Digital Signal Processing Reference
In-Depth Information
time shift involved is called the
phase delay,
and it is a function of the signal fre-
quency. This time delay (or phase shift if we consider the frequency-domain mani-
festation) is a characteristic of all physically realizable filters. Generally, the more
closely a physical filter approximates an ideal filter, the more time delay (negative
phase shift) is present in the output signal.
It is apparent from a comparison of the magnitude frequency spectrum of the
RC
low-pass filter, shown in Figure 6.10, with those of the ideal low-pass filter,
shown in Figure 6.11, that the
RC
low-pass filter is a relatively crude approximation
of the ideal low-pass filter. The magnitude ratio of the
RC
low-pass filter decreases
gradually as the frequency increases, instead of remaining flat, for
Also, the magnitude ratio is finite instead of zero for Engineers who design
analog electronic filters try to achieve a “good-enough” approximation to the ideal
filter by designing for a “flat-enough” frequency response in the passband and a
“steep-enough” roll-off at the cutoff frequency. One of the filter designs that is
commonly used to satisfy these criteria is the
Butterworth filter
.
0
F
v
F
v
c
.
v 7 v
c
.
The general form of the magnitude frequency-response function for the Butterworth
filter is
1
ƒH(v) ƒ =
1 + (v/v
c
)
2N
,
(6.2)
2
where
N
is called the “order” of the filter. In other words,
N
is the order of the dif-
ferential equation needed to describe the dynamic behavior of the filter in the time
domain. By comparing (6.1) and (6.2), we can see that the
RC
low-pass filter is a
first-order Butterworth filter.
Figure 6.12(a) and (b) show
RLC
realizations of second- and third-order
Butterworth filters, respectively. The values of the electrical components are deter-
mined by the desired cutoff frequency,
v
c
,
as indicated in the figure. It is left as an
H
( )
1
Figure 6.11
Frequency spectrum of an
ideal low-pass filter.
/
c
1
1
