# DESIGN OF ALLPASS FILTERSWITH SPECIFIED DEGREES OF FLATNESS AND EQUIRIPPLE PHASE RESPONSES

### Xi Zhang

#### Abstract

This paper proposes a new method for designing allpass filters having the specified degrees of flatness at the specified frequency point(s) and equiripple phase responses in the approximation band(s). First, a system of linear equations are derived from the flatness conditions. Then, the Remez exchange algorithm is used to approximate the equiripple phase responses in the approximation band(s). By incorporating the linear equations from the flatness conditions into the equiripple approximation, the design problem is formulated as a generalized eigenvalue problem. Therefore, we can solve the eigenvalue problem to obtain the filter coefficients, which have the equiripple phase response and satisfy the specified degrees of flatness simultaneously. Furthermore, a class of IIR filters composed of allpass filters are introduced as one of its applications, and it is shown that IIR filters with flat passband (or stopband) and equiripple stopband (or passband) can be designed by using the proposed method. Finally, some examples are presented to demonstrate the effectiveness of the proposed design method.

#### References

- Darlington, S. (Dec. 1978). Filters with chebyshev stopbands, flat passbands, and impulse responses of finite duration. In IEEE Trans. Circuits and Systems. Vol.CAS-25, No.12, pp.966-975.
- Hegde, R. and Shenoi, B. A. (Nov. 1998). Magnitude approximation of digital filters with specified degrees of flatness and constant group delay characteristics. In IEEE Trans. Circuits and Systems II. Vol.45, No.11, pp.1476-1486.
- Laakso, T. I., Valimaki, V., Karjalainen, M., and Laine, U. K. (Jan. 1996). Splitting the unit delay: Tools for fractional delay filter design. In IEEE Signal Processing Mag. Vol.13, No.1, pp.30-60.
- Lang, M. (Sep. 1998). Allpass filter design and applications. In IEEE Trans. Signal Processing. Vol.46, No.9, pp.2505-2514.
- Mitra, S. K. and Kaiser, J. F. (1993). Handbook for Digital Signal Processing. Wiley, New York.
- Regalia, P. A., Mitra, S. K., and Vaidyanathan, P. P. (Jan. 1988). The digital allpass filter: A versatile signal processing building block. In Proc. IEEE. Vol.76, No.1, pp.19-37.
- Selesnick, I. W. (Jan. 1999). Lowpass filters realizable as allpass sums: design via a new flat delay filter. In IEEE Trans. Circuits and Systems II. Vol.46, No.1, pp.40-50.
- Selesnick, I. W. and Burrus, C. S. (June 1998). Generalized digital butterworth filter design. In IEEE Trans. Signal Processing. Vol.46, No.6, pp.1688-1694.
- Selesnick, I. W. and Burrus, C. S. (Sep. 1996). Exchange algorithms for the design of linear phase fir filters and differentiators having flat monotonic passbands and equiripple stopbands. In IEEE Trans. Circuits and Systems II. Vol.43, No.9, pp.671-675.
- Shenoi, B. A. (1999). Magnitude and Delay Approximation of 1-D and 2-D Digital Filters. Springer-Verlag, Berlin, Germany.
- Thiran, J. P. (Nov. 1971). Recursive digital filters with maximally flat group delay. In IEEE Trans. Circuit Theory. Vol.CT-18, No.6, pp.659-664.
- Tseng, C. C. (Sep. 2003). Design of iir digital allpass filters using least pth phase error criterion. In IEEE Trans. Circuits and Systems II. Vol.50, No.9, pp.653-656.
- Vaidyanathan, P. P. (Sep. 1985). Optimal design of linear phase fir digital filters with very flat passbands and equiripple stopbands. In IEEE Trans. Circuits and Systems. VVol.CAS-32, No.9, pp.904-917.
- Zhang, X. and Iwakura, H. (Feb. 1999). Design of iir digital allpass filters based on eigenvalue problem. In IEEE Trans. Signal Processing. Vol.47, No.2, pp.554-559.
- Zhang, X. and Iwakura, H. (June 1996). Design of iir digital filters based on eigenvalue problem. In IEEE Trans. Signal Processing. Vol.44, No.6, pp.1325-1333.

#### Paper Citation

#### in Harvard Style

Zhang X. (2010). **DESIGN OF ALLPASS FILTERSWITH SPECIFIED DEGREES OF FLATNESS AND EQUIRIPPLE PHASE RESPONSES** . In *Proceedings of the International Conference on Signal Processing and Multimedia Applications - Volume 1: SIGMAP, (ICETE 2010)* ISBN 978-989-8425-19-5, pages 205-210. DOI: 10.5220/0002934102050210

#### in Bibtex Style

@conference{sigmap10,

author={Xi Zhang},

title={DESIGN OF ALLPASS FILTERSWITH SPECIFIED DEGREES OF FLATNESS AND EQUIRIPPLE PHASE RESPONSES},

booktitle={Proceedings of the International Conference on Signal Processing and Multimedia Applications - Volume 1: SIGMAP, (ICETE 2010)},

year={2010},

pages={205-210},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0002934102050210},

isbn={978-989-8425-19-5},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the International Conference on Signal Processing and Multimedia Applications - Volume 1: SIGMAP, (ICETE 2010)

TI - DESIGN OF ALLPASS FILTERSWITH SPECIFIED DEGREES OF FLATNESS AND EQUIRIPPLE PHASE RESPONSES

SN - 978-989-8425-19-5

AU - Zhang X.

PY - 2010

SP - 205

EP - 210

DO - 10.5220/0002934102050210