**Modern microwave filter synthesis**

The coupling matrix paradigm is today at heart of microwave filter synthesis in the narrow band domain. It consists in using a coupled resonator circuit model as a coarse model of the filter, and this independently of the technology used to eventually realize the filter (wave guide, planar technologies, LTCC etc…). Derivation of this ideal circuit model is obtained via the computation of classical Tchebychev and quasi-elliptic responses that are in turn realized as coupled resonator circuits. Such circuits are entirely characterised by their coupling matrix (CM for short). The circuital model is then implemented in the target technology using simplified charts, or empirical formulas, relating couplings and physical dimensions. This implementation can also be “virtual” and consist of an electromagnetic simulation on a computer. In either of two cases a fine tuning step is needed to adjust the resonating frequencies of each resonators, the inter-resonator couplings, compensate for the loading effects of the couplings on the resonators, identify parasitic couplings.

**Computer assisted tuning**

De-embedding procedures have therefore been designed in the last decade in order to identify circuits, that is coupling matrices, when starting from scattering measurements or simulations. The approach is complementary to the early design step: it consists to identify a circuit, the response of which, is compatible with the measurements or the EM simulations. Comparing the extracted coupling matrix with the ideal one, allows precise and localised adjustments on the DUT and has proven to drastically speed up the tuning phase. It is this circuital de-embedding task that the matlab toolbox Presto-HF intends to perform by relying on dedicated mathematical tools.

**How does Presto-HF work ?**

The goal of the software Presto-HF is to convert S-measurements in a coupled resonator circuit, represented by its coupling matrix. S-parameters of lumped element circuits, are rational functions of the frequency variable s=jω: part of the de-embedding problem therefore amounts to compute a rational model, of given MacMillan degree, that fits the measurements. The MacMillan degree of the rational model is equal to the number of coupled resonators composing the filter. The measurements entail some non-rational delay components: these are due to the access guides or access lines that induce errors in the determination of the “correct” reference planes of the S-parameters. The algorithmic approach of Presto-HF consists therefore of following steps:

- Removal of the delay components contained in the measurements
- Determination of a stable, rational, high order model of the delay-compensated data
- Computation of a rational model of given MacMillan degree, via Hankel norm approximation by means of a singular decomposition of the Hankel matrix of the high order model
- Gradient based rational approximation (with prescribed MacMillan degree) in order to minimize the L2 distance of the rational model to the measurements. This local optimization procedure is started from the previously obtained rational model and is performed using the software RARL2.

- Determination of a circuital realization of the rational model, yielding a coupling matrix with a canonical coupling topology

Depending on the coupling topology used for the filter’s realisation, the extracted coupling matrix can be further transformed by means of similarity transforms. In practice this can be performed, using for example the tool Dedale-HF.