 2 )

=

1

(j+ 1 )

+

− 1

(j+ 2 )

and the inverse Fourier transform of these terms gives

y(t)=[e−t−e−^2 t]u(t) n

nExample 5.22

Find the Fourier transform of the triangular pulse

x(t)=r(t)− 2 r(t− 1 )+r(t− 2 )

which is piecewise linear, using the derivative property.

Solution

A first derivative gives

dx(t)

dt

=u(t)− 2 u(t− 1 )+u(t− 2 )