Biomedical Engineering Reference
In-Depth Information
for which
|
z
|≥
1 simplifies to
∞
H
(
z
)
≤
n
=0
|
h
(
n
)
|
If there are no poles outside the unit circle, then the magnitude of transfer
function is finite, that is,
∞
H
(
z
)
≤
n
=0
|
h
(
n
)
|
<
∞
and the system is stable.
2.3.5 Discrete Cosine Transform
The discrete cosine transform (DCT) is often used in signal processing for com-
pression purposes, for example, in image processing. Compression reduces the
amount of data to be transmitted, which results in faster data rates, narrower
transmission bandwidths, and more ecient use of transmission power. The
DCT is a transform similar to the DFT, except that it uses only the real part
of the signal. This is possible because the Fourier series of a real and even
signal function has only cosine terms. The standard DFT can be written as
follows for
k
=1
,
2
,...,N
:
N
x
n
e
−
j
2
πnk
X
(
k
)=
(2.55)
N
n
=1
and the DCT can be obtained by taking the transform of the real part
giving
X
c
(
k
)=Re
N
=
x
n
cos
2
πkn
N
N
x
n
e
−
j
2
πnk
(2.56)
N
n
=1
n
=1
A more common form of DCT is given by the following for all
k
=1
,
2
,...,N
:
x
n
cos
πk
(2
n
+1)
2
N
N
X
c
(
k
)=
1
N
(2.57)
n
=1
The inverse of the DCT can be obtained using the following:
x
n
cos
πn
(2
k
+1)
2
N
N
X
c
(
k
)=
1
2
x
0
(2.58)
n
=2
Several other implementations of DCT can improve the computational speed
and algorithm eciencies (Ifeachor and Jervis, 1993).
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