Digital Signal Processing Reference
In-Depth Information
•
it provides a convenient notational framework which unifies the
Fourier transform and the
z
-transform (which we will see shortly).
The DTFT, when it exists, can be inverted via the integral
l
g
r
,
y
i
d
.
,
©
,
L
s
π
1
2
e
j
ω
)
e
j
ω
n
d
x
[
n
]=
X
(
ω
(4.14)
π
−
π
as can be easily verified by substituting (4.13) into 4.14) and using
π
e
−j
ω
(
n−k
)
=
2
πδ
[
n
−
k
]
−
π
In fact, due to the 2
π
-periodicity of the DTFT, the integral in (4.14) can be
computed over
any
2
π
-wide interval on the real line (i.e. between 0 and 2
π
,
e
j
ω
)
for instance). The relation between a sequence
x
[
n
]
and its DTFT
X
(
will be indicated in the general case by
DTFT
←→
e
j
ω
)
x
[
n
]
X
(
While theDFT andDFSwere signal transformations which involved only
a finite number of quantities, both the infinite summation and the real-
valued argument, appearing in the DTFT, can create an uneasiness which
overshadows the conceptual similarities between the transforms. In the fol-
lowing, we start by defining the mathematical properties of the DTFT and
we try to build an intuitive feeling for this Fourier representation, both with
respect to its physical interpretation and to its conformity to the “change of
basis” framework, that we used for the DFT and DFS.
Mathematically, the DTFT is a transform operator which maps discrete-
time sequences onto the space of 2
-periodic functions. Clearly, for the
DTFT to exist, the sum in (4.13) must converge, i.e. the limit for
M →∞
of the partial sum
π
M
e
j
ω
)=
e
−j
ω
n
(
[
]
X
M
x
n
(4.15)
n
=
−
M
must exist and be finite. Convergence of the partial sum in (4.15) is very easy
to prove for
absolutely summable
sequences, that is for sequences satisfying
M
x
]
<
∞
lim
M
→∞
=
[
n
(4.16)
n
=
−
M
since, according to the triangle inequality,
M
M
X
M
(
e
j
ω
)
≤
x
e
−j
ω
n
=
x
]
[
n
]
[
n
(4.17)
=
−
=
−
n
M
n
M
