Digital Signal Processing Reference
In-Depth Information
coefficients
X
[
k
]
. If we try to write the analysis DTFT formula for
x
[
n
]
we
have
∞
l
g
r
,
y
i
d
.
,
©
,
L
s
X
e
j
ω
)=
e
−j
ω
n
(
x
[
n
]
n
=
−∞
1
N
e
j
2
N
nk
e
−j
ω
n
∞
N
−
1
=
X
[
k
]
(4.41)
n
=
−∞
k
=
0
∞
e
j
2
N
nk
e
−j
ω
n
N
−
1
1
N
=
X
[
k
]
(4.42)
n
=
−∞
k
=
0
where in (4.41) we have used the DFS reconstruction formula. Now we rec-
ognize in the last term important to recognize the last terms of (4.42) as
the DTFT of a complex exponential of frequency
(
2
π/
N
)
k
; we can therefore
write
N
k
N
−
1
1
N
2
X
(
e
j
ω
)=
˜
X
[
k
]
δ
ω
−
(4.43)
k
=
0
which is the relationship between the DTFT and the DFS. If we restrict our-
selves to the
interval, we can see that the DTFT of a periodic se-
quence is a series of regularly spaced deltas placed at the
N
roots of unity
andwhose amplitude is proportional to theDFS coefficients of the sequence.
In other words,
the DTFT is uniquely determined by the DFS and vice versa
.
[
−
π
,
π
]
Finite-Support Sequences.
Given a length-
N
signal
x
[
n
]
,
n
=
0,...,
N
−
1andits
N
DFT coefficients
X
[
k
]
, consider the associated finite-support
sequence
x
[
n
]
0
≤
n
<
N
x
[
n
]=
0
other ise
from which we can easily derive the DTFT of
x
as
ω
−
N
k
N
−
1
2
X
e
j
ω
)=
(
X
[
k
]Λ
(4.44)
k
=
0
with
N
−
1
1
N
e
−j
ω
m
Λ(
ω
)=
m
=
0
What the above expression means, is that the DTFT of the finite support
sequence
x
[
n
]
is again
uniquely defined by the N DFT coefficients of the
