Improving accuracy in FFT-based cyclic convolution

When speed considerations are critical, FFT-based implementations of cyclic convolution need to be considered. Unfortunately, such FFT-based implementations of purely integer cyclic convolution can be affected by floating point round-off errors potentially leading to an incorrect integer result. This is particularly true for large length sequences. When an exact integer result is desired we are proposing the use of our sectioned algorithm, which is suitable for parallel-recursive, as well as serial-recursive implementations. Because of the shorter subsections length, the FFTs floating point round-off errors are not as severe as when doing a direct full-length implementation. The attainment of exact integer results was confirmed by coding the proposed algorithm in C++ and MATLAB using the FFTW to perform the shorter cyclic sub-convolutions. The trade-off is an increased computational time compared to the direct, full-length, FFT-based implementation. We confirmed that the proposed sectioned FFT-based approach is still much faster than exact time domain-based implementations or than implementations based on arbitrary precision schemes. Our approach allows handling signal lengths that cannot be tackled when using other length-constrained techniques, which are also fast and give the exact integer result, such as Nussbaumer Convolution or Number Theoretic Transforms.