Digital Signal Processing Reference
In-Depth Information
We also have the approximation function that we derived to meet a specific
set of frequency and attenuation characteristics. The general form of each
quadratic factor in the approximation function is shown in (4.3), where the
subscript
a
is used to designate this quadratic factor as an
approximation
function.
The
a
s and
b
s used as coefficients have already been determined and are
numerical constants.
2
a
⋅
s
+
a
⋅
s
+
a
o
1
2
H
(
s
)
=
a
2
b
⋅
s
+
b
⋅
s
+
b
(4.3)
o
1
2
The implementation process then becomes a matter of equating the two
transfer function factors as shown below. Each
A
and
B
of the circuit transfer
function are equated to the respective
a
and
b
of the approximation function and
results in several equations which must be solved for appropriate
R
and
C
values.
H
(
s
)
=
H
(
s
)
(4.4)
c
a
We will see this common procedure used throughout the next four sections as
we determine the component values needed to implement each of the active filters.
4.2 LOWPASS ACTIVE FILTERS USING OP-AMPS
There are a number of active filter topologies (circuit configurations) that could be
used to implement a lowpass filter. We will limit ourselves to the popular Sallen-
Key filter (shown in Figure 4.1) with a transfer function as described in (4.5). As
indicated by the transfer function, this active filter stage can implement one
second-order factor of a lowpass filter function. This form is very convenient
since it naturally implements the quadratic factor that we have been using for the
description of the approximation function. Of course, several of these circuit
stages in cascade can be used to implement higher-order functions, and with the
addition of a single first-order stage, odd-order filters can be implemented as well.
Figure 4.1
Sallen-Key lowpass active filter stage.
