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A new analysis is given of direct-coupled-resonator filters that results in excellent response at much greater bandwidths than has previously been possible. The method relies on the fact that the coupling elements can be made into perfect impedance inverters, or "quarter-wave" transformers, by the addition of negative elements in lumped-constant circuits, or of short negative lengths of line in waveguide and transmission-line circuits. Specific design formulas are given for filters constructed of lumped-constant elements, waveguide, and strip or other TEM transmission line, and for pass band response functions of the maximally flat and Tchebycheff types. The formulas include a simple frequency transformation that corrects for the frequency sensitivity of the coupling reactances, and thereby greatly improves the design accuracy for both lumpedconstant and microwave types when the bandwidth is more than a few per cent. Exact response curves computed from typical filter designs are compared to the prototype-function response curves, and it is shown that the design formulas give good results with bandwidths of at least 20 per cent in guide wavelength in the case of waveguide filters, or 20 per cent in frequency for TEM-mode transmission-line and lumped-constant filters.