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
206 Transform Domain Processing for Recent Signal and Video Applications
The individual terms in Eq. (7.3) can be written as
Y
(
k
)
= X
(
k
)
H
(
k
)
,
where
X (k) =
N−1
=0
t
k,
x()
and
H (k) =
N−1
n=0
t
k, n
h(n)fork = 0, 1, . . . , N − 1
Hence,
Y (k) =
N−1
=0
t
k,
x
(
)
N−1
n=0
t
k, n
h
(
n
)
=
N−1
n=0
N−1
=0
x
(
)
h
(
n
)
t
k,
t
k, n
However, Y (k) can also be expressed by putting Eq. (7.1) in the
transform domain:
Y (k) =
N−1
m=0
t
k, m
y(m)
=
N−1
m=0
t
k, m
N−1
n=0
x
(
m − n
)
h
(
n
)
Let = m − n;hence, m = + n,where = 0, 1, . . . , N − 1, we
get
Y
(
k
)
=
N−1
n=0
N−1
=0
x
(
)
h(n)t
k, +n
(7.4)
Equation (7.4) must be equivalent to Eq. (7.4);