Analog Filter Approximation Functions - Practical Analog and Digital Filter Design - page 53

some cases, the filter designer might trade the faster transition for the
nondecreasing attenuation.

Table 2.1
Comparison of Filter Gains at 2 rad/sec

Filter Type 3rd Order 4th Order

Butterworth −12.5 dB −18.3 dB

Chebyshev

−22.5 dB −33.8 dB

Inverse Chebyshev

−22.5 dB −33.8 dB

Elliptic −34.5 dB −51.9 dB

Although the magnitude characteristics of a filter are very important, the
phase characteristics of a filter are also crucial in many projects. Whether in audio
networks or data transmission systems, designers are looking for filters with linear
phase response. Nonlinear phase response in an audio network will cause
noticeable phase distortion for the listener that cannot be tolerated, especially in
high-quality systems. In data transmission systems nonlinear phase response
produces group delays that are functions of frequency. This produces distortion in
the pulses sent over the system and can distort edges and levels to the point of
causing errors in the received signal. We can compare the phase responses of
Figures 2.8, 2.14, 2.20, and 2.26 to see the level of phase distortion for each
approximation type. Remember that the transitions from −180 to +180 degrees are
not discontinuities, but rather a function of the display method. (The phase
response is written to a data file in its true form.) Table 2.2 shows the phase angles
for the third-order and fourth-order filters at the passband and stopband edge
frequencies.

Table 2.2
Comparison of Filter Phase at 1 and 2 rad/sec

Filter Type 3rd @ 1 r/s 3rd @ 2 r/s 4th @ 1 r/s 4th @ 2 r/s

Butterworth

−104° −192° −146° −266°

Chebyshev −154° −238° −230° −330°

Inverse Chebyshev −94° −192° −133° −264°

Elliptic

−150° −238° −226° −330°

As Table 2.2 and the phase plots indicate, the filters with the maximally flat
response in the passband (Butterworth and inverse Chebyshev) provide the most
linear response, although the inverse Chebyshev does have phase discontinuities
in the stopband caused by the complex zeros. These are usually not critical
because the filter's magnitude response is very small at these frequencies and the
distortion should be minimal. The phase responses of the standard Chebyshev and