Page 210 - DSP Integrated Circuits

5.4 Quantization In WDFs 195

need a large number range. The number range of fixed-point arithmetic can be
increased by 6 dB by providing just one extra bit in the data word length. Further,
good algorithms tend to utilize the available number range effectively.

5.3 STABILITY

From the previous sections it is evident that stability is a very complicated issue
[12]. Ideally, a nonlinear system should behave as closely as possible to the corre-
sponding linear system so that the outputs from the two systems would be identi-
cal for arbitrary input signals. In other words, the difference between the outputs
of a forced-response stable filter and the ideal linear filter should for arbitrary
input signals tend to zero after a disturbance has occurred. As can be seen from
Figure 5.7, a second-order section in direct form I or II with saturation arithmetic
is not forced-response stable.
An even more stringent stability requirement is that independent of initial
values in the delay elements, the outputs of the nonlinear filter and the linear fil-
ter should become arbitrarily close. Stability, in this sense, can be guaranteed in
wave digital filters using the concept of incremental pseudo-passivity [23]. Large
errors resulting from overflow are often more rapidly suppressed in wave digital
filters as compared to the cascade and parallel forms.
A more complicated stability problem arises when a digital filter is used in a
closed loop [10]. Such loops are encountered in, for example, long-distance tele-
phone systems.

5.4 QUANTIZATION IN WDFs

Parasitic oscillations can, as already discussed, occur only in recursive structures.
Nonrecursive digital filters can have parasitic oscillations only if they are used
inside a closed loop. Nonrecursive FIR filters are therefore robust filter structures
that do not support any kind of parasitic oscillation. Among the IIR filter struc-
tures, wave digital filters, and certain related state-space structures are of major
interest as they can be designed to suppress parasitic oscillations.
Fettweis has shown that overflow oscillations can be suppressed completely in
wave digital filters by placing appropriate restrictions on the overflow characteris-
tic [8, 9]. To show that a properly designed wave digital filter suppresses any para-
1
sitic oscillation, we use the concept of pseudo-power which corresponds to power
in analog networks. Note that the pseudo-power concept is defined only for wave
digital filters and not for arbitrary filter structures.
The instantaneous pseudo-power entering the adaptor network, shown in
Figure 5.9, is defined as

where

!• Pseudo-power corresponds to a Lyapunov function.