In some applications (e.g. [6]), it may be desirable to compensate for the power modulation so that changes in the wave impedances of the waveguides do not affect the power of the signals propagating within.

In [8], three methods are discussed for making signal power
*invariant* with respect to time-varying branch impedances: (1)
The *normalized waveguide* scheme compensates for power modulation
by scaling the signals leaving the delays so as to give them the same
power coming out as they had going in. It requires two additional
scaling multipliers per waveguide junction. (2) The *normalized
wave* approach [4] propagates *rms-normalized waves* in the
waveguide. In this case, each delay-line contains
and
. In
this case, the power stored in the delays does not change when the
wave impedance changes. This is the basis of the *normalized
ladder filter* (NLF) [3,4]. Unfortunately, four
multiplications are obtained at each scattering junction. (3) The
*transformer-normalized waveguide* approach to normalization
changes the wave impedance at the output of the delay back to what it
was at the time it entered the delay using a ``transformer.''

A *transformer* joins two waveguide sections of differing wave
impedance in such a way that signal power is preserved and no
scattering occurs. From Ohm's Law for traveling waves, and from the
definition of power waves, we see that to bridge an impedance
discontinuity with no power change and no scattering requires the
relations

where

The choice of a negative square root corresponds to the

The transformer-normalized DWF junction is shown in Fig.4a. We can now modulate a single junction, even in arbitrary network topologies, by inserting a transformer immediately to the left or right of the junction. Conceptually, the wave impedance is not changed over the delay-line portion of the waveguide section; instead, it is changed to the new time-varying value just before (or after) it meets the junction. When velocity is the wave variable, the coefficients and in Fig.4a are swapped (or inverted).

So, as in the normalized waveguide case, for the price of two extra multiplies per section, we can implement time-varying digital filters which do not modulate stored signal energy. Moreover, transformers enable the scattering junctions to be varied independently, without having to propagate time-varying impedance ratios throughout the waveguide network.

It can be shown [9] that cascade waveguide chains built using
transformer-normalized waveguides are *equivalent* to those
using normalized-wave junctions. Thus, the transformer-normalized DWF
in
Fig.4a and the wave-normalized DWF in
Fig.4b are equivalent. One simple proof is to start
with a transformer and a Kelly-Lochbaum junction, move the transformer
scale factors inside the junction, combine terms, and arrive at
Fig.4b. One practical benefit of this equivalence is
that the normalized ladder filter (NLF) can be implemented with only
three multiplies and three additions instead of four multiplies and
two additions.

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