Next: Wave Digital Elements Up: Wave Digital Elements and Previous: Wave Variables

## Pseudopower and Pseudopassivity

Fettweis [42] defines the instantaneous pseudopower absorbed by a port with port resistance (real) at time step in terms of the discrete input and output wave quantities as

 (2.18)

which, when the transformation (2.14) is inverted, gives

This discrete power definition coincides with the standard definition of power in classical network theory from (2.2), aside from the factor of 4, which is of no consequence if definition (2.18) is applied consistently throughout a wave digital network.

For a real LTI -port, in an exponential state of complex frequency , the steady-state average pseudopower may be written in terms of the vectors and which contain the power-normalized complex amplitudes and , for as

The steady-state reflectance is defined by

and gives

where is the identity matrix. For pseudopassivity [42], we require, then (recalling that the bilinear transform (2.11) maps the right half -plane to the exterior of the unit circle in the plane) that

 for (2.19)

is sometimes called a bounded real matrix. If (2.19) holds with equality for , then it is called lossless bounded real (LBR) [193]. In general, to bounded real matrix reflectances there correspond positive real matrix impedances, and vice versa. In terms of voltage wave quantities, we have for a wave digital -port, that

where is the diagonal square root of the matrix containing the port resistances on its diagonal. We then have

for the voltage wave scattering matrix and thus we require

 for (2.20)

for passivity. For one-ports, the requirements (2.20) and (2.19) are the same.

Also note that we have, by applying the power wave variable definitions (2.16), and the discrete impedance relation (which is identical to the analog relation from (2.1), except that we now have ), that

 (2.21)

If the -port is not LTI, then it is possible to apply a similar idea to the expression for the instantaneous pseudopower, from (2.18) in order to derive a passivity condition [46]; In this case, pseudopassivity has also been called incremental pseudopassivity [125].

Next: Wave Digital Elements Up: Wave Digital Elements and Previous: Wave Variables
Stefan Bilbao 2002-01-22