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130 Chapter 4 Digital Filters

approximations. The Cauer filter, also called an elliptic filter, requires the
smallest order to meet a given magnitude specification.

Bessel The group delay is maximally flat at the origin and monotonically
decreasing in the passband. The magnitude function decreases rapidly in the pass-
band and the stopband attenuation is poor. Analog Bessel filters are not useful
prototypes for designing digital filters.

These filter approximations represent extreme cases since only one property
has been optimized at the expense of other properties. In practice they are often
used directly, but they can also serve as a starting point for an optimization proce-
dure trying to find a solution that simultaneously satisfies several requirements.
It is common to use the following notation to describe standard analog lowpass
filters—for example, COS 1525. The first letter denotes a Cauer filter (P for Butter-
worth, T for Chebyshev I, and C or CC for Cauer filters). There is no letter assigned to
Chebyshev II filters. The first two digits (05) denote the filter order while the second
pair denotes the reflection coefficient (15 %), and the third pair denotes the modular
angle (25 degrees). The latter is related to the cutoff and stopband frequencies by

\ s
We will show later that the reflection coefficient is related to the ripple in the
passband. A Butterworth filter is uniquely described by its order, P05, except for
the passband edge. To describe a Chebyshev I filter we also need the reflection
coefficient—for example, T0710. The Cauer filter requires in addition the modular
angle—for example, C071040.

4.7 DIRECT DESIGN IN THE z-PLANE

IIR filters can be designed by directly placing the poles and zeros in the z-plane
such that the frequency response satisfies the requirement. In principle, approxi-
mation methods, similar to the ones that have been derived for analog filters, can
also be derived for digital filters. Various numerical optimization procedures can
also be used. Numerical optimization procedures are normally used in the design
of FIR filters. However, it is widely recognized that it is advantageous to exploit
knowledge of analog filter synthesis to synthesize digital filters. For example,
widely available programs for the synthesis of analog filters can easily be extended
to the design of digital IIR filters.

4.8 MAPPING OF ANALOG TRANSFER
FUNCTIONS

As mentioned already, the most commonly used design methods capitalize on
knowledge and experience as well as the general availability of design programs for
analog filters. A summary of the classical design process is shown in Figure 4.10.
In the first design step, the magnitude specification is mapped to an equivalent
specification for the analog filter. Several mappings have been proposed: bilinear,
LDI, impulse-invariant, step-invariant, and the matched-z-transform [6]. However,