Nonlinear and adaptive undecimated hierarchical multiresolution analysis for real valued discrete time signals via empirical mode decomposition approach

Abstract

Hierarchical multiresolution analysis is an important tool for the analysis of signals. Since this multiresolution representation provides a pyramid like framework for representing signals, it can extract signal information effectively via levels by levels. On the other hand, a signal can be nonlinearly and adaptively represented as a sum of intrinsic mode functions (IMFs) via the empirical mode decomposition (EMD) algorithm. Nevertheless, as the IMFs are obtained only when the EMD algorithm converges, no further iterative sifting process will be performed directly when the EMD algorithm is applied to an IMF. As a result, the same IMF will be resulted and further level decompositions of the IMFs cannot be obtained directly by the EMD algorithm. In other words, the hierarchical multiresolution analysis cannot be performed via the EMD algorithm directly. This paper is to address this issue by performing a nonlinear and adaptive hierarchical multiresolution analysis based on the EMD algorithm via a frequency domain approach. In the beginning, an IMF is expressed in the frequency domain by applying discrete Fourier transform (DFT) to it. Next, zeros are inserted to the DFT sequence and a conjugate symmetric zero padded DFT sequence is obtained. Then, inverse discrete Fourier transform (IDFT) is applied to the zero padded DFT sequence and a new signal expressed in the time domain is obtained. Actually, the next level IMFs can be obtained by applying the EMD algorithm to this signal. However, the lengths of these next level IMFs are increased. To reduce these lengths, first DFT is applied to each next level IMF. Second, the DFT coefficients of each next level IMF at the positions where the zeros are inserted before are removed. Finally, by applying IDFT to the shorten DFT sequence of each next level IMF, the final set of next level IMFs are obtained. It is shown in this paper that the original IMF can be perfectly reconstructed. Moreover, computer numerical simulation results show that our proposed method can reach a component with less number of levels of decomposition compared to that of the conventional linear and nonadaptive wavelets and filter bank approaches. Also, as no filter is involved in our proposed method, there is no spectral leakage in various levels of decomposition introduced by our proposed method. Whereas there could be some significant leakage components in the various levels of decomposition introduced by the wavelets and filter bank approaches.