DESIGN AND ANALYSIS OF ANALOG FILTERS A Signal ...

A Signal Processing Perspective 225
Example 7.8
Suppose the following specifications are given:
and From the MATLAB function
BESSELOR, N = 9.
7.5 POLE LOCATIONS
Previous filters presented, Butterworth, Chebyshev Type I, Chebyshev Type
II, and elliptic, all have closed-form convenient expressions for the magnitude
frequency response, from which expressions are found for the poles and zeros. The
design of Bessel filters has been presented above, either as a delay network with some
given desired time delay, or as a lowpass filter with given magnitude frequency
response specifications. As mentioned above, no convenient closed-form expression
exists for the magnitude frequency response of a Bessel filter.
The design of a lowpass Bessel filter using the MATLAB function BESSELDE
yields the poles of the transfer function directly. The design of a Bessel time-delay
filter using (7.18) followed by frequency scaling, as in Example 7.6, yields the
transfer function expressed as a constant over a polynomial is s. The poles may be
found by determining the roots of the transfer function denominator polynomial. This
may be accomplished using the MATLAB function ROOTS. All zeros, of course, are
at infinity.
7.6 PHASE RESPONSE, PHASE DELAY, AND GROUP DELAY
A Bessel filter, as seen above, is designed for a maximally-flat group delay
response. The group delay response, normalized for a unit delay, is shown in Figure
7.1 above. The magnitude frequency response, normalized for a 3 dB corner
frequency of unity, is also shown above in Figures 7.3 and 7.4.
The phase response of a Bessel filter, with a normalized and several
values of N, is shown in Figure 7.5. Taking the initial phase slope as a linearphase
reference, deviations from linear phase, for a normalized and for
several values of N, are shown in Figure 7.6. In the figure, solid lines are for even
orders, and dashed lines are for odd orders.
The phase delay, for a filter is defined in (2.80), which is repeated
here for convenience:
The group delay for a filter, is defined by (2.81) and is repeated here
for convenience:
Section 7.6 Phase Response, Phase Delay, and Group Delay