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Spectrum calculation errors caused by truncation of results of mathematical operations in the decimation in time FFT algorithm

Keywords:

O.V. Ershova – Programmer , Supercomputers and Neurocomputers Research Center (Taganrog) E-mail: ershova150681@mail.ru E.V. Kirichenko – Research Scientist, Supercomputers and Neurocomputers Research Center (Taganrog) E-mail: e.v.kirichenko@yandex.ru E.A. Semernikov – Ph. D. (Eng.), Head of Department, Supercomputers and Neurocomputers Research Center (Taganrog) E-mail: semernikov@superevm.ru A.V. Chkan – Research Scientist, Supercomputers and Neurocomputers Research Center (Taganrog) E-mail: chkan_andrey@mail.ru


The errors of FFT calculation with decimation in time without scaling caused by limitation of bit width of the arithmetic multiplication results during fixed-point data processing are considered in the paper. A model of occurrence of the FFT calculation errors caused by both truncation and rounding of the product of data multiplied by phase factors is proposed. Analytical expressions for the mean-square values of FFT calculation errors depending on the spectral component’s number k are obtained. It is shown that the use of truncation of results of the products in the FFT procedure instead of rounding leads to significant errors and should therefore be avoided, despite the advantage in simplicity of implementation. The mathematical experiment was performed to calculate mean-square values of the errors in FFT arithmetic operations for the input sequence as a white Gaussian noise. The graphs of the mean-square values of truncation errors depending on the spectral component’s number and FFT size are presented. The computational modeling has proved matching of experimental and theoretical error levels.
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