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Self-calibration of divided circles on the basis of a prime factor algorithm

A new method for the self-calibration of divided circles is presented which is based on a known prime factor algorithm for the discrete Fourier transform (DFT). The method, called prime factor division (PFD) calibration, is of interest in angle metrology specially for self-calibrating angle encoders, and generally for a significant shortening of the cross-calibration between two divided circles. It requires that the circular division number N can be expressed as a product N = R × S, whereby the factors R and S are relatively prime integer numbers. For the self-calibration of a divided circle, N difference measurements between R angle positions in a regular distribution and one reference angle position determined by S are evaluated by a two-dimensional DFT, yielding the N absolute division errors. The factor R is preferably chosen small, down to a minimum of R = 2, whereas the factor S may be as large as appropriate for the division number N of interest. In the case of a cross-calibration between two divided circles, the PFD method reduces the number of measurements necessary from N2 to (R + 1) × N. Experimental results are demonstrated for the calibrations of an optical polygon with 24 faces (prime factor product 3 × 8) and a gearwheel with 44 teeth (prime factor product 4 × 11).