## Abstract

Time-domain synthesis is used to design continuous-time filters when their time-domain response for a given excitation is known. A case in point is the design of a continuous-time wavelet filter which requires a chosen wavelet as the filter’s impulse response. A key step in the design of a continuous-time wavelet filter is the approximation of its transfer function to a proper rational function. This approximation problem is addressed using methods that can be broadly categorized as closed-form methods and numerical optimization methods. While optimization methods are very effective, closed-form solutions are attractive because they are easy to design. In this work, we propose variants of nonlinear sequence transformation (a closed-form method) that obtain proper rational approximations of wavelet functions which are stable and also perform better in terms of mean square error when compared to those obtained by other closed-form methods. It is shown that the model-order reduction of the proposed variants leads to either similar or better performance compared to \(L_2\) optimization method. These variants are also shown to act as good starting points for optimization methods that use local search routines.

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## References

R. Bhattacharya, D. Roy, S. Bhowmick, Rational interpolation using Levin–Weniger transforms. Comput. Phys. Commun.

**101**(3), 213–222 (1997)A.J. Casson, An analog circuit approximation of the discrete wavelet transform for ultra low power signal processing in wearable sensor nodes. Sensors

**15**(12), 31914–31929 (2015)A.J. Casson, E. Rodriguez-Villegas, A 60 pw \( g_m\)c continuous wavelet transform circuit for portable EEG systems. IEEE J. Solid State Circuits

**46**(6), 1406–1415 (2011)A.J. Casson, E. Rodriguez-Villegas, Nanowatt multi-scale continuous wavelet transform chip. Electron. Lett.

**50**(3), 153–154 (2014)A.J. Casson, E.Rodriguez Villegas, in

*An Inverse Filter Realisation of a Single Scale Inverse Continuous Wavelet Transform,*IEEE International Symposium on Circuits and Systems, ISCAS 2008 (IEEE, 2008), pp. 904–907A.J. Casson, D.C. Yates, S. Patel, E. Rodriguez-Villegas, in

*An Analogue Bandpass Filter Realisation of the Continuous Wavelet Transform,*Engineering in Medicine and Biology Society, 2007. EMBS 2007. 29th Annual International Conference of the IEEE, (IEEE, 2007) pp. 1850–1854C.K. Chui, J. Wang, A cardinal spline approach to wavelets. Proceed. Am. Math. Soc.

**113**(3), 785–793 (1991)I.M. Filanovsky, P.N. Matkhanov, Synthesis of time delay networks approximating the pulse response described by an integer power of a sinusoid over its semi-period. Analog Integr. Circuits Signal Process.

**28**(1), 83–90 (2001)S.A.P. Haddad, W.A. Serdijn, et al., in

*Mapping the Wavelet Transform onto Silicon: The Dynamic Translinear Approach,*International Symposium on Circuits and Systems 2002. ISCAS 2002, vol. 5 (IEEE, 2002) pp. V–621S.A.P. Haddad, R. Houben, W.A. Serdijn, et al., in

*Analog Wavelet Transform Employing Dynamic Translinear Circuits for Cardiac Signal Characterization,*IEEE International Symposium on Circuits and Systems, 2003. ISCAS 2003, vol. 1 (IEEE, 2003) pp. I–121S.A.P. Haddad, N. Verwaal, R. Houben, W.A. Serdijn, et al. in

*Optimized Dynamic Translinear Implementation of the Gaussian Wavelet Transform,*IEEE International Symposium on Circuits and Systems, 2004. ISCAS 2004, vol. 1 (IEEE, 2004) pp. I–145S.A.P. Haddad, S. Bagga, W.A. Serdijn, Log-domain wavelet bases. IEEE Trans. Circuits Syst. I Regul. Pap.

**52**(10), 2023–2032 (2005a)S.A.P. Haddad, J.M.H. Karel, R.L.M. Peeters, R.L. Westra, W.A. Serdijn, et al., in

*Analog complex wavelet filters,*IEEE International Symposium on Circuits and Systems, 2005. ISCAS 2005, (IEEE, 2005b) pp. 3287–3290S.A.P. Haddad, W.A. Serdijn,

*Ultra Low-Power Biomedical Signal Processing: An Analog Wavelet Filter Approach for Pacemakers*(Springer, Berlin, 2009)H. Kamada, N. Aoshima, in

*Analog Gabor Transform Filter with complex First Order System,*SICE’97. Proceedings of the 36th SICE Annual Conference. International Session Papers, (IEEE, 1997) pp.925–930J.M.H. Karel, R.L.M. Peeters, R.L. Westra, S.A.P. Haddad, W.A. Serdijn, in

*An L 2-Based Approach for Wavelet Approximation,*44th IEEE Conference on Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC’05, (IEEE, 2005a) pp. 7882–7887J.M.H. Karel, R.L.M. Peeters, R.L. Westra, S.A.P. Haddad, W.A. Serdijn, in

*Wavelet Approximation for Implementation in Dynamic Translinear Circuits,*Proceedings of 16th IFAC World Congress (IFAC WC05), Prague, Czech republic, vol. 8 (2005b)J.M.H. Karel, S.A.P. Haddad, S. Hiseni, R.L. Westra, W. Serdijn, R.L.M. Peeters et al., Implementing wavelets in continuous-time analog circuits with dynamic range optimization. IEEE Trans. Circuits Syst. I Regul. Pap.

**59**(2), 229–242 (2012)G. Makkena, K.N. Abhilash, M.B. Srinivas, in

*Gaussian filter approximation using Levin’s transformation for implementation in analog domain,*IEEE Asia Pacific Conference on Postgraduate Research in Microelectronics and Electronics, IEEE PrimeAsia, 2013, (IEEE, 2013) pp. 204–207Stephane Mallat,

*A Wavelet Tour of Signal Processing: The Sparse Way*(Academic Press, Cambridge, 2008)J.P. Marmorat, M. Olivi, RARL2: a Matlab based software for \(H^2\) rational approximation (2004)

Y.J. Min, H.K. Kim, Y.R. Kang, G.S. Kim, J. Park, S.W. Kim, Design of wavelet-based ECG detector for implantable cardiac pacemakers. IEEE Trans. Biomed. Circuits Systems

**7**(4), 426–436 (2013)K. Ogata, Y. Yang,

*Modern Control Engineering*(Prentice-Hall, Englewood Cliffs, 1970)L. Pernebo, L.M. Silverman, Model reduction via balanced state space representations. IEEE Trans. Autom. Control

**27**(2), 382–387 (1982)D. Roy, R. Bhattacharya, S. Bhowmick, Rational approximants using Levin–Weniger transforms. Comput. Phys. Commun.

**93**, 159–178 (1996)M. Tuckwell, C. Papavassiliou, An analog Gabor transform using sub-threshold 180-nm CMOS devices. IEEE Trans. Circuits Syst. I Regul. Pap.

**56**(12), 2597–2608 (2009)M. Unser, T. Blu, Cardinal exponential splines: part I theory and filtering algorithms. IEEE Trans. Signal Process.

**53**(4), 1425–1438 (2005)M.A. Unser, in

*Ten Good Reasons for Using Spline Wavelets,*Optical Science, Engineering and Instrumentation’97, International Society for Optics and Photonics (1997) pp. 422–431M. Vucic, G. Molnar, Time-domain synthesis of continuous-time systems based on second-order cone programming. IEEE Trans. Circuits Syst. I Regul. Pap. 55(10), 3110–3118 (2008). doi:10.1109/TCSI.2008.925379

E.J. Weniger, Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Comput. Phys. Rep.

**10**(5), 189–371 (1989)E.J. Weniger, Irregular input data in convergence acceleration and summation processes: general considerations and some special Gaussian hypergeometric series as model problems. Comput. Phys. Commun.

**133**(2), 202–228 (2001)E.J. Weniger, Mathematical properties of a new Levin-type sequence transformation introduced by Čıžek, Zamastil, and Skála. I. Algebraic theory. J. Math. Phys.

**45**(3), 1209–1246 (2004)H. Xu, S. Jain, J. Song, T. Kamgaing, Y.S. Mekonnen, Acceleration of spectral domain immitance approach for generalized multilayered shielded microstrips using the Levins transformation. IEEE Antennas Wirel. Propag. Lett.

**14**, 92–95 (2015)

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Makkena, G., Srinivas, M.B. Nonlinear Sequence Transformation-Based Continuous-Time Wavelet Filter Approximation.
*Circuits Syst Signal Process* **37**, 965–983 (2018). https://doi.org/10.1007/s00034-017-0591-9

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DOI: https://doi.org/10.1007/s00034-017-0591-9

### Keywords

- Wavelet filter
- Nonlinear sequence transformation
- Impulse response
- Rational approximation