Biomedical Engineering Reference
In-Depth Information
2.6 Digital Filters
Digital filters are the counterparts of analog filters used to eliminate noise
from digital signals (Tompkins, 1993; Oppenheim et al., 1999) and they do not
physically attenuate signal amplitudes rather they modify the coecients of
the digital sequences. Their specifications are usually derived from previously
designed and tested analog designs. There are two broad categories of digital
filter, the infinite impulse response (IIR) and the finite impulse response (FIR)
filters. These filters in turn have a variety of practical realizations such as
direct, canonical, and cascade configurations.
2.6.1 Infinite Impulse Response Filters
A linear system can be represented by the following difference equation:
N
M
a
k
y
(
n
−
k
)=
b
j
x
(
n
−
j
)
(2.126)
j
=0
k
=0
or its
z
-transform having the transfer function
j
=0
M
b
j
x
(
n
−
j
)
H
(
z
)=
Y
(
z
)
X
(
z
)
=
(2.127)
k
=0
a
k
y
(
n − k
)
N
The output
y
(
n
) can be obtained by expressing the system in terms of the
previous output and the current and previous input samples giving rise to
a recursive formulation, which is characteristic of IIR-type filters with the
following form:
N
M
a
k
a
0
y
(
n
b
j
a
0
x
(
n
y
(
n
)=
−
−
k
)+
−
j
)
(2.128)
j
=0
k
=0
There are several IIR filter configurations as discussed in the following sections.
2.6.1.1 Direct IIR Filter
The direct IIR filter is the basic digital filter with the configuration as in
Figure 2.14. From Equation 2.126, the direct form of the IIR filter can be
achieved by setting
a
0
= 1, so that Equation 2.126 simplifies to
N
M
1+
a
k
y
(
n
−
k
)=
b
j
x
(
n
−
j
)
(2.129)
k
=1
j
=0
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