A comparison of integer fast Fourier transforms for lossless coding

The lifting scheme-based integer fast Fourier transform (IntFFT), an integer approximation of the FFT, is reversible. When it is used for lossless coding applications, the computational complexity and approximation error increase due to realization of the trivial butterflies by three lifting steps. Since the error appears as a "noise floor" and it limits the lossless coding efficiency, it is desirable to reduce not only the computational complexity but also the noise floor level as much as possible. This survey presents two schemes to realize an improved IntFFT in terms of the number of arithmetic operations and the level of the noise floor. The first scheme is based on employment of two/three lifting step schemes with combined rounding operations, and the second one is the multidimensional lifting (MDL) scheme. The improvement is shown by comparing the number of arithmetic operations and rounding operations to compute the IntFFT and also by comparing levels of the noise floor. In addition, an improvement in lossless coding efficiency due to the reduced noise floor can be predicted by observing the reduced estimated entropy of the IntFFT coefficients.