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The design of broad-band high-frequency transistor amplifiers is a difficult and challenging outstanding problem in active network theory of great practical importance. New and explicit optimum gain-bandwidth limitations of high-frequency transistor amplifiers for arbitrary prescribed transistor gain rolloff characteristics are presented. The transistor is modeled as a reactively constrained active two-port network. The limitations derived and presented are applicable to the design of broad-band small-signal as well as high-power transistor amplifiers. Realization of a class of practical broad-band matching networks are also presented. The explicit gain-bandwidth limitations and the new realization results presented form the basis for a new broadbanding theory for high-frequency transistor amplifiers and represent a significant advancement in the design theory of active networks.