Digital Signal Processing Reference
In-Depth Information
To check that this assertion is consistent, we can now write the DFS re-
construction formula using the DFS values given to us by inserting (4.23)
in (4.10):
l
g
r
,
y
i
d
.
,
©
,
L
s
N
−
1
1
N
e
j
2
N
k
e
j
2
N
nk
x
[
]=
(
)
n
X
(4.24)
k
=
0
By defining
Δ=(
2
π/
N
)
, we can rewrite the above expression as
N
−
1
1
2
e
j
(
k
Δ)
)
e
j
(
k
Δ)
n
x
[
n
]=
X
(
Δ
(4.25)
π
k
=
0
and the summation is easily recognized as the Riemann sum with step
Δ
e
j
ω
)
e
j
ω
n
between 0 and 2
approximating the integral of
f
(
ω
)=
X
(
π
.As
N
goes to infinity (and therefore
x
[
n
]
→
x
[
n
]
), we can therefore write
2
π
1
2
π
e
j
ω
)
e
j
ω
n
d
x
[
n
]
→
X
(
ω
(4.26)
0
which is indeed the DTFT reconstruction formula (4.14).
(5)
4.4.2 The DTFT as a Formal Change of Basis
We now show that, if we are willing to sacrifice mathematical rigor, the DTFT
can be cast in the same conceptual framework we used for the DFT andDFS,
namely as a basis change in a vector space. The following formulas are to be
taken as nothing more than a set of purely symbolic derivations, since the
mathematical hypotheses under which the results are well defined are far
from obvious and are completely hidden by the formalism. It is only fair to
say, however, that the following expressions represent a very handy and in-
tuitive toolbox to grasp the essence of the duality between the discrete-time
and the frequency domains and that they can be put to use very effectively
to derive quick results when manipulating sequences.
One way of interpreting Equation (4.13) is to see that, for any given value
ω
(
)
0
, the corresponding value of the DTFT is the inner product in
of
2
with the sequence
e
j
ω
0
n
; formally, at least, we are still
performing a projection in a vector space akin to
the sequence
x
[
n
]
∞
:
e
j
ω
)=
e
j
ω
n
,
x
]
X
(
[
n
e
j
ω
n
Here, however, the set of “basis vectors”
{
}
ω
∈
is indexed by the real
variable
ω
and is therefore uncountable. This uncountability is mirrored in
(5)
Clearly (4.26) is equivalent to (4.14) in spite of the different integration limits since all the
quantities under the integral sign are 2
π
-periodic and we are integrating over a period.
