Digital Signal Processing Reference
In-Depth Information
exponentials become “denser” between 0 and 2
. We want to show that, in
the limit, we end up with the reconstruction formula of the DTFT.
To do so, let us restrict ourselves to the domain of absolute summable
sequences; for these sequences, we know that the sum in (4.13) exists. Now,
given an absolutely summable sequence
x
[
n
]
, we can always build an
N
-
periodic sequence
x
π
l
g
r
,
y
i
d
.
,
©
,
L
s
[
n
]
as
∞
x
[
n
]=
x
[
n
+
iN
]
(4.20)
i
=
−∞
for any value of
N
(see Example 2.2); this is guaranteed by the fact that the
above sum converges for all
n
∈
(because of the absolute summability of
[
]
) so that all values of
x
[
]
x
n
n
are finite. Clearly, there is overlap between
successive copies of
x
; the intuition, however, is the following: since in
the end we will consider very large values for
N
and since
x
[
n
]
,because
of absolute summability, decays rather fast with
n
, the resulting overlap of
“tails” will be negligible. This can be expressed as
[
n
]
lim
N
x
[
n
]=
x
[
n
]
→∞
Now consider the DFS of
x
[
]
n
:
N−
1
e
−j
2
N
(
n
+
iN
)
k
N
−
1
∞
e
−j
2
N
nk
X
[
k
]=
x
[
n
]
=
x
[
n
+
iN
]
(4.21)
n
=
0
n
=
0
i
=
−∞
where in the last term we have used (4.20), interchanged the order of the
summation and exploited the fact that
e
−j
(
2
π/
N
)(
n
+
iN
)
k
e
−j
(
2
π/
N
)
nk
.We
can see that, for every value of
i
in the outer sum, the argument of the inner
sum varies between
iN
and
iN
=
1, i.e. non-overlapping intervals, so
that the double summation can be simplified as
+
N
−
∞
e
−j
2
N
mk
X
[
k
]=
x
[
m
]
(4.22)
m
=
−∞
and therefore
ω
=
X
e
j
ω
)
[
k
]=
X
(
(4.23)
2
N
k
This already gives us a noteworthy piece of intuition: the DFS coefficients
for the periodized signal are a discrete set of values of its DTFT (here con-
sidered solely as a formal operator) computed at multiples of 2
N
.As
N
grows, the spacing between these frequency intervals narrows more and
more so that, in the limit, the DFS converges to the DTFT.
π/
