Digital Signal Processing Reference
In-Depth Information
Therefore, in its complex Fourier series form, we have
N
−
1
X
p
(k)e
j(
2
π/N)kn
x
p
(n)
=
(3.70)
k
=
0
To find these coefficients, let us multiply both sides by
e
−
jmω
0
k
and sum over
n
from
n
=
0to
(N
−
1
)
:
N
−
1
N
−
1
N
−
1
x
p
(n)e
−
jmω
0
k
X
p
(k)e
j(
2
π/N)kn
e
−
jmω
0
k
=
(3.71)
n
=
0
n
=
0
k
=
0
By interchanging the order of summation on the right side, we get
X
p
(k)
N
−
1
e
j(
2
π/N)k(n
−
m)
N
−
1
(3.72)
k
=
0
n
=
0
It is next shown that [
N
−
1
0
e
j(
2
π/N)k(n
−
m)
] is equal to
N
when
n
=
m
and zero
n
=
m
, the summation reduces to [
N
−
1
n
e
j
0
]
=
for all values of
n
=
m
.When
n
=
=
0
N
,andwhen
n
m
, we apply (3.52) and find that the summation yields zero.
Hence there is only one nonzero term
X
p
(k)N
in (3.72). The final result is
=
N
−
1
1
N
x
p
(n)e
−
jnω
0
k
X
p
(k)
=
(3.73)
n
=
0
Now we notice that
N
−
1
N
−
1
1
N
1
N
x
p
(n)e
−
jnω
0
k
x(n)e
−
j(
2
π/N)nk
=
n
=
0
n
=
0
1
N
X
p
(e
jω
)
ω
k
=
(
2
π/N)k
=
=
X
p
(k)
(3.74)
In other words, when the DTFT of the finite length sequence
x(n)
is evaluated
at the discrete frequency
ω
k
(
2
π/N)k
,(whichisthe
k
th sample when the
frequency range [0
,
2
π
] is divided into
N
equally spaced points) and dividing
by
N
, we get the value of the Fourier series coefficient
X
p
(k)
.
The expression in (3.70) is known as the
discrete-time Fourier series
(DTFS) representation for the discrete-time, periodic function
x
p
(n)
and (3.73),
which gives the complex-valued coefficients of the DTFS is the inverse DTFS
(IDTFS). Because both
x
p
(n)
and
X
p
(k)
are periodic, with period
N
, we observe
=
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