We propose a general approach to the synthesis of cross-coupled resonator filters using an analytical gradient-based optimization technique. The gradient of the cost function with respect to changes in the coupling elements between the resonators is determined analytically. The topology of the structure is strictly enforced at each step in the optimization thereby eliminating the need for similarity transformations of the coupling matrix. For the calculation of group delays, a simple formula is presented in terms of the coupling matrix. A simple recursion relation for the computation of the generalized Chebychev filtering functions is derived. Numerical results demonstrating the excellent performance of the approach are presented.

Synthesis of cross-coupled resonator filters using an analytical gradient-based optimization technique

The paper presents a universal and comprehensive synthesis technique of coupled resonator filters with source/load-multiresonator coupling. The approach is based on repeated analyses of a circuit with the desired topology; no similarity transformation is needed. Restrictions imposed by the implementation on the coupling coefficients such as signs and orders of magnitudes are straightforwardly handled within this technique. The technique is then used to synthesize and design filters with full or almost full coupling matrices by selecting, among the infinite number of solutions, the matrix that corresponds to the actual implementation. In such cases, analytical techniques and those based on similarity transformations cannot be used since they provide no mechanism to constrain individual coupling coefficients in order to discriminate between two full coupling matrices, which are both solutions to the synthesis problem. Using the technique described in this paper, a filter designer can extract the coupling matrix of a filter of arbitrary order and topology while enforcing relevant constraints. There is no need to master all the different existing similarity-transformation-based techniques and the topologies to which they are applicable. For the first time, detailed investigations of parasitic coupling effects, for either compensation or utilization, are made possible. The method is applied to the synthesis of a variety of filters, some of which are then designed and built and their response measured.

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Adaptive synthesis and design of resonator filters with source/load-multiresonator coupling

This paper presents a tutorial overview of the use of coupling between nonadjacent resonators to produce transmission zeros at real frequencies in microwave filters. Multipath coupling diagrams are constructed and the relative phase shifts of multiple paths are observed to produce the known responses of the cascaded triplet and quadruplet sections. The same technique is also used to explore less common nested cross-coupling structures and to predict their behavior. A discussion of the effects of nonzero electrical length coupling elements is presented. Finally, a brief categorization of the various synthesis and implementation techniques available for these types of filters is given.

Cross-coupling in coaxial cavity filters - a tutorial overview

In this article, a process that might be described as "synthesis and approximation" was outlined. Starting from an exact prototype, the desired physical configuration using equivalent circuits and corrections to the resonators due to the influence of the coupling networks was approximated. A process that might be described as "approximation and optimization" was also outlined. Starting from an approximate filter network, optimization can be used to find an exact equal ripple solution. The starting point can be generated using synthesis or narrow-band approximations when appropriate. Applying optimization in an intelligent way allows the designer to circumvent some of the limitations of the classic ladder synthesis method. This article explores some ways to mix approximate filter dimensioning (based on network synthesis) and optimization (based on EM modeling), allowing a fast and accurate design of microwave filters

Microwave filter design by synthesis and optimization

Space mapping is a recognized method for speeding up electromagnetic (EM) optimization. Existing space-mapping approaches belong to the class of surrogate-based optimization methods. This paper proposes a cognition-driven formulation of space mapping that does not require explicit surrogates. The proposed method is applied to EM-based filter optimization. The new technique utilizes two sets of intermediate feature space parameters, including feature frequency parameters and ripple height parameters. The design variables are mapped to the feature frequency parameters, which are further mapped to the ripple height parameters. By formulating the cognition-driven optimization directly in the feature space, our method increases optimization efficiency and the ability to avoid being trapped in local minima. The technique is suitable for design of filters with equal-ripple responses. It is illustrated by two microwave filter examples.

Cognition-Driven Formulation of Space Mapping for Equal-Ripple Optimization of Microwave Filters

A synthesis procedure, using optimization, for multiple coupled resonator filters having general topology and general response is described. The error function for the optimization is based on the values of the characteristic function at its zeros and poles. The optimization is performed directly on the element values of the coupling matrix. Convergence of the optimization is extremely fast and nearly independent of the starting coupling matrix. Examples of the design of practical filters with symmetric or asymmetric responses and topology are presented.

W.A. Atia

Papers

Synthesis of general topology multiple coupled resonator filters by optimization