Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

### A Physical Derivation of Wave Digital Elements

This section provides a ``physical'' derivation of Wave Digital Filters (WDF), which contrasts somewhat with the more formal derivation common in the literature. The derivation is presented as a numbered series of steps (some with rather long discussions):

1. To each element, such as a capacitor or inductor, attach a length of waveguide (electrical transmission line) having wave impedance , and make it infinitesimally long. (Take the limit as its length goes to zero.) A schematic depiction of this is shown in Fig.F.1a. For consistency, all signals are Laplace transforms of their respective time-domain signals. The length must approach zero in order not to introduce propagation delays into the signal path.

Points to note:

• The infinitesimal waveguide is terminated by the element. The element reflects waves as if it were a new waveguide section at impedance , as depicted in Fig.F.1b.

• The interface to the element is recast as traveling-wave components and at impedance . In terms of these components, the physical force on the element is obtained by adding them together: .

• The waveguide impedance is arbitrary because it has been physically introduced. We will need to know it when we connect this element to other elements. The element's interface to other elements is now a waveguide (transmission line) at real impedance .

• The junction is ``parallel'' (cf. §7.2):

• Force (voltage) must be continuous across the junction, since otherwise there would be a finite force across a zero mass, producing infinite acceleration.

• The sum of velocities (currents) into the junction must be zero by conservation of mass (charge).

Subsections
Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work]  [Order a printed hardcopy]  [Comment on this page via email]