Digital Signal Processing Reference
In-Depth Information
L
C
v
1
(
t
)
v
2
(
t
)
R
Figure 6.19
Figure for Example 6.10.
With a little algebraic manipulation, the transfer function can be written as
1
1 + j(2.56 * 10
-4
)v + 1/j(9.75 * 10
-3
)v
.
H
B
(v) =
The
RLC
circuit shown in Figure 6.19 has a frequency response function of the form
For that circuit,
H
B
(v).
V
2
(v)
V
1
(v)
=
R
R + jvL + 1/jvC
=
1
1 + jvL/R + 1/jvRC
.
H(v) =
We now see that the bandpass filter can be realized by the choice of appropriate values for
R
,
L
, and
C
so that the terms of the denominator of
H(v)
match the corresponding terms in the
denominator of
H
B
(v).
For example, if we choose
L = 1H,
we can calculate the other com-
ponent values as
R = 3.9 kÆ
and
C = 2.5 mF.
■
The
ideal
filters considered in Section 6.1 are not physically realizable. However,
the concept of the ideal filter is useful in the initial stages of system analysis and
design.
The filter design process can be viewed as an attempt to approximate the fre-
quency response of an ideal filter with a physical system. Two standard methods of
achieving this approximation are the Butterworth and Chebyschev filter designs.
Physically realizable systems must be
causal;
that is, their impulse response
cannot begin before the impulse occurs. To approximate the amplitude frequency
response of ideal (
noncausal
) filters, the impulse response of the physical filter
must be similar to the impulse response of the ideal filter, but delayed in time.
This time delay results in a negative phase shift in the frequency response of the
physical filter.
6.3
BANDWIDTH RELATIONSHIPS
One of the concerns of an engineer in designing an electronic system is the frequency
bandwidth requirement. The Fourier transform provides a means of determining
the bandwidth of signals and systems. You will recall that we have sometimes called
