Computational and Mathematical Methods in MedicineVolume 2012 (2012), Article ID 365054, 6 pagesdoi:10.1155/2012/365054

Research Article

Approximating Ideal Filters by Systems of Fractional Order

Ming Li1,2

¹School of Information Science & Technology, East China Normal University, No. 500, Dong-Chuan Road, Shanghai 200241, China²Department of Computer and Information Science, University of Macau, Avenue Padre Tomas Pereira, Taipa 1356, Macau, China

Received 16 September 2011; Accepted 10 October 2011

Academic Editor: Carlo Cattani

Copyright © 2012 Ming Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The contributions in this paper are in two folds. On the one hand, we propose a general approach for approximating ideal filters based on fractional calculus from the point of view of systems of fractional order. On the other hand, we suggest that the Paley and Wiener criterion might not be a necessary condition for designing physically realizable ideal filters. As an application of the present approach, we show a case in designing ideal filters for suppressing 50-Hz interference in electrocardiogram (ECG) signals.

1. Introduction

Filters have wide applications in various fields, ranging from medical engineering to electrical engineering; see, for example, Hussain et al. [1], Bhattacharyya et al. [2], Fieguth [3], Bendat and Piersol [4], Gray and Davisson [5], and Li [6], just mentioning a few. In the field, the theory and techniques to approximate ideal filters are desired. There are some methods about approximating ideal filters, such as Butterworth filters, Chebyshev filters, Cauer-Chebyshev filters, and Bessel ones (Wanhammar [7], Mitra and Kaiser [8]).

Recall that the conventional filters of Butterworth type, Chebyshev type, Cauer-Chebyshev type, or Bessel one are discussed in the domain of systems of integer order. More precisely, the frequency response of a filter that is denoted by is a rational function. Both the denominator and the numerator of the rational function are polynomials of integer order; see [7, 8], Vegte [9], Dorf and Bishop [10], and Li [11]. From the point of view of mathematical analysis, conventional filters are in the domain of calculus of integer order.

This paper aims at providing an approach to approximate ideal filters by using frequency responses of fractional-order. The basic idea is like this. Denote by the cutoff frequency of a filter. Then, from a view of ideal filters. In this case, we present the following approximation: where is the amplitude of .

An obvious advantage of the present approach is that the above always holds no matter what the concrete structure of is. However, theoretically speaking, has to be explained from the point of view of fractional calculus.

The remaining paper is organized as follows. Section 2 explains the research background. The problem statement is described in Section 3. The present approximation is given in Section 4. A case study is stated in Section 5, which is followed by conclusions.

2. Research Background

2.1. Glimpse at Ideal Filters

The ideal lowpass filter implies that the amplitude of the frequency response is given by where . One says that is the frequency response of an ideal highpass filter if The ideal bandpass filter has the frequency response expressed by where and are cut-off frequencies. A filter is said to be ideal band stop if its frequency response function is given by

2.2. Paley and Wiener Criterion

For facilitating the discussions, we write where is the phase response of a filter. Note that the condition for to be zero for negative , where implies the inverse of the Fourier transform, is that must be square-integrable. That is, The above implies the causality of a filter; see, for example, Papoulis [12]. A necessary and sufficient condition for to satisfy (7) is explained by Paley and Wiener [13]. That condition is called the Paley and Wiener condition or the Paley and Wiener criterion. It is expressed by

The Paley and Wiener criterion implies that ideal filters are not physically realizable because in a certain frequency range for each type of ideal filters. Therefore, approximations of ideal filters are desired.

2.3. Some Filters of Integer Order for Approximating Ideal Filters

Various methods in the approximations are studied, such as Butterworth filters, Chebyshev’s, Cauer-Chebyshev’s, and Bessel’s filters; see, for example, [2], and Lam [14].

Taking lowpass filtering as an example, the system function of the Butterworth filters of order is given by

Denote the Chebyshev polynomial of the first kind by . Then, The frequency response of the Chebyshev type lowpass filters for is given by

Denote the Chebyshev rational function of degree by . Then, One of the applications of is to design an elliptic filter, which is also known as a Cauer filter, named after Wilhelm Cauer. An elliptic filter has the property of equalized ripple (equiripple) behavior in both the passband and the stopband. The frequency response of the elliptic type lowpass filters for is given by where is the ripple factor, and is the selectivity factor [15, 16].

3. Problem Statement

The Butterworth filters obviously correspond to linear differential equations of integer order [17, 18].

Note that the Chebyshev polynomial of the first kind is the solution to the Chebyshev equation that is the second-order linear differential equation given by Therefore, a consequence we note hereby is that the Chebyshev type filters as well as the elliptic type filters are corresponding to linear differential equations of integer order.

Recently, filters of fractional-order attract much attention in the field of circuits, systems, and signals; see, for example, Podlubny [19], Ortigueira [20], MacHado et al. [21], Lim et al. [22], Chen and Moore [23], and Zhang [24], simply citing a few. However, the literature regarding approximating ideal filters from a view of filters of fractional-order is rarely seen. For that reason, we propose a question like this. May ideal filters be approximated by filters or equations of fractional-order? We will give the affirmative answer to it in the next section.

4. Approximating Ideal Filters by Systems of Fractional Order

A linear filter can be expressed by a linear differential equation given by where is the response and excitation. Denote the Fourier transforms of and by and , respectively. Then, the system function is given by Denote by . Then, . This is the basic principle regarding linear filters. In this case, we say that is the system function or frequency response of a filter of integer order; see, for example, Monje et al. [25].

We now consider a filter of fractional-order presented by Denote Then, Since is similar to , the key difference between and is in the aspect of amplitude response, namely, and .

It can be seen from (19) that In addition,

Denote the 3-dB bandwidth of by where is frequency. Denote the bandwidth for Then, the rectangular coefficient defined by is always ideal for . That is, because of (20).

On the other hand, The expression (26) implies that always has a linear phase response.

Remark 1. Equation (25) does not relate to any concrete forms of . Thus, the present results, namely, (20) and (21), stand for a general approach for approximating ideal filters based on systems of fractional-order.

Remark 2. Let Then, does not satisfy the Paley and Wiener criterion expressed by (8) because That is, Therefore, this remark suggests a theoretical significance that the Paley and Wiener criterion might not be a necessary condition for designing physically realizable ideal filters of fractional-order.

5. Case Study

We consider a finite impulse response filter (FIR) given by where is the sampling period. Figure 1 indicates for .

Figure 1: Amplitude response of filter.

For and , we have Note that Thus, the rectangular coefficient of is The rectangular coefficient of 0.534 exhibits that is not a satisfactory filter in general. Nevertheless, one is able to easily modify it to be such that it is an ideal filter by

Figure 2 shows the approximations of for , and 0.0001, respectively. It exhibits that the present method well approximates the ideal filter. As a matter of fact, in the sense of 0.9994 ≈ 1 for , see Figure 2(d), can be regarded as an ideal filter in practice.

(a)

(b)

(c)

(d)

Figure 2: Approximations of for different values of . (a) . (b) . (c) . (d) .

The following is called a binomial series: where is binomial coefficient [26]. By using binomial series, (34) can be expanded by Therefore, in general, should be taken as a filter of fractional-order from a view of fractional-order systems [25]; see the Appendix for the meaning of in fractional-order systems.

It is worth noting that (34) may yet be an ideal FIR notch filter used for suppressing 50-Hz interference in electrocardiogram (ECG) signals, which is a key component in processing ECG signals in medical engineering; see, for example, Talmaon [27], Levkov et al. [28], Martens et al. [29], Dotsinsky and Stoyanov [30], and Li [31], though is not a satisfactory filter for this purpose. Finally, it is noted that the research though reflected in this paper might be used for studying other topics, such as those in [32–35].

6. Conclusions

We have presented a general approach for approximating ideal filters from a view of fractional-order systems. This approach is based on fractional calculus. The theoretical significance of the present approach is that the Paley and Wiener criterion might be no longer a necessary condition for designing physically realizable ideal filters. We have showed a case that can be used for designing ideal filters for suppressing 50-Hz interference in ECG signals.

Appendix

The fractional derivative of Caputo type of a function is defined by where is the Gamma function [19]. For simplicity, we write by . Without generality losing, we take a system of second-order as a case:

There are two types of fractional-order systems based on (A.2). One is given by (see [36]) Another is expressed by see, for example, [37, 38].

Denote the impulse response function of (A.5) by . Then, where is the Dirac- function.

Denote the Fourier transform of by . Then, we have Therefore, if one denotes the frequency response of (A.2) by , which is a system of 2-order, then, The expression (A.8) gives the explanation of in fractional-order systems discussed in the body text of the paper.

Acknowledgment

This work was supported in part by the 973 Plan under the Project no. 2011CB302802, by the National Natural Science Foundation of China under the Project Grant nos. 61070214 and 60873264.

References

Z. M. Hussain, A. Z. Sadik, and P. O'Shea, Digital Signal Processing: An Introduction with MATLAB and Applications, Springer, New York, NY, USA, 2011.

S. S. Bhattacharyya, R. Leupers, and J. Takala, Eds., Handbook of Signal Processing Systems, Springer, New York, NY, USA, 2010.

P. Fieguth, Statistical Image Processing and Multidimensional Modeling, Springer, New York, NY, USA, 2011.

J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedure, John Wiley & Sons, City, State, USA, 3rd edition, 2000.

R. M. Gray and L. D. Davisson, Introduction to Statistical Signal Processing, Cambridge University Press, Cambridge, Mass, USA, 2004.

M. Li, “Generation of teletraffic of generalized Cauchy type,” Physica Scripta, vol. 81, no. 2, Article ID 025007, 2010.

L. Wanhammar, Analog Filters Using MATLAB, Springer, New York, NY, USA, 2009.

S. K. Mitra and J. F. Kaiser, Handbook for Digital Signal Processing, John Wiley & Sons, New York, NY, USA, 1993.

J. van de Vegte, Fundamentals of Digital Signal Processing, Prentice Hall, New York, NY, USA, 2003.

R. C. Dorf and R. H. Bishop, Modern Control Systems, Prentice Hall, New York, NY, USA, 9th edition, 2002.

M. Li, “Fractal time series—a tutorial review,” Mathematical Problems in Engineering, vol. 2010, Article ID 157264, 26 pages, 2010.

A. Papoulis, The Fourier Integral and Its Applications, McGraw-Hill, New York, NY, USA, 1965.

R. C. Paley and N. Wiener, Fourier Transforms in the Complex Domain, American Mathematical Society Colloquium Publication, New York, NY, USA, 19th edition, 1934.

H. Y.-F. Lam, Analog and Digital Filters: Design and Realization, Prentice-Hall, New York, NY, USA, 1979.

R. W. Daniels, Approximation Methods for Electronic Filter Design, McGraw-Hill, New York, NY, USA, 1974.

M. D. Lutovac, D. V. Tosic, and B. L. Evans, Filter Design for Signal Processing Using MATLAB© and Mathematica©, Prentice Hall, Upper Saddle River, NJ, USA, 2001.

G. Bianchi and R. Sorrentino, Electronic Filter Simulation & Design, McGraw-Hill Professional, New York, NY, USA, 2007.

S. Butterworth, “On the theory of filter amplifiers,” Wireless Engineer, vol. 7, pp. 536–541, 1930.

I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.

M. D. Ortigueira, “An introduction to the fractional continuous-time linear systems: the 21st century systems,” IEEE Circuits and Systems Magazine, vol. 8, no. 3, Article ID 4609961, pp. 19–26, 2008.

J. A. Tenreiro MacHado, M. F. Silva, R. S. Barbosa et al., “Some applications of fractional calculus in engineering,” Mathematical Problems in Engineering, vol. 2010, Article ID 639801, 34 pages, 2010.

S. C. Lim, M. Li, and L. P. Teo, “Langevin equation with two fractional orders,” Physics Letters A, vol. 372, no. 42, pp. 6309–6320, 2008.

Y. Q. Chen and K. L. Moore, “Discretization schemes for fractional-order differentiators and integrators,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 49, no. 3, pp. 363–367, 2002.

X. Zhang, “Maxflat fractional delay IIR filter design,” IEEE Transactions on Signal Processing, vol. 57, no. 8, pp. 2950–2956, 2009.

C. A. Monje, Y.-Q. Chen, B. M. Vinagre, D. Xue, and V. Feliu, Fractional Order Systems and Controls—Fundamentals and Applications, Springer, New York, NY, USA, 2010.

G. Arfken, Mathematical Methods for Physicists, Academic Press, Orlando, Fla, USA, 3rd edition, 1985.

J. L. Talmaon, Pattern recognition of the ECG: a structured analysis, Ph.D. thesis, 1983.

C. Levkov, G. Michov, R. Ivanov, and I. K. Daskalov, “Subtraction of 50 Hz interference from the electrocardiogram,” Medical and Biological Engineering and Computing, vol. 22, no. 4, pp. 371–373, 1984.

S. M. M. Martens, M. Mischi, S. G. Oei, and J. W. M. Bergmans, “An improved adaptive power line interference canceller for electrocardiography,” IEEE Transactions on Biomedical Engineering, vol. 53, no. 11, pp. 2220–2231, 2006.

I. Dotsinsky and T. Stoyanov, “Power-line interference cancellation in ECG signals,” Biomedical Instrumentation and Technology, vol. 39, no. 2, pp. 155–162, 2005.

M. Li, “Comparative study of IIR notch filters for suppressing 60 Hz interference in electrocardiogram signals,” International Journal of Electronics and Computers, vol. 1, no. 1, pp. 7–18, 2009.

C. Cattani, “Harmonic wavelet approximation of random, fractal and high frequency signals,” Telecommunication Systems, vol. 43, no. 3-4, pp. 207–217, 2010.

C. Cattani and G. Pierro, “Complexity on acute myeloid leukemia mRNA transcript variant,” Mathematical Problems in Engineering, vol. 2011, Article ID 379873, 16 pages, 2011.

E. G. Bakhoum and C. Toma, “Specific mathematical aspects of dynamics generated by coherence functions,” Mathematical Problems in Engineering, vol. 2011, Article ID 436198, 10 pages, 2011.

E. G. Bakhoum and C. Toma, “Dynamical aspects of macroscopic and quantum transitions due to coherence function and time series events,” Mathematical Problems in Engineering, vol. 2010, Article ID 428903, 13 pages, 2010.

B. N. N. Achar, J. W. Hanneken, and T. Clarke, “Damping characteristics of a fractional oscillator,” Physica A, vol. 339, no. 3-4, pp. 311–319, 2004.

S. C. Lim and S. V. Muniandy, “Self-similar Gaussian processes for modeling anomalous diffusion,” Physical Review E, vol. 66, no. 2, Article ID 021114, pp. 021114-1–021114-14, 2002.

M. Li, S. C. Lim, and S. Y. Chen, “Exact solution of impulse response to a class of fractional oscillators and its stability,” Mathematical Problems in Engineering, vol. 2011, Article ID 657839, 9 pages, 2011.