082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

24.14 The discrete cosine transform 797
(b) We now apply the inverse d.c.t. toF[k].
The inverse d.c.t. is
f[n] = √ 1 ∑N−1
[ ( π
{F[0] +2 F[k]cos
N N k n + 1 )] }
2
k=1
forn=0,1,2, ... ,N−1
Whenn=0
{
f[0] = √ 1 12
2∑
√ +2
3 3
k=1
= √ 1 [ 12
√ +2× (−2) ×cos
3
[ 3
= √ 1 12
√ −4×
3 3
[ ( π
F[k]cos
3 k 0 + 1 )] }
2
( π
) ]
+2×0
6
√
3
2 +0 ]
=4−2=2
Whenn=1
{
f[1] = √ 1 12
2∑
√ +2
3 3
= 1 √
3
[ 12
k=1
√ +2× (−2) ×cos
3
]
= 4
= 1 √
3
[ 12
√
3
−4×0+0
[ ( π
F[k]cos
3 k 1 + 1 )] }
2
( π
) ]
+2×0
2
Whenn=2
{
f[2] = √ 1 12
2∑
√ +2
3 3
k=1
[ ( π
F[k]cos
3 k 2 + 1 )] }
2
( ) ] 5π
+2×0
6
= √ 1 [ 12
√ +2× (−2) ×cos
3
[ 3
(
= √ 1
√ ) ]
12 3
√ −4× − +0 =4+2=6
3 3 2
So f[n] = 2,4,6, which was the originalsequence as expected.
If a d.f.t. had been performed on the sequence f[n], the output,F[k], would have been
the sequence of complex terms 12, −3 + j1.7321,−3 − j1.7321. Compare this with
the d.c.t. which isF[k] = 12 √
3
,−2, 0. Note that this latter sequence contains only realvalued
terms.