106
THE
DIRAC
&FUNCTION
L..
t
nln
Figure
5.9
The Sinc
Squared
Sequence Function
5.2.2
Defining
S(t)
by
Integral
Operations
In mathematics, the 6-function is defined by how it behaves inside an integral. Any
function which behaves
as
6(t)
in the following equation is
by
dejnition
a 6-function,
t-
<
to
<
t+
otherwise
’
L-“
dt
6(t
-
to)f(t)
=
(5.14)
where
t-
<
t+
and
f
(t)
is
any continuous, well-behaved function. This operation
is sometimes called a
sifiing integral
because it selects the single value
f(to)
out of
Because
this
is a definition, it need not be proven, but its consistency with
our
previous definition and applications
of
the
6-function must be shown.
If
t-
<
to
<
t+
,
the range of the integral can be changed
to
be an infinitesimal region of size
2~
centered
around
to,
without changing the value
of
the integral.
This
is true because
6(t
-
to)
vanishes everywhere except at
t
=
to.
This
means
L-‘+
dt
6(t
-
to)f(t)
=
s,-,
dt
60
-
to)f(t>-
(5.15)
Because
f(t)
is a continuous function, over the infinitesimal region it is effectively a
constant, with the value
f(to).
Therefore,
f
(0.
f0+E
to+€
dt
S(t
-
to)
=
f(to).
(5.16)
This “proves” the first part
of
Equation 5.14. The second part, when
to
is not inside
the range
t-
<
t
<
t+,
easily follows because, in
this
case,
6(t
-
to)
is zero for the
entire range
of
the integrand.
Figure
5.10
shows a representation of the integration in Equation 5.14. The inte-
grand is a product of a shifted 6-function and the continuous function
f(t).
Because
the &function is zero everywhere except at
t
=
to,
the integrand goes to a &function
located at
t
=
to,
with an area scaled by
the
value of
f(to).
s,.
[
dt
60
-
t0)f
(t)
=
f(t0)
