Group Delay of Fractional n+-Order Bessel Filters

Andro Ćoza*, Vesna Županović*, Domagoj Vlah* and Dražen Jurišić* *University of Zagreb/Faculty of electrical engineering and computing, Unska 3, Zagreb, 10000, Croatia

Phone: (+385 1) 612 99 49; Fax: (+385 1) 612 96 52 e-mail: drazen.jurisic@fer.hr

Abstract - In this paper we present the realization of

fractional-order analog filter with Bessel approximation, having linear phase response, which is also usually represented by a constant group delay. There are many applications such as analogue video signal processing, radar and sonar receivers, hard disk drive read channel applications, analog front end of biomedical signal processing, where linear phase response is desirable. Optimally designed Bessel filter can provide better transient response in the passband, it reduces overshoot, ringing and provides minimal phase distortion. In this work we research Bessel approximation with the non-integer order n+; n is integer number and 0<<1. The influence of non-integer filter order to the value of the constant group delay of the filter is presented. It is shown that the group delay value can be additionally tuned by changing the parameter .

Keywords - active filters; analog circuits; circuit synthesis; filtering theory; Bessel approximation; linear-phase response; group delay; fractional-order filters.

I. INTRODUCTION In recent times fractional-order calculus has gained

significant attention by many researchers in modelling biological tissues, describing processes in control theory, materials theory, diffusion theory, robotics, signal processing, oscillators, filters, viscoelasticity, etc. [1][2].

This paper researches advantages of applying fractional order on Bessel approximations (see also [3]). Most of the research on fractional-order filters has been presented for Butterworth and Chebyshev filters [4]−[6]. It is demonstrated that by including of parameter , the group delay can be fine-tuned.

In definition of fractional derivative, two approaches exist: 1) fractional differentiations and integrations are limits of finite differentiations and integrations, and 2) generalization of convolution type representation of repeated integration [1][2].

In the second approach the most popular is the Caputo fractional derivative, defined by:

1 ( )( )

(1 ) ( )

t

a

d fD f t ddt t

=

− − ()

where 0<<1. (gamma) function is generalized factorial ( 1) !p p + = , where p is integer or non-integer number.

The well-known Euler definition of is:

1

0

( ) ; Re 0x pp e t dx p

− − = ()

Fractional derivative (1) by Caputo is useful in the circuit and filter theory because it can use standard (integer-order) initial conditions.

It possesses a Laplace transform defined by

1

1

0( ) ( ) (0)

nk k

a tk

L D f t s F s s D f−

− −

=

= − ()

When the initial conditions are zero, this becomes

( ) ( )L D f t s F s = ()

simplifying substantially analysis in the frequency domain. In the sequel we will use Laplace transformation in the analysis of fractional-order systems.

II. BESSEL FILTERS In this Section we briefly repeat the maximally-flat

constant-phase approximation, which leads to Bessel polynomials [7]. We need this, to compare the results with the new fractional-order (FO) approximation of linear phase to maximally-flat integer-order (IO) approximation.

In approximations of ideal frequency characteristic of Butterworth or Chebyshev filters the emphasis was put on the realization of amplitude-frequency (A-F) characteristic. On the shape of phase-frequency characteristic (P-F) was stated no condition. In real world, it is very often needed to realize electrical systems having linear phase in the pass band. Linear phase is equivalent to the constant group delay. Transfer function, which has constant amplitude characteristic and linear phase in the whole frequency range is defined by:

0( ) sTH s K e−= () This work was fully supported by the Croatian Science Foundation

under project (IP-2016-06-1307) "Fractional analog and mixed systems for signal processing".

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For amplitude-frequency characteristic we have to include s=j into (5) and after application of natural logarithm we have:

0ln ( ) ln ( ) ( )H j K j T j = − = + ()

providing linear phase-frequency characteristic as:

0( ) T = − ()

Group time delay is then defined as:

0( )( )g

dT Td

= − =

()

and it is constant. Note that () in (6) is in Nepers.

In literature (e.g. [7]) it is well-known that the above characteristics is satisfied by Bessel polynomials. They are given by

0

12

23 2

34 3 2

4

( ) 1( ) 1

( ) 3 3

( ) 6 15 15

( ) 10 45 105 105

B sB s sB s s sB s s s s

B s s s s s

=

= +

= + +

= + + +

= + + + +

()

and so on. They can be calculated from recursive equation (starting from known B0 and B1):

21 2( ) (2 1) ( ) ( )n n nB s n B s s B s− −= − + ()

If the Bessel filter transfer function (5) must have the d.c. gain H(0)=1 then, for example, using B3(s) from (9) for the third-order filter it is given by (K=15):

3 3 2

15( )6 15 15

H ss s s

=+ + +

()

This transfer function has group delay T0 equal to unity. If we want to normalize the Bessel filters, that they have magnitudes comparable to those of, for example, Butterworth filters, they must have gain of 1/2 at =1. For this to be accomplished, the group delays T0 differ dependent on the filter order n, and are given in Table I.

TABLE I. CONSTANT T0 VALUES

n 1 2 3 4 5 6 7

T0 1.0 1.362 1.756 2.114 2.424 2.699 2.947

The denormalization of the transfer function is performed by the frequency transformation s→s/0, where the denormalization frequency is 0=1/T0. If it is applied, for example, on our third-order Bessel characteristic, having T03=1.756, its transfer function is given by:

3 3 203 03 03

3 2

15( )( ) 6 ( ) 15 ( ) 15

2.78449 .3.4227 4.88212 2.78449

H sT s T s T s

s s s

=+ + +

=+ + +

()

Phase characteristics is given by

3

3 2

4.86636( ) arctan ,2.77179 3.41749

− = −

− ()

while for the group delay we have:

2 4

3 2 4 6

7.68284 4.73629 1.924654( ) 1.7567.68284 4.73629 1.924654gT + +

= + + +

6

2 4 61.756 17.6828 4.7363 1.92465

= −

+ + + ()

Note that the nth-order Bessel group delay has the form:

2 2( 1)0 2 2( 1)

0 2 2( 1) 20 2 2( 1)

( )n

ngn n n n

n

a a aT T

a a a

−

−

−

−

+ + + =

+ + + +

2

0 2 2( 1) 20 2 2( 1)

1n

n n nn

Ta a a −

−

= −

+ + + +

()

By including =0 into (15) we obtain a value of the group delay Tgn(0)=T0n. This special form of (15) is also known as maximally flat approximation. Note also, although the phase characteristic in (13) is a transcendent function, the group delay is a rational function of two polynomials [7].

Amplitude-frequency characteristics, phase-frequency characteristics and group delays are shown in Figs. 1(a), (b) and (c), respectively for the Bessel functions up to 4th-order. It can be seen that the magnitude functions have all −3dB cut off frequency equal to 1 rad/s, and posses different group delays. Different group delays are visible in Fig. 1(c), and they are given in Table I.

Figure 1. Classical Bessel filters of 1st to 4th-order. (a) Amplitude-frequency characteristics, (b) Phase-frequency characteristics, (c) Group delay.

(a) (b) (c)

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It is not possible to continuously adjust the values of the group delay T0 when using integer order Bessel filters, because only discrete values are possible. In what follows it will be shown that it is possible to obtain other values of group delays in-between the values given in Table I by using fractional-order Bessel filters, only.

III. FRACTIONAL-ORDER BESSEL FILTERS The normalized transfer function of the fractional low-

pass filter of order 1+ is given by

1 1

13 2

( )FOBkH s

s k s k+

+ =

+ + ()

The low-frequency gain is k1/k2.

The 3+-order FO transfer function (which is a fourth-order when =1) is possible by combining (16), with one biquadratic transfer function, because of the stability issues [4]. The obtained transfer function is given by:

3 4 12 1

6 5 3 2

13 2 2 1

7 6 5 4 3 2

( )( )( )

.

FOBa aN sH s

D s s a s a s a s ak

s k s k s k s k s k s k

+

+

+ + +

= = + + + +

=+ + + + + +

()

The values k1 to k7 are the positive constants to be obtained numerically to approximate Bessel passband amplitude-frequency characteristic.

We apply the nonlinear least square routine from Matlab called lsqcurvefit in the frequency range from 0 to 1 to the transfer functions defined by (17) [4][5]. This function uses the trust-region-reflective algorithm with termination tolerances of the function value and the solution set in Matlab by default to 10−6. The routine lsqcurvefit is defined in Matlab as:

( )

21

_ 22

1_1

min ( , ) ( )

min ( , ) ( ) ,

jFOB INT jx

k jFOB i INT j iix

H x H

H x H

− +

− +

=

−

= − ()

where j is the order of the integer-order transfer function, k is the total number of frequency points for the comparison, and x is the filter coefficients (k1 to k7) vector. The lowest bound for the solutions vector x was set to 0.1. In our example, the order of the fractional transfer function is chosen 3.5 (with n=3 and =0.5). The obtained coefficient values of 3.5th-order FO transfer functions are given in Table II on the next page. The corresponding normalized amplitude-frequency characteristics, phase-frequency characteristics, and group time delays for 3rd-, 3.5th- and 4th-order filters are shown in Figs. 2(a) to (c), respectively. We usually have to denormalize the transfer function [e.g. (17)] to some cut-off frequency, e.g. to c and the curves in Fig. 2 to the fc=c/(2). Then we have to apply the frequency transformation s→s/c. Frequency characteristics of fractional-order filters are readily obtainable using FOMCON Matlab toolbox [8].

Besides, A-F, P-F and Tg characteristics of the FO transfer functions, such as (17), can be calculated analytically. We do it by substitution s=j into s, which is defined by:

2( ) cos sin2 2

jj e j

= = +

()

Using a short notation 3/ ( )FOBH N D H j+= = , where N means numerator and D denominator, we obtain the following expressions for the three characteristics:

1

2 2( )

Re Im

kH jD D

=+

()

Im( ) arctan

ReD

HD

= = − ()

2 2

Re Im Im Re( )

Re Img

d dD D D Dd dT

D D

− =

+ ()

Where, after some calculations, we have:

12 3 5

2 2 36 7

Re cos cos (1 )2 2

cos (2 ) cos (3 ) ,2 2

D k k k

k k

+

+ +

= + + +

− + + + +

()

13 4 5

2 37

Im sin sin (1 )2 2

sin (2 ) sin (3 ) .2 2

D k k k

k

+

+ +

= + + +

+ + + +

()

The derivations of (23) and (24) to gives:

23 71

26 5

1Re (2 ) cos2

2 (1 ) (3 ) sin ,2

d D k kd

k k

−

= − +

− + − + + +

()

23 71

24 5

1Im (2 ) sin2

(1 ) (3 ) cos .2

d D k kd

k k

−

= − +

+ + + − +

()

For the FO Bessel filter with =0.5, i.e. for its transfer function of the order 3.5, instead using FOMCON we can plot the FO curves using expressions (20) to (26). Thus, A-F characteristic in (20) is shown in Fig. 2(a) in dB. Also the P-F characteristic in (21) and Tg in (22) of the order 3.5 are in Figs. 2(b) and (c), respectively. All those FO curves are plot as a magenta dotted line.

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Figure 2. Bessel filters of 3rd, 3.5th and 4th-order. (a) Amplitude-frequency characteristics, (b) Phase-frequency characteristics, (c) Group delay.

To calculate the group delay Tg for the fractional-order Bessel filter depending on the fractional parameter we can use (22), in which we also have to substitute (23) to (26). Because, some of those expressions have in the denominator we cannot include =0, as we did above into (15), to obtain group delay Tg(0)=T0. Instead, because all transfer functions are normalized to cut-off frequency equal to 1, we can use a smaller value of =0.1 and we will obtain T03.5=1.9752 from (22). Note that the obtained value for the order 3.5 is between T03=1.756 and T04=2.114, which can also be seen in Fig. 2(c).

If we want to find the dependence of the group delay T0n+ on the value of the fractional parameter , and especially if we want to go the other way around, to find the order n+ for given T0n+ the things are going to be unsolvable in an analytical way. We also have to notice that not only because the expressions (22) to (26) are implicit, and cannot be expressed in the inverse form providing , all the coefficients ki; i=1,...,7 in (17) that are obtained by optimization are determined by the values of and n. In our case those were n=3, and =0.5, and for the new values of n+ we will obtain the new values of the constants ki. Therefore, we must form the nomograph connecting the values of T0n+ and n+, which goes beyond the scope of this paper.

Furthermore, as mentioned above, when denormalizing a filter to a cut-off frequency fc, we can calculate the real group delays T0c of the filter circuit using:

0 00 2c

c c

T TT

f= =

()

E.g., if fc=1kHz from above normalized values of T03=1.756, T03.5=1.9752 and T04=2.114 using (27) we have

03 03.5 04279.4 s; 314.3 s; 336.4 s.c c cT T T= = = ()

Stability check. For the FO system to be stable the angles of all poles in (17) must be greater than the minimum required angle / 2 9W m = (see also [4]). It is obvious from Table II that the presented filter is stable.

TABLE II. COEFFICIENT VALUES OF 3.5TH FO BESSEL FILTER

Order k1 k2 k3 k4

n+=3.5 2.9219 2.9132 0.1000 5.3243

k5 k6 k7 minW

1.2695 2.6296 2.5214 13.2411

IV. FILTER REALIZATION

A. Active-RC filter circuit

In this section we will physically realize 4th-order and 3.5th-order transfer function using active-RC filters with operational amplifiers in Pspice. Consider first Tow-Thomas circuit shown in Fig. 3.

R1

Vin

VBP

R2

R3

R4 R5

R6

V =Vout LP−VLP

C1C2

C

Figure 3. Tow Thomas circuit

If the circuit contains one fractional capacitance C, which replaces the capacitor C2, then the transfer function of the circuit is given by:

3 1 2 3

1 64

1 1 1 2 3 5

1( )

( )1( )

out

in

V s R C C R RH s

RV s R s sR C C C R R R

+

= =

+ +

()

In the case of =1, we can compare (29) to the low-pass transfer function of the second-order in general form, and choosing R5=R6 we can recognize the parameters:

311

2 2 3 41 2 2 3

1 ; ;p pRCq R k

C R R RC C R R = = = ()

If in the design we choose C1=C2=C0, R2=R3=R0, we will have the following simple design expressions:

0 1 0 4 3 00

1 ; ; .pp

R R q R R kR kRC

= = = =

()

The design parameters of the above fourth-order (integer-order) Bessel filter from [7] are given by:

11.43017; 0.521935;

1.60336; 0.80

;

18 ,553 ;pA pA A

pB pB B

q kq k

= = =

= = = ()

(a) (b) (c)

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for the two sections as in Fig. 3, denoted by A and B, and connected in cascade. We then denormalize the cut-off frequency of the filter to the frequency c=2fc; fc=1kHz. We have chosen C0=1nF, by that we have obtained the denormalization impedance R0=(cC0)−1=159.155k. Using above expressions in the design of the fourth-order Bessel filter, i.e. from (32) and using (31) we have obtained the elements given in Table III. We have chosen arbitrarily R5=R6=10k.

B. Realization of constant-phase element (CPE)

The impedance of a constant-phase element (CPE) is defined by:

1( )CPEZ s

s C

= ()

where C is the pseudo-capacitance with units [F/sec1−] and 0<α<1 is the order.

To be able to perform Pspice analysis we must replace CPE by a passive RC network as its approximation. Only a fixed approximation for a specific pseudo-capacitance with value of valid over a bandwidth (around specified center) with acceptable magnitude and phase error is possible. Known methods for approximating the variable s are Carlson’s method, Matsuda’s method, Oustaloup’s method and the continued fraction expansion (CFE) method (see also [9]−[12]). They all form a rational function of “s” as a driving point function, or immittance function (impedances and admittances are referred to as immittances). Among approximation methods, one of the most efficient is the CFE method, and will be used below.

A CFE method approximates the impedance of the fractional capacitance ZCPE(s) in (33) in the following general form:

4 3 2 2 3 4

0 1 0 2 0 3 0 4 04 3 2 2 3 4

0 4 3 0 2 0 1 0 0 0

1( )CPEP s P s P s P s P

Z sC P s P s P s P s P

+ + + +

+ + + + ()

where from [10] we have:

4 3 20

4 3 21

4 22

4 3 23

4 3 24

10 35 50 24,

4 20 40 320 384,

6 150 864,

4 20 40 320 384,

10 35 50 24.

P

PPP

P

= + + + +

= − − + + +

= − +

= − + + − +

= − + − +

()

Note that the chosen order of the approximation in (34) is four, and 0 is the center frequency around which the approximation is performed.

According to classical network theory [7] the rational function of s in (34) can be realized by a network shown in Fig. 4, which is also known as Foster I one-port.

TABLE III. ELEMENTS OF 1KHZ FOURTH-ORDER BESSEL FILTER

Section C1=C2 R1 R2=R3=R4 R5=R6

A 1.0nF 58.083k 111.28k 10.0k

B 1.0nF 79.961k 99.264k 10.0k

Ra Rb Rc Rd

Ca

Cb Cc Cd

C

Re

Figure 4. Constant-phase element simulation by Foster I realisation

The expression of the impedance Z(s) of the Foster I network in Fig. 4, is constructed from passive resistors Ri (i=a, ..., e) and capacitors Cj (j=a, ..., d). They form the following impedance function:

, , ,

1/( )

1/ ( )i

ei a b c d i i

CZ s R

s R C=

= ++

()

Furthermore, the rational function of s in (34) can be given in the form of partial fraction expansion:

4

1 1

( ) i

i

rZ s k

s p=

= +−

()

where k and ri are constant terms, pi are the poles of impedance (34).

The values of passive elements in Fig. 4 readily follow from equating the coefficients of (37) with those derived from the partial fraction expansion of Z(s) in (36). The coefficient k, residues ri and poles pi in (37) are used to calculate the passive elements using the following expressions:

1 1, ,i i ei i i

C R R kr C p

= = = ()

Passive element values for approximating a CPE with C=1nF at 1 kHz (i.e. Z=159.15k), =0.5, using the Foster I topology in Fig. 4, are given by: Ra=40.063k, Ca=526.27pF, Rb=59.906k, Cb=1.8706nF, Rc=142.89k, Cc=3.3414nF, Rd=1.1718 M, Cd=4.3683nF, Re=17.684 k. This is a well-know Foster I network synthesis algorithm.

The frequency dependence of the absolute value of the impedance Z(j) and the phase angle ()=arg[Z(j)] of the CPE are shown in Fig. 5 and 6, respectively.

Figure 5. Impedance-frequency characteristic of CPE in Fig. 4

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Figure 6. Phase-frequency characteristic of CPE in Fig. 4

C. Integer-order and fractional-order filter realization and analysis using Pspice

The integer-order (IO) realization of the fourth-order Bessel filter and fractional-order (FO) Bessel filter of the 3.5th order are analyzed using Pspice.

For the realization of IO Bessel filter the Tow-Thomas section shown in Fig. 3 is used with computed elements in Table II. For the realization of FO Bessel filter Modified-Follow-the-Leader-Feedback (MFLF) realization is used. MFLF is not presented in this paper, because it needs a lot of attention and will be described in details in a follow-up journal publication. A-F characteristics of two filters is shown in Fig. 7. Note that in the case of 4th-order filter, the roll-off at high frequencies is 80dB/decade, while for the 3.5th-order filter, with fractional-step attenuation in the stopband, the roll-off is 70dB/decade. Group delay for two realizations is shown in Fig. 8. It can be noted that the 4th-order filter has T0c=336.6s, whereas the 3.5th-order filter has T0c=313.7s. Note that those simulated values are very close to, and confirm, the calculated values (28).

Figure 7. Amplitude-frequency characteristics IO and FO Bessel filters

Figure 8. Group delay characteristics IO and FO Bessel filters

We can conclude that the fine adjustment of the group delay T0 value is impossible with IO Bessel filter design, only. The FO approach provides more degrees of freedom in the design. In doing so besides the fine-tuning of the group delay, we can keep the same amplitude frequency characteristics in the pass-band.

V. CONLUSION In this paper we present the design of maximally-flat,

fractional-order Bessel filter. We have first stated that as a consequence of normalizing the cut-off frequencies of integer-order Bessel filters to 1rad/s (−3dB magnitude), we have only discrete and fixed values of group delays. To obtain more degrees of freedom in the design, we must construct Bessel filters with fractional order. In this way we can fine-tune group delays, and it is possible to obtain more freedom in the design of theirs values, that can be between initial discrete values. To design fractional-order Bessel filters we have used the least square optimization routine in Matlab. With different group delays, we succeed to keep the same amplitude frequency characteristics in the pass-band. The group delay of the fractional-order Bessel filters is increasing with increasing order (i.e. by both increasing IO order n and FO order ).

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[8] A. Tepljakov, E. Petlenkov, and J. Belikov, “FOMCON: Fractional- order modeling and control toolbox for Matlab,” in Proc. of the 18th International Conference MIXDES 2011, Gliwice, Poland, June 16-18 2011, pp. 684–689.

[9] A. Oustaloup, F. Levron, B. Mathieu, and F. M. Nanot, “Frequency-band complex noninteger differentiator: characterization and synthesis,” IEEE Trans Circuits Syst– I: Fundam Theory App 2000; vol. 47, no. 1, pp. 25–39.

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