VFO FIR differintegrator, pure differentiator and pure integrator examples

The design example 1 deals with a VFO FIR differintegrator with N =40, M =5, ω_s =0.05π,

f 0.95

ω = π , p_s= −0.5, p_f =0.5 and ^W

( )

^{ω =1}. Figs. 5-2(a) and (b) display the variable magnitude response and the absolute error of variable frequency response, respectively. For example 2, a pure VFO differentiator is designed with N =30, M =6, ω_s =0, ω_f =0.9π, , and . The variable magnitude response and the absolute error of variable frequency response are shown in Figs. 5-3(a) and (b). Then, consider the design of a VFO integrator

with , ,

Similarly, Fig. 5-4(a) presents the variable magnitude response and Fig. 5-4(b) displays the absolute error of variable frequency response. Notice that, this technique can also be applied when is odd.

N

(a)

(b)

Fig. 5-2 Design of a VFO differintegrator with

f 0.95

ω = π, p_s = −0.5, and p_f =0.5 (a) Variable magnitude response (b) Absolute error of variable frequency response.

, , M =5 , ω_s =0.05π 40

N =

(b)

Fig. 5-3 Design of a VFO differentiator with N =30, M =6, ω_s =0, ω_f =0.9π, 1p_s = , (a)

and 2p_f = (a) Variable magnitude response (b) Absolute e y respo

rror of variable frequenc nse.

(a)

(b)

Fig. 5-4 Design of a VFO integrator with N =60, M =6 , ω_s =0.05π , ω_f =0.9π ,

s 1.5

p = − , and p_f = −0.5 (a) Variable magnitude resp nse (b) bsolute erro of variableo A r frequency response.

5.4 Conclusions

In this section, the design of the variable fractional-order FIR differintegrators is summarized. The technique of Weighted Least-Squares Approach was proposed for designing the VFO FIR differintegrators. Besides, the symmetric and antisymmetric characteristics were also applied to the coefficients of the VFO FIR filters. As a result, not only even N but also odd N can be used by this method. Finally, some design examples, including a VFO FIR differintegrator, a pure VFO differentiators and a pure VFO integrator, were presented to demonstrate the design effectiveness.

Chapter 6

Design of Variable Fractional-Order IIR Differintegrators

6.1 Introduction

In this chapter, the design of the VFO IIR differintegrators based on IIR-typed Farrow structure is pr

zed as follows: problem formulation is derived step by step in section 6.2.

Se

6.2 Problem Formulation

For designing the VFO differintegrator, the desired response is given by

oposed. For designing the IIR digital filters, the issue of the stability is significant concerned [39]-[40]. By incorporating a constrained function into the objective error function, the stability of the VFO IIR digital filters can be overcome. To achieve the results which satisfy the stable condition and simplify the calculations, a quadratic method is successfully used to deal with this highly nonlinear problem [41].

This chapter is organi

ction 6.3 gives three examples, containing a VFO IIR differintegrator, a pure VFO IIR differentiator, and a pure VFO IIR integrator. At last, the conclusions are summarized in section 6.4.

(

^,

) ( )

^ˆ

(

^,

)

ω > , it is a VFO differintegrator. The subscripts _s and _f denotes the start points and final of the adjustable region and the designed band.

Let

where means a sign function.

oxima e used transfer function of VFO IIR filter is

( )

sgn .

To appr te the desired response, th

( ) ( )

ubstituting (6.4a) and (6.4b) into (6.3) can yield

( ) ( )

( ) ( )

otice that, (6.5) can be implemented by IIR-typed Farrow structure as shown in Fig. 6-1.

( ) ( )

Fig. 6-1 IIR-typed Farrow structure

As a result, the frequency response of the desired filter can be formulated into

( ) ⁽ ₍ ⁾ ₎

here the superscript ^T denotes a transpose operator, w

( )

^{1, 0 ,...,}

(

a^{, 0 ,...,}

) (

^1,

)

^,...,

(

a^,

)

^{T (6.8 )}

Thus, the approximation error function can be gotten by

( ) ( ) ( )

oncerning the stability of the VFO IIR digital filter, it has been shown in [41] that C

of

∑

ust be small enough to satisfy the stable condition. The con rained function is def

( ) ( )

herefore, the objective error function of the VFO IIR digital filter can be given by T

li or function is expressed as

To simp fy the calculation, the objective err

(

,

)

,

(

,

)

,

( )

(6.15 ) Then, (6.13) becomes

( ) ( ) ( )

Additionally, the integrand of the constrained function can be formulated into

( )

²

Significantly, the constrained function would become

( )

So the original nonlinear problem in (6.12) can be converted into a quadratic problem, and the objective error function is indicated as

( )

,

( )

,

( )

Furthermore, by differentiating with respect to , and setting the result to zero, the solution can be obtained as below:

( )

k

6.3 VFO FIR differintegrator, pure differentiator and pure integrator examples

Three examples are designed in this section. For example 1, it deals with a VFO IIR differintegrator with I =19, N_a =11, N_b =29 , M =5, ω_s =0.05π , ω_f =0.95π , 0.5p_s = − and , and shows the results in Figs. 6-2(a) and (b), which are the magnitude response and absolute error of variable frequency response. A pure VFO IIR differentiator with

f 0.5 presents the magnitude response of a pure VFO IIR differentiator. Besides, Fig. 6-3(b) displays the absolute error of variable frequency response. Then, consider the design of a pure VFO IIR integrator with I =28, N_a =15, N_b =45, 4M = , ω_s =0.05π , ω_f =0.9π , , and in the example 3. Similarly, the magnitude response and the absolute error of variable frequency response of a pure VFO IIR integrator are illustrated in Figs. 6-4(a) and (b).

s 1.5 p = −

= −0.5 pf

(a)

(b)

Fig. 6-2 Design of a N_a =11, N_b =29, M =5, ω_s =0.05π, ω_f =0.95π, ,

, and

s 0.5 p = −

f 0.5

p = α =1.77827294×10⁻⁵ I =19 VFO IIR differintegrator. (a) Magnitude response (b) Absolute error of variable frequency response.

(a)

(b)

(b) Absolute error of variable frequency response.

Fig. 6-3 Design of a N_a =7, N_b =23, M =3, ω_s =0, ω_f =0.9π , , p_s =1 p_f = , 2 and VFO IIR differentiator (a) Magnitude response 1.77827294 10 4

α = × ⁻ I =14

(a)

(b)

Fig. 6-4 Design of a N_a =15, N_b =45, 4M = , ω_s =0.05π , ω_f =0.9π , ,

, and

s 1.5 p = −

f 0.5

p = − α =5.62341325×10⁻² I =28 VFO IIR integrator (a) Magnitude response (b) Absolute error of variable frequency response.

6.4 Conclusions

The design of the VFO IIR differintegrator, including the pure VFO IIR differentiator and the pure VFO IIR integrator, is summarized in this section. To implement it, IIR-typed Farrow structure was proposed. A quadratic method was used to overcome the highly nonlinear problem of the objective error function and simplify the calculations. However, the relative weighting constant α was different in all design examples because it must be properly chosen in order to ensure the stability of the designed filters. To show the performance, three examples were presented. In these three examples, the frequency ω and the variable parameter p were sampled similarly, which were and

(

, respectively. Finally, the results of this method are illustrated in Figs. 6-2, 6-3, and 6-4.

(

^{ωf −ωs}

)

^{/ 200 pf − ps}

)

^{/ 200}

Chapter 7

Conclusions and Future works

In this thesis, it is focused on investigating the various designs of variable digital filters. These designs are all discussed in detail. First, the research topic is concentrated on the variable fractional-delay (VFD) filters. The VFD FIR, VFD IIR, and VFD Allpass filters are designed in chapters 2, 3, and 4. Weighted Least-Squares Approach (WLS) has been successfully used to achieve the desired results, including the magnitude response, the group-delay response, the absolute error of variable frequency response, and the absolute group-delay error. As for the IIR and Allpass filters, they are more difficult to realize than the FIR filters because their stability problem must be considerable attention. Fortunately, the desired VFD filter systems can be implemented by the Farrow structure.

Second, the variable fractional-order (VFO) differintegrators are also concerned. The VFO FIR differintegrators and the VFO IIR differintegrators are proposed to design in chapters 5 and 6, respectively. Besides, both the pure VFO differentiator and the pure VFO integrator can be realized by choosing the proper range of the parameter p . Similarly, the results can be obtained to demonstrate the performance as well. In addition, the VFO IIR differintegrators also must satisfy the stable condition. In the design procedure of this thesis, many techniques are applied to achieve the desired goals. Binomial Series Expansion and Taylor Series Expansion helpfully evaluate the related vectors or matrices in closed-forms. The symmetric or antisymmetric relationships of the filter coefficients are also widely used. When the highly nonlinear formulations occur, quadratic method can overcome them to simplify the calculations and complexity.

In the future, the research of the variable digital filters will be more developed due to their wide applications. Based on many past achievements in regard to this aspect, more and more methods and criteria will appear. So, this issue is still worthy of further study.

Reference

[1] S.-C. Pei and C.-C. Tseng, “A comb filter design using fractional-sample delay”, IEEE Trans.Circuits Syst. I, Analog Digit. Signal Process., vol. 45, pp. 649-653, June 1998.

[2] R. Sobot, S. Stapleton and M. Syrzycki, “Tunable continuous-time bandpass ∑ Δ modulators with fractional delays”, IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 53, pp. 264-273, Feb. 2006.

[3] K.-J. Cho, J.-S. Park, B.-K. Kim, J.-G. Chung and K. K. Parhi, “Design of a sample-rate converter from CD to DAT using fractional delay allpass filter”, IEEE Trans. Circuits Syst.

II, Exp. Brief, vol. 54, pp. 19-23, Jan. 2007.

[4] H.-M. Lehtonen, V. Valimaki and T. I. Laakso, “Canceling and selecting partials from musical tones using fractional-delay filters”, Computer Music Journal, vol. 32, pp. 43-56, Feb. 2008.

[5] C. W. Farrow, “A continuously variable digital delay element”, in Proc. ISCAS, 1998, pp. 2641-2645.

[6] C.-C. Tseng, “Design and applications of variable fractional order differentiator”, in IEEE Asia-Pacific Conference on Circuits and Systems, Dec. 2004, pp. 405-408.

[7] Y.Q. Chen and B.M. Vinagre, “A new IIR-type digital fractional order differentiator”, Signal Processing, vol. 83, pp. 2359-2365, 2003.

[8] P. Ostalczyk, “Fundamental properties of the fractional-order discrete-time integrator”, Signal Processing, vol.83, pp. 2367-2376, 2003.

[9] Y.Q. Chen and K.L. Moore, “Discretization schemes for fractional-order differentiators and integrators”, IEEE Trans. Circuits Syst. I, vol. 49, pp. 363-367, Mar. 2002.

[10] C.-C. Tseng, “Design of variable and adaptive fractional order differentiators”, Signal Processing, vol. 86, pp. 2554-2566, 2006.

[11] H. Zhao, G. Qin, L. Yao and J. Yu, “Design of fractional order digital FIR differentiators using frequency response approximation”, in Proceedings of the 2005 International Conference on Communications, Circuits and Systems, May 2005, pp. 1318-1321.

[12] G. Maione, “Concerning continued fraction representation of nonlinear order digital differentiators”, IEEE Signal Process. Lett., vol. 13, pp. 725-728, Dec. 2006.

[13] D. Xue, C. Zhao and Y.Q. Chen, “A modified approximation method of fractional order system”, in Proceeding of the 2006 IEEE International Conference on Mechatronics and Automation, June 2006, pp. 1043-1048.

[14] T.-B. Deng, “Symmetric structures for odd-order maximally flat and weighted-least-squares variable fractional-delay filters”, IEEE Trans. Circuit Syst. I. Reg. Papers, vol.54, no.12, pp.

2718-2732, Dec. 2007.

[15] J.-C. Liu and S.-J. You, “Weighted least squares near-equiripple approximation of variable fractional delay FIR filters”, IET Signal Process., vol.1, no.2, pp. 66-72, 2007.

[16] C.Y. Chi and Y.T. Kou, “A new self-initiated optimum WLS approximation method for the design of linear phase FIR digital filters”, in Proc. ISCAS, 1991, pp. 168-171.

[17] T.-B. Deng, “High-resolution image interpolation using two-dimensional Lagrange-type variable fractional-delay filter,” in IEICE Technical Report, Hirosaki, Japan, Mar. 2005, vol.

SIS2004-60, pp. 27–30.

[18] C.-C. Tseng, “Design of variable fractional delay FIR filters using symmetry”, in Proc. 2004 IEEE Int. Symp. Circuits and Systems, Vancouver, Canada, May 23–26, 2004, vol. III, pp.

477–480.

[19] M. O. Ahmad and J.-D. Wang, “An analytical least square solution to the design problem of two-dimensional FIR filters with quadrantally symmetric or antisymmetric frequency response”, IEEE Trans. Circuits Syst., vol. 36, no. 7, pp. 968-979, Jul. 1989.

[20] T.-B. Deng and Y. Lian, “Weighted-least-squares design of variable fractional-delay FIR filters using coefficient symmetry”, IEEE Trans. Signal Process., vol. 54, no. 8, pp . 3023-3038, Aug. 2006.

[21] H. Zhao, H.K. Kwan, L. Wan and L. Nie, “Design of 1-D stable variable fractional delay IIR filters using finite impulse response fitting”, in Proc. ICCCAS, 2006, pp. 201-205.

[22] H. Zhao and H.K. Kwan, “Design of 1-D stable variable fractional delay IIR filters”, IEEE Trans. Circuit Syst. II, Exp. Briefs, vol. 54, pp. 86-90, Jan. 2007.

[23] K. Rajamani and Y.-S. Lai, “A novel method for designing allpass digital filters”, IEEE Signal Processing Letters, vol. 6, pp. 207-209, Aug. 1999.

[24] W.R. Lee, L. Caccetta and V. Rehbock, “Optimal design of all-pass variable fractional-delay digital filters”, IEEE Transactions on Circuits Systems I, Regular Papers, vol. 55, pp.

1248-1256, May 2008.

[25] J.-J. Shyu and S.-C. Pei, “A new approach to the design of complex allpass IIR digital filters”, Signal Processing, vol.40, pp. 207-215, Sep. 1994.

[26] S.-C. Pei and P.-H. Wang “Closed-form design of all-pass fractional delay filters”, IEEE Signal Processing Letters, vol. 11 pp. 788-791, Oct. 2004.

[27] B. Mbodje and G. Montseny, “Boundary fractional derivative control of the wave equation”, IEEE Trans. Automat. Control, vol. 40, pp. 378-382, Feb. 1995.

[28] N. Engheta, “On fractional calculus and fractional multipoles in electromagnetism”, IEEE Trans. Antennas Propag., vol. 44, pp.554-556, Apr. 1996.

[29] R. Hilfer (Ed.), “Application of Fractional Calculus in Physics”, World Scientific, River Edge, NJ, 2000.

[30] B.M. Vinagre, I. podlubny, A. Hernandez and V. Feliu, “Some approximation of fractional order operators used in control theory and applications”, J.Fractional Calculus Appl. Anal., vol. 4, pp.47-66, 2001.

[31] B.Mathieu, P. Melchior, A. Ousaloup and C. Ceyral, “Fractional differentiation for edge detection”, Signal Processing, vol. 83, pp. 2421-2432, 2003.

[32] C.-C. Tseng, “Design of fractional order digital FIR differentiators”, IEEE Signal Process.

Lett., vol. 8, pp. 77-79, Mar. 2001.

[33] S. Samadi, M.O. Ahmad and M.N.S. Swamy, “Exact fractional-order differentiators for polynomial signals”, IEEE Signal Process. Lett., vol. 11, pp. 529-532, June, 2004.

[34] C.-C. Tseng, “Improved design of digital fractional-order differentiators using fractional sample delay”, IEEE Trans. Circuits Systems I, vol. 53, pp. 193-203, Jan. 2006.

[35] C.-C. Tseng, “Design of FIR and IIR fractional order Simpson digital integrators”, Signal Processing, vol. 87, pp. 1045-1057, 2007.

[36] C.K.S. Pun, S.C. Chan, K.S. Yeung and K.L. Ho, “On the design and implementation of FIR and IIR digital filters with variable frequency characteristics”, IEEE Trans. Circuits Systems II, vol. 49, pp. 689-703, Nov. 2002.

[37] T.-B. Deng, “Design of arbitrary-phase variable digital filters using SVD-based vector-array decomposition”, IEEE Trans. Circuits Systems I, vol. 52, pp. 148-167, Jan. 2005.

[38] T.I. Laakso, V. Valimaki, M. Karjalainen and U.K. Laine, “Splitting the unit delay: tools for fractional delay filter design”, IEEE Signal Process. Mag., vol.13, pp. 30-60, 1996.

[39] M. Makundi, T.I. Laakso and V. Valimaki, “Efficient tunable IIR and allpass filter structures”, Electronics Lett., vol. 37, pp. 344-345, June 2001.

[40] H. Zhao and H.K. Kwan, “Design of 1-D stable variable fractional delay IIR filters”, in Proc.

IEEE ISPACS, 2005, pp. 517-520.

[41] K. M. Tsui, S. C. Chan and H.K. Kwan, “A new method for designing causal stable IIR variable fractional delay filters”, IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 54, pp.

999-1003, Nov. 2007.

[42] W. -S. Lu and T.-B. Deng, “An improved weighted least-squares design for variable fractional delay FIR filters”, IEEE Trans. Circuits Syst. I, Analog Digit. Signal Process., vol. 46, no. 8, pp. 1035-1040, Aug. 1999.

[43] T.-B. Deng, “Discretization-free design of variable fractional-delay FIR filters”, IEEE Trans.

Circuits Syst. II, Analog Digit. Signal Process., vol. 48, no.6, pp. 637-644, Jun. 2001.

[44] C. K. S. Pun, Y. C. Wu, S. C. Chan and K. L. Ho, “On the design and efficient implementation of the farrow structure”, IEEE Signal Process. Lett., vol. 10, no. 7, pp. 189-192, Jul. 2003.

[45] H. Zhao and J. Yu, “A simple and efficient design of variable fractional-delay FIR filters”, IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 53, no. 2, pp. 157-160, Feb. 2006.

[46] T. -B. Deng and Y. Lian, “Weighted-least-squares design of variable fractional-delay FIR filters using coefficient symmetry”, IEEE Trans. Signal Process., vol. 54, no. 8, pp. 3023- 3038, Aug. 2006.

[47] J. T. Olkkonen and H. Olkkonen, “Fractional delay filter based on the B-spline transform”, IEEE Signal Process. Lett., vol. 14, no. 2, pp. 97-100, Feb. 2007.

[48] J. T. Olkkonen and H. Olkkonen, “Fractional time-shift B-spline filter”, IEEE Signal Process.

Lett., vol. 14, no. 10, pp. 688-691, Oct. 2007.

[49] E. Hermanowicz and H. Johansoon, “On designing minimax adjustable wideband fractional delay FIR filters using two-rate approach”, in Proc. Eur. Conf. Circuit Theory and Design, 2005, pp. III/437-III/440.

[50] P. Lowenborg and H. Johanson, “Minimax design of adjustable -bandwidth linear-phase FIR filters”, IEEE Trans. Circuits Syst. I, Reg. Paper, vol. 53, no. 2, pp. 431-439, Feb. 2006.

[51] H. H. Dam, A. Cantoni, K. L. Teo, and S. Norholm, “Variable digital filter with least-square criterion and peak gain constraints”, IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 54, no. 1, pp. 24-28, Jan. 2007.

[52] S. -C. Pei and J. -J. Shyu, “Efficient fast design of three- and higher-dimensional FIR digital filters by analytical least squares method”, IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 43, no. 4, pp. 339-344, Apr. 1996.

[53] J.-J. Shyu, S. -C. Pei, C. -H. Chan and Y. -D. Hung, “Minimax design of variable fractional delay FIR digital filters by iterative weighted least-squares approach”, IEEE Signal Process.

Lett., vol. 15, pp. 693-696, 2008.

[54] C. -C. Tseng, “Design of 1-D and 2-D variable fractional delay allpass filters using weighted least-squares method”, IEEE Trans. Ciecuits Syst. I, Fundam. Theory Appl., vol. 49, no. 10, pp. 1413-1422, 2002.

[55] J. Yli-Kaakinen and T. Saramaki, “An algorithm for the optimization of adjustable fractional delay allpass filters”, in Proc. ISCAS, 2004, pp. III. 153-III. 156.

[56] T. -B. Deng, “Noniterative WLS design of allpass variable fractional-delay digital filters”, IEEE Trans. Circuits Syst. I, Reg. Paper, vol. 55, no. 5, pp. 1248-1256, 2008.

[57] H. K. Kwan and A. Jiang, “Design of IIR variable fractional delay digital filters”, in Proc.

ISCAS, 2007, pp. 2714-2717.

[58] H. Johansson and P. Lowenborg, “On the design of adjustable fractional delay FIR filters”, IEEE Trans. Circuits Syst. II, Analog and Digital Signal Process., vol. 50, pp. 164-169, Apr. 2003.

[59] J.-J. Shyu, S. -C. Pei, C. -H. Chan and Y. -D. Huang, “A generalized approach to the design of variable fractional-delay FIR digital filters”, Signal Processing, vol. 88, pp. 1428-1435, 2008.

[60] M. Makundi, V. Valimaki and T. I. Laakso, “Closed-form design of tunable fractional-delay allpass filter structures”, in Proceedings of the 2001 IEEE International Symposium on Circuits and Systems, May, 2001, pp. 434-437.

[61] C.-C. Tseng, “Eigenfilter approach for the design of variable fractional delay FIR and allpass filters”, IEE Proceedings of the Visual Image Signal Processing, vol. 149, pp. 297-303, May, 2002.

[62] T. -B. Deng, “Noniterative WLS design of allpass variable fractional-delay digital filters”, IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 53, pp. 358-371, Feb. 2006.

[63] H. Hacihabiboglu, B. Gunel and A. M. Kondoz, “Analysis of root displacement interpolation method for tunable allpass fractional-delay filters”, IEEE Trans. Signal Processing, vol. 55, pp. 4896-4906, Oct. 2007.

[64] Z. Jing, “A new method for digital all-pass filter design”, IEEE Transactions on Acoustic, Speech and Signal Processing, vol. 35, pp. 1557-1564, Nov. 1987.

[65] M. Lang and T. I. Laakso, “Simple and robust method for the design of allpass filters using least-squares phase error criterion”, IEEE Processing, vol. 41, pp. 40-48, Jan. 1994.

[66] T. Q. Nguyen, T. I. Laakso and R. D. Koilpillai, “Eigenfilter approach for the design of allpass filters approximating a given phase response”, IEEE Trans. Signal Processing, vol. 42, pp.

2257-2263, Sep. 1994.

[67] J. -J. Shyu, S. -C. Pei and C. -H. Chan, “An iterative method for the design of variable fractional-order FIR differintegrators”, Signal Processing, vol. 89, pp. 320-327, 2009.

[68] Y. Q. Chen, B. M. Vinagre and I. Podlubny, “Continued fraction expansion approaches to discretizing fractional order derivatives–an expository review”, Nonlinear Dynamics, vol.

38, pp. 155-170, 2004.

[69] H. H. Dam, A. Cantoni, K. L. Teo and S. Nordholm, “Variable digital filter with least-square criterion and peak gain constraints”, IEEE Trans. Circuits Systems II, vol. 54, pp. 24-28, Jan. 2007.

[70] C. -C. Tseng, S. -C. Pei and S. -C. Hsia, “Computation of fractional derivatives using Fourier transform and digital FIR differentiators”, Signal Processing, vol. 80, pp. 151-159, 2000.

[71] X. Yuan, Y. Wei and J. Yu, “Sinc function and generalized digital fractional differentiators”, in Proceedings of the 2004 International Conference on Communications, Circuits and Systems, June 2004, pp. 1414-1418.

[72] R.S. Barbosa, J.A.T. Machado and M.F. Silva, “Time domain design of fractional differintegrators using least-squares”, Signal Processing, vol. 86, pp.2567-2581, 2006.

在文檔中 可調式數位濾波器之設計 (頁 43-0)