Analog Filter Approximation Functions - Practical Analog and Digital Filter Design - page 34

Figure 2.14 Chebyshev third-order and fourth-order phase responses.
2.4 INVERSE CHEBYSHEV NORMALIZED APPROXIMATION
FUNCTIONS
The inverse Chebyshev approximation function, also called the Chebyshev type II
function, is a rational approximation with both poles and zeros in its transfer
function. This approximation has a smooth, maximally flat response in the
passband, just as the Butterworth approximation, but has ripple in the stopband
caused by the zeros of the transfer function. The inverse Chebyshev
approximation provides better transition characteristics than the Butterworth filter
and better phase response than the standard Chebyshev. Although the inverse
Chebyshev has these features to recommend it to the filter designer, it is more
involved to design.
2.4.1 Inverse Chebyshev Magnitude Response
The development of the inverse Chebyshev response is derived from the standard
Chebyshev response. We will discuss the methods needed to determine the inverse
Chebyshev approximation function while leaving the intricate details to the
reference works. The name “inverse Chebyshev” is well-deserved in this case
since we will see that many of the computations are based on inverse or reciprocal
values from the standard computations. Let's begin with the definition of the
magnitude frequency response function as shown in (2.34).
The first observation concerning (2.34) is that it indeed has a numerator
portion that allows for the finite zeros in the transfer function. Upon closer
inspection, we find the use of ε i in place of ε. Equation (2.35) indicates ε i , the