Arithmetic errors in the sub-band FFT: Derivation of error equations and simulation results

Two types of numerical errors occur in the implementation of the fast Fourier transform (FFT): coefficient errors and arithmetic errors. This paper deals with the second type, due to the finite word-length used in all operations. The signal-flow graph of the sub-band DFT (SB-DFT, or SB-FFT as realized in the same efficient structure) consists of two parts: the Hadamard part (which contains only additions and subtractions) and the recombination part (which contains also multiplications). The outputs of all these mathematical operations must be scaled. Results for the two's complement fixed-point arithmetic errors of the classical radix-2, "Cooley-Tukey type" (CT-) FFT are known from various publications. Especially, three radix-2 DIT butterflies were defined and studied, which are normally used in most integrated DSP realizations and which differ in the quantizer locations. Three corresponding butterfly structures are now defined for the recombination network of the SB-FFT; they are analysed theoretically, and error equations are derived from a suitable error model. Real input data are assumed in the analysis of arithmetic errors in SB-FFT with both rounding and truncation scaling in the Hadamard and the recombination parts. Monte-Carlo simulations are included in the analysis of the arithmetic errors in SB-FFT. The results of a thorough evaluation are to be presented, yielding insights into the mechanisms of scaling and multiplier-output quantizations and allowing for a comparison between the SB-FFT and CT-FFT. For the partial-band versions of the SB-FFT, the arithmetic errors are compared with the aliasing components inherent in those approximated versions. Both half-band and quarter-band SB-FFT are considered in this study. Conclusions are drawn for the necessary internal wordlengths of fixed-point realizations.