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### A Physical Derivation of Wave Digital Filters

• To each element, such as a capacitor or inductor, attach a length of transmission line at impedance , and make it infinitesimally long. (Take the limit as the length of the transmission line goes to zero.)

• The infinitesimal transmission line is terminated by the element.

• The line impedance is arbitrary because it has been physically introduced.

• If two such line-augmented elements are connected together by their transmission lines, scattering will clearly be induced at the junction in the usual way.

• Calculate the reflectance of the terminated line. That is, find the Laplace transform of the return wave divided by the Laplace transform of the input wave going into the line:

• For a capacitor (impedence ), we get the reflectance , which simplifies to

• For an inductor , we get , or

• For a resistor , we get , or

• Note that both the capacitor and inductor reflectances are stable allpass filters, as they must be. Also, the resistor reflectance is always less than 1, no matter what line impedance we choose.

• Observe that there is a natural choice for each transmission-line impedance which will give us a normalized, universal reflectance for each element:

• For the capacitor,

• For the inductor,

• And for the resistor,

• Going to discrete time via the bilinear transform means making the substitution

where c is some arbitrary positive constant, usually taken to be .

• Solving for gives

• In this case, we see that setting further simplifies our universal reflectances in the digital domain:

Equivalently, we may obtain the same results by setting in the bilinear transform (which defines a frequency-scaling) and take the transmission-line (port) impedances to be instead for the inductor, and for the capacitor (thereby compensating the frequency scaling).

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