563489578934

Sec. 2–2 Fourier Transform and Spectra 53
In this definition, the variable x could be time or frequency, depending on the application.
An alternative definition for d(x) is
and
(2–46a)
d(x) = e q, x = 0
(2–46b)
0, x Z 0
where both Eqs. (2–46a) and (2–46b) need to be satisfied. The Dirac delta function is not a
true function, so it is said to be a singular function. However, d(x) can be defined as a function
in a more general sense and treated as such in a branch of mathematics called generalized
functions and the theory of distributions.
From Eq. (2–45), the sifting property of the d function is
w(x) d(x - x 0 ) dx = w(x 0 )
(2–47)
L-q
That is, the d function sifts out the value w(x 0 ) from the integral.
In some problems, it is also useful to use the equivalent integral for the d function,
which is
q
d(x) = e ;j2pxy dy
(2–48)
L
where either the + or the - sign may be used as needed. This assumes that d(x) is an even
function: d(-x) = d(x). Equation (2–48) may be verified by taking the Fourier transform of
a delta function
and then taking the inverse Fourier transform of both sides of this equation; Eq. (2–48) follows.
(For additional properties of the delta function, see Appendix A.)
Another function that is closely related to the Dirac delta function is the unit step function.
DEFINITION.
L
The unit step function u(t) is
u(t) = e 1, t 7 0
(2–49)
0, t 6 0
Because d( l) is zero, except at l = 0, the Dirac delta function is related to the unit step function
by
L
where E 7 0 and P : 0. Consequently,
q
q
-q
L
d(t)e -2pft dt = e 0 = 1
t+P
- q
q
-q
d(l) dl = u(t)
du(t)
dt
d(x) dx = 1
-q
= d(t)
(2–50)
(2–51)