Transfer-function realisation of a class of doubly terminated two-variable lossless networks and their application in two-dimensional analogue and digital filter design

The method used in the design of two-dimensional wave digital filters is to employ appropriate analogue reference filters. The analogue filters often used for this purpose are doubly terminated two-variable lossless two-port networks, since they are well known for their low sensitivity under coefficient quantisations. In this paper, necessary and sufficient conditions are given under which a two-variable rational function can be realised as the transfer function of a doubly terminated cascade of p1- and p2-variable lossless two-ports, each two-port having all of its transmission zeros at the origin or infinity. Using these conditions, the transfer function of a two-variable analogue reference filter is generated. Parameters of this transfer function are then used as variables of optimisation to minimise the least-mean-square error between the amplitude response of the ideal and designed filters. The discrete version of the filter is obtained by the application of a bilinear transformation to the designed analogue reference filter. To illustrate the method an example is given.