Nonlinear, Time-Varying DWNs

The DWN paradigm is highly useful for constructing time-varying nonlinear digital
filters which are guaranteed *stable* in the
Lyapunov sense [59,25]. Lyapunov
stability analysis involves finding a positive-definite ``Lyapunov
function'' (which is like a norm defined on the filter state), which
decreases each time step (given zero inputs) in the presence of linear or nonlinear
computations (such as numerical round-off). Whenever each state variable
(delay element) in a filter computation has a physical interpretation
as an independent wave variable, a Lyapunov function can be defined simply as the sum of
wave energies associated with the state variables. By choosing the
scattering-junction round-off rules to be passive (e.g., by using magnitude
truncation), the total stored energy in the filter decreases or remains the
same relative to an infinite-precision implementation. As long as the
infinite-precision filter is asymptotically stable, the quantized filter
must be also. Thus, the total stored energy in any ``physical'' digital
filter is precisely what is needed for a Lyapunov function.

Since many ladder, lattice, and wave digital filters are equivalent to DWNs, possibly after some delay manipulations [51,92,93,75], it is clear that DWNs can be used to realize a wide class of transfer functions. In particular, every filter transfer function can be realized as a linear combination of DWNs (in a manner analogous to using ``tap parameters'' on a ladder or lattice filter [57,12]), and linear combinations are well behaved in the presence of nonlinearities and time-varying parameters.

In practice, Lyapunov stability in a DWN is normally
guaranteed by ensuring that wave-variables are never *amplified* by
nonlinear operations or time-varying gains. For example, magnitude
truncation on each output of an extended-precision scattering junction
suffices to ensure passivity. Non-amplifying nonlinearities and varying
gains are sufficient for passivity but sometimes too dissipative,
especially in waveguide mesh applications
[122]. A more refined rule is obtained by summing
rounded-off energy (or an approximation to it) in an
accumulator and changing the direction of rounding so as to servo the
cumulative energy error to zero [93].

In the normalized case, the energy associated with each signal sample in a
DWN is given by the square of that sample,^{8} so the total energy stored in a DWN is given simply as
the sum of squares of all the samples contained in all the delay elements.
In the unnormalized case, the energy of the sample in
waveguide , at time , is given by
, where
is the wave impedance of the th waveguide, and the total energy at
time in the DWN can be written as

where is the length of waveguide in samples. The DWN is stable in the presence of time-varying nonlinearities so long as (the Lyapunov functional for the DWN) is not increased when all inputs are zero, i.e.,

Thus,

The above general result is applied in the following specific example (which is related to a waveguide sitar model):

*Proof:*
It suffices to show that (a) the arrangement of
Fig. 4 is numerically equivalent to
a DWN, and (b) is a wave variable
of the DWN corresponding to a unit impedance.
Consider the redrawing of Fig. 4 as
in Fig. 5. The delay line is broken into two halves
to form a digital waveguide. (If the delay line length is odd, a sample of
delay can be moved in cascade with the allpass filter.) It is well known
that an arbitrary order allpass filter can be realized in the form of a
digital ladder filter [57] with no ``taps,'' and it is
straightforward to show that such a ladder filter can be manipulated into a
digital waveguide filter by splitting each unit delay into two half-delays
and commuting half of the half-delays around the ladder so that there is a
half-delay between each scattering junction on both the upper and lower
rails of the lattice [92].^{9} In this configuration, also shown in
Fig. 5, the ladder filter is a DWN consisting of
a cascade of digital waveguides, where the input and output of the
allpass are the incoming and outgoing wave variables at the left of
the leftmost waveguide of the allpass, and the rightmost waveguide is terminated
on the right by a perfect reflection (infinite wave impedance, given
pressure or force waves). Without loss of generality, we may choose the wave
impedance of the bidirectional delay line to be . The return
signal from the allpass is a wave variable at wave impedance 1, and its energy at
time is . After scaling by the nonlinearity, its energy
becomes
, and the energy lost each sample
interval is therefore
, which is nonnegative,
and the total energy lost is simply the sum over time of these
per-sample losses.

Of course, by keeping track of the energy gained or lost, as
illustrated in the above theorem, and by controlling the
gain/nonlinearity so that energy gained or lost is compensated at a
later time, a wider class of nonlinear and/or time-varying systems may
be defined which are lossless *on average*. An example is the
exactly lossless (on average) DWN which servos round-off error energy
to zero mentioned above. It is also possible to approximate lossless
nonlinear behavior within the allpass filter
[67,127].

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