Infinite Impulse Response Filters - Introduction to Digital Signal Processing and Filter Design - page 212

4.2.8 Elliptic Function Approximation

There is another type of filter known as the elliptic function filter or the Cauer
filter . They exhibit an equiripple response in the passband and also in the stop-
band. The order of the elliptic filter that is required to achieve the given spec-
ifications is lower than the order of the Chebyshev filter, and the order of the
Chebyshev filter is lower than that of the Butterworth filter. Therefore the elliptic
filters form an important class, but the theory and design procedure are complex
and beyond the scope of this topic. However, in Section 4.11 we will describe
the use of MATLAB functions to design these filters.

4.3 ANALOG FREQUENCY TRANSFORMATIONS

Once we have learned the methods of approximating the magnitude response
of the ideal lowpass prototype filter, the design of filters that approximate the
ideal magnitude response of highpass, bandpass, and bandstop filters is easily
carried out. This is done by using well-known analog frequency transformations
p

g(s) that map the magnitude response of the lowpass filter H(j) to that of
the specified highpass, bandpass, or bandstop filters H(jω) . The parameters of
the transformation are determined by the cutoff frequency (frequencies) and the
stopband frequency (frequencies) specified for the highpass, bandpass, or band-
stop filter so that frequencies in their passband(s) are mapped to the passband of
the normalized, prototype filter, and the frequencies in the stopband(s) of the high-
pass, bandpass, or bandstop filters are mapped to the stopband frequency of the
prototype filter. After the normalized prototype lowpass filter H(p) is designed
according to the methods discussed in the preceding sections,the frequency trans-
formation p

=

g(s) is applied to H(p) to calculate the transfer function H(s) of
the specified filter. With this general outline, let us consider the design of each
filter in some more detail.

=

4.3.1 Highpass Filter

It is easy to describe the design of a highpass filter by choosing an example.
Suppose that a highpass filter with an equiripple passband ω p ∞ is

specified, along with a stopband frequency ω s . The magnitude at ω p ,whichis
the cutoff frequency of the passband, and also the magnitude at ω s (or A p and A s )
are given. The lowpass-highpass (LP-HP) frequency transformation p

≤ | ω | <

= g(s)

to be used in designing the highpass (HP) filters is

ω p
s

p = (4.75)

It is seen that when s = jω p , the value of p =− j and when s =− jω p ,the

value of p j . It can also be shown that under this transformation, all frequen-

cies in the passband of the highpass filter map into the passband frequencies

=