可調式數位濾波器之設計

全文

(1)國立高雄大學電機工程學系碩士班碩士論文. 可調式數位濾波器之設計 Design of Variable Digital Filters. 研究生：黃韻達撰指導教授：徐忠枝博士. 中華民國 97 年 7 月.

(2)

(3) 可調式數位濾波器之設計指導教授：徐忠枝博士國立高雄大學電機工程學系. 學生：黃韻達國立高雄大學電機工程學系碩士班. 摘要. 在本篇論文中，我們將對可調式數位濾波器之設計做深入的研究。此研究因為它在通訊及影像處理上有著廣泛的應用，在近十年來相當受到重視。它擁有可調參數，使得濾波器得以立即線上的改變頻率響應，而不需重新設計一個新的濾波器。依可調的頻率響應，我們可將設計的濾波器分為延遲響應可調滤波器及振幅響應可調濾波器二大類。首先在延遲響應可調濾波器方面，我們提出係數關係的方法來設計分數階延遲可調式 (Variable Fractional-Delay) FIR 濾波器，將此方法與最新的方法比較，我們的設計降低了將近一半的設計係數並且在高階的濾波器下，有更高的精準度。此外，我們也介紹了另一個以 IIR 濾波器設計分數階延遲可調式濾波器的方法。接著在振幅響應可調濾波器方面，我們提出了二項式級數展開(Binomial Series Expansion)及泰勒級數展開(Taylor Series Expansion)兩種方法，來實現分數階可調式微分器及分數階可調式積分器(Variable fractional-order differentiators and integrators)之設計。而在二維可調式數位濾波器設計中，我們運用了 McClellan 轉換及提出的架構來實現各種二維可調式 FIR 濾波器。最從我們也發展出了一個可調的視窗(Variable window)，在此視窗下，我們可以實現傳輸頻寬可調式 FIR 濾波器。在設計的過程中，我們會以各種的級數展開來將相關矩陣的元素化為 closed-form，此動作可避免數值積分的運算，並且以權重式最小平方差逼近法(Weighted least-squares approach)來求得最佳解(Optimal Solution)。關鍵字：分數階延遲可調式 FIR 濾波器、分數階延遲可調式 IIR 濾波器、分數階可調式微分器、分數階可調式積分器、McClellan 轉換、傳輸頻寬可調式 FIR 濾波器、權重式最小平方差逼近法。. i.

(4) ii.

(5) Design of Variable Digital Filters Advisor: Dr. Jong-Jy Shyu Institute of Electrical Engineering National University of Kaohsiung. Student: Yun-Da Huang Institute of Electrical Engineering National University of Kaohsiung. ABSTRACT. Variable digital filters have received considerable attention in the last decade due to their wide usages in communication systems and image processing. Their important advantage is that they contain a number of parameter in transfer function, which can be used to tune the frequency response immediately on-line without redesigning a new filter. The design of variable digital filters can be classified into two categories, the variable delay response filters and variable magnitude response filters. First, in terms of variable delay response filters, this thesis proposes a new coefficient relationship method to design variable fractional-delay (VFD) FIR filters, which can achieved higher designed performance in large order subfilters and less number of designed coefficient than the latest method. Then, the allpass IIR filters are also introduced to design VFD filters. Second, in the variable magnitude response filters, we propose two methods, binomial series expansion and Taylor series expansion, to design the variable fractional-order differentiators and integrators. Furthermore, in the design of 2D variable digital filters, the variable 2-D subfilters and variable 1-D prototype filters are designed with the same parameter to construct all kinds of variable 2-D FIR filters by using McClellan transformation. They can also be implemented by the proposed structures. Finally, the variable window is also generated which can be applied to design variable transition bandwidth FIR filters. In the thesis, several series expansions are used such that elements of relative matrices can be evaluated in closed-forms and annoying numerical integrals can be all avoided. Then, to find the optimal solution, the weighted least-squares (WLS) approach is used.. iii.

(6) Keywords: Variable Fractional-Delay FIR Filter, Variable Fractional-Delay IIR Filter, Variable Fractional-Order Differentiator, Variable Fractional-Order Integrator, McClellan Transformation, Variable Transition Bandwidth FIR Filter, Weighted Least-Squares Approach.. iv.

(7) 誌謝. 經過了二年多的努力，終於完成了這篇碩士論文。首先，我必需感謝我的指導教授徐忠枝老師，您在過去的四年多來，教導了我許多。在研究上，讓我瞭解了虛心受教，紮實研究的道理，有您的扶持，我的研究方向不會徬徨，您力求完美的個性，更培養我做事積極細心的態度，當我遇到瓶頸時，也總能說出最衝擊內心的鼓勵，這亦是我堅持下去的動力。不但如此，在做人處事上，您一直以謙遜及負責的態度感動旁人，讓我學會感恩及珍惜身邊的一切。我很高興成為您的學生，並以此自豪著。其次我要感謝黃祥哲老師，您在學業上給了我莫大的幫助與鼓勵，讓我有信心於接受未來更大的挑戰。再來感謝口試委員曾建誠老師、郝敏忠老師及魏清煌老師，您們對於論文的指點及建議，都使得這篇論文更具嚴謹與充實。此外還要感謝貝蘇章老師，有您的肯定，讓我有明確的目標，並能下定決心不顧一切地向前邁進。接著，我要感謝實驗室的學長閔涵，同學宗益、冠喻、一成、信宏，學弟信豪、世昌、崑益，在生活上給了我許多的協助及歡笑，並特別感謝系上辦公室的助理姿蓉及羿婷，在學校事務上的幫忙。最後，最要感謝的就是我的家人，有你們在背後的支持與鼓勵，我才能不顧一切地追求夢想，特別是媽媽，您二十多年來對我細心的栽培及無怨無悔的付出，讓我體會到不管外面的風雨再大，都有您溫暖的臂膀可以讓我依靠並重新出發，您對我用心的教養，更是我人格養成上的重要推手，另外還要感謝我的女友雅璟，這兩年來你一路相陪，患難與共，毫無怨言的包容，你真心的支持與不求回報的付出，都是我堅持下去的意義。僅以此論文獻給以上我所要感謝的人，沒有你們，就沒有今日的韻達，我願將我的成就與你們分享。黃韻達於高雄大學 2008 年 7 月. v.

(8) vi.

(9) Contents Abstract ( In Chinese )............................................................................................................. i Abstract ( In English ) ........................................................................................................... iii Acknowledgement ( In Chinese ) ............................................................................................v Contents ................................................................................................................................. vii List of Figures......................................................................................................................... xi List of Tables......................................................................................................................... xiii. Chapter 1. Introduction ................................................................................... 1 Chapter 2. Design of Variable Fractional-Delay FIR Filters ....................... 5 2.1. Introduction .............................................................................................................5. 2.2. Design of Variable Fractional-Delay FIR Filters Using Coefficient Symmetric..................................................................................6. 2.3. Design of Variable Fractional-Delay FIR Filters Using Coefficient Relationship .............................................................................10. 2.4. Comparison............................................................................................................13. 2.5. Conclusions ............................................................................................................17. Chapter 3. Design of Allpass Variable Fractional-Delay Digital Filters ... 19 3.1. Introduction ...........................................................................................................19. 3.2. Problem Formulation for the Design of Allpass Variable Fractional-Delay Filters ........................................................................................20. 3.3. Design Method by Using Weighted Least-Squared Approach ..........................23. 3.4. Numerical Example...............................................................................................28. 3.5. Conclusions ............................................................................................................29 vii.

(10) Chapter 4. Variable Fractional-Order Differintegrators ........................... 31 4.1. Introduction ...........................................................................................................31. 4.2. Formulation for the Design of Fractional-Order Differintergrators by Using Binomial Series ......................................................32. 4.3. Formulation for the Design of Fractional-Order Differintergrators by Using Taylor Series Expansion........................................37. 4.4. Numerical Examples .............................................................................................39. 4.5. Conclusions ............................................................................................................42. Chapter 5. Design of Variable 2-D FIR Digital Filters by McClellan Transformation ............................................................................ 43 5.1. Introduction ...........................................................................................................43. 5.2. Structure of Variable 2-D FIR Filters by McClellan Transformation..............44. 5.3. Design of Variable 2-D FIR Fan Filters...............................................................46 5.3.1. Determination of the Cut-Off frequency Orbit Function ........................48. 5.3.2. Design of Variable 2-D Transformation Subfilters ...................................49. 5.3.3. Design of Variable 1-D Prototype Low-Pass Filters .................................51. 5.3.4. Derivation of Variable 2-D Fan Filters ......................................................52. 5.4. Design of Variable 2-D Circularly Symmetric Filters ........................................54. 5.5. Design of Variable 2-D Elliptically Symmetric Filters .......................................58. 5.6. Conclusions ............................................................................................................62. Chapter 6. Design of Variable Transition Bandwidth FIR Filters Using Kaiser Window ............................................................................ 63 6.1. Introduction ...........................................................................................................63 viii.

(11) 6.2. Design of Lowpass Filters Using Window Functions .........................................64. 6.3. Design of Variable Transition Bandwidth FIR Filters .......................................67. 6.4. 6.3.1. Generation of Variable Kaiser Window.....................................................67. 6.3.2. Design of Variable Transition Bandwidth FIR Filters with Simplest Finite Impulse Response .....................................................70. 6.3.3. Design of Variable Transition Bandwidth FIR Filters with WLS Impulse Response ......................................................................71. Conclusions ............................................................................................................75. Chapter 7. Conclusions and Future works................................................... 77 References ......................................................................................................... 79. ix.

(12) x.

(13) List of Figures Fig. 1-1 Farrow structure for variable FIR digital filters..........................................................2 Fig. 2-1 The design of VFD FIR filter from the conventional and proposed methods respectively, when N = 23 , M = 4 , α = 0.9 .(a) Magnitude responses. (b) VFD responses. (c) Absolute VFD errors. .........................................................................14 Fig. 2-2 The design of VFD FIR filter from the conventional and proposed methods respectively, when N = 33 , M = 6 , α = 0.9 .(a) Magnitude responses. (b) VFD responses. (c) Absolute VFD errors. .........................................................................15. Fig. 3-1 Variable Fractional Delay of the designed filter using WLS approach when α = 0.9 , N = 35 and M = 5 . ..................................................................................................28 Fig. 4-1 Magnitude response of the designed variable fractional-order differentiator by using binomial series, when N = 40 , M = 5 , ω p1 = 0 , ω p2 = 0.95π , p1 = 0 and p2 = 1 ........................................................................................................................39. Fig. 4-2 Magnitude response of the designed variable fractional-order integrator by using binomial series, when N = 40 , M = 5 ω p1 = 0.05π , ω p2 = 0.95π , p1 = −1 and p2 = 0 .......................................................................................................................40. Fig. 4-3 The magnitude response of the designed variable fractional-order differentiator by using Taylor series expansion, when N = 13 , M = 6 , ω p1 = 0 , ω p2 = 0.95π , p1 = 0 and p2 = 1 . .................................................................................................41. Fig. 4-4 The magnitude response of the designed variable fractional-order integrator by using Taylor series expansion, when N = 13 , M = 6 , ω p1 = 0.05π , ω p2 = 0.95π , p1 = −1 and p2 = 0 .................................................................................................41. Fig. 5-1 (a) Structure of the proposed variable 2-D FIR filters.( N = 5 ) (b) Variable coefficient generator for both variable 1-D prototype filters and variable 2-D subfilters. ( c p ( i ) may be a p ( i ) , t p ( i, j ) or s p ( i, j ) ) .......................................46. xi.

(14) Fiq. 5-2 The cut-off frequency orbit of 1-D prototype low-pass filters for designing variable 2-D fan filters. ( and solid line: individual design, × and dotted line: variable design).......................................................................................................................48 Fig. 5-3 Design of variable 2-D fan filter. (a) The isopotential cut-off edge contours for different integer inclination degrees from 30 to 45 . (b) Magnitude responses of variable 1-D prototype low-pass filter. (c) Magnitude responses of variable 2-D fan filter for p = tan ( 30 ) , tan ( 35 ) , tan ( 40 ) and tan ( 45 ) . ...............................53 Fig. 5-4 Design of variable 2-D circularly low-pass filter. (a) The cut-off frequency orbit of 1-D prototype low-pass filters. ( and solid line: individual design, × and dotted line: variable design) (b) The isopotential cut-off edge contours for different passband radiuses from 0.35π to 0.6π . (c) Magnitude responses of variable 1-D prototype low-pass filter. (d) Magnitude responses of variable 2-D circularly low-pass filter for p = 0.35π , 0.45π , 0.55π and 0.6π . .....................................56 Fig. 5-5 Design of variable 2-D elliptically symmetric low-pass filter. (a) The cut-off and solid line: individual frequency orbit of 1-D prototype low-pass filters. ( design, × and dotted line: variable design) (b) The isopotential cut-off edge contours for different inclination angle θ from −35 to 35 . (c) Magnitude responses of variable 1-D prototype low-pass filter. (d) Magnitude responses of variable 2-D elliptically symmetric low-pass filter for θ = −35 , 0 , 20 and 35 . ...................................................................................................................................60 Fig. 6-1 The magnitude response of lowpass filter using rectangular window when N =13 and ωc = 0.45π . ........................................................................................................65 Fig. 6-2 The magnitude response of lowpass filter using Kaiser window when N =13 , As = 22 and ωc = 0.45π . ..........................................................................................67 Fig. 6-3 The magnitude response of the designed variable transition bandwidth filter with simplest finite impulse response when N =13 , M = 4 , ωc = 0.45π , ω p1 = 0.35π and ω p2 = 0.44π . ......................................................................................................70 Fig. 6-4 The magnitude response of the designed variable transition bandwidth filter with WLS impulse response when N =13 , M = 4 , ωc = 0.45π , ω p1 = 0.35π and. ω p = 0.44π . ..............................................................................................................74 2. xii.

(15) List of Tables Table 2-1 Least-squares error, maximum absolute group delay error and coefficients number of proposed and conventional methods for various N with M = 6 , α = 0.9 . ..... ................................................................................................................................16 Table 2-2 Least-squares error, maximum absolute group delay error and coefficients number of proposed and conventional methods for various M with N = 33 , α = 0.9 . ... ................................................................................................................................16 Table 4-1 The coefficients bk ,i in the binomial series expansion when K = 5 ...................35 Table 5-1 Key parameters and coefficients of the design examples in sections 5.3, 5.4 and 5.5...........................................................................................................................62. xiii.

(16)

(17) Chapter 1 Introduction. The variable digital filter design has been developed for a few years, which contains a number of parameters in transfer function that can be used to tune the frequency response immediately and easily without redesigning a new filter. Hence, the variable filter is useful for on-line tuning in signal processing. From the variable characteristic of frequency response, the variable digital filter can be classified into two major categories. The first one deals with the variable magnitude response filters [1]–[7], such variable feature is useful in implementing digital filters with variable banks or tunable magnitude responses for signal processing. The second category deals with the variable fractional delay (VFD) filters which have the characteristic of tunable fractional phase-delay or fractional group-delay [8]–[17]. Generally speaking, there are two techniques for designing variable digital filters. One is variable FIR filter design, and the other is variable allpass filter design. In the variable FIR cases, the transfer function of the variable FIR filter in this thesis is characterized as. H ( z, p ) =. N. ∑ h ( p) z. n =− N. −n. (1.1). n. where p is a variable parameter for the designed filter and the variable impulse response coefficients hn ( p ) are expressed as the polynomials of the parameter p as. M. hn ( p ) = ∑ a ( n, m ) p m ,. for − N ≤ n ≤ N .. m=0. 1. (1.2).

(18) Substituting (1.2) into (1.1), we can obtain the variable transfer function. N. M. ∑ ∑ a ( n, m ) p. H ( z, p ) =. m −n. z. n =− N m = 0. M ⎡ N ⎤ = ∑ ⎢ ∑ a ( n, m ) z − n ⎥ p m m = 0 ⎣ n =− N ⎦ M. = ∑ Gm ( z ) p m ,. (1.3). m =0. in which subfilters Gm ( z ) are. Gm ( z ) =. N. ∑ a(n,m) z. −n. for 0 ≤ m ≤ M .. ,. (1.4). n=− N. After formulating the transfer function of the variable FIR filter, the design problem is reduce to find the optimal subfilter coefficients a ( n, m ) such that the frequency response of (1.3) can approach the objective desired frequency response as well as possible. In the thesis, the weighted least-squares (WLS) approach is used to solve this problem. Then, the designed filter in (1.3) can be implemented in Farrow structure proposed by [1] as shown in Fig. 1-1.. Fig. 1-1 Farrow structure for variable FIR digital filters. 2.

(19) In contrast to FIR filter, the variable allpass filter design is much more difficult and complicated because the stability and nonlinear optimization problems must be taken into account. However, in the performance, the variable allpass filters can achieve higher design accuracy than variable FIR filters in terms of smaller frequency response errors, the design and implementation of variable allpass filters have received more attention recently. In the thesis, the design of both variable FIR and allpass filters are introduced. In chapter 2, the design of VFD FIR filters is investigated. A new coefficient relationship is proposed, such that the number of the designed coefficients is about a half of the latest method [17]. Moreover, due to the reduction of the size for related vectors and matrices, the coefficient relationship method presents more accurate performance than the latest one. Contrast to chapter 2, chapter 3 deals with the design of VFD allpass filters. To avoid nonlinear optimization and iterative procedures in designing variable allpass filters, the noniterative WLS method is introduced as in [24]. Furthermore, in the procedure of finding optimal coefficients, the closed-form solution can be easily obtained by exploiting some identities of trigonometric expansion. In chapter 4, the topic of fractional calculus would be researched. We use variable FIR filters to design the variable fractional-order differintegrators (a combined word of differentiators and integrators). In deriving the objective error function, two methods would be proposed to conquer the annoying integrals. In chapter 5, McClellan transformation is applied to design the variable 2-D FIR filters. We propose the method and structure for designing both 2-D transformation subfilters and 1-D prototype filter, so that their frequency characteristics are adjustable in the same parameter. Several examples such as variable fan filters, variable circularly symmetric filters and variable elliptically symmetric filters with arbitrary orientation are presented to demonstrate the effectiveness and flexibility of the proposed method. In chapter 6, the design of variable transition bandwidth FIR filters is proposed. Based on the adjustable Kaiser window, we first generate variable window and exploit this window to design variable transition bandwidth filters. Then, to improve the performance of the designed filter, WLS approach is applied so that the optimal impulse response coefficients can be achieved. Finally, in chapter 7, the conclusions of designing variable digital filters would be given and the future works would also be discussed in the chapter.. 3.

(20) 4.

(21) Chapter 2 Design of Variable Fractional-Delay FIR Filters. 2.1 Introduction The VFD filters have the property for on-line tuning the fractional delay. In many applications of signal processing, there is a need for this property such as comb filter design, time adjustment in digital receivers, music instrument modeling, sampling rate conversion, discrete time signal interpolation and time delay estimation [8]–[14]. So far, several methods have been developed to design VFD FIR filters such as Lagrangetype method [12], WLS method [17], Farrow structure [1], constrained minimax optimization [14], differentiator bank method [15] and coefficient symmetry method [16]–[18], etc. Among these methods, Lagrange-type method is simple, but the obtained frequency response at high frequency band is less accurate than that at low frequency band. To solve this problem, the WLS techniques have been developed for optimizing the frequency responses of VFD FIR filters in the whole band. During recent years, the coefficient symmetry method has been successfully applied into the design of VFD digital filters [17], which results in reduction of filter complexity and better response characteristic. In this chapter, two methods are presented for designing VFD FIR filters. First method which uses coefficient symmetry in WLS approach would be presented in section 2.2, this useful and latest technique was proposed by [17]. Then, in section 2.3, a method which uses new coefficient relationship to design VFD FIR filters would be proposed such that the number of coefficients to be designed is reduced to about a half of that in section 2.2. In 5.

(22) section 2.4, some design examples are presented to demonstrate the effectiveness of these two methods. Moreover, to illustrate the improvement of the proposed method, some design errors and coefficient number of these two methods are tabulated for comparison. Finally, the conclusions of designing VFD FIR filters are given in section 2.5.. 2.2 Design of Variable Fractional-Delay FIR Filters Using Coefficient Symmetry In this section, the conventional WLS design of VFD filters through exploiting coefficient symmetry along with an imposed coefficient constraint would be reviewed. The desired frequency response of VFD filters is given by. ω ∈ [ -απ , απ ]. D (ω , p ) = e − jω p ,. (2.1). where α is a fixed number for specifying in the passband and the parameter p is a variable fractional group delay in the range [ −0.5, 0.5]. The transfer function of the variable FIR filter is characterized as. N. H ( z, p ) =. ∑ h ( p) z. n =− N. −n. (2.2). n. where the coefficients hn ( p ) are expressed as the polynomials of parameter p as. M. hn ( p ) = ∑ a ( n, m ) p m .. (2.3). m =0. In this chapter, M is chosen to be an even number. Substituting (2.3) into (2.2), the transfer function can be rewritten as. H ( z, p ) =. N. M. ∑ ∑ a ( n, m ) p. m −n. n =− N m = 0. 6. z. (2.4).

(23) and frequency response is. (. ) ∑ ∑ a ( n, m ) p. H e jω , p =. N. M. m − jnω. e. .. (2.5). n =− N m = 0. In [18], it is shown that the coefficients a ( n, m ) s in (2.5) have symmetric property as ⎧⎪a ( n, m ) , a ( −n, m ) = ⎨ ⎪⎩−a ( n, m ) ,. for even m ,. (2.6). for odd m .. To further reduce the number of the VFD filter coefficients, the coefficient constraint has been developed in [17] as ⎧⎪a ( 0, 0 ) = 1 , ⎨ ⎪⎩a ( n, 0 ) = 0,. (2.7). for n ≠ 0.. by (2.6) and (2.7), the frequency response of the VFD filter (2.5) can be expressed as I N ⎡ ⎤ H e jω , p = 1 + ∑ ⎢ a ( 0, 2i ) + ∑ 2a ( n, 2i ) cos ( nω ) ⎥ p 2i i =1 ⎣ n =1 ⎦. (. ). (2.8). ⎡N ⎤ − j ∑ ⎢ ∑ 2a ( n, 2i − 1) sin ( nω ) ⎥ p 2i −1 i =1 ⎣ n =1 ⎦ I. where I = M 2 . Defining A c = ⎡⎣ a ( 0,2 ) , a (1, 2 ) ,. , a ( N , M ) , a (1,1) , a ( 2,1) ,. , a ( N , M − 1) ⎤⎦. t. (2.9a). and Cc ( ω, p ) = ⎡⎣ p 2 , 2 p 2 cos ( ω) ,. , 2 p M cos ( N ω) ,. − j 2 p sin ( ω) , − j 2 p sin ( 2ω) ,. , − j 2 p M −1 sin ( N ω) ⎤⎦. 7. t. (2.9b).

(24) where. t. denotes transpose operator, Eq. (2.8) can be rewritten in vector product form as. (. ). H e jω , p = 1 + A c t Cc (ω , p ) = 1 + Cct (ω , p ) A c .. (2.10). In order to approach the desired frequency response D (ω , p ) , the objective error function is defined by. e( Ac ) =. απ 0.5 −. ∫ ∫ W (ω , p ) D (ω , p) − H (e απ. jω. ). −0.5. απ 0.5. =. ∫απ ∫ W (ω , p ) ⎡⎣D (ω, p ) − 1 − A *. −. 2. , p dpdω H c. −0.5. Cc* (ω , p )⎤⎦ ⎡⎣ D (ω , p ) − 1 − A ct Cc (ω , p )⎤⎦ dpdω. = sc + A ct Pc + A c H Pc* + A c H Q c A c = sc + 2A ct Re ( Pc ) + A ct Q c A c. (2.11). where A c is real due to the symmetric specification, W (ω, p ) is a nonnegative weighting function, Re ( ⋅) stands for the real part of a complex number, operator,. H. *. denotes complex conjugate. denotes Hermitian transpose operator and απ 0.5. sc = −. ∫ ∫ W (ω , p ) ( e απ. jω p. ) (e. − jω p. ). -1 dpd ω ,. (2.12a). -1 Cc (ω , p ) dpd ω. (2.12b). -1. −0.5. απ 0.5. Pc = − −. ∫απ ∫ W (ω , p ) ( e. jω p. ). −0.5. and απ 0.5. Qc =. ∫ ∫ W (ω , p ) C (ω , p ) C (ω , p ) dpdω . απ *. c. −. t. c. (2.12c). −0.5. For detail, the elements of relative vector Pc and matrix Q c can be evaluated in closed forms as. 8.

(25) ⎧ απ 0.5 jω p m if i < ( N + 1) I and n = 0, ⎪− ∫ ∫ W (ω , p ) ( e -1) p dpdω , ⎪ −απ −0.5 ⎪ απ 0.5 ⎪ Pc = ⎨− ∫ ∫ W (ω , p ) ( e jω p -1) 2 p m cos ( nω )dpdω , if i < ( N + 1) I and n ≠ 0, ⎪ −απ −0.5 ⎪ απ 0.5 ⎪ jω p m−1 ⎪ j ∫ ∫ W (ω , p ) ( e -1) 2 p sin ( nω )dpdω , otherwise, ⎩ −απ −0.5 for. 0 ≤ i ≤ NM + I − 1,. ⎧2 ⎪⎪ n=⎨ m=⎨ ⎪2 ⎩⎪mod (i − ( N + 1) I , N ) + 1, if i ≥ ( N + 1) I , ⎩⎪ ⎧⎪mod (i, N + 1) ,. if i < ( N + 1) I ,. (⎢⎢⎣ (⎢⎣⎢. ). ⎥ +1 , ⎦ N + 1⎥ i. if i < ( N + 1) I ,. ),. i − ( N + 1) I ⎥ +1 ⎦⎥ N. (2.13a) if i ≥ ( N + 1) I ,. and ⎧ απ 0.5 m+ mˆ ⎪ ∫ ∫ W (ω , p) p dpdω , ⎪−απ −0.5 ⎪ απ 0.5 ⎪ m+ mˆ ⎪ ∫ ∫ W (ω , p)2 p cos(nˆω )dpdω , ⎪−απ −0.5 ⎪ απ 0.5 ⎪ m+ mˆ ⎪ ∫ ∫ W (ω , p)2 p cos(nω )dpdω , Qc = ⎨−απ −0.5 ⎪ απ 0.5 ⎪ m+ mˆ ⎪ ∫ ∫ W (ω , p)4 p cos(nω )cos(nˆω )dpdω , ⎪−απ −0.5 ⎪ απ 0.5 ⎪ W ω , p)4 pm+mˆ −2 sin (nω )sin (nˆω )dpdω , ⎪∫ ∫ ( ⎪−απ −0.5 ⎪0, ⎩. if i, l < ( N +1) I and n = nˆ = 0, if i, l < ( N +1) I and n = 0, nˆ ≠ 0, if i, l < ( N +1) I and n ≠ 0, nˆ = 0, if i, l < ( N +1) I and n ≠ 0, nˆ ≠ 0, if i, l ≥ ( N +1) I , otherwise,. for 0 ≤ i, l ≤ NM + I −1, ⎧⎪mod(i, N +1),. if i < ( N +1) I ,. n=⎨ ⎪⎩mod(i − ( N +1) I , N ) +1, if i ≥ ( N +1) I ,. if l < ( N +1) I , ⎧⎪mod(l , N +1), nˆ = ⎨ ⎩⎪mod(l − ( N +1) I , N ) +1, if l ≥ ( N +1) I ,. ⎧ ⎛⎢ i ⎥ ⎞ if i < ( N +1) I , ⎪2⎜⎝ ⎢⎣ N +1⎥⎦ +1⎟⎠, ⎪ m=⎨ ⎪ ⎛ ⎢ i − ( N +1) I ⎥ ⎞ ⎪⎩2⎜⎝ ⎢⎣ N ⎥⎦ +1⎟⎠, if i ≥ ( N +1) I , ⎧ ⎛⎢ l ⎥ ⎞ +1 , if l < ( N +1) I , ⎪2⎜ ⎢ ⎥ ⎟ ⎪ ⎝ ⎣ N +1⎦ ⎠ mˆ = ⎨ ⎪ ⎛ ⎢ l − ( N +1) I ⎥ ⎞ ⎪⎩2⎝⎜ ⎣⎢ N ⎦⎥ +1⎟⎠, if l ≥ ( N +1) I . 9. (2.13b).

(26) To minimize e ( A c ) , Eq. (2.11) is differentiated with respect to the unknown vector A and set the result to zero, ∂e ( A c ) ∂A c. = 2 Re ( Pc ) + 2Q c A c = 0 .. (2.14). Then, the optimal coefficient vector A c is given by A c = −Q c -1Re ( Pc ). (2.15). and the minimal error emin ( A c ) can be obtained by substituting (2.12) and (2.15) into (2.11). In this conventional method, the number of the designed filter coefficients is. NM +. M . 2. (2.16). 2.3 Design of Variable Fractional-Delay FIR Filters Using Coefficient Relationship In this section, the novel coefficient relationship would be developed to design the VFD FIR filter and the formulation of the proposed method will be derived in detail. In (2.1), the response e− jω p can be separated into real part and imaginary part by using Euler formula. Then, corresponding with (2.8), there are two approximative equations would be obtained as I N ⎡ ⎤ 1 + ∑ ⎢ a ( 0, 2i ) + ∑ 2a ( n, 2i ) cos ( nω ) ⎥ p 2i → cos (ω p ) i =1 ⎣ n =1 ⎦. I. ⎡. N. ⎤. ∑ ⎢⎣∑ 2a ( n, 2i − 1) sin ( nω )⎥⎦ p i =1. 2 i −1. → sin (ω p ). (2.17) (2.18). n =1. where “ → ” means “approximate”. To find the relationship between (2.17) and (2.18), (2.17) 10.

(27) is differentiated with respect to ω , which results in. I. N. ∑∑ 2na ( n, 2i ) sin ( nω ) p. 2 i −1. → sin (ω p ) .. (2.19). i =1 n =1. Comparing (2.19) with (2.18), we can find the coefficient relationship as a ( n, 2i − 1) ≅ na ( n, 2i ) , for n = 1, 2,. , N,. i = 1, 2,. ,I .. (2.20). Substituting (2.20) into (2.8), the frequency response can be rewritten as I N ⎧ ⎫ H e jw , p = 1 + ∑ ⎨a ( 0, 2i ) p 2i + ∑ a ( n, 2i ) ⎡⎣ 2 p 2i cos ( nω ) − j 2np 2i −1 sin ( nω ) ⎤⎦ ⎬ i =1 ⎩ n =1 ⎭ t = 1 + A pCp (ω , p ). (. ). (2.21). where A p = ⎡⎣ a ( 0, 2 ) a (1, 2 ) ⋅ ⋅ ⋅ a ( N , M ) ⎤⎦. t. (2.22a). and Cp (ω , p ) = ⎣⎡ p 2 , 2 p 2 cos (ω ) − j 2 p sin (ω ) ,. , 2 p M cos ( N ω ) − j 2 Np M −1 sin ( N ω ) ⎦⎤ . (2.22b) t. To find the coefficient vector A p , we substitute (2.21) into (2.11), the objective error function can be represented as. e ( A p ) = sp + 2A p t Re ( Pp ) + A p t Q p A p. (2.23). where sp = sc ,. απ 0.5. Pp = − −. ∫ ∫ W (ω , p ) ( e απ. jω p. ). -1 Cp (ω , p ) dpd ω. −0.5. and 11. (2.24a).

(28) απ 0.5. Qp =. ∫ ∫ W (ω , p ) C (ω , p ) C (ω , p ) dpdω . απ *. −. (2.24b). t. p. p. −0.5. In (2.23), the scalar sp and vector Pp can be evaluated by applying the Taylor series expansion of e jω p or e− jω p as that in [17], such that the numerical computation can be avoided. Also, only the first K terms of Taylor series are considered, and the remaining ones are truncated. Then, the elements of vector Pp and matrix Q p are given by. ⎧ K j k απ 0.5 W (ω , p ) ω k p m + k dpdω , if n = 0, ⎪−∑ ∫ ∫ ! k = 1 k −απ −0.5 ⎪⎪ Pp ( i ) ≅ ⎨ ⎪ K j k απ 0.5 ⎪−∑ W (ω , p ) ω k ⎣⎡ 2 p m + k cos ( nω ) − j 2np m + k −1 sin ( nω ) ⎦⎤ dpdω , otherwise, ∫ ∫ k ! ⎪⎩ k =1 −απ −0.5 for 0 ≤ i ≤ ( N + 1). M 2. − 1,. n = mod ( i, N + 1) ,. ⎛⎢ i ⎥ ⎞ m = 2⎜ ⎢ ⎥ + 1⎟ ⎝ ⎣ N + 1⎦ ⎠. (2.25a). and ⎧ απ 0.5 m + mˆ ⎪ ∫ ∫ W (ω , p ) p dpdω , ⎪ −απ −0.5 ⎪ απ 0.5 ⎪ m mˆ ˆ mˆ −1 sin ( nˆω ) ⎤⎦dpd ω , ⎪ ∫ ∫ W (ω , p ) p ⎡⎣ 2 p cos ( nˆω ) − j 2np ⎪ −απ −0.5 ⎪ Q p ( i, l ) = ⎨ απ 0.5 ⎪ ∫ ∫ W (ω , p ) ⎡⎣ 2 p m cos ( nω ) + j 2np m −1 sin ( nω ) ⎤⎦ p mˆ dpd ω , ⎪ −απ −0.5 ⎪ ⎪ απ 0.5 m m −1 ⎪ ∫ ∫ W (ω , p ) ⎡⎣ 2 p cos ( nω ) + j 2np sin ( nω ) ⎤⎦ ⎪ −απ −0.5 ⎡⎣ 2 p mˆ cos ( nˆω ) − j 2np ˆ mˆ −1 sin ( nˆω ) ⎤⎦ dpdω , ⎪⎩ for 0 ≤ i, l ≤ ( N + 1). M 2. − 1,. if n = nˆ = 0, if n = 0, nˆ ≠ 0, if n ≠ 0, nˆ = 0,. otherwise,. n = mod ( i, N + 1) ,. ⎛⎢ i ⎥ ⎞ m = 2⎜ ⎢ ⎥ + 1⎟ , ⎝ ⎣ N + 1⎦ ⎠. nˆ = mod ( l , N + 1) ,. ⎛⎢ l ⎥ ⎞ mˆ = 2 ⎜ ⎢ ⎥ + 1⎟ , ⎝ ⎣ N + 1⎦ ⎠. (2.25b). which all can be represented in closed-forms after some mathematical manipulation when W (ω , p ) is separable. Once Pp and Q p are evaluated, the coefficient vector A p can be. obtained as below. 12.

(29) A p = −Q p -1Re ( Pp ) .. (2.26). Notice that the total number of the designed filter coefficients in the proposed method is. ( N + 1). M 2. (2.27). which is nearly a half of that in Section 2.2.. 2.4 Comparison To demonstrate the effectiveness of the proposed method, some design examples with different filter lengths and subfilter orders are presented in this section. Moreover, some errors are tabulated for comparison between the conventional approach and the proposed one in Section 2.2 and 2.3. For convenience, we set W (ω, p ) = W1 ( p )W2 (ω ) = 1 in this section. Fig. 2-1(a), (b) and (c) depict the magnitude responses, VFD responses and absolute VFD errors of designed filter from conventional and proposed methods respectively, when N = 23 ,. M = 4 , α = 0.9 . And then, Fig. 2-2(a), (b), (c) show the cases for N = 33 , M = 6 ,. α = 0.9 To evaluate the performance, the maximum absolute group delay error ετ is defined by. ετ. {. = max τ (ω , p ) − p , ω ∈ [−απ , απ ], p ∈ [−0.5, 0.5]. }. (2.28). where τ (ω , p ) is the actual fractional group-delay of the designed filter. To compute the error, the frequency ω is uniformly sampled at the step size π. 500. p is uniformly sampled at the step size 1 . 40. and the fractional delay. In Table 2-1, the coefficient number,. least-squares error emin and maximum absolute errors ετ of the proposed and conventional methods for various N from 18 to 38 are listed with M = 6 , α = 0.9 . Moreover, the cases for various M from 2 to 8 are listed in Table 2-2 with N = 33, α = 0.9 .. 13.

(30) (a). (b). (c) Fig. 2-1 The design of VFD FIR filter from the conventional and proposed methods respectively, when N = 23 , M = 4 , α = 0.9 .(a) Magnitude responses. (b) VFD responses. (c) Absolute VFD errors. 14.

(31) (a). (b). (c) Fig. 2-2 The design of VFD FIR filter from the conventional and proposed methods respectively, when N = 33 , M = 6 , α = 0.9 .(a) Magnitude responses. (b) VFD responses. (c) Absolute VFD errors. 15.

(32) Table 2-1 Least-squares error, maximum absolute group delay error and coefficients number of proposed and conventional methods for various N with M = 6 , α = 0.9 . least-squares error emin. N. 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38. Proposed method 4.3772e-007 2.2147e-007 1.1237e-007 5.7182e-008 2.9195e-008 1.4972e-008 7.7293e-009 4.0341e-009 2.1457e-009 1.1792e-009 6.8379e-010 4.295e-010 2.9882e-010 2.3159e-010 1.9695e-010 1.791e-010 1.6989e-010 1.6513e-010 1.6266e-010 1.6139e-010 1.6073e-010. Conventional method 4.3768e-007 2.2145e-007 1.1236e-007 5.7178e-008 2.9193e-008 1.4971e-008 7.7282e-009 4.0333e-009 2.145e-009 1.1785e-009 6.8314e-010 4.2887e-010 2.9821e-010 2.31e-010 1.9645e-010 1.7876e-010 1.71e-010 1.6576e-010 1.7141e-010 1.9357e-010 2.8558e-010. Maximum absolute group delay error ετ. Pemin 9.7617e-005 8.9312e-005 8.3719e-005 8.1866e-005 8.5934e-005 0.00010032 0.00013421 0.00020304 0.00033332 0.00056654 0.00094748 0.0014771 0.0020543 0.002554 0.0025635 0.0018942 -0.0065366 -0.0038218 -0.051004 -0.16622 -0.43717. Proposed method 0.14067 0.11307 0.090498 0.07214 0.057293 0.045345 0.035773 0.028138 0.02207 0.017264 0.013471 0.010485 0.0081425 0.0063086 0.0048765 0.0037608 0.0028933 0.0022202 0.0016989 0.0012959 0.00098529. Conventional method 0.1414 0.11364 0.090932 0.072475 0.057551 0.045544 0.035927 0.028255 0.02216 0.017333 0.013524 0.010526 0.0081742 0.0063332 0.0048959 0.0037765 0.0029043 0.0022303 0.0017105 0.0013053 0.00098687. Number of designed coefficients. Peτ -0.0051019 -0.0049352 -0.004779 -0.0046333 -0.0044984 -0.0043742 -0.0042612 -0.0041599 -0.0040715 -0.0039969 -0.0039392 -0.0038924 -0.0038739 -0.0038894 -0.0039593 -0.0041626 -0.0037876 -0.0045016 -0.0067366 -0.0072104 -0.0016002. Proposed method 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99 102 105 108 111 114 117. Conventional method 111 117 123 129 135 141 147 153 159 165 171 177 183 189 195 201 207 213 219 225 231. Table 2-2 Least-squares error, maximum absolute group delay error and coefficients number of proposed and conventional methods for various M with N = 33 , α = 0.9 . least-squares error emin. M. 2 4 6 8. Proposed method 0.0036052 1.4969e-006 1.791e-010 1.9086e-011. Conventional method 0.0035818 1.4889e-006 1.7876e-010 1.409e-010. Maximum absolute group delay error ετ. Pemin 0.006529 0.0053703 0.0018942 -0.86454. Proposed method 0.071953 0.0059139 0.0037608 0.0038021. Conventional method 0.075553 0.0060949 0.0037765 0.0038145. Number of designed coefficients. Pemin -0.047649 -0.029698 -0.0041626 -0.0032551. Proposed method 34 68 102 136. Conventional method 67 134 201 268. In the tables 5-1 and 5-2, Pemin and Peτ are defined by. Pemin =. emin ( Ap ) − emin ( Ac ). (2.29). emin ( Ac ). and Peτ =. ετ ,p −ετ ,c ετ ,c. (2.30). respectively, where ετ ,p and ετ ,c represent the maximum absolute group delay errors of the 16.

(33) proposed and conventional methods respectively. In Table 2-1, we can find that Pemin is negative when N is larger than 33, which means that the accuracy of the proposed method in least-squares error is higher than the conventional one. This feature is also found in Table 2-2 when M is large than 6. In addition, the maximum absolute group delay errors of the proposed method are smaller than those of the conventional method for all cases in Table 2-1 and 2-2, due to the reduction of size for the related matrices or vectors.. 2.5 Conclusions In this chapter, the conventional WLS method which is using coefficient symmetric technique for designing the VFD filters has been reviewed. Then, a new coefficient relationship is proposed and conventional WLS method. From the comparison, it has been shown that the number of the filter coefficients which need to be designed in proposed method is only nearly a half of that in conventional method. Moreover, from the error tables of two methods, the performance of the proposed method has better frequency characteristic for larger filter length and subfilter order. So, this novel technique is the best method for designing the VFD filters up to the present.. 17.

(34) 18.

(35) Chapter 3 Design of Allpass Variable Fractional-Delay Digital Filters. 3.1 Introduction After detailing the design of VFD FIR filter in the previous chapter, this chapter introduces the design of allpass VFD filters. So far, there have been developed many methods to design VFD FIR filters as introduced in section 2.1. In contrast, allpass VFD filters are much more difficult to design because the stability problem must be taken into account and the design of allpass VFD filters usually involves nonlinear optimization or iterative procedures. However, since allpass VFD filters have two major advantages comparing with FIR VFD filters so that the design and implementation of allpass VFD filters have received more attention recently [19]–[22]. First, the allpass VFD filters can achieve higher design accuracy than FIR VFD filters in terms of smaller frequency response errors [11]. Second important advantage is that allpass VFD filters always have unity magnitude responses in the entire frequency band, which is the utile property when the applications of unity gain are required to design. This chapter presents a latest WLS method for designing allpass VFD filters. First, each coefficient of the design variable allpass filter is expressed as the polynomial of the fractional delay parameter. Then, the WLS method developed in [23] would be used to find the optimal polynomial coefficients of variable allpass filters by minimizing the phase response errors. To further reducing the filter complexity, the constraint on the VFD filter coefficients proposed by [24] would be imposed in WLS approach. As a result, in deriving the objective error functions, all elements of the relative vectors and matrix can be evaluated in 19.

(36) closed-forms and without any iterative calculus procedures by exploiting the identities of trigonometric expansions. In section 3.2, the problem formulation for the design of allpass VFD filters would be derived and, in section 3.3, WLS minimizing approaching method would be used to obtain the optimal solution. Then, section 3.4 givens the design example to illustrate the effectiveness of the WLS design method. Finally section 3.5 givens some conclusions of designing VFD allpass filters.. 3.2 Problem Formulation for the Design of Allpass Variable Fractional-Delay Filter For the design of allpass VFD filter, the desired frequency response is given by. ω ∈ [ -απ , απ ] ,. D ( ω , p ) = e − j ( p + I )ω ,. (3.1). where I is a prescribed integer group-delay, α is a fixed number for specifying in the passband and the parameter p is a variable fractional group delay in the range [ −0.5, 0.5]. The transfer function of variable IIR allpass filter is characterized as aN ( p ) + + a2 ( p ) z − N +2 + a1 ( p ) z − N +1 + z − N H ( z, p) = 1+ a1 ( p ) z −1 + a2 ( p ) z −2 + + aN ( p ) z − N = Z−N ⋅. (3.2). A( z , p) , A( z, p ) −1. which is chosen to approximate the desired frequency response D (ω , p ) , and the denominator of (3.2) is defined by A( z , p ) =1+ a1 ( p ) z −1 + a2 ( p ) z −2 + N. =1+ ∑ an ( p) z. + aN ( p ) z − N (3.3). −n. n =1. 20.

(37) where the coefficients an ( p ) are expressed as the polynomial of parameter p as. M. an ( p ) = ∑bnm pm .. (3.4). m =0. Substituting (3.4) into (3.3) yields. N. M. A( z , p ) =1+ ∑∑b( n,m) z − n pm .. (3.5). n =1 m=0. To further reduce the number of the designed coefficients b( n,m) , the coefficient constraint would be imposed in (3.5) as proposed by [24]. Considering p = 0 in (3.1) and (3.5), the desired frequency response of the allpass VFD filter is D(ω ,0)= e− jIω. (3.6). and the frequency response of A( z ,0) can be expressed as. N. A(e jω ,0) =1+ ∑b( n,0)e− jnω .. (3.7). n=1. If we let N =I. and bn 0 = 0,. for n =1,2, , N ,. (3.8). Eq. (3.7) can be simplified as A(e jω ,0) =1.. (3.9) 21.

(38) Then, by substituting (3.9) into (3.2), the frequency response of design VFD allpass filter at. p = 0 would equal to the desired frequency response D(ω ,0) as H (e jω ,0) = e− jIω = D(ω ,0).. (3.10). Notice that imposing the constraint (3.8) on the designed VFD allpass filter coefficients b( n,m) not only can reduce the total number of b( n,m) by N but also makes H (e jω , p ) equals to D(ω , p ) with no distortion occurring at p = 0 . By substituting (3.8) into (3.5), the frequency response of A( z , p ) can be rewritten as. N. M. A(e jω , p) =1+ ∑∑b( n,m) e− jnω pm n =1 m=1 N. M. N. M. =1+ ∑∑b( n,m)cos( nω ) pm − j ∑∑b( n,m)sin ( nω ) pm n =1 m=1. n =1 m=1. =1+ Bt C(ω , p) − jBt S(ω , p ). (3.11). where B = ⎡⎣b(1,1),. , b(1, M ), b( 2,1),. , b( 2, M ), , b( N , M )⎤⎦. t. (3.12a). C(ω , p ) = ⎡⎣ p cos(ω ), , pM cos(ω ), p cos( 2ω ), , pM cos( 2ω ), , pM cos( Nω ) ⎤⎦. t. (3.12b). and S(ω , p ) = ⎡⎣ psin (ω ), , pM sin (ω ), psin ( 2ω ), , pM sin ( 2ω ), , pM sin ( Nω ) ⎤⎦ . t. (3.12c). Now, after representing A(e jω , p ) into matrix form as (3.11), the problem for designing VFD allpass filter reduces to determine the optimal coefficient b( n,m) which were the elements of vector B such that H (e jω , p ) can approach D(ω , p ) as well as possible at the difference values of parameter p during the range [ −0.5, 0.5]. In the following section, the WLS approach would be introduces to solve this problem. 22.

(39) 3.3 Design Method by Using Weighted Least-Squared Approach To formulate the objective error function, the further analysis would be presented between the frequency responses H (e jω , p ) and D(ω , p ) . Based on A(e jω , p ) in (3.11), the frequency response of the actual VFD allpass filter (3.2) can be rewritten as. H (ω , p ) = e. − jNω. where. *. ⋅. A* (e jω , p ). (3.13). A(e jω , p ). denotes the complex conjugate operator. By correspondence between (3.1) and. (3.13), the approximative equation can be obtained as. − jNω. e. ⋅. A* (e jω , p ) A(e jω , p ). ≈ e− j( N + p)ω .. (3.14). After some manipulations, the further approximative equation would be achieved as A(e jω , p)e− j( pω 2) ≈ A* (e jω , p )e j( pω 2) .. (3.15). By means of the identity A* (e jω , p )e j( pω 2) = ⎡⎣ A(e jω , p )e− j( pω 2) ⎤⎦ , *. (3.16). the approximation in (3.15) can be rewritten as A(e jω , p )e− j( pω 2) ≈ ⎡⎣ A(e jω , p )e− j( pω 2) ⎤⎦ ,. (3.17). Im ⎡⎣ A(e jω , p )e− j( pω 2) ⎤⎦ ≈ 0. (3.18). *. and then. which means the objective weighted squared error function can be formed as 23.

(40) e( B ) =. 0.5 απ. 2. − j ( pω 2) ⎤ dωdp ∫ W (ω , p) Im ⎡⎣ A(e jω , p)e ⎦. ∫. −0.5 −απ. ∫ ∫ W (ω , p) −sin( 2 )⎣⎡1+ B C(ω , p)⎤⎦ − cos( 2 )B S(ω , p) dωdp. 0.5 απ. =. ωp. 2. ωp. t. t. −0.5 −απ. ∫ απ∫ W (ω , p)⎢⎣sin ( 2 ) + 2sin ( 2 )B C(ω , p) + sin ( 2 )B C(ω, p)C (ω , p)B. 0.5 απ. =. ⎡. −0.5 −. 2. ωp. 2. ωp. ωp. 2. t. t. t. ( )cos( )B S(ω, p)+ cos ( )B S(ω, p)S (ω, p)B +2sin ( )cos( )B C(ω , p )S (ω , p )B⎤ dωdp ⎥⎦ +2sin. ωp. ωp. 2. 2. ωp. ωp. 2. 2. 2. t. ωp. t. t. 2. t. t. = s + Bt Pc + Bt QcB + Bt Ps + Bt Qs B + Bt Qcs B. (3.19). where W (ω , p ) is a nonnegative weighting function, and. ∫ απ∫ W (ω , p)sin ( 2 ) dω dp,. 0.5 απ. s=. 2. ωp. (3.20a). −0.5 −. ∫ απ∫ W (ω , p)2sin ( 2 )C(ω , p)dω dp,. (3.20b). ∫ απ∫ W (ω , p)sin ( 2 )C(ω , p)C (ω , p)dω dp,. (3.20c). ∫ απ∫ W (ω , p)2sin( 2 )cos( 2 )S(ω , p)dω dp,. (3.20d). ∫ απ∫ W (ω , p)cos ( 2 )S(ω , p)S (ω , p)dω dp. (3.20e). ∫ απ∫ W (ω , p)2sin( 2 )cos( 2 )C(ω , p)S (ω , p)dω dp.. (3.20f). 0.5 απ. Pc =. 2. ωp. −0.5 −. 0.5 απ. Qc =. 2. ωp. t. −0.5 −. 0.5 απ. Ps =. ωp. ωp. −0.5 −. 0.5 απ. Qs =. 2. ωp. t. −0.5 −. and 0.5 απ. Qcs =. ωp. ωp. −0.5 −. 24. t.

(41) Notice that, in (3.20), Qc and Qs are symmetric matrices. To achieve the optimal coefficient vector B such that the error function e(B) would be minimized, e(B) in (3.19) is differentiated with respect to the unknown vector B and set the result to zero, ∂e(B) = Pc + 2Qc B + Ps + 2Qs B + (Qcs + Qcst )B = 0. ∂B. (3.21). Then, the coefficient vector B can be obtained as t ⎡ ⎤ B = − 1 ⎢Q + Q + (Qcs + Qcs ) ⎥ c s 2⎣ 2 ⎦. −1. (Pc + Ps ).. (3.22). By substituting (3.20a) to (3.20f) and (3.22) into (3.19), the weighted least-squared error e(B) can be obtained. Moreover, in the following text of this section, the elements of (3.20b). to (3.20f) would be calculated as detail as possible. First, by assuming the independence between ω and p , the weighted function W (ω , p ) could be separated into W1 ( p )W2 (ω ) . Then, by considering three identities of trigonometric expansions. 1). 2sin. 2. ( ) ωp 2. ( −1)k −1 (ω p ) = ⎡⎣1− cos(ω p )⎤⎦ = ∑ , ( 2k )! k =1. 2). 2sin. ( )cos( ). 3). cos2. ( ). ωp. ωp. 2. 2. ωp 2. 2k. ∞. ( −1)k −1 (ω p ) = sin (ω p ) = ∑ ( 2k −1)! k =1 ∞. 2 k −1. ,. 2k ∞ ⎡ ( )k ω p ⎤ = 1 ⎡⎣1+ cos(ω p )⎤⎦ = 1 ⎢2 + ∑ −1 ( ) ⎥, 2 2 ⎣ k =1 ( 2k )! ⎦. The elements of vectors Pc , Ps and matrices Qc , Qs , Qcs would be evaluated as. 25.

(42) ∫ απ∫ W (ω , p)2sin ( 2 ) p. 0.5 απ. Pc (i ) =. ωp. 2. m. cos( nω ) dω dp. −0.5 −. ( −1)k −1 (ω p) m = ∫ ∫ W (ω , p )∑ p cos( nω ) dω dp ( 2k )! k =1 −0.5 −απ 0.5 απ. 2k. ∞. απ. 0.5. K ( )k −1 W1 ( p ) p2 k +m dp ∫ W2 (ω )ω 2 k cos( nω ) dω dp ≈ ∑ −1 ∫ ! ( ) 2 k k =1 −0.5 −απ. n = ⎢ i ⎥ +1, ⎣⎢ M ⎦⎥. 0 ≤ i ≤ NM −1,. Qc (i,l ) =. ∫ απ∫ W (ω , p)sin ( 2 ) p. 0.5 απ. 2. ωp. m. m = mod(i, M ) +1,. (3.23a). cos( nω ) pmˆ cos( nˆω ) dω dp. −0.5 −. ( −1)k −1 (ω p) m+mˆ 1 p ⎡cos(( n − nˆ )ω ) + cos(( n + nˆ )ω )⎦⎤ dω dp 2( 2k )! 2⎣ k =1. 0.5 απ. =. ∫. 2k. ∞. ∫ W (ω , p)∑. −0.5 −απ. απ. 0.5. ( −1)k −1 ≈∑ W1 ( p ) p2 k +m+mˆ dp ∫ W2 (ω )ω 2 k ⎡⎣cos(( n − nˆ )ω ) + cos(( n + nˆ )ω )⎤⎦ dω ∫ ( ) 4 ! 2k −0.5 k =1 −απ K. 0 ≤ i ≤ NM −1,. n = ⎢ i ⎥ +1, ⎣⎢ M ⎦⎥. m = mod(i, M ) +1,. 0 ≤ l ≤ NM −1,. nˆ = ⎢ l ⎥ +1, ⎢⎣ M ⎥⎦. mˆ = mod(l , M ) +1,. Ps (i ) =. ∫ απ∫ W (ω , p)2sin( 2 )cos( 2 ) p. 0.5 απ. ωp. ωp. m. (3.23b). sin ( nω ) dω dp. −0.5 −. ( −1)k −1 (ω p ) = ∫ ∫ W (ω , p )∑ ( 2k −1)! k =1 −0.5 −απ 0.5 απ. ∞. 2 k −1. pm sin ( nω ) dω dp απ. 0.5. K ( )k −1 ≈ ∑ −1 W1 ( p ) p2 k +m−1dp ∫ W2 (ω )ω 2 k −1 sin ( nω ) dω ∫ ( ) ! 2 k − 1 k =1 −0.5 −απ. 0 ≤ i ≤ NM −1,. n = ⎢ i ⎥ +1, ⎣⎢ M ⎦⎥. m = mod(i, M ) +1,. 26. (3.23c).

(43) Qs (i,l ) =. ∫ απ∫ W (ω , p)cos ( 2 ) p. 0.5 απ. 2. ωp. m. sin ( nω ) pmˆ sin ( nˆω ) dω dp. −0.5 −. 2k ∞ ( −1)k (ω p ) ⎤ m+mˆ 1 1⎡ W , p ω ( ) 2 + ⎥ p 2 ⎡⎣cos(( n − nˆ )ω ) − cos(( n + nˆ )ω )⎤⎦ dω dp ∫ ∫ 2 ⎢⎣ ∑ ( 2k )! ⎦ −0.5 −απ k =1 0.5 απ. =. απ. 0.5. ≈ 1 ∫ W1 ( p ) pm+mˆ dp ∫ W2 (ω ) ⎡⎣cos(( n − nˆ )ω ) − cos(( n + nˆ )ω )⎤⎦ dω + 2 −0.5 −απ απ. 0.5. ( −1)k W1 ( p ) p2 k +m+mˆ dp ∫ W2 (ω )ω 2 k ⎡⎣cos(( n − nˆ )ω ) − cos(( n + nˆ )ω )⎤⎦ dω ∑ ∫ 4( 2k )!−0.5 k =1 −απ K. 0 ≤ i ≤ NM −1,. n = ⎢ i ⎥ +1, ⎣⎢ M ⎦⎥. m = mod (i, M ) +1,. 0 ≤ l ≤ NM −1,. nˆ = ⎢ l ⎥ +1, ⎣⎢ M ⎦⎥. mˆ = mod (l , M ) +1,. (3.23d). and. Qcs (i,l ) =. ∫ απ∫ W (ω , p)2sin( 2 )cos( 2 ) p. 0.5 απ. ωp. ωp. m. cos( nω ) pmˆ sin ( nˆω ) dω dp. −0.5 −. ( −1)k −1 (ω p ) = ∫ ∫ W (ω , p )∑ ( 2k −1)! k =1 −0.5 −απ 0.5 απ. ∞. 2 k −1. pm+mˆ 1 ⎡⎣sin (( n + nˆ )ω ) − sin (( n − nˆ )ω )⎤⎦ dω dp 2 απ. 0.5. ( −1)k −1 W1 ( p ) p2 k +m+mˆ −1dp ∫ W2 (ω )ω 2 k −1 ⎡⎣sin (( n + nˆ )ω ) − sin (( n − nˆ )ω )⎤⎦ dω ≈∑ ∫ ( ) 2 ! 2k −1 −0.5 k =1 −απ K. 0 ≤ i ≤ NM −1,. n = ⎢ i ⎥ +1, ⎢⎣ M ⎥⎦. m = mod(i, M ) +1,. 0 ≤ l ≤ NM −1,. nˆ = ⎢ l ⎥ +1, ⎢⎣ M ⎥⎦. mˆ = mod(l , M ) +1.. 27. (3.23e).

(44) 3.4 Numerical Example This section presents an example to illustrate the effectiveness of the WLS technique. The parameter K =10 is chosen and take the weighting functions as.. W1 ( p ) =1,. for p∈[−0.5,0.5]. (3.24a). W2 (ω ) =1,. for ω ∈[−απ ,απ ]. (3.24b). and. Fig. 3-1 shows the actual variable fractional group-delay of the designed allpass filter when. α = 0.9 , N = 35 and M = 5 .. Fig. 3-1 Variable Fractional Delay of the designed filter using WLS approach when α = 0.9 , N = 35 and M = 5 .. 28.

(45) 3.5 Conclusions The design of the allpass variable fractional group-delay filter is introduced in this chapter. The WLS method is used to achieve the optimal solution in the whole band. In the procedure of formulating objective error function, the technique of Taylor series expansion is used such that all elements of relative vectors and matrices can be calculated in closed-forms without any annoying iterative integral. Moreover, a coefficient constraint is exploited in the WLS design. It not only can reduce the complexity in terms of the total number of VFD filter coefficients but also assure no distortion of frequency response at p = 0 . The example has shown the effectiveness of the WLS method. In chapter 2 and 3, the designs of VFD digital filters are detail presented in both FIR and IIR cases. The methods mentioned in these two chapters are the latest techniques for designing VFD filters in the world. Based on WLS method, the optimal solution could be obtained such that the frequency response of designed filter would have high performance in the whole band.. 29.

(46) 30.

(47) Chapter 4 Variable Fractional-Order Differentiators and Integrators. 4.1 Introduction Fractional calculus is an important topic in mathematical analysis which deals with derivatives and integrals of arbitrary order. The theory of fractional-order derivative and integral was developed in the nineteenth century. However, just during the last three decades, the concept of fractional calculus start to be investigated in different areas of engineering applications such as electromagnetic theory, fluid flow, automatic control and electrical networks [25]-[32]. In signal processing, it has also drawn considerable attention in implementing digital fractional-order differentiators and integrators [33]-[38]. So far, several methods have been developed to design digital fractional-order differenttiators, which can be classified into time domain method [33], [34] and frequency domain method [35]-[38]. Among these methods, the FIR approximation methods which were proposed by Tseng [33], [35] are simple, but the frequency response at high frequency band is less accurate than that at low frequency band. To solve this problem, the analysis of fractional calculus in frequency domain was generated in [36]. In this chapter, the design of variable fractional-order differintegrators which contain differentiators and integrators is proposed by using weighted least-squares approach. To avoid numerical integrals, the closed-from error function is derived by using two proposed methods, one is the binomial series expansion and the other one is Taylor series expansion. Both of them can obtain nearly optimal solution. This chapter is organized as follows. In section 4.2, the design formulation of variable 31.

(48) fractional-order differintegrators by using binomial series is derived step by step. Then, the same design approach by using the Taylor series expansion would be introduced in section 4.3. Section 4.4 provides some design examples to demonstrate the effectiveness of these two proposed methods. Finally, conclusions of designing variable fractional-order differintegrators are given in section 4.5.. 4.2 Formulation for the Design of Fractional-Order Differintegrators by Using Binomial Series For the design of variable fractional-order differintegrators, the desired frequency response is given by. D (ω , p ) = ( jω ). p. (4.1). where the parameter p is a variable fractional number in the range [ p1 , p2 ] . Notice that the designed filter is named differentiator when p is positive and named integrator when. p is negative. The variable transfer function is characterized as. N. ∑ h ( p) z. H ( z, p ) =. n =− N. −n. (4.2). n. where the coefficients hn ( p ) are generally expressed as the polynomials of p as. M. hn ( p ) = ∑ a ( n, m ) p m .. (4.3). m =0. So the frequency response of the designed system can be represented as. H ( e jω , p ) =. N. M. ∑ ∑ a ( n, m ) p. m − jnω. e. n =− N m = 0. Defining. 32. .. (4.4).

(49) A = ⎡⎣ a ( − N ,0 ) ,. , a (−N , M ),. E (ω , p ) = ⎡⎣e jN ω ,. , p M e jN ω ,. , a ( N ,0 ) ,. , a ( N , M ) ⎤⎦. t. (4.5a). and. where. t. , e − jN ω ,. , p M e − jN ω ⎤⎦. t. (4.5b). (. denotes transpose operator, the frequency response H e jω , p. ). can be represented. in the vector product form as. H ( e jω , p ) = A t E (ω , p ) = Et (ω , p ) A .. (4.6). In order to approach the desired response D (ω , p ) , the objective error function is defined by. e( A) =. ∫. p2. jω ∫ W (ω , p ) D (ω, p ) − H ( e , p ) dpdω. 2. (4.7). R p1. where W (ω , p ) is a nonnegative weighting function and R represents the region of passband which is chosen as [−ω p2 , − ω p1 ] ∪ [ω p1 , ω p2 ] . Substituting (4.6) into (4.7), e ( A ) can be written as e ( A ) = s + 2 A t P+A t QA. (4.8). where ω p2 p2. s=2∫. ∫ W (ω , p ) D (ω , p ) D (ω , p ) dpd ω , *. (4.9a). ω p1 p1. ⎡ω p2 p2 ⎤ P = −2 Re ⎢ ∫ ∫ W (ω , p ) D (ω , p ) E* (ω , p ) dpd ω ⎥ ⎢⎣ ω p1 p1 ⎥⎦. (4.9b). ⎡ω p2 p2 ⎤ Q = 2 Re ⎢ ∫ ∫ W (ω , p ) E* (ω , p ) Et (ω , p ) dpd ω ⎥ ⎢⎣ ω p1 p1 ⎥⎦. (4.9c). and. in which. *. denotes complex-conjugate operator and Re [⋅] stands for the real part of a. complex number. To minimize e ( A ) , Eq. (4.8) is differentiated with respect to the unknown 33.

(50) vector A and set the result to zero as ∂e ( A ) ∂A. = 2P+ 2QA = 0 ,. (4.10). then the optimal coefficient vector A is obtained as. A = −Q -1 P .. (4.11). The elements of Q can be represented in closed-forms easily, but those of P are not so. In this section, the binomial series method is proposed, such that closed-forms of elements in. P can be derived and the performance of the nearly optimal filter is satisfactory. For simplification, we set. W (ω , p ) = 1 in this chapter. By (4.9b), the elements of P are given. by ⎡ω p2 P ( i ) = −2 Re ⎢ ∫ ⎢⎣ ω p1. ⎤ p m jnω ⎥, p e j ω dpd ω ( ) ∫ ⎥⎦ p1 ⎢ i ⎥ − N , m = mod(i, M + 1) . 0 ≤ i ≤ ( 2 N + 1)( M + 1) − 1, n = ⎢ ⎣ M + 1 ⎥⎦ p2. (4.12). Let ω = 2π (1 − δ ) , Eq. (4.12) can be rewritten as. ⎡1− ω p1 ⎢ 2π P ( i ) = −4π Re ⎢ ∫ ⎢1− ω p2 ⎣ 2π. p2. ∫ p ( j 2π ) (1 − δ ) m. p. p1. p. ⎤ ⎥ e − j 2π nδ dpdδ ⎥ . ⎥ ⎦. (4.13). By applying the binomial series. (1 − δ ). p. ∞. = 1+ ∑. p ( p − 1). k =1. ( p − k + 1) k!. which is substituted into (4.13), then 34. ( −δ ). k. (4.14).

(51) ⎡1− ω p1 ⎢ 2π P ( i ) = −4π Re ⎢ ∫ ⎢1− ω p2 ⎣ 2π. p2. ∫ p ( j 2π ) m. p. p1. ⎤ ∞ ⎡ ⎥ p ( p − 1) ( p − k + 1) k ⎤ − j 2π nδ dpdδ ⎥ . ( −δ ) ⎥ e ⎢1 + ∑ k! ⎣ k =1 ⎦ ⎥ ⎦. (4.15). In practice, only the first K + 1 terms in the above series are needed to be considered, and the remaining terms are truncated. Hence, the elements of P can be approximated as ωp ⎡p 1− 1 2π 2 ⎢ p P ( i ) ≅ −4π Re ⎢ ∫ p m ( j 2π ) dp ∫ e − j 2π nδ dδ ωp ⎢ p1 1− 2 2π ⎣ K. +∑ k =1. ( −1) k!. k. p2. ∫. (. p m bk ,1 p k +. p1. ω p1. ⎤ ⎥ p + bk , k p ( j 2π ) dp ∫ δ k e− j 2π nδ dδ ⎥ ωp ⎥ 1− 2 2π ⎦ 1−. ). 2π. (4.16). where the coefficients bk ,i in the above series satisfy the following recursive formula ⎧bk ,i = bk −1,i − ( k − 1) bk −1,i −1 , ⎪ ⎨ ⎪b = 0 , ⎩ k ,i. if i ≤ k , k = 1,2,. ,K , (4.17a). if i > k , k = 1, 2,. ,K. and bk ,1 = 1 ,. k = 1, 2,. , K.. (4.17b). For example, Table 4-1 lists the coefficients bk ,i when K = 5 .. Table 4-1 The coefficients bk , i in the binomial series expansion when K = 5 . k. bk ,1. bk ,2. bk ,3. bk ,4. bk ,5. 1. 1. 0. 0. 0. 0. 2. 1. -1. 0. 0. 0. 3. 1. -3. 2. 0. 0. 4. 1. -6. 11. -6. 0. 5. 1. -10. 35. -50. 24. 35.

(52) Moreover, the integrals in (4.16) can be computed by using the following formulas. p2. ∫p. p1. m. −1) m! p2m − l ( j 2π ) − p1m − l ( j 2π ) ( ⋅ ( j 2π ) dp = ∑ ln l +1 ( j 2π ) l = 0 ( m − l )! p. m. l. p2. p1. (4.18). and x2. k ax ∫ x e dx =. x1. x2 k e ax2 x1k e ax1 k − − a a a. x2. ∫. x k −1e ax dx. (4.19). x1. where x , x1 , x2 and a correspond to δ , 1 −. ωp. 2. 2π. , 1−. ωp. 1. 2π. and − j 2π ( n − I ) in (4.16),. respectively. By substituting (4.17)~(4.19) into (4.16), the vector P can be evaluated in the closed-forms. Furthermore, the elements of matrix Q can also be evaluated in closed-forms as ⎡ω p2 Q ( i, l ) = 2Re ⎢ ∫ ⎢⎣ ω p1. p2. ∫. p1. ⎤ p me jnω p mˆ e− jnˆω dpdω ⎥ ⎥⎦. ⎧ p2 m + mˆ +1 − p1m + mˆ +1 (ω p2 − ω p1 ) ⎪2 m + mˆ + 1 ⎪ =⎨ ⎪ p2 m + mˆ +1 − p1m + mˆ +1 sin ⎡⎣( n − nˆ ) ω p2 ⎤⎦ − sin ⎡⎣( n − nˆ ) ω p1 ⎤⎦ ⎪2 m + mˆ + 1 n − nˆ ⎩. 0 ≤ i, l ≤ ( 2 N + 1)( M + 1) − 1,. if n = nˆ otherwise. ⎢ i ⎥ n=⎢ − N, ⎣ M + 1 ⎥⎦. m = mod(i, M + 1),. ⎢ i ⎥ nˆ = ⎢ − N, ⎣ M + 1 ⎥⎦. mˆ = mod(l , M + 1).. (4.20). Once P and Q are computed, the nearly optimal coefficient vector A can be obtained by (4.11).. 36.

(53) 4.3 Formulation for the Design of Fractional-Order Differintegrators by Using Taylor Series Expansion In this section, the second method would be proposed to derive the objective error function in closed-form. This method is using the technique of Taylor series expansion such that the relative vector and matrix can be formulated easily and the nearly optimal solution can be also obtained. As the same analysis in section 4.2, the objective error function (4.8) in the normal least-squares design, W (ω , p ) = 1 , can be written as e ( A ) = s + 2 A t P+A t QA. (4.21). where ω p2 p2. s=2∫. ∫ D (ω, p ) D (ω , p ) dpdω , *. (4.22a). ω p1 p1. ⎡ω p2 P = −2 Re ⎢ ∫ ⎢⎣ ω p1. ⎤ * ⎥ D ω , p E ω , p dpd ω ( ) ( ) ∫p ⎥⎦ 1. (4.22b). ⎤ t * ⎥. E ω , p E ω , p dpd ω ( ) ( ) ∫ ⎥⎦ p1. (4.22c). p2. and ⎡ω p2 Q = 2Re ⎢ ∫ ⎢⎣ ω p1. p2. To avoid annoying numerical integrals in (4.22b), P can be evaluated by using the Taylor series expansion of D* (ω , p ) as. ∞. D* (ω , p ) = ( − jω ) = ∑ p. k =0. ln k ( − jω ) k!. pk .. (4.23). In practice, only the first K + 1 terms in the above series are considered as in [17]. Hence,. 37.

(54) by substituting (4.23) into (4.22b), the elements of P can be approximated as ⎡ ω p2 P ( i ) = −2 Re ⎢ ∫ ⎣⎢ −ωc. p2. ∫ ( − jω ). p. p1. ⎤ p me − jnω dpd ω ⎥ ⎦⎥. ω p2 ⎡ K 1 p2 ⎤ k +m ≅ −2 Re ⎢ ∑ ∫ p dp ∫ ln k ( − jω ) e − jnω dω ⎥ ⎢⎣ k = 0 k ! p1 ⎥⎦ ω p1 ⎢ i ⎥ − N , m = mod(i, M + 1) , for 0 ≤ i ≤ ( 2 N + 1) , n = ⎢ ⎣ M + 1 ⎥⎦. (4.24). and the latter integral of (4.24) can be computed by using the Taylor series expansion of e − jnω as ⎧ ω p2 − ω p1 , ⎪ ⎪ ⎪ ⎪ 1 e− jnωp2 − e− jnωp1 , ⎪ ( − jn) ω p2 ⎪ ⎪ − jnω k ω p2 ∫ω ln (− jω )e dω = ⎨ k k ⎪ p1 ω ln − jω p2 −ω p1 ln − jω p1 − k ∫ lnk −1 ( − jω )dω , ⎪ p2 ω p1 ⎪ ⎪ ⎪ ωp2 ∞ ( − jnω )l ⎪ ∫∑ lnk ( − jω )dω , l ! ⎪⎩ ωp1 l =0. (. ). (. ). (. ). if k = 0, n = 0, if k = 0, n ≠ 0,. if k ≠ 0, n = 0,. otherwise. (4.25). Moreover, in the last equation of (4.25), only the first L + 1 terms are considered, and the following formula is applied to evaluate this equation ω p2. ∫ω. ω p1. l. ln ( − jω ) dω = k. ωp. 2. l +1. (. ln k − jω p2 l +1. )−ω. l +1 p1. (. ln k − jω p1 l +1. )−. ωp. k 2 l k −1 ω ln ( − jω ) dω . l + 1 ω∫p 1. (4.26) By substituting (4.26) and (4.25) into (4.24), all elements of P can be represented in closed forms after some mathematical manipulations. In addition, the elements of Q can also be evaluated in closed forms easily as (4.20). 38.

(55) 4.4 Numerical Examples To illustrate the effectiveness of these two proposed method, some examples of designing variable fractional-order differintegrators with different variable ranges are presented in this section. The parameter K = 20 and L = 120 are chosen, and it the obtained results are satisfied. Fig. 4-1 illustrates the magnitude response of the designed variable fractional-order differentiator by using the binomial series, when N = 40 , M = 5 , ω p1 = 0 , ω p2 = 0.95π , p1 = 0 and p2 = 1 . Also, the binomial series method is applied to the design of variable fractional-order integrator, and the obtained magnitude response is shown in Fig. 4-2 with N = 40 , M = 5 , ω p1 = 0.05π , ω p2 = 0.95π , p1 = −1 and p2 = 0 . Notice that the value. of frequency edge ω p1 must be larger than zero when the integrator is designed .. Fig. 4-1 Magnitude response of the designed variable fractional-order differentiator by using binomial series, when N = 40 , M = 5 , ω p1 = 0 , ω p2 = 0.95π , p1 = 0 and p2 = 1 .. 39.

(56) Fig. 4-2 Magnitude response of the designed variable fractional-order integrator by using binomial series, when N = 40 , M = 5 ω p1 = 0.05π , ω p2 = 0.95π , p1 = −1 and p2 = 0 .. As usual, some examples would be presented by using the method of Taylor series expansion. Fig. 4-3 presents the magnitude response of the designed variable fractional-order differentiator when the parameters N = 13 , M = 6 , ωc = 0.9π , p1 = 0 and p2 = 1 . Also, the magnitude response of the designed variable fractional-order integrator is shown in Fig. 4-4, when N = 13 , M = 6 , ω p1 = 0.05π , ω p2 = 0.95π , p1 = −1 and p2 = 0 .. 40.

(57) Fig. 4-3 The magnitude response of the designed variable fractional-order differentiator by using Taylor series expansion, when N = 13 , M = 6 , ω p1 = 0 , ω p2 = 0.95π , p1 = 0 and p2 = 1 .. Fig. 4-4 The magnitude response of the designed variable fractional-order integrator by using Taylor series expansion, when N = 13 , M = 6 , ω p1 = 0.05π , ω p2 = 0.95π , p1 = −1 and p2 = 0 .. 41.

(58) 4.5 Conclusions. Two methods have been proposed to design the FIR variable fractional-order differintegrators in this chapter. First one is the binomial series expansion method and the other one is Taylor series expansion. By using these two expansions, the annoying difficult numerical integrals can be totally avoided in the procedures of evaluating filter coefficients and all elements of calculated matrices can be obtained through using the corresponding closed-form integrals. To achieve the optimal solution, the WLS approach technique was used in the design of FIR variable fractional-order differintegrators as introduced in the chapters 2 and 3. Some examples have shown the performance and effectiveness of these two methods.. 42.

(59) Chapter 5 Design of Variable 2-D FIR Digital Filters by McClellan Transformation. 5.1 Introduction Among the existing methods, it is no doubt that the McClellan transformation [39]–[47] is one of the most powerful and popular techniques for designing 2-D FIR digital filters. The purpose of transformation is mapping 1-D prototype filters into 2-D filters by a change of variables. The procedure of design can be divided into two steps. First, a low-order 2-D subfilter, which is the kernel of transformation, is designed such that the contour of the cut-off edge can meet the requirement for the designed 2-D filter. And then, a high-order 1-D prototype filter can be designed easily by the existing methods, for example the Remez exchange algorithm [48]. Furthermore, these 2-D filters can be implemented with highly structured architecture [41][42]. The McClellan transformation not only can be used to design 2-D fan-type, circularly symmetric, elliptically symmetric and diamond-shape filters, but also complex coefficient filters [47]. Recently, variable 2-D digital filters receive considerable attention for their wide usages in communication systems and image processing where the frequency characteristics need to be adjustable [49]–[53]. Among them, only [53] deals with the application of McClellan transformation for designing variable 2-D filters. In this chapter, the application of McClellan transformation for designing variable 2-D filters is investigated again. Comparing with [53], the proposed method designs both variable 2-D subfilters and variable 1-D prototype filters. Moreover, they are tunable by the same variable parameter, so that the variable characteristics of 1-D prototype filters are compatible with those of 2-D subfilters. 43.