| Adaptive signal decomposition is a time-frequency analysis technique which characterizes arbitrary complex and variable signals as the superposition of multiple components,and can be realized without pre-defined kernel function.On the contrary,conventional signal decomposition methods like Fourier analysis and wavelet analysis need to predefine kernel functions.However,a predefined kernel function works well towards a signal while it is not effective for some other signals.Therefore,adaptive signal decomposition plays an important role in extracting effective information of observed signals in biomedical signal processing,seismic analysis,data compression and digital communication systems.These time-frequency analysis techniques based on adaptive decomposition are kinds of method to study the change of signal spectrum over time.In this thesis,properties of empirical mode decomposition(EMD)and singular spectrum analysis(SSA)that are kinds of adaptive signal decomposition are studied.On the other hand,suppressing the noise and recovering the clean signal from the corrupted signal while preserving its important structures remains a challenging issue in digital signal processing.In addition,fractional delay is one of the most important topic of digital signal processing to estimate unknown response between two consecutive integer time index values in a discrete time signal.Hence,this thesis applied EMD based hierarchical multiresolution analysis and adaptivity of SSA to signal denoising and fractional delay,respectively.The main contents of this thesis are as follows:(1)This thesis proposes a strategy for suppressing noise via EMD based hierarchical multiresolution analysis.First,an intrinsic mode function(IMF)index that can differentiate the noise dominant IMFs and the information dominant IMFs in the first level of decomposition is determined by the conventional EMD based denoising method.Next,the EMD is applied to those discarded IMFs in the first level of decomposition to obtain the IMFs in the second level of decomposition.Then,the detrended fluctuation analysis(DFA)is introduced to define the IMF index separating the noise dominant IMFs and the information dominant IMFs in the second level of decomposition.The summing of Information dominant IMFs in both the first and the second level of decompositions forms reconstructed signal.The potential of our proposed method is proved by simulation results in some cases especially at high input SNR levels.Also,our proposed method can extract information from more than one level of decomposition to reconstruct the denoised signal compared to only one level of decomposition in the conventional EMD based denoising method.(2)In this thesis,the application of thresholding and permutation processing in EMD denoising based on hierarchical multiresolution analysis is studied.In order to further improve performance of hierarchical multiresolution analysis based EMD denoising,the thesis combines hierarchical multiresolution analysis based EMD,thresholding operation and averaging operation together.The first IMF in the second level of decomposition is chosen as the noise component.For each realization,this noise component is segmented into various pieces and these segments are permutated.By summing up this permutated IMF to the rest of IMFs in both the first level of decomposition and the second level of decomposition,new realization of the noisy signal is obtained.For original signal and each realization of newly generated noisy signal,the EMD denoising method based on hierarchical multiresolution analysis is performed.After the EMD denoising method based on hierarchical multiresolution analysis,the noisy IMFs in the second level of decomposition corresponding to each noisy IMFs in the first level of decomposition and the information dominated IMFs of the first level of decomposition and the second level of decomposition are obtained.The noisy IMFs in the second level of decomposition corresponding to each noisy IMFs in the first level of decomposition are summed together and then thresholded.These thresholded components are added to the information components in both the first layer and the second layer to obtain the denoised signal.Finally,the above procedures are repeated and several realizations of denoised signals are obtained.Then,denoised signal obtained by applying thresholding to each realization are averaged together to obtain final denoised signal.The extensive computer numerical simulations are conducted and the results illustrate that our proposed method is effective.(3)This thesis proposes a fractional singular spectrum analysis based method for performing the fractional delay.First,the input sequence is divided into two overlapping sequences with the first sequence being the input sequence without its last point and the second sequence being the input sequence without its first point.Then,the singular value decompositions are performed on the trajectory matrices constructed based on these two sequences.Next,the designs of both the left unitary matrix and the right unitary matrix for generating the new trajectory matrix are formulated as the quadratically constrained quadratic programing problems.The analytical solutions of these quadratically constrained quadratic programing problems are derived via the singular value decomposition approach.Finally,the fractional singular spectrum analysis components are obtained via performing the diagonal averaging operation and the fractional delay sequence is obtained by summing up all the fractional singular spectrum analysis components together.Since the fractional singular spectrum analysis operations are nonlinear and adaptive,our proposed method is a kind of nonlinear and adaptive approach for performing the fractional delay.Besides,by discarding some fractional singular spectrum analysis components,the joint fractional delay operation and the denoising operation can be performed simultaneously.The extensive computer numerical simulations are conducted and the results present that both of these processes are beneficial to improve the denoising performance.(4)SSA has been drawn a lot of attentions in the signal processing community because of its excellent properties such as achieving the exact perfect reconstruction based on both the full length SSA components and the SSA components with the shorter lengths as well as retaining the linear phase property of the SSA components for the linear phase signals.The existences of these nice properties of the SSA are due to the formation of the trajectory matrix based on the Hankelization with the rectangular window.That is,each segment of the signal is multiplied by the rectangular window and the windowed segments are put into the columns of the trajectory matrix.The thesis generalizes rectangular window into generalized window and explores whether generalized window based SSA still has the above properties.In order to guarantee the exact perfect reconstruction based on the full length SSA components,the thesis proposes weighted diagonal averaging to perform e-Hankelization.Also,for the exact perfect reconstruction based on the SSA components with the shorter lengths,the first row and last column of two-dimensional SSA components obtained by SVD decomposition are used to reconstruct the de-hanklization process to ensure the conversion between two-dimensional SSA components and one-dimensional SSA components.In addition,the diagonal matrix is generated according to generalized window coefficient and is used to multiply the trajectory matrix based on generalized window to ensure the conversion between trajectory matrix based on generalized window and original signal.On the other hand,this thesis proves that the linear phase property of the SSA with the generalized linear phase window can be retained for the linear phase signals.As the rectangular window is a special case of the generalized window,some system performances can be achieved better.The computer numerical simulation results validate these properties of the SSA with the generalized window. |
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