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This work describes a new general approach to the analysis of periodic tape helix circuits and its application in practical computer programs for numerical design. The procedure is based on transforming the stationary periodic tape helix itself to Fourier space, in the same manner as the RF fields. The interaction between the RF field harmonics and the stationary circuit harmonics is conceptually a convolution process. The mathematical description of this process is done through the intermediary of a convoluting circuit matrix created from the geometrical harmonics of the stationary metallic circuit, representing its Fourier space transform. Similar representations are made of the external dielectric rod and metallic vane system so that the entire configuration is analyzed by the same approach. The procedure separates the overall fields into two orthogonal distributions belonging to the space and the range of the convolution matrices of the relevant circuit regions. All relevant data such as the dispersion relation and fundamental and space harmonic impedances are obtained as part of the space distribution alone. The distributions belonging to the range have the character of independent idler fields playing no part in the solution process. The presented theory is exact for all configurations of tape helix structures and of anisotropic dielectric rods and metallic vanes. In order to benefit from reduced programming efforts, the exact theory is approximated by disregarding certain second-order effects, and implemented into a comprehensive computer program for design of tape helix circuits for traveling wave tubes. A set of computer-generated numerical data compared with experimental results are presented at the end.