CONCLUSION

In this paper, we have presented a concept of IntFFT, which can be used to construct FFT with integer coefficients. It provides a new method for approximating the DFT without using any multiplication and can simply be applied to the case of large-size DFT. Unlike the FxpFFT, which is the fixed-point arithmetic version of FFT, the IntFFT is reversible when the coefficients are quantized. Its inverse IntIFFT can be computed with the same computational cost as that of the forward transform. The new transform is suitable for mobile computing and any handheld devices that run on batteries since it is adaptable to available power resources. Specifically, the coefficients appearing in the proposed structures can be quantized directly for different resolutions, i.e., different computational costs, while preserving the reversibility property. Although a large class of FFT structures such as radix-2, radix-4, and split-radix can be approximated by this approach, the split-radix structure is used to illustrate the technique. An analysis of the dynamic range of the internal nodes is presented. Using an appropriate choice of lifting factorizations, it is proven that lifting approximation of a complex multiplier can increase the resolution of the input by at most one bit. An upper bound of the dynamic range of the internal nodes is estimated and confirmed by a simulation for the case of the split-radix FFT. The computational complexity of the resulting transform is calculated by the numbers of real additions and shifts. A method for minimizing the number of additions is presented and used to compare the computational costs of IntFFT and FxpFFT. According to the simulation, the complexity of IntFFT is lower than that of FxpFFT by a significant margin. The accuracy of the transforms is compared experimentally. It is evident from the simulations that both IntFFT and FxpFFT have approximately the same distortion from ideal FFT when the resolution of the input Ni is greater than Nc . When Nc
Ni ,
the FxpFFT yields 3 dB higher accuracy. The results are different when testing with the convolutional rule, where the IntFFT and the FxpFFT provide essentially the same accuracy for any value of Nc . When the inverse transform is performed after the forward transform, fixed-point arithmetic approach results in reconstruction error, whereas the proposed approach can reconstruct the input perfectly for any fixed resolution of the coefficients. Finally, these two implementations are tested in noise reduction application. At low power, i.e., the coefficients are quantized to low resolution, the IntFFT yields significantly better results than the FxpFFT, and they yield similar results at high power.