Digital Signal Processing Reference
In-Depth Information
undefined for
t
0. This interpretation,
together with (4.27), defines the so-called
sifting property
of the Dirac delta;
this property allows us to write (outside of the integral sign):
=
0 and to have a value of
∞
at
t
=
l
g
r
,
y
i
d
.
,
©
,
L
s
δ
(
−
τ
)
(
)=
δ
(
−
τ
)
(
τ
)
t
f
t
t
f
(4.31)
k
=
6
6
k
=
5
k
=
4
4
k
=
3
k
=
2
2
k
=
1
0
-1
-0.5
0.5
1
Figure 4.12
The Dirac delta as the limit of a family of rectangular functions.
The physical interpretation of the Dirac delta is related to quantities ex-
pressed as continuous
distributions
for which the most familiar example is
probably that of a probability distribution (pdf ). These functions represent
a value which makes physical sense only over an interval of nonzero mea-
sure; the punctual value of a distribution is only an abstraction. The Dirac
delta is the operator that extracts this punctual value from a distribution, in
a sense capturing the essence of considering smaller and smaller observa-
tion intervals.
To see how the Dirac delta applies to our basis expansion, note that
equation (4.27) is formally identical to an inner product over the space of
functions on the real line; by using the definition of such an inner product
we can therefore write
∞
δ
(
)
δ
(
(
)=
−
τ
)
(
−
τ
)
τ
f
t
s
,
f
s
t
d
(4.32)
−∞
which is, in turn, formally identical to the reconstruction formula of Sec-
tion 3.4.3. In reality, we are interested in the space of 2
-periodic functions,
since that is where DTFTs live; this is easily accomplished by building a 2
π
π
-
periodic version of the delta as
∞
˜
δ
(
ω
)=
2
π
δ
(
ω
−
2
π
k
)
(4.33)
=
−∞
k
