Properties of approximated empirical mode decomposition and optimal design of its system kernel matrix for signal decomposition

Abstract

An approximated empirical mode decomposition generates a set of approximated intrinsic mode functions via a linear, nonadaptive but iterative approach. The decomposition was found to be very useful for a content-independent pattern recognition application. As the process is characterized by a system kernel matrix and performed iteratively, the approximated intrinsic mode functions can be understood as the original signals processed by a set of mask operations. Here, some properties of the decomposition are studied and an optimal design of the system kernel matrix is proposed. It is found that there is only one approximated intrinsic mode function if the exact perfect reconstruction condition is satisfied. Obviously, the approximated intrinsic mode function is the original signal. Therefore, the decomposition is practically not meaningful. To address this issue, the infinite number of iterations in the algorithm is truncated to a finite number of iterations. Also, the design of the system kernel matrix is formulated as an optimization problem. In particular, the exact perfect reconstruction error between the sum of the approximated intrinsic mode functions and the original signal is minimized and the total absolute sum of the difference between any two different eigenvalues of the iterative matrix is maximized subject to a stability condition. Here, the stability condition refers to the eigenvalues of the iterative matrix being between zero and one. Since the optimization problem is nonsmooth and nonconvex, a genetic algorithm is employed for finding its near global optimal solution. Compared to the conventional approximated empirical mode decomposition, computer numerical simulations show that our proposed approach can achieve more than one approximated intrinsic mode function with each approximated intrinsic mode function corresponding to an output of a more meaningful mask operation.