Quantization errors in the fast Fourier transform

When a fast Fourier transform (FFT) is implemented on a digital machine, quantization errors will arise due to finite word lengths in the digital system. The magnitudes and characteristics of these errors must be known if an FFT is to be designed with the minimum word lengths needed for acceptable performance. Two forms of FFT quantization, coefficient rounding and floating point arithmetic quantization, are analyzed in this paper. A theory is presented from which several new results can be obtained. The error characteristics of FFT's using exact and truncated values for the coefficients 1 and -j are found to be roughly equivalent. The accuracy of the theory is tested by computer simulations. Using the models introduced in this paper, new and accurate models can be derived to model quantization errors in high-speed convolution filters.