July 2015
Intermediate to advanced
600 pages
17h 41m
English
Content preview from Trends in Digital Signal Processing
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June 16, 2015 13:36 PSP Book - 9in x 6in 07-Yong-Ching-Lim-c07
Discrete Cosine Transform 221
This convolution sum can be computed by NTT. Now let us use an
FNT to make the calculation. Let M = F
4
= 2
16
+ 1, α = 2
8
and of
course N = 4. Hence
⎡
⎢
⎢
⎣
x
0
x
1
x
2
x
3
⎤
⎥
⎥
⎦
= <
⎡
⎢
⎢
⎣
1111
12
8
−1 −2
8
1 −11−1
1 −2
8
−12
8
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
x(3)
x(4)
x(2)
x(1)
⎤
⎥
⎥
⎦
>
M
(7.54)
⎡
⎢
⎢
⎣
W
0
W
1
W
2
W
3
⎤
⎥
⎥
⎦
= <
⎡
⎢
⎢
⎣
1111
12
8
−1 −2
8
1 −11−1
1 −2
8
−12
8
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
W
1
0
W
2
0
W
1
∗
0
W
2
∗
0
⎤
⎥
⎥
⎦
>
M
(7.55)
where W
0
= e
−j (2π/5)
In order to use modulo arithmetic, the W
0
terms have to be
normalized to integer values. Multiplying these terms by 90 and
rounding off the results to integers, we obtain
⎡
⎢
⎢
⎣
W
0
W
1
W
2
W
3
⎤
⎥
⎥
⎦
=
⎡
⎢
⎢
⎣
−90 + j 0
0 + j 38229
202 ...