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Signal, Image and Video Processing

, Volume 8, Issue 8, pp 1451–1457 | Cite as

A new approach to the design of biorthogonal triplet half-band filter banks using generalized half-band polynomials

  • Amol D. RahulkarEmail author
  • Bhushan D. Patil
  • Raghunath S. Holambe
Original Paper

Abstract

This paper presents a novel approach to design a class of biorthogonal triplet half-band filter banks based on the generalized half-band polynomials. The filter banks are designed with the help of three-step lifting scheme (using three kernels). The generalized half-band polynomial is used to construct these three kernels by imposing the number of zeros at \(z=-1\). The maximum number of zeros imposed for the three kernels is half of the order of half-band polynomial (\(K/2\) for \(K\) order polynomial). The three kernels give a set of constraints on the coefficients of half-band polynomial by imposing the zeros. In addition to structural perfect reconstruction and linear phase, the proposed filter banks provide better frequency selectivity, more similarity between analysis and synthesis filters (measure of near-orthogonality), and good time–frequency localization. The proposed technique offers more flexibility in the design of filters using two degrees of freedom. Some examples have been presented to illustrate the method.

Keywords

Wavelets and filter bank Regularity  Triplet half-band filter bank Generalized half-band polynomial Perfect reconstruction 

List of symbols

\(H_0 (z)\)

Analysis LPF

\(G_0 (z)\)

Synthesis LPF

\(H_1 (z)\)

Analysis HPF

\(G_1 (z)\)

Synthesis HPF

\(P(z)\)

Product polynomial

\(h_0 {[n]}\)

Analysis LPF coefficients

\(h_1 {[n]}\)

Analysis HPF coefficients

\(g_0 {[n]}\)

Synthesis LPF coefficients

\(g_1 {[n]}\)

Synthesis HPF coefficients

\(\phi (t)\)

Scaling function

\(\psi (t)\)

Wavelet function

\(T_0 (z^{2}), T_1 (z^{2})\) and \(T_2 (z^{2})\)

Three kernels belong to the class of THFB

\(P_1 (z), P_2 (z)\), and \(P_3 (z)\)

Three half-band polynomials

\(c_j \)

Constant which can be expressed as functions of independent parameters

\(M_1 , M_2 ,\) and \(M_3 \)

Number of zeros

\(N_1 , N_2 ,\) and \(N_3 \)

Length of the polynomials

\(a_k \)

Independent parameters where \(k\in \,\mathbb Z \)

\(R(z)\)

Remainder polynomial

\(\Delta t^{2}\)

Time localization

\(\Delta \omega ^{2}\)

Frequency localization

\(E\)

Energy of the ripples

Notes

Acknowledgments

The authors are very much thankful to the editors and anonymous reviewers for insightful comments and valuable suggestions to improve the quality, which have been incorporated in this manuscript.

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Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  • Amol D. Rahulkar
    • 1
    Email author
  • Bhushan D. Patil
    • 2
  • Raghunath S. Holambe
    • 3
  1. 1.Department of Instrumentation and Control EngineeringAISSMS Institute of Information TechnologyPuneIndia
  2. 2.Imagination Technologies India Pvt. LtdPuneIndia
  3. 3.Department of Instrumentation EngineeringSGGS Institute of Engineering and TechnologyVishnupuri, NandedIndia

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