Digital Signal Processing Reference
In-Depth Information
signal) is going to be the only non-zero component of the DFT
output X
k
.
Starting with X
k
¼
P
N
1
i
¼
0
x
i
e
j2
p
ki
=
N
and setting N
¼
8 and all
x
i
¼
1:
X
k
¼
P
i
¼
0
1
e
j2
p
ki
=
8
, and setting k
¼
0 (recall that e
0
¼
1)
X
0
¼
X
7
i
¼
0
1
1
¼
8
Next, evaluate for k
¼
1:
X
1
¼
X
7
i
¼
0
1
e
j2
p
i
=
8
¼
1
þ
e
j2
p=
8
þ
e
j4
p=
8
þ
e
j6
p=
8
þ
e
j8
p=
8
þ
e
j10
p=
8
þ
e
j12
p=
8
þ
e
j14
p=
8
X
1
¼
1
þð
0
:
7071
j0
:
7071
Þ
j
þð
0
:
7071
j0
:
7071
Þ
1
þð
0
:
7071
þ
j0
:
7071
Þþ
j
þð
0
:
7071
þ
j0
:
7071
Þ
X
1
¼
0
The eight terms of the summation for X
1
cancel out. This makes
sense because it's a sum of eight equally spaced points about the
origin on the unit circle of the complex plane. The summation of
these points must equal the center, in this case zero.
Next, evaluate for k
¼
2:
X
2
¼
X
7
i
¼
0
1
e
j2
p
i
=
8
¼
1
þ
e
j
p=
2
þ
e
j
p
þ
e
j3
p=
2
þ
e
j2
p
þ
e
j5
p=
2
þ
e
j3
p
þ
e
j7
p=
2
X
2
¼
1
j
1
þ
j
þ
1
j
1
þ
j
¼
0
We will find out similarly that X
3
,X
4
,X
5
,X
6
,X
7
are also zero.
Each of these will represent eight points equally spaced about the
unit circle. X
1
has its points spaced at
45
increments, X
2
has its
points spaced at
90
increments, X
3
has its points spaced at
135
increments, and so forth (the points may wrap multiple
times around the unit circle in the frequency domain). So, as we
expected, the only non-zero term is X
0
, which is the DC term.
There is no other frequency content of the signal.
Now, let us use the IDFT to get the original sequence back:
N
X
N
1
k
¼
0
X
k
e
þ
j2
p
ki
=
N
for N
¼
8 and X
0
x
i
¼
1
=
¼
8
;
all other X
k
¼
0
