Optimal Design Of Multirate Filter Banks

Filtering theory plays an important role in the field of modern information technology.It is an indispensable technology for the acquisition,transmission,processing and storage of signals.Through the optimization algorithm to design the filter,the actual demand index and filter performance are improved to a great extent,which provides a new idea for the optimal time-frequency analysis and other related fields.However,the optimal design of filter banks is an optimization problem with infinite constraints.How to propose a fast and effective optimization algorithm to design digital filters and filter banks,and ensure that these infinite constraints are satisfied is the core and key problem in this field.In this paper,several problems on the optimal design of filter banks are studied.It includes the structure design,solving the optimization problem and the application of the filter.The whole paper includes three aspects:(1)Aiming at the problem that the traditional maximum decimation M-channel mirror paraunitary filter bank can't get the optimal solution in the frequency domain,the paraunitary condition is redefined and the optimal solution is obtained by the optimization method;(2)The block sampling method is adopted to solve the problem that the traditional sampling incompatible non-uniform filter banks cannot achieve perfect reconstruction.Further more,the optimal design of incompatible non-uniform filter banks is realized and verified in multiplexers.(3)Aiming at solve the problem that the structures of many filter banks are symmetrical and can not be reconstructed completely,a new structure design of filter banks based on periodic window function is studied,whichextends the design of multiple structure filter banks.The main contributions of this dissertation are summarized as follows.1)An optimal design method of maximum decimated M-channel mirror paraunitary matrix linear phase FIR filter banks is proposed.The traditional maximum sampling M-channel mirror paraunitary matrix linear phase FIR filter banks are defined in the frequency domain.Since the frequency domain is a continuous set,and the frequency response of the filter is expressed as a set of high-order polynomials of a set of trigonometric functions,it is difficult to find the gradient of the objective function of the optimization problem.Therefore,the problem of optimal filter bank design is not traceable,which brings difficulties to filter optimization design in terms of computational complexity and frequency selectivity.To solve these problems,we propose a new solution.Firstly,this condition is expressed as a finite number of discrete function equations at the sampling frequency points.The problem of finding the optimal rotation angle is transformed into the problem of optimal frequency response sampling of the filter.The number of sampling points for the amplitude response of the filter is required to be greater than the filter length.The filter design problem is transformed into an optimization problem of minimizing the total mirror paraunitary moment position error under L1 norm.However,this optimization problem is a non convex problem,which is difficult to solve.Therefore,the norm relaxed sequential quadratic programming method is used to find the local optimal solution.By iterating the above process with different initial conditions,an approximate global optimal solution is obtained.The numerical simulation results show that the design scheme is better than the existing state-of-art methods.2)An optimal design method of single input single output linear time invariant nonuniform multiplexing filter banks based on block sampling is presented.This method can achieve approximately perfect reconstruction.The traditional sampling based incompatible nonuniform filter banks can not achieve perfect reconstruction.They can only be designed by multiple inputs multiple outputs linear time invariant filters or single input single output linear time-varying systems.However,this design method is difficult to balance the implementation cost and the perfect reconstruction error.In this dissertation,the perfect reconstruction conditions of single input single output linear time invariant filter and non-uniform multiplexing filter based on block sampling are described,and a perfect reconstruction error under L1 norm is established based on this condition.By solving this optimization problem,the design scheme of the non-uniform multiplexer with single input and single output linear time invariant filter based on block sampling is obtained.The numerical simulation results show that the designed non-uniform multiplexer based on block sampling has strong robustness to channel noise and small reconstruction error.3)A filter bank structure design with periodic window function is proposed to solve the problem that the block sampling nonuniformity analysis filter bank and the block sampling nonuniformity synthesis filter bank can not achieve accurate and perfect reconstruction.That is because their sampling structures are symmetrical.Firstly,a window filter bank is proposed,which consists of a group of analysis filters,a group of analysis periodic windows,a group of comprehensive filters and a group of comprehensive periodic windows.When the comprehensive periodic window function is constant,the window filter is composed of the traditional sampling non-uniform filter bank and the block sampling non-uniform filter bank.When the periodic function is time-varying,the integrated periodic window function filter bank is a generalization of asymmetric block nonuniform filter banks.Thus,the signal can be decomposed into subband coefficients by a more flexible time-frequency decomposition method?Secondly,the filter banks with constant comprehensive periodic window function and time-varying comprehensive periodic window function are studied,respectively,and their exact perfect reconstruction conditions are derived.Further more,the paraunitary conditions of periodic window functions are further studied,and some representative examples are given to verify them.In summary,this dissertation studied several key problems of multirate filter bank design,through the optimization method to carry out the research.These contents are the innovative research on the intersection of information science and mathematical science.It is believed that it has important research significance for discovering characteristics and laws and solving practical application problems.