Analysis of errors in the computation of Fourier coefficients using the arithmetic Fourier transform (AFT) and summation by parts (SBP)

The computational complexity and the effects of quantization and sampling instant errors in the arithmetic Fourier transform (AFT) and the summation by parts discrete Fourier transform (SBP-DFT) algorithms are examined. The relative efficiency of the AFT and SBP-DFT algorithms is demonstrated by comparing the number of multiplications, additions, memory storage locations, and input signal samples as well as the latency time and level of parallelism of these two methods with that of more conventional single-output DFT and multiple-output fast Fourier transform (FFT) routines. The error response of the kth Fourier bin of these algorithms is analyzed as a function of increasing levels of input signal sampling errors in the AFT and coefficient quantization errors in the SBP-DFT. It is demonstrated that invalid assumptions on the bandwidth of the input signal will cause aliasing errors to occur in the AFT spectrum that are different from the aliasing errors that occur in the DFT