Research On Constrained Optimal Designs Of Variable Fractional Delay FIR Digital Filters

Variable digital filters have found lots of applications in image processing,voice signal processing,medical signal processing,radar signal processing and so on.The variable fractional delay(VFD)finite impulse response(FIR)digital filter is one type of variable digital filters and its group delay can be fractionally adjusted on line.Because of VFD FIR digital filters'characteristic and broad applications,their optimal design problems have attracted wide attention from scholars at home and abroad.The optimal design problem of VFD FIR digital filters is essentially a bivariate function approximation problem,similar to the two-dimensional digital filter design problem.The quantity of unknown design parameters of VFD FIR digital filters is significantly large.Since the approximation theory of the bivariate function is incomplete,the design problem of VFD FIR digital filters is extremely complicated.At present,most VFD FIR digital filter design methods are vector-based,which arrange design parameters to vectors and use conventional optimal algorithms to solve them.This results in high complexity and occupies lots of computer memory.Although computer hardware technology develops fast,it is meaningful and challenging to find fast and stable VFD FIR digital filter optimal design algorithms.This thesis studies the constrained optimal design problems of VFD FIR digital filters in the Least Squares(LS)sense,containing amplitude error constraints,group delay error constraints,and both amplitude and group delay error constraints,respectively.First,the thesis establishes a mathematical model of the optimal design problem of VFD FIR digital filters in matrix form.By a series of derivations,the optimal design problem of VFD FIR digital filters is transformed into solving two unknown parameter matrices.The elements of the unknown parameter matrices are related to the impulse responses of VFD FIR filters.Next,this thesis obtains the optimal solution of the unknown parameter matrices in the LS sense without constraints.For the optimal design problem of VFD FIR digital filters with amplitude error constraints,this thesis proposes an effective matrix-based iterative algorithm.The thesis establishes a mathematical model of the amplitude error Constrained Least Squares(CLS)optimal design problem in matrix form.In the model,the amplitude error function is highly nonlinear.By taking some variables as constants,the amplitude error constraint function is linearized.Then,by linearizing the amplitude error function repeatedly,the amplitude error CLS optimal design problem of VFD FIR digital filters is transformed into a series of linear inequality iterative CLS optimal design subproblems.For these CLS optimal design subproblems,the thesis proposes an efficient matrix-based active-set algorithm and introduces the Lagrange function to solve them.Eventually,the proposed algorithm can efficiently solve the VFD FIR digital filter amplitude error CLS optimal design problem with fixed error bounds,and thus the amplitude errors are constrained into the given values.The thesis analyzes the rapidity and convergence of the algorithm finallyFor the optimal design problem of VFD FIR digital filters with group delay error constraints,the thesis proposes an efficient matrix-based iterative algorithm First,the thesis establishes a mathematical model of the group delay error CLS optimal design problem in matrix form.In the model,the group delay error constraint function is highly nonlinear and non-convex.Since the design problem is a non-convex optimal problem,solving it directly will require the use of complicated non-convex optimization techniques.Similarly,by taking some variables as constants,the group delay error constraint function can be linearized.As a result,the original design problem is transformed into a series of linear inequality CLS subproblems.For these CLS subproblems,the thesis also uses the efficient matrix-based active-set algorithm to solve them.Eventually,the proposed algorithm can efficiently solve the group delay error CLS optimal design problem of VFD FIR digital filters with fixed error bounds,so that the group delay errors are constrained into the given values.In addition,the thesis analyzes the rapidity and convergence of the algorithm.For the optimal design problem of VFD FIR digital filters with both amplitude and group delay error constraints,the thesis proposes a matrix-based algorithm.First,the thesis takes the amplitude error constraint optimal design algorithm to obtain a solution of coefficient matrices.Then,taking this solution as the initial iterative matrices,the group delay error constraint algorithm is used to solve the problem and thus can obtain an approximate solution.Eventually,the proposed algorithm can solve the amplitude and group delay error CLS optimal design problem of VFD FIR digital filters with fixed error bounds.As a result,the amplitude errors and group delay errors are constrained into the approximate given values at the same time.Simulation results demonstrate that the proposed matrix-based iterative CLS algorithms can efficiently solve the CLS optimal design problems of VFD FIR digital filters with fixed error bounds.The proposed algorithms are more computationally efficient compared with existing methods,and can more accurately control the frequency characteristics of VFD FIR digital filters.As a result,amplitude errors and group group delay errors can be reduced efficiently.In addition,the matrix-based algorithms can be further extended to other designs of VFD FIR digital filters.