Digital Signal Processing Reference
In-Depth Information
In (4.18) when k is even, we can write k
¼
2m for m
¼
0to7:
¼
X
þ
X
7
15
X
even
k
x
n
W
n2m
x
n
W
n2m
16
16
n
¼
0
n
¼
8
¼
X
þ
X
7
15
x
n
W
nm
x
n
W
nm
ð
4
:
19
Þ
8
8
n
¼
0
n
¼
8
!
!
X
X
7
7
x
m
W
nm
x
8
þ
m
W
nm
¼
þ
:
ð
4
:
20
Þ
8
8
m
¼
0
m
¼
0
top
and
x
bot
, then (4.20) has the structure
If we divide the sequence x
n
into two halves as
x
even
top
bot
S
¼ W
8
x
þW
8
x
top
bot
¼ W
8
½
x
þ x
ð
4
:
21
Þ
Note that even values of the spectrum are obtained by dividing the given sequence
into two halves and adding them. Then we take the DFT of this new sequence,
which is only half the original sequence. In (4.18) when k is odd, we can write
k
¼
2m
þ
1 for m
¼
0to7:
¼
X
þ
X
7
15
x
n
W
n
ð
2m
þ
1
Þ
x
n
W
n
ð
2m
þ
1
Þ
X
odd
k
16
16
n
¼
0
n
¼
8
¼
X
þ
X
7
15
W
nm
8
W
nm
8
x
n
W
16
x
n
W
16
ð
4
:
22
Þ
n
¼
0
n
¼
8
!
!
X
7
X
7
W
nm
8
W
nm
8
x
m
W
16
x
m
þ
8
W
m
þ
8
16
¼
þ
ð
4
:
23
Þ
m
¼
0
m
¼
0
Looking at (4.23) in detail, we have created two new sequences by doing a point-
by-point multiplication with another constant sequence given as
top
x
m
W
16
...
bot
x
m
þ
i
W
m
þ
i
x
new
¼½...
for
i
¼
0 to 7
; x
new
¼½...
...
for
i
¼
0to7
:
16
ð
4
:
24
Þ
With this notation the remaining part of the spectrum is given as
:
odd
top
bot
new
S
¼ W
8
½
x
new
þ x
ð
4
:
25
Þ
Why are we complicating things with so many variables ? The reason is that it gives
us some structure. We summarise the equations as follows:
1. Given the sequence we generate one more extra sequence by multiplying term
by term with a complex quantity W
i
N
¼
cos
ð
n
Þþ
j sin
ð
n
Þ
2. These two sequences are divided into two halves.
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