Research On Mathematical Theory Of Empirical Mode Decomposition

Signal analysis is used to research and represent the basic property of the signal. In traditional view, the different representations of the signal are made by the signal's extention onto different orthogonal basises. However, the basis function is fixed in an application so that it can not accommodate the special need of the practical signal. Empirical Mode Decomposition (EMD) is just to solve the problem. The signal decomposites into Intrinsic Mode Functions (IMFs) not by some orthogonal basis but by some iterative rule. The new method throws off the chains of the basis and finds the feature of the signal adaptively. However, the flexibility is at the cost of the complexity of the iterative rule. Therefore, the mathematical theory research of EMD focuses on the analysis to the property of iterative rule.The iterative process in EMD is called the sifting process, which has the following steps: finding the extrema of the signal, constructing the upper and lower envelopes of the signal by the extrema and subtracting the mean of the two envelopes from original signal. The steps are repeated until the mean envelope is close to zero and the residue signal is called IMF. From the steps, the two key parts are the location of the extream and the construction of the envelope. This dissertation focuses on these two points and proposes a method to locate the extrema and a theory to design the envelopes.The location algorithm is based on the interpolation theory including algebraic interpolation and trigonometric interpolation. Algebraic interpolation mainly composes of cubic interpolation and piecewise Hermite interpolation, and their interpolation formula and errors are introduced. For the trigonometric interpolation, the interpolation error is discussed and its the upper bound is also given.The envelope design defines the property of the undetermined envelope, and then constructs the envelope by quadratic polynomial, cubic polynomial and n-order polynomial. The important property of these envelopes is that the fluctuation of the envelope will be decreased in the iterative process and the iterative process will converge to a constant-envelope signal. Therefore, the EMD using the new envelope is a method that decomposites the signal into the constant-envelope signals. In the analysis of the envelopes, we find the envelopes pass through some special points and propose an equivalent construction method by piece-wise Hermite interpolation.The dissertation intends tofill the gaps in the theoretical research of Empirical Mode Decomposition. We solve the extrema location problem under low sampling rate and propose an envelope construction method that leads to a constant-envelope signal decomposition. Simulations show that the new algorithm is better than the traditional algorithm in the constant-envelope signal decomposition. Therefore, the works of this dissertation have strong theoretical and practical significance.