Biomedical Engineering Reference
In-Depth Information
2
Single channel (univariate) signal
2.1 Filters
Signals may contain many frequency components. Some of them may be of in-
terest like frequency bands of EEG rhythms; others like power line artifacts may be
undesired. To remove the unwanted part of spectral contents of the signal one usu-
ally uses filters. In order to be implementable the filter must be
causal
.Thatis,the
filter response must only depend on the current and past inputs. If the filter has to
depend also on the past outputs it still can be causal by introducing the feedback
with appropriate delays.
We shall discuss here only the digital filters implemented in software (not the
hardware electronic devices used, e.g., as anti-aliasing filters before sampling). Most
of the digital filters perform the following action: they compute a linear combination
of a number
n
b
of past inputs
x
and a number of
n
a
of past outputs
y
to evaluate
current output
y
[
n
]
:
y
[
n
]=
b
(
1
)
∗
x
[
n
]+
b
(
2
)
∗
x
[
n
−
1
]+
...
+
b
(
n
b
+
1
)
∗
x
[
n
−
n
b
]
(2.1)
−
a
(
2
)
∗
y
[
n
−
1
]
−...−
a
(
n
a
+
1
)
∗
y
[
n
−
n
a
]
The maximum delay, in samples, used in creating each output sample is called the
order
of the filter. In the difference-equation representation (2.1), the order is the
larger of numbers
n
b
and
n
a
. There are three possibilities:
•
If
n
a
0 then the filter is called finite impulse response (FIR),
which means that once the filter is fed with one non-zero pulse the response
will have finite duration. After
n
b
time intervals the output sets back to zero.
This filter sometimes is called moving average, MA.
=
0and
n
b
=
•
If
n
b
0thefilter has an infinite impulse response (IIR). After a
single non-zero sample the filter could produce some output forever. It is also
called recursive or autoregressive (AR) filter.
=
0and
n
a
=
•
If
n
b
0 this is the most general filter type. It is also called IIR or
ARMA—autoregressive moving average.
=
0and
n
a
=
The IIR filters usually have lower orders than FIR filters with the same attenuation
properties.
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