Digital Signal Processing Reference
In-Depth Information
Figure 2.26
Elliptic third-order and fourth-order phase responses.
2.6 COMPARISON OF APPROXIMATION METHODS
Now that we have discussed the four approximation methods and displayed third-
order and fourth-order magnitude and phase plots, we are in a position to compare
the results. First, we look at the magnitude plots of Figures 2.7, 2.13, 2.19, and
2.25. Table 2.1 shows the gains achieved at the stopband edge frequency of 2
rad/sec for each normalized filter type and order. (Each filter was designed with a
passband gain of −1 dB.) Obviously, if attenuation characteristics in the stopband
are the primary concern, an elliptic filter would have to be the choice.
It provides 12 dB more attenuation than the Chebyshev types and 22 dB more
attenuation than the Butterworth filter for the third-order case. In the fourth-order
case, the differences increase to over 18 and 33 dB compared to the Chebyshev
and Butterworth filters. The Chebyshev filter types themselves afford better
stopband characteristics when compared to the Butterworth filter. They provide 10
and 15 dB more attenuation for the third-order and fourth-order cases. Although
the table only lists the gains for third-order and fourth-order filters, the same trend
continues for higher-order filters.
Although the Chebyshev and inverse Chebyshev filters provide the same
gains at the passband and stopband edge frequencies, their responses are not
identical. If we were to take a close look at the frequency response in the
passband, we would find that the inverse Chebyshev provides a better
approximation to the ideal response except at frequencies very near to 1 (the
normalized passband edge frequency). In that case, the standard Chebyshev
produces a tighter fit. In the transition band, the standard Chebyshev response
provides a more rapid transition. And in the stopband, the standard Chebyshev's
response continues to increase the attenuation as the frequency increases, while
the inverse Chebyshev's response alternates between small gains and
a
stop
. In
