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4.11 Reference Filters 139

for the filter in Figure 4.19. The filter is a fifth-order LC ladder structure with
Chebyshev I characteristic.
Figure 4.18 also shows the
variations in the attenuation for
very large changes (±20% from the
nominal value) of one of the induc-
tances in the ladder structure. The
passband ripple has been chosen to
be a very large value, A max = 3 dB,
in order to demonstrate the low Figure 4.18 Double resistively terminated
sensitivity. reactance network
A counterintuitive property of
these filters is that sensitivity in
the passband is reduced if the rip-
ple is reduced. It follows from Fett-
weis-Orchard's argument that the
attenuation is bounded from below
(the magnitude function is bounded
from above) by the maximum power
transfer constraint. This lower
value is usually normalized to 0 dB.
Hence, if any element value in the
lossless network deviates from its Figure 4.19 Attenuation in the passband for a
nominal value, the attenuation fifth-order LC filter with nominal
must necessarily increase and can- element values and ±20%
deviation in one of the inductors
not decrease independently of the
sign of the deviation. Thus, for these
frequencies the sensitivity is zero, i.e.,

where e represents any element in the lossless network—for example, an induc-
tance. These filters are optimal for frequency-selective filters with "flat" pass-
bands. The sensitivity in the stopband depends on the internal structure of the
reactance network. Usually, lattice or ladder networks are used.
Many schemes for simulating doubly terminated LC networks have been
developed for active RC filters [24]. Unfortunately, the problem of simulating an
analog reference filter using a digital network is a nontrivial problem. Generally, a
nonsequentially computable algorithm is obtained (see Chapter 6). However,
Alfred Fettweis [7—10] has developed a comprehensive theory, the wave digital fil-
ter theory, which solves the problem. Indeed, it is not only a theory for design of
digital filters, it is also a general theory describing the relationships between cer-
tain discrete-time networks and certain classes of lumped and distributed element
2
networks. Furthermore, it inherently contains an energy concept which can be
used to guarantee stability in nonlinear digital networks.

2
- There exists a Lyanuponov function for wave digital filter structures.