Design Methods Of Modulated Filter Bank And Graph Filter Bank

As the core of multirate signal processing,filter bank has become a popular topic and has been widely applied in communication and image signal processing,etc.As a special kind of filter bank,modulated filter bank has gained much popularity due to its simple design and implementation.Among the modulated filter banks,the discrete Fourier transfrom(DFT)modulated filter bank is suitable to process the complex-valued signal and construct two-dimensional(2D)directional filter bank.Therefore,it is of prominent importance to investigate the theory and design methods of the DFT-modulated filter bank.On the other hand,the filter bank has recently been extended to the irregular domain,resulting in the graph filter bank.It is employed to process irregular and dynamic network data,such as wireless sensor network,social network,and smart grid,etc.The graph filter bank is a powerful tool to represent and extract the characteristic of network data by using its multiresolution property.Therefore,it is of great practical significance to study the structure and design of the graph filter bank.Recently,one of the main research topics is the cosine-modulated filter bank in the family of the one-dimensional(1D)nonuniform modulated filter bank.There is lack of methods to design the nonuniform DFT-modulated filter bank with good overall performance.Meanwhile,there exists few algorithm to design the 2D critically sampled modified DFT-modulated filter bank.Furthermore,2D DFT-modulated filter bank with arbitrary sampling requires deep analysis.The perfect reconstruction(PR)condition has not been derived yet.On the other hand,as a new research area,the work on the graph filter bank is still on primary stage.And there are lots of open problems unresolved.In summary,this thesis will focus on the design of 1D nonuniform DFT-modulated filter bank,the 2D critically sampled modified DFT-modulated filter bank,the theory and design of the 2D DFT-modulated filter bank with arbitrary sampling,as well as the design of the graph filter bank.Specifically,the work includes:(1)Design of 1D nonuniform DFT-modulated filter bankFirstly,the structure of the 1D nonuniform DFT-modulated filter bank is analyzed.Then,two design methods are proposed.One is an indirect design method and the other is a direct one.In the indirect method,the nonuniform 1D filter bank is obtained from the uniform one.Different from the nonuniform cosine-modulated 1D filter bank,it is infeasible to obtain nonuniform DFT-modulated 1D filter bank with good performance by simply merging the adjacent subbands of the uniform one.To overcome this deficiency,a phase-modulated technique is employed.The relationship between the nonuniform and uniform 1D filter bank is theoretically built,which is verified by the numerical results.In the direct method,all subband filters of the nonuniform DFT-modulate 1D filter bank is obtained by sampling and modulating a prototype filter in the time domain.By representing the transfer and aliasing distortion as the functions of the prototype filter,we formulate the design problem of the nonuniform 1D filter bank into an unconstrained optimization problem that minimizes the weighted sum of transfer distortion,aliasing distortion and stopband energy.With the derived gradient vector of the objective function,an iterative algorithm is proposed to solve the problem.Numerical results show the effectiveness of the proposed method.(2)Design of 2D critically sampled modified DFT-modulated filter bankTwo design methods are proposed in this section.In the first method,the spatial form of the PR condition is derived.Based on the spatial condition,the design problem is formulated into an unconstrained optimization problem,whose objective function consists of the reconstruction error of the 2D filter bank and stopband energy of the prototype filter.An iterative algorithm is deduced by using the gradient vector of the objective function.Simulation results show that the proposed method can lead to 2D critically sampled modified DFT-modulated filter bank with good frequency selectivity and reconstruction property.In the second method,based on the spatial PR condition,the design problem is casted as a constrained optimization problem that minimizes the stopband energy subject to the PR constraint.By employing the Lagrange multiplier method,an iterative method is proposed to solve the problem,where the block LU decomposition technique is used to reduce the computational complexity.Simulation results demonstrate that the second method is with lower computational cost and better performance than the first one.(3)Design of 2D fully oversampled DFT-modulated filter bankFirst of all,based on the polyphase decomposition of the prototype filter,the PR condition of the 2D fully oversampled DFT-modulated filter bank is derived and the polyphase structure of the 2D filter bank is built.For design convenience,the PR condition is transferred into the spatial form.Based on the spatial condition,the design problem is formulated as an unconstrained optimization problem that minimizes the reconstruction error of 2D filter bank and the stopband energy of prototype filter.A gradient-based method is proposed to iteratively solve the problem.Numerical results show that the designed 2D filter banks possess good frequency selectivity and small reconstruction error.On the other hand,a fast method is proposed to design the prototype filter,which is suitable for designing 2D large-scale DFT-modulated filter bank.In this method,the fast inversion of the block toeplitz matrix and a matrix inversion lemma are exploited to dramaticlly reduce the computational complexity at each iteration.Beside,the unit step size is employed to further reduce the complexity.Numerical results demonstrate the effectiveness of the fast method.(4)Design of graph filter bankThis part focuses on the design of the two-channel biorthogonal and M-channel oversampled graph filter banks.The two-step based and lifting-based methods are proposed for designing the two-channel biorthogonal graph filter bank.In the first method,the analysis filter is first optimized subject to 3d B constraint to acheive good spectral selectivity.Then the synthesis filters is optimized by constrained optimization problem where the reconstruction error of the graph filter bank is controlled.In the second method,the PR condition is parameterized by the lifting structure.The obtained two-channel biorthogonal graph filter bank is of structurally perfect reconstruction.Numerical results and comparison are included to show that the second method has better PR property.On the other hand,the SCA-based method is proposed to design the M-channel oversampled graph filter bank.The design of the analysis filters is formulated as a non-convex quadratically constrained quadratic program(QCQP)that minimizes the stopband energy of the analysis filters subject to the reconstruction error constraint.The non-convex QCQP problem is typically NP-hard.In order to overcome this challenge,we apply the successive convex approximation(SCA)to successively solve the problem.At each iteration,the non-convex QCQP is approximately transformed into a convex one by linearizing the constraint.Numerical examples and comparison are included to show that the proposed approach can lead to the oversampled graph filter bank with satisfactory overall performance,particularly good spectral selectivity.