Linear-phase Finite Impulse Response Filters Research Articles

A comparison of LTE channelizers implemented with linear phase m-path recursive filters and FIR filters

Cell sites repeaters may receive a composite signal containing a mix of long term evolution channels with bandwidths of 5, 10, 15, and 20 MHz and be required to rearrange the frequency plan of the channels or to drop and insert specific channels prior to transmitting the altered composite signal. The straight forward approach to this task is to down-convert, and down-sample each channel in the mix and then up-sample and up-convert and merge the new traffic mix. The filters applied to the up and down conversion task as well as the up and down sampling task would likely be linear phase finite impulse response (FIR) filters because of the ease with which the resampling task can be embedded in the filtering task. We present an alternate filter structure formed from linear phase recursive filters and compare their performance and computational complexity with their FIR filter counterparts. We show that the recursive filter version of the channel extractor requires significantly few arithmetic operations and actually outperforms the non-recursive version as demonstrated by the error vector magnitude of the two options.

Matrix-Based Algorithms for Constrained Least-Squares and Minimax Designs of 2-D Linear-Phase FIR Filters

The impulse response coefficients of a two-dimensional (2-D) finite impulse response (FIR) filter are in a matrix form in nature. Conventional optimal design algorithms rearrange the filter's coefficient matrix into a vector and then solve for the coefficient vector using design algorithms for one-dimensional (1-D) FIR filters. Some recent design algorithms have exploited the matrix nature of the 2-D filter's coefficients but not incorporated with any constraints, and thus are not applicable to the design of 2-D filters with explicit magnitude constraints. In this paper, we develop some efficient algorithms exploiting the coefficients' matrix nature for the constrained least-squares (CLS) and minimax designs of quadrantally symmetric 2-D linear-phase FIR filters, both of which can be formulated as an optimization problem or converted into a sequence of subproblems with a positive-definite quadratic cost and a finite number of linear constraints expressed in terms of the filter's coefficient matrix. Design examples and comparisons with several existing algorithms demonstrate the effectiveness and efficiency of the proposed algorithms.

Theory and Design of Linear-Phase Minimax FIR Filters with Mixed Constraints in the Frequency Domain

The alternation theorem is the core of efficient approximation algorithms for the minimax design of finite-impulse response (FIR) filters. In this paper, an extended alternation theorem with additional mixed constraints, i.e., equality-and-inequality constraints, is obtained. Then, an efficient multiple-exchange algorithm based on the extended theorem is presented for designing linear-phase FIR filters with frequency mixed constraints in the minimax sense. Further, convergence of the algorithm is established. Several design examples and comparisons with existing techniques are presented, and the simulation results show that the proposed algorithm is numerically more efficient and guaranteed to converge to the optimal solution.

Recursive FIR filter structures on FPGA

A new approach to piecewise-polynomial approximation and recursive implementation structures for linear-phase Finite Impulse Response (FIR) filters have been recently proposed. In this paper, we describe hardware prototype implementations of the new structures for all four types of linear-phase FIR filters using a Field Programmable Gate Array (FPGA) based platform. Narrowband lowpass filters and narrowband differentiators are used as design examples to demonstrate the functionality and efficiency of the implementations. The required wordlength and resource usage is analyzed.

Design of Discrete-Valued Linear Phase FIR Filters in Cascade Form

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.

Implementation of Linear-Phase FIR Filters for a Rational Sampling-Rate Conversion Utilizing the Coefficient Symmetry

This paper proposes an efficient structure for implementing a linear-phase finite-impulse-response (FIR) filter of an arbitrary order <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$N$</tex></formula> for the sampling-rate conversion by a rational factor of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$L/M$</tex> </formula> , where <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$L(M)$</tex> </formula> is the integer upsampling (downsampling) factor to be performed before (after) the actual filter. In this implementation, the coefficient symmetry of the linear-phase filter is exploited as much as possible and the number of delay elements is kept as low as possible while utilizing the following facts. When increasing (decreasing) the sampling rate by a factor of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$L(M)$</tex></formula> , only every <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$L$</tex></formula> th input sample has a nonzero value (only every <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$M$</tex> </formula> th output sample has to be evaluated). In this way, the number of required multiplications per output sample is reduced approximately by a factor of two compared with the conventional polyphase implementation. The proposed implementation is first illustrated using two examples. Based on these examples, guidelines are then given on how to efficiently realize an <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$N$</tex></formula> th-order linear-phase FIR filter for a sampling-rate converter having an arbitrary rational conversion factor <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$L/M$</tex></formula> . Finally, the implementation complexity of the proposed approach is evaluated and some examples are included, showing the efficiency of the proposed implementation compared with other existing ones.

A nearly optimum linear-phase digital FIR filters design

A new simple method to design linear-phase finite impulse response (FIR) digital filters, based on the steepest-descent optimization method, is presented in this paper. Starting from the specifications of the desired frequency response and a maximum approximation error a nearly optimum digital filter is obtained. Tests have shown that this method is alternative to other traditional ones such as Frequency Sampling and Parks–McClellan, mainly when other than brick wall frequency response is required as a desired frequency response.

Design of Linear Phase FIR Filters With High Probability of Achieving Minimum Number of Adders

In this paper, an algorithm is proposed for the design of low complexity linear phase finite impulse response (FIR) filters with optimum discrete coefficients. The proposed algorithm, based on mixed integer linear programming (MILP), efficiently traverses the discrete coefficient solutions and searches for the optimum one that results in an implementation using minimum number of adders. During the searching process, discrete coefficients are dynamically synthesized based on a continuously updated subexpression space and, most essentially, a monitoring mechanism is introduced to enable the algorithm's awareness of optimality. Benchmark examples have shown that the proposed algorithm can, in most cases, produce the optimum designs using minimum number of adders for the given specifications. The proposed algorithm can be simply extended for the optimum design with the maximum adder depth constraint.

Group Lifting Structures for Multirate Filter Banks I: Uniqueness of Lifting Factorizations

Group lifting structures are introduced to provide an algebraic framework for studying lifting factorizations of two-channel perfect reconstruction finite-impulse-response (FIR) filter banks. The lifting factorizations generated by a group lifting structure are characterized by Abelian groups of lower and upper triangular lifting matrices, an Abelian group of unimodular gain scaling matrices, and a set of base filter banks. Examples of group lifting structures are given for linear phase lifting factorizations of the two nontrivial classes of two-channel linear phase FIR filter banks, the whole- and half-sample symmetric classes, including both the reversible and irreversible cases. This covers the lifting specifications for whole-sample symmetric filter banks in Parts 1 and 2 of the ISO/IEC JPEG 2000 still image coding standard. The theory is used to address the uniqueness of lifting factorizations. With no constraints on the lifting process, it is shown that lifting factorizations are highly nonunique. When certain hypotheses developed in the paper are satisfied, however, lifting factorizations generated by a group lifting structure are shown to be unique. A companion paper applies the uniqueness results proven in this paper to the linear phase group lifting structures for whole- and half-sample symmetric filter banks.

A Design Method for Variable Linear-Phase FIR Filters with Changing Multifactors for Checkweighers

Digital signal processing requires digital filters with variable frequency characteristics. A variable digital filter (VDF) is a filter whose frequency characteristics can be easily and instantaneously changed. In this paper, we present a design method for variable linear-phase finite impulse response (FIR) filters with multiple variable factors and a reduction method for the number of polynomial coefficients. The obtained filter has a high piecewise attenuation in the stopband. The stopband edge and the position and magnitude of the high piecewise stopband attenuation can be varied by changing some parameters. Variable parameters are normalized in this paper. An optimization methodology known as semidefinite programming (SDP) is used to design the filter. In addition, we present that the proposed VDF can be implemented using the Farrow structure, which suitable for real time signal processing. The usefulness of the proposed filter is demonstrated through examples.

Multiple damage assessment in composite laminates using a Doppler-effect-based fiber-optic sensor

In this paper, carbon fiber-reinforced plastic (CFRP) laminates are addressed for the purpose of multiple damage assessment. Doppler-effect-based fiber-optic (FOD) sensors were used to capture guided waves propagating in the CFRP laminates. Characteristics of the fundamental symmetric (S0) and anti-symmetric (A0) Lamb waves in captured guided-wave signals were extracted by taking advantage of linear-phase finite impulse response filter and Hilbert transform, so as to systematically investigate the influence of delaminations on guided-wave propagation. Both dispersive characteristics of multi-mode Lamb waves and features of the Lamb wave-excited fundamental shear horizontal (SH0) guided wave were applied for damage evaluation and multiple damage identification. Results demonstrate that the FOD sensor is effective in multiple damage identification for composite laminates because it is omnidirectional in ultrasonic detection.

Synthesis of Wideband Linear-Phase FIR Filters with a Piecewise-Polynomial-Sinusoidal Impulse Response

A method is presented to synthesize wideband linear-phase finite-impulse-response (FIR) filters with a piecewise-polynomial-sinusoidal impulse response. The method is based on merging the earlier synthesis scheme proposed by the authors to design piecewise-polynomial filters with the method proposed by Chu and Burrus. The method uses an arbitrary number of separately generated center coefficients instead of only one or none as used in the method by Chu–Burrus. The desired impulse response is created by using a parallel connection of several filter branches and by adding an arbitrary number of center coefficients to form it. This method is especially effective for designing Hilbert transformers by using Type 4 linear-phase FIR filters, where only real-valued multipliers are needed in the implementation. The arithmetic complexity is proportional to the number of branches, the common polynomial order for each branch, and the number of separate center coefficients. For other linear-phase FIR filter types the arithmetic complexity depends additionally on the number of complex multipliers. Examples are given to illustrate the benefits of this method compared to the frequency-response masking (FRM) technique with regard to reducing the number of coefficients as well as arithmetic complexity.

Low-Complexity Design of Variable Bandedge Linear Phase FIR Filters With Sharp Transition Band

This paper presents a very low-complexity design of variable bandedge linear phase finite-impulse-response (FIR) filters with fixed sharp transition width. The idea is to first decompose the input signal into several channels in the frequency domain. The channel(s) involved with the transition band of the variable filter due to the variation of the bandedge is (are) shaped to produce the required transition band, and then summed up with the channels involved with the passband of the variable filter to produce the required frequency response. The proposed variable filter has extremely low complexity when the transition band is sharp, if compared with other techniques such as the Farrow structure. It is possible that the computational complexity of the variable filter is even lower than that of a corresponding fixed filter with the same transition width and ripple specifications implemented in its direct form.

A Generic Open-Loop Algorithm for Three-Phase Grid Voltage/Current Synchronization With Particular Reference to Phase, Frequency, and Amplitude Estimation

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This paper presents a new open-loop architecture for three-phase grid synchronization based on moving average and predictive filters, where accurate measurements of phase, frequency, and amplitude are carried out in real time. Previous works establish that the fundamental positive sequence vector of a set of utility voltage/current vectors can be decoupled using Park's transformation and low-pass filters. However, the filtering process introduces delays that impair the system performance. More specifically, when the input signal frequency is shifted above the nominal, a nonzero average steady-state phase error appears in the measurements. To overcome such limitations, a suitable combination of predictive and moving average finite impulse response (FIR) filters is used by the authors to achieve a robust synchronization system for all input frequencies. Moving average filters are linear phase FIR filters that have a constant time delay at low frequencies, a characteristic that is exploited to good effect to design a predictive filter that compensates such time delays, enabling zero steady-state phase errors for shifted input frequencies. In summary, the main attributes of the new system are its good frequency adaptation, good filtering/transient response tradeoff, and the fact that its dynamics is independent of the input vector amplitude. Comprehensive experimental results validate the theoretical approach and the high performance of the proposed synchronization algorithm. </para>

Synthesis of Narrowband Linear-Phase FIR Filters With a Piecewise-Polynomial Impulse Response

Classes of linear-phase finite-impulse response (FIR) filters with a piecewise-polynomial impulse response are proposed for the four types of linear-phase FIR filters. In addition, very efficient recursive structures to implement these filters in a straightforward and consistent manner are proposed. The desired impulse response is created by using a parallel connection of several filter branches. Only one branch has an impulse response of the full filter length, whereas the impulse responses are shorter for the remaining branches but the center is at the same location. The arithmetic complexity of these filters is proportional to the number of branches and the common polynomial order for each branch, rather than the actual filter order. In order to generate the overall piecewise-polynomial impulse response the polynomial coefficients are found, with the aid of linear programming, by optimizing the responses in the minimax sense, for both narrowband conventional filters and narrowband differentiators. The generation of these structures is based on the use of accumulators so that after using an accumulator, the resulting impulse response is divided into two parts. The first part follows the desired polynomial form, and the second part is what is left after the division, i.e., the nonpolynomial part. This same procedure can be used for all the following accumulators. Several examples are included, illustrating the benefits of the proposed filters, in terms of a reduced number of unknowns used in the optimization and the reduced number of multipliers required in the actual implementation.

Chebyshev Design of Linear-Phase FIR Filters With Linear Equality Constraints

By using basis transformation, the Chebyshev approximation of linear-phase finite-impulse response (FIR) filters with linear equality constraints can be converted into an unconstrained one defined on a new function space. However, since the Haar condition is not necessarily satisfied in the new function space, the alternating property does not hold for the solution to the resulted unconstrained Chebyshev approximation problem. A sufficient condition for the best approximation is obtained in this brief, and based on this condition, an efficient single exchange algorithm is derived for the Chebyshev design of linear-phase FIR filters with linear equality constraints. Simulations show that the proposed algorithm can converge to the optimal solution in most cases and to a near-optimal solution otherwise. Design examples are presented to illustrate the performance of the proposed algorithm.

On Chebyshev design of linear-phase FIR filters with frequency inequality constraints

The alternation theorem is the basis of the Remez algorithm for unconstrained Chebyshev design of finite-impulse response (FIR) filters. In this paper, we extend the alternation theorem to the inequality-constrained case and present an improved Remez algorithm for the design of minimax FIR filters with inequality constraints in frequency domain. Compared with existing algorithms, the presented algorithm has faster convergence rate and guaranteed optimal solutions.

Design of FIR digital filters with prescribed flatness and peak error constraints using second-order cone programming

This paper studies the design of digital finite impulse response (FIR) filters with prescribed flatness and peak design error constraints using second-order cone programming (SOCP). SOCP is a powerful convex optimization method, where linear and convex quadratic inequality constraints can readily be incorporated. It is utilized in this study for the optimal minimax and least squares design of linear-phase and low-delay (LD) FIR filters with prescribed magnitude flatness and peak design error. The proposed approach offers more flexibility than traditional maximally-flat approach for the tradeoff between the approximation error and the degree of design freedom. Using these results, new LD specialized filters such as digital differentiators, Hilbert Transformers, Mth band filters and variable digital filters with prescribed magnitude flatness constraints can also be derived.

Design of linear phase FIR filters via linear complementarity problem approach

In this paper, the design of constrained linear phase finite impulse response (FIR) filters is considered. The problem is formulated as a linear complementarity problem (LCP), which is solved using Lemke's algorithm. The LCP is a refined mathematical formalism with useful theoretical results. The digital filters presented meet efficiently the specifications of the magnitude response error. The used algorithm is a direct one and therefore, there is no need for matrix inversion. However, in the iterative methods that are frequently used, the bulk of the design computation is concerned with the evaluation of matrix inversion in order to solve a system of equations. Examples to illustrate the proposed method are presented. Copyright © 2005 John Wiley & Sons, Ltd.

Derivative-controlled design of linear-phase FIR filters via waveform moments

A new method for designing linear-phase finite impulse response (FIR) filters is proposed by using the blockwise waveform moments. The proposed method yields linear-phase FIR filters whose magnitude response and its derivatives to a certain order take the prescribed values at equally spaced frequency points. The design procedure only needs to solve a system of linear equations, whose size is slightly smaller than the degree of the resulting filter. In addition, the inversion of the linear equations can be essentially precomputed. Therefore, the proposed design method is computationally efficient. In particular, for some important cases, i.e., the maximally flat R-regular L/sup th/-band FIR filters, a closed-form formula can be obtained. It is also shown that the resulting R-regular L/sup th/-band FIR filters have the zero intersymbol interference property.