Research On The Design Algorithms Of Sparse FIR Filters

In modern communication and digital signal processing,FIR digital filters are often used because of their intrinsic stability according to non-recursion form and the linear phase that is easy to implement in their design.However,higher orders are required when it comes to design an FIR with better performance,thus inducing more designing complexity and more sophisticated computation.Besides,if the filter order is high,when using hardware circuit to implement FIR filter,it may need more storage units and arithmetic unit,leading to a series of problems such as large power consumption,high cost and low efficiency.In this paper,the above mentioned problems are resolved by designing sparse FIR filters.The mainly purpose of sparse FIR filter designs is to reduce the number of non-zero valued filter coefficients as many as possible subject to some constraints on the filter performance.The design of sparse FIR filters can be described as minimizing the 0-norm of its coefficient vector subject to the filter performance coefficients.This is an NP-hard problem whose optimal solution is very difficult to find.After a brief introduction to some basic concepts of digital filters and their optimal designs as well as sparse reconstruction algorithms in compressive sampling,the design algorithms for sparse FIR filters are studied in this paper.The main works of this paper includes:(1)By combining the iterative reweighted 1-norm minimization algorithm and a binary search,we propose a linear-phase sparse FIR filters design algorithm.It uses the iterative reweighted1-norm minimization method to design a filter with many zero and/or small coefficients and then applies a binary search to finally determine how many and which of those smallest ones can be set to zero while not violating the magnitude constraints on the frequency response.(2)In practical applications,we may just need approximate rather than exact linear-phase filters rather than linear-phase.Under the same filter order and magnitude error,the coefficients of an approximate linear-phase FIR filter may be sparser than those of an exact linear-phase FIR filter.So we propose a method of designing approximate linear-phase sparse FIR filters,by imposing constrains on the magnitude of the frequency response error.(3)The filter designed by the above method which imposes merely constraints on the frequency response error may have a comparatively large phase error.In order to solve this problem,this paper proposes another approximate linear-phase sparse FIR filter method thatby imposing constraints simultaneously on the magnitude error and phase error respectively.The filter designed by this method has fewer non-zero coefficients than the corresponding exact linear-phasefilter and smaller phase error than that obtained by the method imposing constraints merely on the frequency response error.