Improvement for Characteristic of Multirate Discrete Fourier Transform

Abstract

Multirate discrete Fourier transform (MR-DFT) has been proposed as one of the DFT algorithms by us. The feature of the MR-DFT is to use only one multiplication for each Fourier coefficient. Therefore the number of multiplications can be decreased comparing with the conventional DFT. In this paper, for applications of the MR-DFT, we consider how the noise affects the estimation performance of the MR-DFT. Then we propose two improved algorithms of the estimation performance, that is, a shifted multirate discrete Fourier transform (SMR-DFT) and a parallel shifted multirate discrete Fourier transform (PSMR-DFT). The SMR-DFT prevents an aliasing of the additive noise by a frequency modulation. The PSMR-DFT performs the parallel processing of the SMR-DFT with different sample points using the periodicity of an input signal. The estimation performances of the proposed algorithms are compared with MR-DFT by the computer simulations. The SMR-DFT could prevent the degradation of the estimation performance of the MR-DFT in the lower components caused by the wide bandwidth noise. Hence its estimation variance could become 0.61 times of the MR-DFT. Furthermore, the PSMR-DFT could improve the estimation variance of the SMR-DFT to 0.46 times. Comparing the number of operations of the proposed algorithms with the MR-DFT, the number of additions has increased, however, the number of multiplications has the same complexity.