In this paper, a detailed analysis on the operating mechanism and parameter properties of free electron laser amplifier (FELA) in the case of small signal is presented. The results show that FELA is in essence the interaction between the electron beam and the fast cyclotron wave. The cyclotron resonance condition of electron motion in the uniform axial and periodic transverse static magnetic fields significantly influences the interaction process. We have calculated numerically the central operating frequency, band width, small signal gain and wave number of the amplified electromagnetic wave as functions of the parameters of the amplifier (such as the voltage and current density of the electron beam, the intensities of axial and transverse static magnetic field). The way to optimize the design of FELA is also suggested. It is possible to obtain a small signal gain of 10-2 cm-1 with enough band width in the millimeter and submillimeter range of weve length following a proper design.