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A practical one-port parametric amplifier is analyzed to determine the conditions under which minimum effective noise temperature and maximum gain-bandwidth product can be obtained. The analysis considers the effects of resistive loss and stray parasitic reactance in the junction-diode that provides the essential nonlinear reactance. It is shown that the conditions necessary for minimum noise temperature are compatible with those necessary for maximum gain-bandwidth product only if the diode has a high self-resonant frequency (the frequency at which the average diode capacitance resonates with the diode lead inductance). It is also shown that minimum noise temperature is always achieved when the diode loss alone is used as the idler circuit loading (regardless of the temperature of any additional idler loading), that there is a characteristic figure of merit for the diode, and that there is an optimum pump frequency. Based on the equations derived, universal curves have been drawn that enable the design of an optimum amplifier when the signal frequency and diode characteristics are specified. In conclusion, it is shown that if junction-diode parametric amplifiers operated at room temperature are to seriously challenge the low-noise performance of the maser at microwave frequencies, a substantial improvement in the diode figure of merit is required.