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Unusually low noise, combined with high stable gain over fairly wide bands, has been obtained with electron beam amplifiers of a new kind. This paper explains how this performance is achieved by the action of a transverse quadrupole field upon a fast cyclotron wave. The first two sections give a qualitative description of the device and of the amplifying mechanism. A physical picture of the fast cyclotron wave is used to explain the interchange of signal and noise in the input coupler and the mechanism of parametric amplification in the quadrupole region. The third section presents a detailed analysis of the amplification process. It shows that the fast cyclotron wave is amplified in accordance with a cosh function of distance traveled through the quadrupole, and that a new cyclotron wave at idler frequency (difference between pump and signal frequencies) is generated which grows as a sinh function of distance. The fourth section describes experimental tubes built to date. These operate on frequency bands between 400 and 800 mc. Typical bandwidth is 40 to 50 mc independent of gain, which may be adjusted to 20 or 30 db. Residual noise temperature of the electron beam in good specimens within this experimental lot is 70Â°K; input coupler loss raises this figure to about 100Â°K. This is equivalent to a noise figure of 1.4 db if the device is used, for instance, in radio astronomy.