research-article

A Robust and Scalable Implementation of the Parks-McClellan Algorithm for Designing FIR Filters

Author:

Silviu-Ioan FilipAuthors Info & Claims

ACM Transactions on Mathematical Software (TOMS), Volume 43, Issue 1

Article No.: 7, Pages 1 - 24

https://doi.org/10.1145/2904902

Published: 13 August 2016 Publication History

Abstract

With a long history dating back to the beginning of the 1970s, the Parks-McClellan algorithm is probably the most well known approach for designing finite impulse response filters. Despite being a standard routine in many signal processing packages, it is possible to find practical design specifications where existing codes fail to work. Our goal is twofold. We first examine and present solutions for the practical difficulties related to weighted minimax polynomial approximation problems on multi-interval domains (i.e., the general setting under which the Parks-McClellan algorithm operates). Using these ideas, we then describe a robust implementation of this algorithm. It routinely outperforms existing minimax filter design routines.

References

[1]

W. A. Abu-Al-Saud and G. L. Stüber. 2004. Efficient wideband channelizer for software radio systems using modulated PR filterbanks. IEEE Transactions on Signal Processing, 52, 10, 2807--2820.

Digital Library

[2]

J. W. Adams and A. N. Wilson. 1985. On the fast design of high-order FIR digital filters. IEEE Transactions on Circuits and Systems 32, 9, 958--960.

[3]

M. Ahsan and T. Saramäki. 2012. Two Novel Implementations of the Remez Multiple Exchange Algorithm for Optimum FIR Filter Design. Retrieved July 16, 2016, from http://www.intechopen.com/download/pdf/39374

[4]

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen. 1999. LAPACK Users' Guide. Vol. 9. SIAM.

Digital Library

[5]

A. Antoniou. 1982. Accelerated procedure for the design of equiripple nonrecursive digital filters. IEE Proceedings G: Electronic Circuits and Systems 129, 1, 1--10.

[6]

A. Antoniou. 1983. New improved method for the design of weighted-Chebyshev, nonrecursive, digital filters. IEEE Transactions on Circuits and Systems 30, 10, 740--750.

[7]

A. Antoniou. 2005. Digital Signal Processing: Signals, Systems, and Filters. McGraw-Hill Education.

[8]

J.-P. Berrut and L. N. Trefethen. 2004. Barycentric Lagrange interpolation. SIAM Review 46, 3, 501--517.

Digital Library

[9]

F. Bonzanigo. 1982. Some improvements to the design programs for equiripple FIR filters. In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'82), Vol. 7. 274--277.

[10]

L. P. Bos, J.-P. Calvi, N. Levenberg, A. Sommariva, and M. Vianello. 2011. Geometric weakly admissible meshes, discrete least squares approximations and approximate Fekete points. Mathematics of Computation 80, 275, 1623--1638.

[11]

L. P. Bos and N. Levenberg. 2008. On the calculation of approximate Fekete points: The univariate case. Electronic Transactions on Numerical Analysis 30, 377--397.

[12]

J. P. Boyd. 2006. Computing the zeros, maxima and inflection points of Chebyshev, Legendre and Fourier series: Solving transcendental equations by spectral interpolation and polynomial rootfinding. Journal of Engineering Mathematics 56, 3, 203--219.

[13]

J. P. Boyd. 2013. Finding the zeros of a univariate equation: Proxy rootfinders, Chebyshev interpolation, and the companion matrix. SIAM Review 55, 2, 375--396.

Digital Library

[14]

J. P. Boyd and D. H. Gally. 2007. Numerical experiments on the accuracy of the Chebyshev-Frobenius companion matrix method for finding the zeros of a truncated series of Chebyshev polynomials. Journal of Computational and Applied Mathematics 205, 1, 281--295.

Digital Library

[15]

J.-P. Calvi and N. Levenberg. 2008. Uniform approximation by discrete least squares polynomials. Journal of Approximation Theory 152, 1, 82--100.

Digital Library

[16]

A. Çivril and M. Magdon-Ismail. 2009. On selecting a maximum volume sub-matrix of a matrix and related problems. Theoretical Computer Science 410, 47--49, 4801--4811.

Digital Library

[17]

E. W. Cheney. 1982. Introduction to Approximation Theory. AMS Chelsea Publications.

[18]

L. Dagum and R. Menon. 1998. OpenMP: An industry standard API for shared-memory programming. IEEE Computational Science Engineering 5, 1, 46--55.

Digital Library

[19]

D. Day and L. Romero. 2005. Roots of polynomials expressed in terms of orthogonal polynomials. SIAM Journal on Numerical Analysis 43, 5, 1969--1987.

Digital Library

[20]

F. De Dinechin, M. Istoan, and A. Massouri. 2014. Sum-of-product architectures computing just right. In Proceedings of the 2014 IEEE 25th International Conference on Application-Specific Systems, Architectures, and Processors (ASAP'14). 41--47.

[21]

T. A. Driscoll, N. Hale, and L. N. Trefethen. 2014. Chebfun Guide. Pafnuty Publications, Oxford, UK.

[22]

A. Dutt, M. Gu, and V. Rokhlin. 1996. Fast algorithms for polynomial interpolation, integration, and differentiation. SIAM Journal on Numerical Analysis 33, 5, 1689--1711.

Digital Library

[23]

S. Ebert and U. Heute. 1983. Accelerated design of linear or minimum phase FIR filters with a Chebyshev magnitude response. IEE Proceedings G: Electronic Circuits and Systems 130, 6, 267--270.

[24]

M. Embree and L. N. Trefethen. 1999. Green's functions for multiply connected domains via conformal mapping. SIAM Review 41, 4, 745--761.

Digital Library

[25]

M. S. Floater and K. Hormann. 2007. Barycentric rational interpolation with no poles and high rates of approximation. Numerische Mathematik 107, 2, 315--331.

Digital Library

[26]

L. Fousse, G. Hanrot, V. Lefèvre, P. Pélissier, and P. Zimmermann. 2007. MPFR: A multiple-precision binary floating-point library with correct rounding. ACM Transactions on Mathematical Software 33, 2, 13.

Digital Library

[27]

G. Guennebaud and B. Jacob. 2010. Eigen v3. Available at http://eigen.tuxfamily.org.

[28]

N. J. Higham. 2004. The numerical stability of barycentric Lagrange interpolation. IMA Journal of Numerical Analysis 24, 547--556.

[29]

L. J. Karam and J. H. McClellan. 1995. Complex Chebyshev approximation for FIR filter design. IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing 42, 3, 207--216.

[30]

L. J. Karam and J. H. McClellan. 1996. Design of optimal digital FIR filters with arbitrary magnitude and phase responses. In Proceedings of the IEEE International Symposium on Circuits and Systems (ISCAS'96), Vol. 2. 385--388.

[31]

D. M. Kodek. 2012. LLL algorithm and the optimal finite wordlength FIR design. IEEE Transactions on Signal Processing 60, 3, 1493--1498.

Digital Library

[32]

E. Levin and E. B. Saff. 2006. Potential theoretic tools in polynomial and rational approximation. In Harmonic Analysis and Rational Approximation. Lecture Notes in Control and Information Science, Vol. 327. Springer, 71--94.

[33]

W. F. Mascarenhas. 2014. The stability of barycentric interpolation at the Chebyshev points of the second kind. Numerische Mathematik 128, 2, 265--300.

Digital Library

[34]

W. F. Mascarenhas and A. Camargo. 2014. On the backward stability of the second barycentric formula for interpolation. Dolomites Research Notes on Approximation 7, 1--12.

[35]

J. H. McClellan and T. W. Parks. 2005. A personal history of the Parks-McClellan algorithm. IEEE Signal Processing Magazine 22, 2, 82--86.

[36]

J. H. McClellan, T. W. Parks, and L. Rabiner. 1973. A computer program for designing optimum FIR linear phase digital filters. IEEE Transactions on Audio and Electroacoustics 21, 6, 506--526.

[37]

J. M. Muller, N. Brisebarre, F. de Dinechin, C. P. Jeannerod, V. Lefèvre, G. Melquiond, N. Revol, D. Stehlé, and S. Torres. 2009. Handbook of Floating-Point Arithmetic. Birkhäuser, Boston, MA.

Digital Library

[38]

A. V. Oppenheim and R. W. Schafer. 2010. Discrete-Time Signal Processing. Prentice Hall.

Digital Library

[39]

R. Pachón and L. N. Trefethen. 2009. Barycentric-Remez algorithms for best polynomial approximation in the Chebfun system. BIT Numerical Mathematics 49, 4, 721--741.

[40]

T. W. Parks and J. H. McClellan. 1972. Chebyshev approximation for nonrecursive digital filters with linear phase. IEEE Transactions on Circuit Theory 19, 2 (March 1972), 189--194.

[41]

D. Pelloni and F. Bonzanigo. 1980. On the design of high-order linear phase FIR filters. Digital Signal Processing 1, 3--10.

[42]

M. J. D. Powell. 1981. Approximation Theory and Methods. Cambridge University Press, Cambridge, MA.

[43]

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. 2007. Numerical Recipes: The Art of Scientific Computing (3rd ed.). Cambridge University Press, Cambridge, MA.

Digital Library

[44]

E. Z. Psarakis and G. V. Moustakides. 2003. A robust initialization scheme for the Remez exchange algorithm. IEEE Signal Processing Letters 10, 1, 1--3.

[45]

L. Rabiner, J. Kaiser, and R. W. Schafer. 1974. Some considerations in the design of multiband finite-impulse-response digital filters. IEEE Transactions on Acoustics, Speech, and Signal Processing 22, 6, 462--472.

[46]

H.-J. Rack and M. Reimer. 1982. The numerical stability of evaluation schemes for polynomials based on the Lagrange interpolation form. BIT Numerical Mathematics 22, 1, 101--107.

[47]

E. Remes. 1934. Sur le calcul effectif des polynômes d'approximation de tchebichef. Comptes Rendus Hebdomadaires Des séances De l'Académie Des Sciences 199, 337--340.

[48]

H. Rutishauser. 1976. Vorlesungen Über Numerische Mathematik. Birkhäuser Basel, Stuttgard. English translation, 1990. Lectures on Numerical Mathematics, W. Gautschi (Ed.). Birkhäuser, Boston, MA.

[49]

T. Saramäki. 1992. An efficient Remez-type algorithm for the design of optimum IIR filters with arbitrary partially constrained specifications. In Proceedings of the IEEE International Symposium on Circuits and Systems (ISCAS'92), Vol. 5. 2577--2580.

[50]

T. Saramäki. 1994. Generalizations of classical recursive digital filters and their design with the aid of a Remez-type algorithm. In Proceedings of the IEEE International Symposium on Circuits and Systems (ISCAS'94), Vol. 2. 549--552.

[51]

I. W. Selesnick. 1997. New exchange rules for IIR filter design. In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'97), Vol. 3. 2209--2212.

Digital Library

[52]

I. W. Selesnick, M. Lang, and C. S. Burrus. 1994. Magnitude squared design of recursive filters with the Chebyshev norm using a constrained rational Remez algorithm. In Proceedings of the 6th IEEE Digital Signal Processing Workshop. 23--26.

[53]

J. Shen, G. Strang, and A. J. Wathen. 2001. The potential theory of several intervals and its applications. Applied Mathematics and Optimization 44, 1, 67--85.

Digital Library

[54]

D. J. Shpak and A. Antoniou. 1990. A generalized Remez method for the design of FIR digital filters. IEEE Transactions on Circuits and Systems 37, 2, 161--174.

[55]

A. Sommariva and M. Vianello. 2009. Computing approximate Fekete points by QR factorizations of Vandermonde matrices. Computers and Mathematics with Applications 57, 8, 1324--1336.

Digital Library

[56]

A. Sommariva and M. Vianello. 2010. Approximate Fekete points for weighted polynomial interpolation. Electronic Transactions on Numerical Analysis 37, 1--22.

[57]

G. Strang. 1999. The discrete cosine transform. SIAM Review 41, 1, 135--147.

Digital Library

[58]

L. N. Trefethen. 2013. Approximation Theory and Approximation Practice. SIAM.

Digital Library

[59]

L. Veidinger. 1960. On the numerical determination of the best approximation in the Chebyshev sense. Numerische Mathematik 2, 1, 99--105.

Digital Library

[60]

M. Webb, L. N. Trefethen, and P. Gonnet. 2012. Stability of barycentric interpolation formulas for extrapolation. SIAM Journal on Scientific Computing 34, 6, A3009--A3015.

[61]

P. Zahradnik and M. Vlcek. 2008. Robust analytical design of equiripple comb FIR filters. In Proceedings of the IEEE International Symposium on Circuits and Systems (ISCAS'08). 1128--1131.

Cited By

Ma TLi YZhou ZMeng J(2023)Wrinkle Detection in Carbon Fiber-Reinforced Polymers Using Linear Phase FIR-Filtered Ultrasonic Array DataAerospace10.3390/aerospace1002018110:2(181)Online publication date: 15-Feb-2023
https://doi.org/10.3390/aerospace10020181
Kumm MVolkova AFilip S(2023)Design of Optimal Multiplierless FIR Filters With Minimal Number of AddersIEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems10.1109/TCAD.2022.317922142:2(658-671)Online publication date: Feb-2023
https://doi.org/10.1109/TCAD.2022.3179221
Ranjan RSahana B(2023)Automated Alzheimer’s Disease Diagnosis Using Norm Features Extracted from EEG Signals2023 14th International Conference on Computing Communication and Networking Technologies (ICCCNT)10.1109/ICCCNT56998.2023.10413583(1-6)Online publication date: 6-Jul-2023
https://doi.org/10.1109/ICCCNT56998.2023.10413583
Show More Cited By

Index Terms

A Robust and Scalable Implementation of the Parks-McClellan Algorithm for Designing FIR Filters

Recommendations

Design of two-channel linear phase IIR PRQMF banks based on the approximation of FIR filters
ASILOMAR '95: Proceedings of the 29th Asilomar Conference on Signals, Systems and Computers (2-Volume Set)

This paper presents an algorithm to design two-channel linear phase IIR PRQMF (LP IIR PRQMF) banks. This algorithm is based on the approximation of the FIR by IIR digital filters matching a finite portion of the impulse response and the autocorrelation ...
On arbitrary-length, M-channel linear-phase FIR filter banks
ASILOMAR '95: Proceedings of the 29th Asilomar Conference on Signals, Systems and Computers (2-Volume Set)
Tree-structured IIR/FIR uniform-band and octave-band filter banks with very low-complexity analysis or synthesis filters

This paper introduces new tree-structured uniform-band and octave-band digital filter banks (FBs). These FBs make use of half-band IIR filters in the analysis FBs and FIR filters in the synthesis FBs. The resulting FBs are asymmetric in the sense that ...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Mathematical Software

ACM Transactions on Mathematical Software Volume 43, Issue 1

March 2017

202 pages

ISSN:0098-3500

EISSN:1557-7295

DOI:10.1145/2987591

Editor:
Michael A. Heroux
Sandia National Laboratories, USA

Issue’s Table of Contents

Copyright © 2016 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 13 August 2016

Accepted: 01 March 2016

Revised: 01 March 2016

Received: 01 March 2015

Published in TOMS Volume 43, Issue 1

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article
Research
Refereed

Funding Sources

French National Agency for Research (ANR)
FastRelax

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

17
Total Citations
View Citations
277
Total Downloads

Downloads (Last 12 months)31
Downloads (Last 6 weeks)3

Reflects downloads up to 13 Nov 2024

Other Metrics

View Author Metrics

Citations

Cited By

Ma TLi YZhou ZMeng J(2023)Wrinkle Detection in Carbon Fiber-Reinforced Polymers Using Linear Phase FIR-Filtered Ultrasonic Array DataAerospace10.3390/aerospace1002018110:2(181)Online publication date: 15-Feb-2023
https://doi.org/10.3390/aerospace10020181
Kumm MVolkova AFilip S(2023)Design of Optimal Multiplierless FIR Filters With Minimal Number of AddersIEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems10.1109/TCAD.2022.317922142:2(658-671)Online publication date: Feb-2023
https://doi.org/10.1109/TCAD.2022.3179221
Ranjan RSahana B(2023)Automated Alzheimer’s Disease Diagnosis Using Norm Features Extracted from EEG Signals2023 14th International Conference on Computing Communication and Networking Technologies (ICCCNT)10.1109/ICCCNT56998.2023.10413583(1-6)Online publication date: 6-Jul-2023
https://doi.org/10.1109/ICCCNT56998.2023.10413583
Brisebarre NFilip S(2023) Towards Machine-Efficient Rational L ∞ -Approximations of Mathematical Functions 2023 IEEE 30th Symposium on Computer Arithmetic (ARITH)10.1109/ARITH58626.2023.00029(119-126)Online publication date: 4-Sep-2023
https://doi.org/10.1109/ARITH58626.2023.00029
Rao JZeng LLiu MFu H(2023)Ultrasonic defect detection of high-density polyethylene pipe materials using FIR filtering and block-wise singular value decompositionUltrasonics10.1016/j.ultras.2023.107088134(107088)Online publication date: Sep-2023
https://doi.org/10.1016/j.ultras.2023.107088
Duan SWu XWang JZou YJiang LWei Y(2023)Defect Characterization Method for Bridge Cables Based on Topology of Dynamical Reconstruction of Magnetostrictive Guided Wave Testing SignalsJournal of Nondestructive Evaluation10.1007/s10921-023-00940-242:2Online publication date: 28-Mar-2023
https://doi.org/10.1007/s10921-023-00940-2
de Dinechin FKumm Mde Dinechin FKumm M(2023)Arithmetic in the Design of Linear Time-Invariant FiltersApplication-Specific Arithmetic10.1007/978-3-031-42808-1_23(669-705)Online publication date: 23-Aug-2023
https://doi.org/10.1007/978-3-031-42808-1_23
Ranjan RNeeti Sahana B(2022)Automatic Detection of Mental Health Status using Alpha Subband of EEG Data2022 IEEE International Symposium on Medical Measurements and Applications (MeMeA)10.1109/MeMeA54994.2022.9856586(1-6)Online publication date: 22-Jun-2022
https://dl.acm.org/doi/10.1109/MeMeA54994.2022.9856586
Ghuli AEdla DTavares J(2022)Epileptic seizure endorsement technique using DWT power spectrumThe Journal of Supercomputing10.1007/s11227-021-04196-378:6(8604-8624)Online publication date: 1-Apr-2022
https://dl.acm.org/doi/10.1007/s11227-021-04196-3
Lian LTian Z(2022)FIR digital filter design based on improved artificial bee colony algorithmSoft Computing10.1007/s00500-022-07506-w26:24(13489-13507)Online publication date: 1-Oct-2022
https://doi.org/10.1007/s00500-022-07506-w
Show More Cited By

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Issue’s Table of Contents