A fast matrix iterative algorithm for WLS design of 2-D linear-phase FIR filters

This correspondence presents a novel approach for the design of two channel perfect reconstruction (PR) linear phase finite-impulse response (FIR) filter banks. A broad class of even and odd filter length PR filter bank cases can be obtained by simply reordering an appropriate cascade form synthesizing the polyphase components of a linear phase FIR filter.

High computational complexity is a major problem encountered in the optimal design of two-dimensional (2-D) finite impulse response (FIR) filters. In this paper, we present an iterative matrix solution with very low complexity to the weighted least square (WLS) design of 2-D quadrantally symmetric FIR filters with two-valued weighting functions. Firstly, a necessary and sufficient condition for the WLS design of 2-D quadrantally symmetric filters with general nonnegative weighting functions is obtained. Then, based on this optimality condition, a novel iterative algorithm is derived for the WLS design problem with a two-valued weighting function. Because the filter parameters are arranged in their natural 2-D form and the transition band is not sampled, the computation amount of the proposed algorithm is reduced significantly, especially for high-order filters. The exponential convergence of the algorithm is established, and its computational complexity is estimated. Design examples demonstrating the convergence rate and solution accuracy of the algorithm, as well as the relation between the iteration number of the algorithm and the size and transition-band width of the filter are given.

This paper considers the design of computationally efficient narrow-band linear-phase finite-impulse response (FIR) filters with the transfer function of the form H(z) = F(z/sup M/)G(z), where F(z) is a linear-phase FIR transfer function. G(z) is constructed as a cascade of two linear-phase multiplier-free FIR filter building blocks. The first blocks are of the form 2/sup -P//sub r/[1-z/sup -K//sub r//(1-z/sup -1/) with different values of the integers K/sub r/. In these blocks the P/sub r/'s are integers. The second blocks are of the form ( 1/2 )(1 + z/sup -L//sub r/) with different values of the integer L/sub r/. Given the filter specifications, the problem is to find these building blocks, the order of F(z), and the integer M in F(z/sup M/) and to optimize the impulse-response values of F(z) so that the arithmetic complexity is minimized. It is shown how F(z) can be conveniently optimized with the aid of the fast Remez multiple exchange algorithm. For multiplier-free designs, the overall filter should exceed the given criteria in order to enable one to quantize the coefficient values of F(z/sup L/) to have two powers-of-two representations. Two examples taken from the literature show that the optimized proposed filters outperform other existing filter designs meeting the same criteria.

Recursive structures are introduced for implementing long linear-phase finite impulse response (FIR) filters using a very small number of multipliers. The implementation uses the principle of switching and resetting between two identical copies of the same infinite impulse response (IIR) filter. The impulse response of the resulting filters is a truncated and shifted version of the response of a filter G(z)G(z/sup -1/) where G(z) is a stable IIR stable and G(z/sup -1/) is the corresponding unstable IIR filter. The filters are implemented as a parallel connection of several branches, each branch generating a truncated response corresponding to a complex conjugate pole pair and its mirror-image pair. The truncation is performed using a feedforward term which provides pole-zero cancellations. To stabilize the pole-zero cancellations, and to avoid the quantization error from growing too much, the branch filters are implemented by applying the principle of switching and resetting. It is shown, by means of an example, that by using this approach a nearly optimum FIR filter of order larger than 500 using just 17 adjustable parameters can be designed. >

A novel procedure that makes use of the maximum entropy method (MEM) for the design of linear-phase FIR (finite impulse response) digital filters is described. It is shown that, by applying MEM to the inverse of the desired gain function or its square root function, it is possible to generate linear-phase FIR filters that match the given gain function to any desired degree of accuracy. Moreover, an iterative algorithm makes the design procedure very efficient, since the higher-order filters can be recursively generated from the lower-order ones. To minimize the effect of any residual passband distortion, a final averaging scheme on the lower-order filters generates a class of passband distortion-free linear-phase FIR filters. The new filters are seen to be more robust to changes in transition bandwidth as well as stopband level. Simulation results that compare the present procedure with other well-known methods are also presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Digital filters play a central role in modern Digital Signal Processing (DSP) systems. Finite Impulse Response (FIR) filters can provide solutions with guaranteed stability and linear phase. However, the main disadvantage of conventional FIR filter designs is that they become computationally complex, especially in applications demanding narrow transition bandwidths. Therefore, designing FIR filters with very stringent specifications and a low complexity is currently an important challenge. In this chapter, a review of the recent methods to efficiently design low-complexity linear-phase FIR filters is presented. The chapter starts with an introduction to linear-phase FIR digital filters. Then, an overview of the design methods that have been developed in literature to design low-complexity FIR filters is presented. Finally, the most common and recent of these methods along with their corresponding special structures are explained.

Digital filters play a central role in modern digital signal processing (DSP) systems. Finite impulse response (FIR) filters can provide solutions with guaranteed stability and linear phase. However, the main disadvantage of conventional FIR filter designs is that they become computationally complex, especially in applications demanding narrow transition bandwidths. Therefore, designing FIR filters with very stringent specifications and a low complexity is currently an important challenge. In this chapter, a review of the recent methods to efficiently design low-complexity linear-phase FIR filters is presented. The chapter starts with an introduction to linear-phase FIR digital filters. Then, an overview of the design methods that have been developed in literature to design low-complexity FIR filters is presented. Finally, the most common and recent of these methods along with their corresponding special structures are explained.

The Cook-Toom algorithm is widely used in short-length linear convolution, which is the building block of large points convolution algorithms. This paper proposes improved parallel finite impulse response (FIR) filter structures for linear-phase FIR filter, which is based on the Cook-Toom algorithm. In the proposed structures, Cook-Toom algorithm is used to reduce the number of sub-filters, and the symmetric properties of the linear-phase FIR filter's coefficients is used to further reduce the number of multipliers in sub-filters. Compared with the reported FFA and ISCA parallel FIR filter structures, the proposed method can substantially reduce the computational complexity. Specifically, for a 8-parallel 144-tap filter, the proposed design saves 18 multipliers (5%), 45 adders (7.9%) compared with the structure based on Winograd convolution algorithm [7].

ABSTRACTAn original analytical method, based on modified Christoffel–Darboux formula, is used in the paper in order to synthesise a linear-phase band-pass finite impulse response (FIR) filter function that can have an effect of Hilbert transformer. New structure of the band-pass FIR filter in recursive realisation, together with the corresponding difference equation, is presented providing the efficient filter solution without multipliers. Several examples of filter types for different parity of two real free integer parameters, including a particular solution of Hilbert transformer, are considered in terms of required number of adders and values of cut-off frequencies of the pass and stop bands. A comparison of the proposed band-pass filter characteristics with those of a classical filter solution is provided in the paper.

The design of finite impulse response (FIR) filters in one or two-dimensions can be performed with good computational efficiency using weighted least square (WLS) design. This paper presents an improved WLS technique for the design of 2-D FIR filters. We introduce an updating desired frequency response, which implicitly includes the weighting function such that the sum of weighted square errors to be minimized can be represented in a 2-D matrix form. This makes it possible to keep all filter parameters in their natural 2-D form, thereby reducing the computational complexity from O(N/sup 6/) to O(N/sup 3/). It is confirmed through design examples that our approach is computationally efficient and leads to nearly optimal approximations.

The design of low complexity finite impulse response (FIR) filters in a single-stage direct form or transposed direct form has reached a certain limitation since various multiplierless techniques have been proposed and explored. In addition, further investigations have shown that, instead of minimizing the number of adders used to implement the filter, it makes more sense to reduce the adder depth (AD) of the multiplier block (MB) from a power consumption point of view. Therefore, filters in cascade form deserve more attention because of the possibility of achieving even lower AD within each subfilter. In this paper, an algorithm is proposed for the design of linear phase FIR filters in cascade form with discrete coefficients. The proposed algorithm designs the coefficients of the subfilters simultaneously in integer space and finds all the possible discrete valued solutions. We show by an example that it is possible to design a filter using even smaller effective wordlength (EWL) and lower AD, whereas the total number of adders used to implement the overall filter is also kept minimum.

To enhance the efficiency of designing finite impulse response (FIR) filters with a controllable cut-off frequency that possess excellent transfer characteristics, this paper proposes a closed-form filter design based on transfer characteristic compensation. First, a novel filter design based on a convolution window is presented, and the relationship between the spectrum of this window and the filter performance is elaborated. We then derive a three-stage filter design scheme that describes the design of an irregular filter, design of a compensation filter and filter summation. This scheme can be simplified into a closed-form design characterized by two analytic formulas by merging the intermediate steps. The configuration of a vital Kaiser window parameter is also derived. Numerical results show that the proposed closed-form design accurately controls the cut-off frequencies and exhibits a transfer performance comparable to the Remez design and the closed-form weighted least square (WLS) design. Moreover, our method is more efficiency than the closed-form WLS method for the design of high-order FIR filters.

The computational complexity of linear phase finite impulse response (LPFIR) filters used in the channelizer of a wideband receiver is dominated by the number of adders (subtracters) employed in the multipliers. Common subexpression elimination (CSE) is a well-known technique for minimizing the number of adders in LPFIR filters. An improved CSE method is proposed in this paper, which is used to implement the channel filters of a filter bank channelizer (FBC). In the FBC, each modulated bandpass filter extract one channel from the input wideband signal. The reduction in number of adders is obtained by eliminating redundant multiplications of common subexpressions that exist among the channel filters of the FBC with the input signal. Design example of the channel filters employed in the digital advanced mobile phone system (D-AMPS) show that the proposed method offers considerable reduction in the number of full adders when compared with conventional CSE methods.

PurposeTraditionally, each time a new design for a finite impulse response (FIR) filter is required, a new algorithm has to be developed specially for the FIR filter. Furthermore, corresponding hardware architecture must be designed specially to meet the requirement of the FIR specifications. The purpose of this paper is to propose an arithmetic logic unit (ALU)‐based universal FIR filter suitable for realization in field programmable gate arrays (FPGA), where various FIR filters can be implemented just by programming instructions in the ROM with identical hardware architecture.Design/methodology/approachRather than multiplier‐accumulator‐based architecture for conventional FIR, the proposed ALU architecture implements the FIR functions by using accumulators and shift‐registers controlled by the instructions of ROM. Furthermore, time division multiplexing access (TDMA) technique is employed to reduce the chip size. In addition, the proposed FIR architecture is verified in a SOC hardware and/or software co‐emulation system.FindingsAn ALU‐based universal FIR filter suitable for realization in FPGA is designed and verified in a SOC hardware/software co‐emulation system with example of a 64‐tap FIR filter design.Originality/valueA software‐based design method as well as TDMA scheme for the ALU‐based FIR filter are introduced, making FIR filter architecture universal, programmable, and consuming less FPGA resources.

Digital filters in cascade form enjoy many advantages over their equivalent single-stage realizations in that lower coefficient sensitivity, higher throughput, reduced computational and smaller implementation cost can be achieved. However, the numerical design and optimization of such structure are of much more difficulty than the single-stage case if the filter coefficients are restricted to be of discrete values. This is mainly due to the non-convexity of the constraints, which rules out the possibility of employing sophisticated convex optimization techniques as well as the guaranteed global optimality. In this work, a general-purpose algorithm is proposed for the design of linear phase finite impulse response (FIR) filters in cascade form with discrete coefficients. The proposed algorithm decomposes the overall filter into subfilters during the traverse of a tree search of the overall filter. Discrete-valued linear phase FIR filters are able to be searched and decomposed into both symmetric and non-symmetric subfilters. The optimization complexity is of the same order as the single-stage filter optimization. Design examples have shown that the proposed algorithm is capable of achieving notable reduction in both implementation cost and adder depth compared with their single-stage optimum designs.