Using Weighted Least-Squares Approach to design VFD IIR filters

Chapter 3. Design of Variable Fractional-Delay IIR Filters

3.3 Using Weighted Least-Squares Approach to design VFD IIR filters

In this section, the WLS Approach would be introduced to solve the previous problem. The equency response of the desired VFD IIR filter can be represented by

VFD IIR filters

where the superscript denotes a transpose operator,

Thus, the approximation error function can be gotten as

( )

which can be rewritten as

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

( )

Concerning the stability of the VFD IIR filter, it has been shown in [22] that

( ) ( ( ) ) ( ) ( ( ) ) ( )

∑

a be small enough to satisfy the stable condition. The constrained function is defined

( )

^0.5_0.5 ²

( )

tin (3.14) ai ould be achieve

By substitu g (3.13) and into (3.15), the constr ned function w d as

( )

^0.5_0.5 ^T ^T ^T ^(3.16)

Then, the objective error function of the stable VFD IIR filter is gotten by

( )

^T ^T ^(3.18)

e x =e x +αe x = +s x r+x Qx+αxQ x_p

where α is a relative weighting constant.

By differentiating (3.18) with respect to x and setting the result to zero can yield

( )

1 (3.19)

2 α ^p

= − +

x Q Q r

3.4 Numerical Examples

For example 1, it deals with the design of a N_a =14, 55N_b = , M =5, ω_p =0.9π, I =27, VFD IIR filter. Fig. 3-2(a) presents the magnitude response. Fig. 3-2(b) displays the absolute error of variable frequency response. Besides, the group delay response and the absolute error of variable group-delay response are shown in Figs. 3-2(c) and (d), respectively. About example 2, a VFD IIR filter is designed with N_a =N_b = =I 35, M =5, ω_p =0.9π, and ^W

( )

^ω ⁼¹. Figs. 3-3(a) and (b) illustrate the magnitude response and the absolute error of variable frequency response. The group-delay response is shown in Fig. 3-3(c). At last, the absolute error of variable group-delay response is presented in Fig. 3-3(d).

(d) (b) (a)

(c)

Fig. 3-2 Design of a N_a =14, N_b =55, M =5, ω_p = 0.9π , I =27 VFD IIR filter.

(a) Magnitude response (b) Absolute error of variable frequency response (c) Group-delay response (d) Absolute error of variable group-delay response.

(b)

(d) (a)

(c)

(a) Magnitude response (b) Absolute error of variable frequency response (c) Group-delay response (d) Absolute error of variable group-delay response.

, V F D I I R f i l t e r .

p 0.9

ω = π

, Fig. 3-3 D e s i g n o f a N_a = N_b = I = 3 5 , M = 5

3.5 Conclusions

Weighted Least-Squares Approach was successfully used to design the VFD IIR filter in this chapter. To demonstrate the effectiveness of the proposed method, two design examples were presented. Notice that, the constrained value is needed to be less than 0.5 because of the stable condition must be satisfied. So, the stable problem of the VFD IIR filter can certainly be overcome.

Finally, both the absolute error of variable frequency response and the absolute error of variable group-delay are significantly small.

Chapter 4 Design of Variable Fractional-Delay Allpass Filters

4.1 Introduction

In this chapter, the VFD Allpass filter is designed by Weighted Least-Squares Approach (WLS) method. As for the allpass filter design, the issue of stability is the principal problem [23]-[26].

However, the VFD Allpass filter would be stable if the variable parameter could be properly tuned between -0.5 and 0.5. In particular, each coefficient can be expressed as a polynomial-form, and all elements of relative vectors and matrices can be evaluated in closed-forms.

First, the problem formulation is derived in section 4.2. Then, WLS method is introduced in section 4.3. Section 4.4 gives a VFD Allpass design example. Finally, some conclusions are offered in section 4.5

4.2 Problem Formulation

As for the VFD Allpass filter, the desired response can be characterized by

(

)

^{j N}⁽ ^p⁾ ^,

d p

H ω p =e⁻ ⁺ ^ω ω ω≤ (4.1)

where N is the order of the desired filter and ^p^{∈ −}

[

^{0.5, 0.5}

]

is a variable parameter. When the desired fractional-delay is needed, p is used to tune it. In order to approximate (4.1), the used transfer function is given by

( ) ( )

then, (4.2) yields

( ) ( )

which can be implemented by the Farrow structure as shown in Fig. 4-1.

Fig. 4-1 Farrow structure of an allpass VFD digital filter (N =5,M = ) 4

According to (4.5), the frequency response would be express as

For the Allpass digital filter design, the problem only is focused on the phase approximation. Hence, the phase of (4.6)

( )

( ) ( ⁽ ⁾ )

arg H e^j^ω,p = −Nω−2 arg A e^j^ω,p (4.7)

will approximate the phase of (4.1)

( )

arg H_d ω,p = −Nω−pω (4.8)

Therefore, the design problem can be expressed by

( ) ( )

where “→” means “approximate”.

Because of (4.9) is highly nonlinear, it can be converted into

( ) ( )

which can be further formulated into

( ) ( ) ( ) ( )

4.3 Using Weighted Least-Squares Approach to design VFD Allpass filters

After previous discussion, the further analysis would be continued. By using Weighted Least-Squares Approach (WLS) method, the objective error function of the VFD Allpass filter can be represented by where the superscript ^T denotes the transpose operator,

( )

^{1,1 ,} ^,

(

^{,1 ,}

)

(

^1,

)

^, ^,

(

)

^T ^{(4.13 )}

( )

Finally, s, r, and Q are approximated by s , _K r , _K Q , respectively, _K

4.4 Numerical Example

In this example, an N =30, M =5, ω_p =0.9π, ^p^{∈ −}

[

^{0.5, 0.5}

]

allpass VFD filter is designed.

Notice that, ω and p are uniformly sampled at step sizes ω_p/ 200 and 1/60. Fig. 4-2(a) displays the variable group-delay response. Then, the absolute group-delay error is illustrated in Fig.

4-2(b).

(a)

(b)

[

^{0.5, 0.5}

]

p∈ − 30

N = M =5

Fig. 4-2 Design of an allpass VFD filter with , , ω_p =0.9π , (a) Variable group-delay response (b) Absolute delay errors

4.5 Conclusions

The design of the VFD Allpass filter is summarized in this section. Weighted Least-Squares Approach (WLS) method was applied to achieve this design. Besides, the technique of Taylor series expansion was used such that all relative vectors and matrices can be calculated in closed-forms.

Furthermore, the performance of the desired VFD Allpass filter would be obviously influenced because of the variable parameter p . It was also found that the variable parameter can be properly tuned to overcome the stability problem. At last, an example was offered to show the effectiveness of this proposed method.

Chapter 5 Design of Variable Fractional-Order FIR Differintegrators

5.1 Introduction

During the last decades, the concept of fractional calculus has been investigated in different aspects of engineering applications, such as electromagnetic theory, automatic control and signal processing [27]-[31]. Therefore, several methods have been developed for designing variable fractional-order differintegrators, integrators and differentiators [32]-[38]. Recently, this topic has been significant concerned and frequently discussed. Weighted Least-Squares Approach (WLS) can also be successfully applied to design the VFO FIR differintegrators, the pure VFO integrators and the pure VFO differentiators.

In this chapter, the technique of WLS is proposed. The problem formulation is derived step by step in section 5.2. Then, section 5.3 provides some design examples to demonstrate the effectiveness. At last, the conclusions are given in section 5.4.

5.2 Problem Formulation

For the design of the VFO differintegrator, the desired response is given by

(

)

^jI

( )

^p^, s f^, s f ^(5.1)

D ω p =e⁻ ^ω jω p ≤ ≤p p ω ≤ ω ω≤

where I is a prescribe delay and p is the variable order. When p_s ≥0 and ω_s ≥ , it is a pure 0 VFO differentiator. If p_f ≤0 and ω_s > , it is a pure VFO integrator. As for 0 p_s < <0 p_f and

s 0

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

Assume

In order to approach the desired response, the used transfer function is

( ) ( )

Substituting (5.5) into (5.4) can get

( ) ( ) ( ) ( )

Notice that, Eq. (5.6) can be implemented by FIR-typed Farrow structure as shown in Fig. 5-1.

Fig. 5-1 FIR-typed Farrow structure

In order to simplify the design, only even is used now. However, odd is similarly.

According to the symmetric and antisymmetric characteristics of (5.2), the coefficients are separated into even part and odd part by

N N

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

Moreover, the objective error function is used as below:

( ) ( ) ( ) ( )

Additionally, (5.14) can be expressed in matrix form as

( ) ( ) ( ) ( ) ( )

where ^tr

( )

^. denotes a trace operator, the superscript ^T denotes a transpose operator,

( )

( ) ( )

Then set the result to zero, the coefficient matrix A can be gotten as

(

)

⁻¹ ^T

( )

^T ⁻¹ ^(5.20)

= _A

A C C C D P P P

The coefficient matrix B can also be gotten by the similar calculation as

( )

^T ⁻¹ ^T

( )

^T ⁻¹ ^(5.21)

= _B

B S S S D P P P

5.3 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.

(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.

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

∑

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

( )

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:

( )

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.

(

^ω^f ⁻^ω^s

)

^{/ 200} ^p^f ⁻ ^p^s

)

^{/ 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.

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