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.

( ) [ ]

1 1, for 0.5,0.5

W p = p∈ − (3.24a)

and

( )

[ ]

2 1, for ,

W ω ⁼ ω∈ −απ απ (3.24b)

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.

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= . The example has 0 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.

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 different- tiators, 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

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 differ- integrators 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

(

^,

) ( )

^p

D ω p = jω (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} n

( )

ⁿ So the frequency response of the designed system can be represented as

( ) ^{( )}

(

^{,0 , ,}

) (

^,

)

^{, ,}

(

^{,0 ,}

)

^,

(

^,

)

^t passband which is chosen as

2 1 1 2 complex number. To minimize ^{e A}

( )

, Eq. (4.8) is differentiated with respect to the unknown

vector A and set the result to zero as

then the optimal coefficient vector A is obtained as

= − ^-1 .

A Q 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 applying the binomial series

( ) ( ) ( ) ( )

which is substituted into (4.13), then

( )

^{1 2}

( ) ( ) ( )( )

In practice, only the first K+ terms in the above series are needed to be considered, and 1 the remaining terms are truncated. Hence, the elements of P can be approximated as

( ) ( )

where the coefficients b in the above series satisfy the following recursive formula _{k i,}

( )

Moreover, the integrals in (4.16) can be computed by using the following formulas 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

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

To avoid annoying numerical integrals in (4.22b), P can be evaluated by using the Taylor series expansion of ^D*

(

^ω,p

)

^as

In practice, only the first K + terms in the above series are considered as in [17]. Hence, 1

by substituting (4.23) into (4.22b), the elements of P can be approximated as

and the latter integral of (4.24) can be computed by using the Taylor series expansion of e⁻jn^ω as

Moreover, in the last equation of (4.25), only the first L+ terms are considered, and the 1 following formula is applied to evaluate this equation

( ) ( ) ( ) _{( )}

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).

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, fractional-order integrator, and the obtained magnitude response is shown in Fig. 4-2 with

40 of frequency edge

p1

ω 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

40

Fig. 4-2 Magnitude response of the designed variable fractional-order integrator by using binomial series, when N =40,

5 M =

1 0.05

ωp = π ,

2 0.95

ωp = π , p₁ = − and 1 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, 0.9ω =_c π , p₁ = and 0 p₂ = . Also, 1 the magnitude response of the designed variable fractional-order integrator is shown in Fig.

4-4, when N =13, M =6,

1 0.05

ωp = π,

2 0.95

ωp = π , p₁= − and 1 p₂ = . 0

Fig. 4-3 The magnitude response of the designed variable fractional-order differentiator by using Taylor series expansion, when N =13 , M =6 ,

1 0

ωp = ,

2 0.95

ωp = π , p₁ = and 0

2 1

p = .

Fig. 4-4 The magnitude response of the designed variable fractional-order integrator by using Taylor series expansion, when

13

N = , M =6 , ω_p1 =0.05π , ω_p2 =0.95π , p₁= − and 1

2 0

p = .

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.

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.

This chapter is organized as follows. Section 5.2 deals with the variable structure of 2-D filters designed by McClellan transformation, which are modified from the Farrow structure [1]. In section 5.3, the procedure of designing variable 2-D fan filters is presented in several subsections. First, in section 5.3.1, a cut-off orbit function is found by solving an overdetermined system; and then both variable 2-D subfilters and variable 1-D prototype filters are designed by least-squares approach in section 5.3.2 and 5.3.3, respectively. Finally, in section 5.3.4, the coefficients of variable 2-D filters can be obtained by using the recurrence relations for Chebyshev polynomial. Following the steps in section 5.3, the designs of variable circularly symmetric filters and variable elliptically symmetric filters with arbitrary orientation are presented in section 5.4 and section 5.5, respectively. Finally, the conclusions are given in section 5.6.

5.2 Structure of Variable 2-D FIR Filters by McClellan Transformation

For a variable zero-phase FIR digital filter, its frequency response is represented by

( ) ( ) ( ) Following the McClellan transformation [39]–[47] and substituting the variable 2-D

transformation subfilter

into (5.3), we can obtain the frequency response of the desired variable 2-D filter

(

1 2

)

^{( )}

(

1 2

)

In this chapter, the variable 2-D subfilter is also designed such that its variable characteristics can be controlled by the same parameter as the variable 1-D prototype filter, i.e. the parameter p as in (5.1) and (5.2). For simplification, M =M₁ is used in this paper.

Moreover, due to the recurrence relations for Chebyshev polynomial as follows,

0( ) 1 ,

the structure of the designed variable 2-D FIR filters is illustrated in Fig. 5-1(a), and the variable coefficient generator for both variable 1-D prototype filters and variable 2-D subfilters is shown in Fig. 5-1(b).

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. ( _cp( )_i may be _ap( )_i , tp

( )

i j^, or sp

( )

i j^, )

5.3 Design of Variable 2-D FIR Fan Filters

For designing 2-D fan filters, the original substitution

( )

( ) ( ) ( )

1

( ) ( )

2

( ) ( ) ( )

1 2

cos ω =t 0,0 +t 1,0 cos ω +t 0,1 cos ω +t 1,1 cos ω cos ω (5.7)

for McClellan transformation is used. To avoid scaling problem, it is desired to give the following constraints: (i) the 1-D frequency origin, ω =0 , is mapped into the

(

0,π point

)

of the 2-D frequency plane, and (ii) the point ω = π of the 1-D frequency is mapped into the

(

π,0

)

point of the 2-D frequency plane, which result in

( ) ( )

0,0 1,1 ,

t =t (5.8a) and

( )

^{1,0 1}

( )

^{0,1 .}

t = +t (5.8b) (b)

(a)

p

(, )

c i M c i M

(

^, −¹

)

c i M

(

, −2

)

p( ) c i

( )

,1

c i c i

( )

,0

Hence the transformation becomes

( )

( ) ( ( ) ( )

1 2

) ^{( )} ( ( )

1

( )

2

) ( )

1

cos ω =t 1,1 1 cos+ ω cos ω +t 0,1 cos ω +cos ω +cos ω (5.9) .

To find the proper cut-off frequency ω of 1-D prototype low-pass filter and the _c transformation coefficients, the following objective error function is defined as in [46],

( )

( ) ( )

1

( )

1 ² 1 inclination angle of the 2-D fan filter, and

, 0 45 ,

When different values of inclination angle θ are given in the 2-D fan filter, the optimal cut-off frequency ω of the 1-D prototype low-pass filter is also changed according to _c θ and it can be achieved by the method in [44]. To illustrate the relationship between θ and

ω , Fig. 5-2 shows the cut-off frequency orbit for individual designing integer inclination c

angles in the range 30 ≤ θ ≤45 marked by “ ”.

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)

5.3.1 Determination of the Cut-Off Frequency Orbit function

To further design variable 2-D subfilters and variable 1-D prototype filters, a function

( )

c p

ω , which can represent the cut-off frequency orbit shown in Fig. 5-2, should be determined. In this chapter, the method for solving least-squares problems, which occur in overdetermined systems [54], is applied as follows.

Suppose the variable range of inclination angle θ for designing a variable 2-D fan filter is

[

θ θ1, 2

]

where θ , and ₁ θ are integer degrees for simplification, and the variable parameter ₂ p is defined by

tan( ),

p= θ (5.14)

then the corresponding overdetermined system can be represented by

( ) ( ) ( )

which can be expressed in matrix form as

= .

5.3.2 Design of Variable 2-D Transformation Subfilters

For designing variable 2-D fan filters, the variable transformation

( )

( ) ( ( ) ( )

1 2

) ^{( )} ( ( )

1

( )

2

) ( )

1

cos ω =t_p 1,1 1 cos+ ω cos ω +t_p 0,1 cos ω +cos ω +cos ω (5.19)

is applied, in which

As the same formula in (5.10), the corresponding objective error function of (5.19) can be represented by

So the transformation coefficient vector T can be obtained by _p contours for different integer inclination degrees from 30 to 45 are shown in Fig. 5-3(a).

5.3.3 Design of Variable 1-D Prototype Low-Pass Filters

By Eqs. (5.1) and (5.2), the frequency response of a variable zero-phase 1-D prototype low-pass filter is characterized by

( )

( )

^{( )} which is used to approach the desired variable frequency response

( )

( )

where ω represents the width of transition band. Defining _T

( ) ( )

0,0 , 1,0 , ,

(

,0 ,

)

,

(

0,

) (

, 1,

)

, ,

(

,

)

^t,

and following section 5.3.2, the objective error function is defined by

( ) ^{( )}

( ( ) ) ( ) ( )

Also, differentiating (5.28) with respect to A and setting the result to zero, the optimal solution is given by

magnitude responses of the variable 1-D prototype low-pass filter are shown in Fig. 5-3(b).

5.3.4 Derivation of Variable 2-D Fan Filters

Once the variable 2-D subfilter and the variable 1-D prototype filter have been designed, we can obtain the coefficients of variable 2-D fan filters by (5.5) after some mathematic manipulations. Fig. 5-3(c) presents the obtained magnitude responses of variable 2-D fan filter for _{p=tan 30}

( )

_{, tan 35 ,}

( )

tan 40 and

( )

_{tan 45 .}

( )

(a)

(b)

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 .

( )

(c)

5.4 Design of Variable 2-D Circularly Symmetric Filters

In this section, the transformation of (5.7) is used for designing 2-D circularly symmetric filters. Also, there are three constraints to be considered: (i) ω =0 is mapped into

( )

0,0 , (ii)

ω = π is mapped into

(

π π , and (iii) ,

)

t

( )

1,0 =t⁽0.1⁾ due to the symmetric contour, which result in

( )

^0,0

( )

^1,1

t = −t (5.31a) and

( ) ( )

1,0 0,1 ¹.

t =t =2 (5.31b)

So the transformation becomes

( )

( )

1

( )

2

( ) ( ( ) ( )

1 2

)

1 1

cos cos cos 1,1 cos cos 1

2 2 t

ω = ω + ω + ω ω − (5.32)

in which only one transformation coefficient needs to be determined.

Following the steps in section 5.3, we can obtain the cut-off frequency orbit shown in Fig.

5-4(a) (marked by “ ”) for individual design when the passband radius of the circularly low-pass filter varies from ω =_r1 0.35π to ω =_r2 0.6π in the step size ^{2 1}

40

r r

ω − ω

Δ = . To

find the cut-off orbit function ωc

( )

p , the variable parameter p is defined by

p= ω (5.33) r

which is same as the variable passband radius, and the corresponding overdetermined system is given by

( ) ( )

Also, the variable 2-D subfilter can be designed, which is similar to section 5.3.2, and the objective error function is given by

( )

²

( ^{( )} ) ( ) ( ) ^{( )}

Fig. 5-4(b) presents the isopotential cut-off edge contours when the passband radius varies from ω =_r1 0.35π to ω =_r2 0.6π , when the step ^{2 1}

20

r r

ω − ω

Δ = and M =5 are used.

For the design of variable 1-D prototype low-pass filter, it is similar to the section 5.3.3 except that the cut-off orbit function ωc

( )

p in (5.28) and (5.29) is replaced by which shown in Fig. 5-4(a). For example, the magnitude response of variable 1-D prototype low-pass filter is shown in Fig. 5-4(c) when N =17, 0.1ω =_T π , p₁=0.35π and p₂ =0.6π . Also, the final magnitude responses of variable 2-D circularly low-pass filter for p=0.35π , 0.45π,

0.55π and 0.6π are illustrated in Fig. 5-4(d).

(a)

(b)

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.

(c)

(d)

Fig. 5-4 Cont.

5.5 Design of Variable 2-D Elliptically Symmetric Filters

In this section, the modified McClellan transformation

( )

( ) ( ) ( ) ( ) ( )

is used to design the variable 2-D elliptically symmetric filters. For an ellipse rotated by an angle θ with respective to ω -axis, it can be described by the curve ₂ L

where a and b are semiminor axis and semimajor axis respectively. For designing a 2-D elliptically low-pass filter with arbitrary orientation, there are two constraints to be considered: (i) ω =0 is mapped into

( )

0,0 and (ii) ω = π is mapped into

(

^π,0

)

^{, which}

Hence the transformation (5.37) becomes

To design a variable 2-D elliptically symmetric filter, we first obtain the cut-off frequency orbit for individual design shown in Fig. 5-5(a) (marked by “ “) when the rotated angle θ varies from θ = −₁ 35 to θ =₂ 35 , and a=0.25π, b=0.5π. In this section, the variable parameter p is defined by

180.

p= π θ (5.42)

Like (5.15) and (5.34), the corresponding overdetermined system can be formulated as

( ) ( ) ( )

¹

To design the variable 2-D subfilter, the objective error function is defined by

( )

²

( ^{( )} ) ( ) ( ) ( )

(

0,0,0 ,

)

, 0,0,

( ) (

, 1,0,0 ,

)

, 1,0,

( ) (

, 1,1,0 ,

)

, 1,1,

( )

^t ,

denotes a line integral along the curve of (5.38) for a given variable parameter p . Fig. 5-5(b) presents the isopotential cut-off edge contours when the variable parameter p varies from p to ₁ p and ₂ M =5. As to the design of variable 1-D prototype low-pass filter, the magnitude responses are shown in Fig. 5-5(c) with N =17 and ω =_T 0.1π . The final magnitude responses of variable 2-D elliptically symmetric low-pass filter for θ = −35 , 0 , 20 and 35 are shown in Fig. 5-5(d). Finally, for convenience, all of the key parameters and coefficients for the above design examples are tabulated in Table 5-1.

(a)

Fig. 5-5 Design of variable 2-D elliptically symmetric 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 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 .

(b)

(c)

(d)

Fig. 5-5 Cont.

Table 5-1

Key parameters and coefficients of the design examples in sections 5.3, 5.4 and 5.5.

Filter type

bm -2.043223 32.207234 -88.050827 113.505986 -72.157955 18.109558

(1,1, )

t m 3.023753 -21.956686 57.367132 -73.255295 46.470939 11.649826 Fan filter

(^0,1, )

t m -1.037892 2.339133 -5.528215 7.226146 -4.654126 1.154979 bm -2.196294 8.268790 -9.898812 6.564644 -2.161814 0.282677 Circularly

low-pass filter t(1,1,m) -2.514849 9.281784 -12.348944 8.166986 -2.677448 0.348823

bm 0.785005 0 0.268140 0 0.126633 0

(0, 0, )

t m -0.049102 0 -0.006968 0 -1.163051 0

(1, 0, )

t m 0.756657 0 -0.832384 0 0.660701 0

Elliptically low-pass filter

(1,1, )

s m 0 -0.791090 0 0.1140548 0 -0.188982

5.6 Conclusions

In this chapter, the technique of conventional McClellan transformation has successfully been extended to design variable 2-D FIR digital filters. Once the cut-off orbit function is determined, both variable 2-D transformation subfilter and variable 1-D prototype filters can be designed and are adjustable by the same variable parameter. From the numerical examples, the effectiveness and flexibility of the proposed method have been fully illustrated by the presented figures.

Chapter 6

Design of Lowpass Variable Transition Bandwidth FIR Filters Using Kaiser Window

6.1 Introduction

Window function is used to truncate and smooth the impulse response of an ideal zero-phase infinite-impulse-response filter. The frequency response of window function consists of a main lobe in the middle of the spectrum, and several side lobes located on both sides of the main lobe. The desirable window function should satisfy the two requirements: (a) the width of the main lobe be as narrow as possible, and (b) the maximum level of the side lobes be as small as possible. However, these two requirements are contradictory.

So far, several famous window functions are proposed such as rectangular, triangular, Hanning, Hamming, Blackman and Kaiser window [55]-[58]. And 2-D window functions have also been developed during the past three decades [59]-[64]. Furthermore, some modified window functions are proposed to improve and optimize the performance in the particular situations [65]-[73].

A new technique is proposed to generate a variable window in this chapter. Using this technique, the design of lowpass variable transition bandwidth FIR filters can be realized. In section 6.2, the general design of lowpass filter using rectangular and Kaiser window functions would be reviewed. In section 6.3, the design of variable transition bandwidth FIR

A new technique is proposed to generate a variable window in this chapter. Using this technique, the design of lowpass variable transition bandwidth FIR filters can be realized. In section 6.2, the general design of lowpass filter using rectangular and Kaiser window functions would be reviewed. In section 6.3, the design of variable transition bandwidth FIR

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