Abstract
Fractional delay filters modeling non-integer delays are digital filters that ideally have flat group delays. This paper proposes a simple design method of fractional delay FIR filter based on binomial series expansion theory. First, the design technique is based on the binomial series expansion method which is applied to a discrete fractional system to obtain a closed form FIR digital filter which approximates the digital fractional delay operator z−m\( (m \in \Re^{ + } ) \). Then, the principal differentiation is used to design fractional delay FIR filter with a broader group delay bandwidth. Finally, numerical examples of fractional delay FIR filter design show that the proposed approach yields better performance compared to the existing techniques.
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References
T. Bensouici, A. Charef, Approximate realization of digital fractional forward operator using digital IIR filter. Signal Image Video Process. J. 6(3), 411–420 (2012)
T. Bensouici, A. Charef, Fractional Euler analog-to-digital transform. AEÜ Int. J. Electron. Commun. 69(4), 730–735 (2015)
T. Bensouici, A. Charef, I. Assadi, A new approach for the design of fractional delay by an FIR filter. ISA Trans. (2018). https://doi.org/10.1016/j.isatra.2018.03.021
A. Charef, T. Bensouici, Digital fractional delay implementation based on fractional order system. IET Proc. Signal Process. 5(6), 547–556 (2011)
A. Charef, T. Bensouici, Design of digital FIR variable fractional order integrator and differentiator. Signal Image Video Proc. J. 6(4), 679–689 (2012)
H.H. Dam, Design of variable fractional delay filter with fractional delay constraints. IEEE Signal Process. Lett. 21(11), 1361–1364 (2014)
H.H. Dam, Design of allpass variable fractional delay filter with powers-of-two coefficients. IEEE Signal Process. Lett. 22(10), 1643–1646 (2015)
T.B. Deng, W. Qin, Improved bi-equiripple variable fractional-delay filters. Sig. Process. 94(5), 300–307 (2014)
T.B. Deng, P. Soontornwong, Delay-error-constrained minimax design of all-pass variable fractional delay digital filters. Sig. Process. 120, 438–447 (2016)
R.L. Graham, D.E. Knuth, O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd edn. (Addison-Wesley, Reading, 1994)
X. Huang, B. Zhang, H. Qin, W. An, Closed-form design of variable fractional-delay FIR filters with low or middle cutoff frequencies. IEEE Tran. Circuits Syst. I 65(2), 628–637 (2018)
H. Johansson, A. Eghbali, Two polynomial FIR filter structures with variable fractional delay and phase shift. IEEE Trans. Circuits Syst. I 61(5), 1355–1365 (2014)
M. Kumar, T.K. Rawat, Optimal fractional delay-IIR filter design using cuckoo search algorithm. ISA Trans. 59, 39–54 (2015)
T.I. Laakso, V. Valimaki, M. Karjalainen, U.K. Laine, Splitting the unit delay: tool for fractional delay filter design. IEEE Signal Process. Mag. 13(1), 30–60 (1996)
P. Mohindru, R. Khanna, S.S. Bhatia, New tuning model for rectangular windowed FIR filter using fractional Fourier transform. Signal Image Video Proc. J. 9(4), 761–767 (2015)
M. Olsson, H. Johansson, P. Lowenborg, Delay estimation using adjustable fractional delay all-pass filters, in Proc. 7th Nordic Signal Processing Symposium. Reykjavík, Iceland, June 7–9 (2006), pp. 346–349
P. Murphy, A. Krukowski, A. Tarczynski, An efficient fractional sampler delayer for digital beam steering, in Proc. IEEE Int. Conference on Acoustics, Speech and Signal Processing. Munich, Germany, April 21–24 (1997), pp. 2245–2248
J. Shi, X. Liu, N. Zhang, Multiresolution analysis and orthogonal wavelets associated with fractional wavelet transform. Signal Image Video Proc. J. 9(1), 211–220 (2015)
C.C. Tseng, S.L. Lee, Design of fractional delay filter using discrete Fourier transform interpolation method. Sig. Process. 90(4), 1313–1322 (2010)
C.C. Tseng, S.L. Lee, Designs of fixed fractional delay filters using fractional derivative constraints. IEEE Trans. Circuits Syst. II 59(10), 683–687 (2012)
V. Välimäki, H.-M. Lehtonen, T.I. Laakso, Musical signal analysis using fractional delay inverse comb filters, in Proc. 10th Int. Conference on Digital Audio Effects. Bordeaux, France, September 10–15 (2007), pp. 261–268
V. Valimaki, A. Haghparast, Fractional delay filter design based on truncated Lagrange interpolation. IEEE Signal Process. Lett. 14(11), 816–819 (2007)
J. Vesma, T. Saramiki, Interpolation filters with arbitrary frequency response for all-digital receivers, in Proc. IEEE Int. Symp. Circuits Syst. Atlanta, GA, USA, May 12–15 (1996), pp 568–571
M.M.J. Yekta, Half-band FIR fractional delay filters with closed-form coefficient formulas and modular implementation based on Lagrange interpolators. Sig. Process. 88(12), 2913–2916 (2008)
M.M.J. Yekta, Wideband maximally flat fractional delay allpass filters. Electron. Lett. 46(10), 722–723 (2010)
J.Y. Yu, W.J. Xu, Investigation on the optimization criteria for the design of variable fractional delay filters. IEEE Trans. Circuits Syst. II 60(8), 522–526 (2013)
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Bensouici, T., Charef, A. & Imen, A. A Simple Design of Fractional Delay FIR Filter Based on Binomial Series Expansion Theory. Circuits Syst Signal Process 38, 3356–3369 (2019). https://doi.org/10.1007/s00034-018-1000-8
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DOI: https://doi.org/10.1007/s00034-018-1000-8