Classical Complex Valued Variational Mode Decomposition Based on Quaternion Valued Embedding Via Quaternion Valued Fourier Transform

For applying the time frequency analysis methods to process the complex valued signals, the quaternion valued embedding (QE) often fails to produce the meaningful time frequency representations of the multi-component signals and lacks of the ability to decompose the signal into a single component. On the other hand, applying the classical Fourier domain based methods to the complex valued signals results to the mismatch of the center frequencies of the components due to the asymmetric spectra. To address these issues, this paper proposes a QE based complex valued variational mode decomposition (CVMD) (CVMD-QE) method for performing the time frequency analysis and the time frequency representation of the multi-component complex valued signals. This method is based on the symmetry of the complex valued signals in the quaternion valued Fourier transform (QFT) domain and allows the classical CVMD method to be applied within a QE based QFT framework. By formulating the decomposition of a multi-component signal as a quaternion valued optimization problem and finding the optimal solution of the optimization problem in the real augmented domain, the CVMD-QE method can separate the multi-component signal into the different modes. Then, the Euler decomposition is performed in the QE form of each mode to obtain the instantaneous parameters of the original multi-component signal. Different from the other time frequency representations such as the bilateral Hilbert spectrum which is based on the analytical form of the signals for representing the complex valued signals, this paper proposes to construct the unilateral Hilbert spectrum of the signals based on the QE framework. Our proposed method enjoys the advantages of having the symmetric spectrum of the complex valued signals in the QFT domain and the adaptive filtering property of the CVMD. Therefore, it can tackle the difficulties in the existing methods such as being not effectively represented the time frequency information in the multi-component signals in the QE framework, requiring to pre-define the total number of the modes before performing the decomposition and occurring the mode mixing phenomenon due to the asymmetric spectrum of the signals.