Approximating Ideal Filters by Systems of Fractional Order

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 𝐻(𝜔) 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 𝜔_𝑐 the cutoff frequency of a filter. Then, 𝐻(𝜔_𝑐)=0 from a view of ideal filters. In this case, we present the following approximation:lim_𝑟→0|𝐻(𝜔)|^𝑟=1,𝜔≠𝜔_𝑐,0,𝜔=𝜔_𝑐, (1) where |𝐻(𝜔)| is the amplitude of 𝐻(𝜔).

An obvious advantage of the present approach is that the above always holds no matter what the concrete structure of 𝐻(𝜔) is. However, theoretically speaking, 𝐻^𝑟(𝜔) 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|𝐻(𝜔)|=1,𝜔<𝜔_𝑐0,elsewhere, (2) where 𝐻(𝜔_𝑐)=0. One says that 𝐻(𝜔) is the frequency response of an ideal highpass filter if|𝐻(𝜔)|=1,𝜔>𝜔_𝑐,0,elsewhere. (3) The ideal bandpass filter has the frequency response expressed by|𝐻(𝜔)|=1,𝜔_cl<𝜔<𝜔_ch,0,elsewhere, (4) where 𝜔_cl and 𝜔_ch are cut-off frequencies. A filter is said to be ideal band stop if its frequency response function is given by|𝐻(𝜔)|=0,𝜔_cl<𝜔<𝜔_ch,1,elsewhere. (5)

2.2. Paley and Wiener Criterion

For facilitating the discussions, we write𝐻(𝜔)=𝑅(𝜔)+𝑗𝑋(𝜔)=𝐴(𝜔)𝑒^{−𝑗𝜗(𝜔)}, (6) where 𝜗(𝜔) is the phase response of a filter. Note that the condition for 𝐹⁻¹[𝐻(𝜔)]=ℎ(𝑡) to be zero for negative 𝑡, where 𝐹⁻¹ implies the inverse of the Fourier transform, is that 𝐴(𝜔) must be square-integrable. That is,^∞_−∞𝐴²(𝜔)𝑑𝜔<∞. (7) The above implies the causality of a filter; see, for example, Papoulis [12]. A necessary and sufficient condition for 𝐴(𝜔) 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^∞_−∞|ln𝐴(𝜔)|1+𝜔²𝑑𝜔<∞. (8)

The Paley and Wiener criterion implies that ideal filters are not physically realizable because 𝐴(𝜔)=0 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 𝑛 is given by𝐴(𝜔)=11+𝜔/𝜔_𝑐^2𝑛,𝑛=1,2,…. (9)

Denote the Chebyshev polynomial of the first kind by 𝐶_𝑛(𝜔). Then,𝐶_𝑛(𝜔)=cos𝑛cos⁻¹𝜔𝜔_𝑐,|𝜔|≤𝜔_𝑐,ch𝑛ch⁻¹𝜔𝜔_𝑐,|𝜔|>𝜔_𝑐,𝑛=1,2,…. (10) The frequency response of the Chebyshev type lowpass filters for 𝜀>0 is given by𝐴(𝜔)=11+𝜀²𝐶²_𝑛(𝜔),𝑛=1,2,…. (11)

Denote the Chebyshev rational function of degree 𝑛 by 𝑅_𝑛(𝜔). Then,𝑅_𝑛(𝜔)=𝐶_𝑛𝜔−1𝜔+1,𝑛=1,2,…. (12) One of the applications of 𝑅_𝑛(𝜔) 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 𝜀>0 is given by𝐴(𝜔)=11+𝜀²𝑅²_𝑛𝜉,𝜔/𝜔_𝑐,𝑛=1,2,…, (13) where 𝜀 is the ripple factor, and 𝜉 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 by1−𝑥²𝑑²𝑦𝑑𝑥²−𝑥𝑑𝑦𝑑𝑥+𝑛²𝑦=0. (14) 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^𝑝_𝑖=0𝑎_𝑖𝑑^𝑝−𝑖𝑦(𝑡)𝑑𝑡^𝑝−𝑖=^𝑞_𝑖=0𝑏_𝑖𝑑^𝑞−𝑖𝑒(𝑡)𝑑𝑡^𝑞−𝑖, (15) where 𝑦(𝑡) is the response and 𝑒(𝑡) excitation. Denote the Fourier transforms of 𝑦(𝑡) and 𝑒(𝑡) by 𝑌(𝜔) and 𝐸(𝜔), respectively. Then, the system function is given by𝐻(𝜔)=𝑌(𝜔)𝐸(𝜔). (16) Denote |𝐻(𝜔)| by 𝐴(𝜔). Then, 𝐻(𝜔)=𝐴(𝜔)𝑒^{−𝑗𝜗(𝜔)}. This is the basic principle regarding linear filters. In this case, we say that 𝐻(𝜔) 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 by𝐻₁(𝜔)=[𝐻(𝜔)]^𝑟=[𝐴(𝜔)]^𝑟𝑒^{−𝑗𝑟𝜗(𝜔)}(𝑟>0). (17) Denote𝐻₁(𝜔)=𝐴₁(𝜔)𝑒^{−𝑗𝜗1(𝜔)}. (18) Then,𝐴₁(𝜔)=[𝐴(𝜔)]^𝑟,𝜗₁(𝜔)=𝑟𝜗(𝜔). (19) Since 𝜗₁(𝜔) is similar to 𝜗(𝜔), the key difference between 𝐻(𝜔) and 𝐻₁(𝜔) is in the aspect of amplitude response, namely, 𝐴(𝜔) and 𝐴₁(𝜔).

It can be seen from (19) thatlim_𝑟→0𝐴₁(𝜔)=lim_𝑟→0[𝐴(𝜔)]^𝑟=1for𝐴(𝜔)≠0. (20) In addition,lim_𝑟→0𝐴₁𝜔_𝑐=lim_𝑟→0𝐴𝜔_𝑐^𝑟=0if𝐴𝜔_𝑐=0. (21)

Denote 𝐵_0.7 the 3-dB bandwidth of 𝐻₁(𝜔) by𝐴₁𝑓_𝑓=𝐵0.7=0.707, (22) where 𝑓=𝜔/2𝜋is frequency. Denote 𝐵_0.1 the bandwidth for𝐴₁𝑓_𝑓=𝐵0.1=0.1. (23) Then, the rectangular coefficient defined by𝐵Rec=^0.7𝐵_0.1 (24) is always ideal for 𝐴₁(𝜔). That is,𝐵Rec=^0.7𝐵_0.1=1for𝐴₁(𝜔), (25) because of (20).

On the other hand,lim_𝑟→0𝜗₁(𝜔)=0. (26) The expression (26) implies that 𝐻₁(𝜔) always has a linear phase response.

Remark 1. Equation (25) does not relate to any concrete forms of 𝐻₁(𝜔). 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 𝐻₀(𝜔)=lim_𝑟→0[𝐻(𝜔)]^𝑟,𝐴₀(𝜔)=lim_𝑟→0[𝐴(𝜔)]^𝑟. (27) Then, 𝐴₀(𝜔) does not satisfy the Paley and Wiener criterion expressed by (8) because 𝐴₀(𝜔)=1if𝐴₀(𝜔)≠0,0if𝐴₀(𝜔)=0. (28) That is, ^∞_−∞ln𝐴₀(𝜔)1+𝜔²𝑑𝜔=∞. (29) 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 by𝐻𝑓=1+cos2𝜋𝑓𝑇−𝑗sin2𝜋𝑓𝑇2=121+exp−𝑗2𝜋𝑓𝑇,𝑗=√−1, (30) where 𝑇 is the sampling period. Figure 1 indicates 𝐴(𝑓) for 𝑇=0.01.

Figure 1 

Amplitude response of filter.

For 𝐴(𝑓)=|𝐻(𝑓)| and 𝑇=0.01, we have𝐴𝑓_𝑓=50=0. (31) Note that𝐴𝑓_𝑓=25𝐴𝑓=0.707,_𝑓=46.8=0.1, (32) Thus, the rectangular coefficient of 𝐻(𝑓) is𝐵_0.7𝐵_0.1=2546.8=0.534. (33) The rectangular coefficient of 0.534 exhibits that 𝐻(𝑓) is not a satisfactory filter in general. Nevertheless, one is able to easily modify it to be such that it is an ideal filter bylim_𝑟→0𝐻𝑓^𝑟=lim_𝑟→01+exp−𝑗2𝜋𝑓𝑇2^𝑟=1,𝑓<50,0,𝑓=50. (34)

Figure 2 shows the approximations of [𝐻(𝑓)]^𝑟 for 𝑟=0.1,0.01,0.001, 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 [𝐻(𝑓)]^𝑟 for 𝑟=0.0001, see Figure 2(d), can be regarded as an ideal filter in practice.

(a)

(b)

(c)

(d)

Figure 2 

Approximations of [𝐻(𝑓)]𝑟 for different values of 𝑟. (a) 𝑟=0.1. (b) 𝑟=0.01. (c) 𝑟=0.001. (d) 𝑟=0.0001.

The following is called a binomial series:(1+𝑥)^𝜈=^∞_𝑘=0𝜈𝑘𝑥^𝑘for|𝑥|<1, (35) where (^𝜈_𝑘)=Γ(𝜈+𝑘)/Γ(𝜈)Γ(1+𝑘) is binomial coefficient [26]. By using binomial series, (34) can be expanded bylim_𝑟→0𝐻𝑓^𝑟=lim_𝑟→01+exp−𝑗2𝜋𝑓𝑇2^𝑟=12lim^∞_{𝑟→0𝑘=0}Γ(𝑟+𝑘)Γ(𝑟)Γ().1+𝑘exp−𝑗𝑘2𝜋𝑓𝑇 (36) Therefore, in general, [𝐻(𝑓)]^𝑟 should be taken as a filter of fractional-order from a view of fractional-order systems [25]; see the Appendix for the meaning of [𝐻(𝑓)]^𝑟 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 𝐻(𝑓) 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 𝑓(𝑡) is defined by𝐷^𝜈_𝑡𝑓(𝑡)=1Γ(𝑛−𝜈)^𝑡₀𝑓^(𝑛)(𝑢)𝑑𝑢(𝑡−𝑢)^{𝜈−𝑛+1},𝑛−1≤𝛼≤𝑛, (A.1) where Γ is the Gamma function [19]. For simplicity, we write ₀𝐷^𝑣_𝑡 by 𝐷^𝑣. Without generality losing, we take a system of second-order as a case:𝑑²𝑑𝑡²+𝜔²₀𝑥(𝑡)=𝑒(𝑡),𝜔₀>0. (A.2)

There are two types of fractional-order systems based on (A.2). One is given by (see [36])𝑑^2−𝜀𝑑𝑡^2−𝜀+𝜔²₀𝑥(𝑡)=𝑒(𝑡),0<𝜀<1. (A.3) Another is expressed by𝑑²𝑑𝑡²+𝜔²₀^𝛽𝑥(𝑡)=𝑒(𝑡),𝛽>0; (A.4) see, for example, [37, 38].

Denote the impulse response function of (A.5) by ℎ(𝑡). Then,𝐷²+𝜔²₀^𝛽ℎ(𝑡)=𝛿(𝑡), (A.5) where 𝛿(𝑡) is the Dirac-𝛿 function.

Denote the Fourier transform of ℎ(𝑡) by 𝐻(𝜔). Then, we have𝐻(𝜔)=1𝜔²+𝜔²₀^𝛽. (A.6) Therefore, if one denotes the frequency response of (A.2) by 𝐻₀(𝜔), which is a system of 2-order,𝐻₀(𝜔)=1𝜔²+𝜔²₀, (A.7) then,𝐻(𝜔)=𝐻₀(𝜔)^𝛽. (A.8) The expression (A.8) gives the explanation of [𝐻(𝑓)]^𝑟 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.