Digital Signal Processing Reference
In-Depth Information
question. Generally speaking, high power and high frequency dictates that passive
filters be used. Active filters (with op-amps) have a maximum frequency range
and power range where they can be used successfully. (Section 4.7 discusses the
frequency issue in more detail.) Passive filters are also a bit more of a challenge to
design because they do not exhibit the near ideal qualities of input and output
impedances that op-amps have.
Another choice for filter implementation is the transconductance-C (Gm-C)
filter that extends the effective frequency range beyond that of the typical active
filter. Finally, switched-capacitor (SC) filters, like the Gm-C filters, use MOSFET
technology and switching signals to implement analog filters. In that way the SC
filter uses technology that bridges the gap between continuous-time and discrete-
time systems.
However, it is not possible to treat the complete field of analog filter design in
one chapter. Instead, we will concentrate on a more traditional approach to
implementing our transfer functions using a single form of active filter. The
Sallen-Key filter, which will be discussed in more detail in the following sections,
has the advantage of implementing a second-order factor with a single op-amp
stage. There are many other topologies (circuit configurations) that could be
chosen, but the Sallen-Key is a tried and true configuration. It is important to see
one method of implementing our continuous-time transfer function before
beginning the discussion of discrete-time systems.
Each of the active filters discussed will be composed of several stages of
electronic circuitry consisting of a single operational amplifier (op-amp) and a
number of electronic components called resistors (
R
s) and capacitors (
C
s). The
fact that active filters can implement complex poles without the use of inductors
(
L
s
)
is a key point in their favor. Inductors have the disadvantages of being large,
heavy, costly, and generators of spurious magnetic fields. Therefore, being able to
implement an active filter with components that can be miniaturized is a big
advantage.
Each of these stages of electronic filtering will have a transfer function that
characterizes the relationship of the output voltage to the input voltage, as
indicated in (4.1). More specifically, each quadratic factor of the transfer function
will have the form as shown in (4.2), where each
A
and
B
in the transfer function
will be a function of the
R
and
C
used in the circuit. (The subscript
c
is used to
indicate that these transfer functions are describing the
circuit
response.)
H
(
s
)
=
V
(
s
)
V
(
s
)
c
o
i
(4.1)
2
A
⋅
s
+
A
⋅
s
+
A
o
1
2
H
(
s
)
=
c
2
B
⋅
s
+
B
⋅
s
+
B
(4.2)
o
1
2
