The Optimal Design And Application Of FIR Filter With Nonlinear Phase

Digital filter is the core of digital signal processing system,and its design and implementation method has always been an important research field of signal processing.Compared with linear phase FIR filter(LPFF),nonlinear phase FIR filter(NLPFF)can obtain a wider performance tradeoff space,achieving better amplitude performance and lower group delay.While,its design method and optimization is more complicated,there is still much space for improvement.Therefore,this paper takes nonlinear phase FIR filter as the research object,and its optimal design and application are studied.Firstly,to achieve the performance optimization of NLPFF,a gradient method based on dynamic step size is proposed,which is an unconstrained optimization.On the other hand,in the constrained optimization,new constraint methods are adopted to improve the existing constraint model.To settle the problems in Minimax design,namely that the Cremez algorithm may not make NLPFF quasi-equiripple and the iteratively re-weighted least square algorithm has high computational complexity,a gradient method is utilized in which dynamic step-size is introduced rather than constant setting.This dynamic step-size is achieved by iteratively estimating the optimal cost function,in which two strategies are adopted and both can assist the gradient method to realize quasi-equiripple of NLPFF.Besides,by extending the cost function,this paper can realize tradeoffs between different NLPFF performances,including the tradeoff between complex frequency response error(CFRE)and stopband error energy or group delay response error(GRE).Aiming at the feasible domain problem and the constraint scale problem in the existing constrained optimization model,new constraint methods are adopted,including the constraints on coefficients symmetry and improved amplitude constraints.In the constrained minimax(CMM)and constrained least square(CLS)designs,the individual symmetric constraint and the global symmetric constraint are used respectively,they can perform similarly to the phase constraint in optimization.Combined with the improved amplitude constraint,the obtained constrained optimization method can not only avoid the amplitude and phase performance limitation in CFRE,but also effectively reduce the size of constraints.In the simulations of NLPFF designs,compared with the existing constraint mode,a relatively wider performance tradeoff can be obtained by the proposed method,and the optimization solving by second-order cone programming(SOCP)is enhanced in terms of efficiency.Secondly,in order to realize the constrained optimization of NLPFF with narrow transition band,and effectively reduce the filter implementation complexity,the interpolated finite impulse response(IFIR)and the frequency response masking(FRM)are designed by complementary optimization method.Because of nonlinear phase,the implementation complexity of NLPFF is much higher than that of LPFF,and the original IFIR and FRM methods in LPFF cannot be directly applied.To solve this problem,the IFIR and FRM for NLPFF will apply complementary optimization between sub-filters designs.In the IFIR design,based on the analysis of the frequency response performance,two sub-filter design approaches are established: the sub-filters are all NLPFF and the sub-filters are LPFF and NLPFF respectively.When designing sub-filters,the complementary optimization method is used,in which the model filter is optimized after the masking filter is firstly designed.In the FRM design of NLPFF,the same sub-filters design to IFIR approach one is adopted,but due to the additive relationship between the model filter and its complementary filter,the performance distribution becomes much more difficult.Thus,the alternative complementary optimization of masking filters and model filer is applied instead of the design method under performance allocation.When applying the proposed IFIR and FRM methods,the model filter can make up for the performance insufficiency of the masking filter,thus the performance requirements of the masking filter can be greatly relaxed and its filter length can be reduced.The simulation results show that the length sum of subfilters is much smaller than that of the singly designed NLPFF,which can effectively reduce the implementation complexity.Simultaneously,the performance requirements of the synthesis filter can also be effectively obtained through the constrained optimization design in the complementary optimization method.Finally,the application of NLPFF in filter-bank multicarrier(FBMC)is studied,in which the NLPFF optimization problem with Nyquist constraint is solved and the NLPFF selection mode based on signal-to-interference plus noise ratio(SINR)is established,then the NLPFF can be used as a prototype filter for FBMC.According to the signal transmission modeling and analysis of FBMC system,the SINR is expressed in discrete-time and directly related to the FBMC prototype filter.Compared to the signal to interference ratio(SIR)in literature,the discrete-time and noise are additionally considered in this SINR.In the constrained optimization of NLPFF,to fit the requirement of FBMC prototype filter,namely the Nyquist condition(NYQ),the original time domain representation is equivalently converted to the one in frequencydomain,then the complicated constraints are replaced by the easy-to-use constraints in frequency-domain.Through the NLPFF optimization,the performance tradeoffs can be achieved,and better stopband characteristics than that of PHYDYAS filters can be obtained,especially in the vicinity of the transition band.However,different phase characteristics(including group delay and phase fluctuation),under different relaxation of NYQ,the designed NLPFFs result in different SINRs of FBMC,which may be comparable to or far worse than that of the PHYDYAS filter.Therefore,under the guidance of SINR,a constrained optimization of NLPFF is established,in which the phase fluctuation is controllable and the group delay is adjustable.When employing the designed NLPFF as the FBMC prototype filter,it is possible to achieve SINR and BER results comparable to that of the PHYDYAS filter,while achieving better stopband characteristics and lower group delay.