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.

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.

The concept of the conventional coupling matrix is extended to include designs of dual-and triple-band filters. The multiband response is created by either placing transmission zeros within the bandwidth of a wideband filter or using higher order resonances. Realizable topologies both in planar and waveguide technologies can be imposed and associated coupling coefficients enforced during optimization. The design process is verified by measurements and comparison with results of commercially available field solvers. Index Terms-Dual-band filters, filters, microstrip filters, waveguide filters. I. INTRODUCTION R ECENT developments in microwave filters have focused on many different topics, one of them being the design of dual-and triple-band filters. They are in heavy demand due to applications in modern wireless and satellite systems. Usually, theoretical design techniques are related to individual filter technologies. For instance, lumped-element or stepped-impedances approaches are used in low-temperature co-fired ceramic (LTCC) applications, e.g., [1], [2]. So-called dual-behavior resonators [3], [4] create attenuation poles at specific frequencies in order to separate individual passbands. The coupling between two modified open-loop resonators is used to create a dual-band filter in microstrip technology [5]. Polynomial approaches and coupling matrices are applied to the design of dual-band bandpass [6] and bandstop filters [7] in waveguide technology.Common to all such design procedures is the fact that they cannot immediately be used if the filter topology changes. Moreover, many approaches are limited with respect to the number of transmission zeros and locations over the frequency band of interest.Therefore, this paper focuses on the design of dual-and tripleband filters by employing the coupling matrix and the optimization of its entries. One of the major advantages of this approach is that topologies and certain coupling elements can be controlled from the onset [8]. The basic approach of this method was introduced in [9]. However, measurements failed to confirm the transmission zeros between individual passbands.

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