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This paper begins by presenting a powerful method which is easy to apply to many broad-band circuit design problems. Anyone with a broad-band design problem, which in the narrow-band case amounts to finding a point inside of a certain circle (say on the Smith chart), might find the method here very useful (Section I). Gain equalization problems fall into this category and the main subject of this paper is a conceptually appealing, highly practical, and very flexible theory of gain equalization. The clever matching theory developed by Fano and Voula in principle handles passive one-ports well except for some difficulty in computing gain-bandwidth limitations. It converts the main problem into computing solutions to a system of nonlinear equations which are in practice so formidable that typical text book treatments ,  never address the issue of solving them systematically. Also classical theory requires the load and gains to be specified as rational functions. Our theory does a good job on gain-bandwidth limitations, reduces all problems to ones of finding eigenvalues and eigenvectors of a given matrix, and only requires the load and gains be specified as data on a frequency band. Our theory is highly effective for multiports and so settles the old impedance matching problem for passive multiport circuits. The concrete results which we present here are: (1) Two numerically efficient ways to determine theoretical gain bandwidth limitations for one-ports and n-ports; (2) For one-ports a quick way to compute the frequency response function for the optimal coupling circuit directly from the answer obtained in (1). The recent advance of broad-band microwave technology has produced a need for more general and more flexible theories of gain equalization. The type of theory called for is based on measured data and avoids rational functions and spectral factorizations until late in the design process. One typically specifies a desired gain profile and then wants to find the largest multiple of it which is realizable. The procedure described herein is well suited to these needs since is requires only measured data and since determination of is automatic. A very different method for broadbanding which - fills these needs was developed by Carlin . It is a clever approach with quite a few compromises. One possible use of the lengthier rigorous procedure here would be to check the accuracy of Carlin's method. In addition to quantitative results we present some (much more easily learned) qualitative properties which every circuit (passive or active) designed to optimize gain possesses. They might be of considerable practical use in that any designer can learn them instantly and thereby obtain a certain (small) amount of general orientation very cheaply. The section on qualitative results, Section Ill, can be read independently of the rest of the paper (except for Fig. 1.1 and environs) and that might be best for some practical designers who have little taste for theory. Also in this paper we describe a certain viewpoint to the matching problem itself. From this perspective the matching problem is an elegant mathematical problem which fits solidly into a long line of classical mathematics. The classical mathematics underlies the computation of "prediction error" in Wiener's prediction theory. Our contention is that the matching problem is a very natural nonlinear analog of the classical prediction error problem. This formulation might broaden the appeal of impedance matching theory since it is easy to remember and is intriguing to the many systems theorists who are schooled in linear prediction theory (Section IV).