Digital Signal Processing Reference
In-Depth Information
the inversion formula (4.14), in which the usual summation is replaced by
an integral; in fact, the DTFT operator maps
onto
L
2
[
−
π
π
]
which
2
(
)
,
is a space of 2
-periodic, square integrable functions. This interpretation
preserves the physical meaning given to the inner products in (4.13) as a
way to measure the frequency content of the signal at a given frequency; in
this case the number of oscillators is infinite and their frequency separation
becomes infinitesimally small.
To complete the picture of the DTFT as a change of basis, we want to
show that, at least formally, the set
π
l
g
r
,
y
i
d
.
,
©
,
L
s
e
j
ω
n
{
}
ω
∈
constitutes an orthogonal
.
(6)
In order to do so, we need to introduce a quirky mathe-
matical entity called the Dirac delta functional; this is defined in an implicit
way by the following formula
∞
“basis” for
(
)
2
δ
(
t
−
τ
)
f
(
t
)
dt
=
f
(
τ
)
(4.27)
−∞
where
f
(
t
)
is an arbitrary integrable function on the real line; in particular
∞
δ
(
t
)
f
(
t
)
dt
=
f
(
0
)
(4.28)
−∞
While no ordinary function satisfies the above equation,
can be inter-
preted as shorthand for a limiting operation. Consider, for instance, the
family of parametric functions
(7)
δ
(
t
)
r
k
(
t
)=
k
rect
(
kt
)
(4.29)
which are plotted in Figure 4.12. For any continuous function
f
(
t
)
we can
write
∞
k
1
/
2
k
−
(
γ
)
γ
∈
[
−
1
/
2
k
,1
/
2
k
]
r
k
(
t
)
f
(
t
)
dt
=
f
(
t
)
dt
=
f
(4.30)
−∞
1
/
2
k
wherewehaveusedtheMeanValuetheorem.Now,as
k
goes to infinity, the
integral converges to
f
; hence we can say that the limit of the series of
functions
r
k
(
t
)
converges then to the Dirac delta. As already stated, the delta
cannot be considered as a proper function, so the expression
(
0
)
outside of
an integral sign has no mathematical meaning; it is customary however to
associate an “idea” of function to the delta and we can think of it as being
δ
(
t
)
(6)
You can see here already why this line of thought is shaky unsafe: indeed,
e
j
ω
n
∈
2
(
)
!
(7)
The rect function is discussed more exhaustively in Section 5.6 its definition is
1f r
2
0 t rwie
|
x
|≤
1
/
rect
(
x
)=
