Direct Single Frequency Synthesis from a Prescribed Scattering Matrix

This paper presents two techniques for synthesizing an n-port (at a single frequency) directly from its normalized scattering matrix without recourse to its associated impedance matrix. Both methods depend on standard matrix canonic forms. In the first one, all the loss is extracted immediately in the form of n resistors and the problem is reduced to the synthesis of a lossless 2n-port. The second makes use of the Jordan decomposition of a matrix and leads to a hybrid mixture of ideal transformers, reactances, gyrators and resistors. The only elements required for a complete synthesis are ideal transformers, lossless capacitors, inductors, and gyrators, and positive and negative resistors. It is shown that synthesis from a prescribed scattering matrix requires, in general, irrational operations (computation of eigenvectors, eigenvalues, etc.) whereas synthesis from a prescribed impedance matrix (if it exists) can be achieved with rational operations alone.