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

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## 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.

## References

- 1.Patil, B.D., Patwardhan, P.G., Gadre, V.M.: On the design of FIR wavelet filter banks using factorization of a halfband polynomial. IEEE Signal Process. Lett.
**15**, 485–488 (2008)CrossRefGoogle Scholar - 2.Daubechies, I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)CrossRefzbMATHGoogle Scholar
- 3.Cohen, A., Daubechies, I., Feauveau, J.C.: Biorthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math.
**45**(5), 485–560 (1992)CrossRefzbMATHMathSciNetGoogle Scholar - 4.Ansari, R., Guillemot, C., Kaiser, J.F.: Wavelet construction using lagrange halfband filters. IEEE Trans. Circuits Syst. Express Lett.
**38**(9), 1116–1118 (1991)CrossRefGoogle Scholar - 5.Sweldens, W.: The lifting scheme: a custom-design construction of biorthogonal wavelets. Appl. Comput. Hormon. Anal.
**3**(2), 186–200 (1996)CrossRefzbMATHMathSciNetGoogle Scholar - 6.Phoong, S.M., Kim, C.W., Vaidyanathan, P.P., Ansari, R.: A new class of two-channel biorthogonal filter banks and wavelet bases. IEEE Trans. Signal Process.
**43**(3), 649–665 (1995)CrossRefGoogle Scholar - 7.Ansari, R., Kim, C.W., Dedovic, M.: Structure and design of two-channel filter banks derived from triplet of halfband filtres. IEEE Trans. Circuits Syst. II Analog Digit. Signal Process.
**46**(12), 1487–1496 (1992)CrossRefGoogle Scholar - 8.Tay, D.B.H., Palaniswami, M.: A novel approach to the design of the class of triplet halfband filterbanks. IEEE Trans. Circuits Syst. II Express Briefs
**51**(7), 378–383 (2004)CrossRefGoogle Scholar - 9.Tay, D.B.H.: ETHFB: a new class of even-length biorthogonal wavelet filters for hilbert pair design. IEEE Trans. Circuits Syst. I Regul. Pap.
**55**(6), 1580–1588 (2008)CrossRefMathSciNetGoogle Scholar - 10.Chan, S.C., Yeung, K.S.: On the design and multiplierless realization of perfect reconstruction triplet-based FIR filterbanks and wavelet bases. IEEE Trans. Circuits Syst. I
**51**(8), 1476–1491 (2004)CrossRefMathSciNetGoogle Scholar - 11.Kha, H.H., Tuan, H.D., Nguyen, T.Q.: Optimal design of FIR triplet halfband filter bank and application in image coding. IEEE Trans. Image Process.
**22**(2), 586–591 (2011)CrossRefMathSciNetGoogle Scholar - 12.Eslami, R., Radha, H.: Design of regular wavelets using a three-step lifting scheme. IEEE Trans. Signal Process.
**58**(4), 2088–2101 (2010)CrossRefMathSciNetGoogle Scholar - 13.Kovacevic, J., Sweldens, W.: Wavelet families of increasing order in arbitrary dimensions. IEEE Trans. Image Process.
**9**(3), 480–496 (2000)CrossRefzbMATHGoogle Scholar - 14.Vaidyanathan P.P.: Multirate Systems and Filter Banks. Prentice-Hall, Englewood Cliffs (1993)Google Scholar
- 15.Vetterli, M., Kovacevic, J.: Wavelets and Subband Coding. Prentice-Hall, Englewood Cliffs (1995)zbMATHGoogle Scholar
- 16.Strang, G., Nguyen, T.: Wavelets and Filter Banks. Wellesley-Cambridge, New York (1996)zbMATHGoogle Scholar
- 17.Tay, D.B.H.: Balanced spatial and frequency localized 2-D nonseparable wavelet filters. Proc. IEEE Int. Symp. Circuits Syst.
**2**, 489–492 (2001)Google Scholar - 18.Monro, D.M., Sherlock, B.G.: Space frequency balance in biorthogonal wavelets. Proc. IEEE Int. Conf. Image Proc.
**1**, 624–627 (1997)CrossRefGoogle Scholar