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The electron beam interaction in a novel slow-wave structure (SWS) called dielectric-lined azimuthally periodic circular waveguide (DLAP-CW) is analyzed in a linear frame. Moreover, the linear gain characteristics of the DLAP-CW are obtained by the self-consistent relativistic field theory. Analytical solutions for the hot dispersion characteristics are derived, and the complicated dispersion equations have been numerically solved with MATLAB. The small-signal growth rate is calculated for dimensions of the improved SWS and the parameters of the electron beam. It is shown that selecting the appropriate thickness and location of the metal rods increases the small-signal gain (dielectric constant held fixed). In addition, the gain of the DLAP-CW increases as the beam current increases, and the beam voltage not obviously influences the small-signal gain. Furthermore, a comparison of the small-signal gain of this structure with a conventional dielectric-lined circular waveguide (DL-CW) is made, and the results validate that the novel SWS has an advantage over the DL-CW on the electron efficiency, potentially resulting in a higher gain traveling-wave-tube circuit.