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Power-Normalized Waveguide Filters

Above, we adopted the convention that the time variation of the wave impedance did not alter the traveling force waves $ f^\pm _i$. In this case, the power represented by a traveling force wave is modulated by the changing wave impedance as it propagates. The actual power becomes inversely proportional to wave impedance:

$\displaystyle {\cal I}_i(t,x)
= {\cal I}^{+}_i(t,x)+{\cal I}^{-}_i(t,x)
= \frac{[f^{+}_i(t,x)]^2-[f^{-}_i(t,x)]^2}{R_i(t)}

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 $ {\tilde f}^{+}_i(t,x) =
f^{+}_i(t,x)/\sqrt{R_i(t)}$ and $ {\tilde f}^{-}_i(t,x) = f^{-}_i(t,x)/\sqrt{R_i(t)}$. 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

$\displaystyle \frac{[f^{+}_i]^2}{R_i(t)} = \frac{[f^{+}_{i-1}]^2}{R_{i-1}(t)} \qquad\qquad
\frac{[f^{-}_i]^2}{R_i(t)} = \frac{[f^{-}_{i-1}]^2}{R_{i-1}(t)}

Therefore, the junction equations for a transformer [1] can be chosen as

$\displaystyle f^{+}_i= g_i(t) f^{+}_{i-1}\qquad\qquad f^{-}_{i-1}= g_i^{-1}(t) f^{-}_i$ (1)


$\displaystyle g_i(t) \isdef \sqrt{\frac{R_i(t)}{R_{i-1}(t)}} = \sqrt{\frac{1+k_i(t)}{1-k_i(t)}}$ (2)

The choice of a negative square root corresponds to the gyrator [1]. The gyrator is equivalent to a transformer in cascade with a dualizer [9]. A dualizer is a direct implementation of Ohm's Law for traveling waves (to within a scale factor): the forward path is unchanged while the reverse path is negated. On one side of the dualizer there are force waves, and on the other side there are velocity waves. Ohm's law can thus be interpreted as a gyrator in cascade with a transformer whose scale factor equals the wave admittance.

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 $ g_i$ and $ g_i^{-1}$ in Fig.4a are swapped (or inverted).

Figure 4: a) Transformer-normalized waveguide digital filter section, for transformer on left of junction. b) Normalized ladder filter section. The two are equivalent.

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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``Waveguide Filter Tutorial'', by Julius O. Smith III, Proceedings of the International Computer Music Conference (ICMC-87, Champaign-Urbana), pp. 9-16, Computer Music Association, 1987..
Copyright © 2010-11-22 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University