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 filters would be derived step by step. First, in subsection 6.3.1, the variable window based on Kaiser window [57] is proposed. Then, in subsection 6.3.2, this window is applied to design objective filters with simplest impulse response. In subsection 6.3.3, the WLS approach [17], [24] is used to find the optimal impulse response coefficients such that the better performance of the designed filter can be achieved. Finally, some conclusions are given in section 6.4.

6.2 Design of Lowpass Filters Using Window Functions

For designing FIR filters, the most straightforward method is to use various designed window functions to truncate and modify the ideal infinite impulse response. Taking the lowpass filter with cutoff frequency ω_c for example, its ideal frequency response is given by

( ) ^{1 , 0 ,}

By using the Fourier series expansion, the corresponding impulse response coefficients of lowpass filter (6.1) can be expressed as

( ) sin( )

As can be seen from (6.2), the coefficient length of lowpass impulse response is doubly infinite and therefore unrealizable. By using designed window function, the realizable finite-length of impulse response can be achieved. For example, the simplest rectangular window

is used to multiply by impulse response (6.2), then the simplest finite length approximative response with length 2N+1 can be obtained as

To show the performance of truncated impulse response, Fig. 6-1 plots the magnitude response of (6.4). when N=13 and ω_c=0.45π

Fig. 6-1 The magnitude response of the lowpass filter using rectangular window when N=13 and ω_c=0.45π .

From Fig. 6-1, the large ripples can be found near the cutoff frequency ω_c. This is named Gibbs phenomenon. To reduce this phenomenon, some windows have been proposed by various authors. Among these windows, the most widely used adjustable window is the Kaiser window [57] and the function is given by

( ) function which can be expressed in the simple power series expansion as

( )

( )

²

In practice, only first 20 terms of (6.6) are chosen and the approximative value of I x is ₀( ) satisfied. In [57], the parameter α is developed to control the peak stopband ripples δ_s and is computed as

where the minimum attenuation A is _s

10

( )

20log .

s s

A = − δ (6.8)

Moreover, the filter order parameter N is estimated using the formula as

7.95 14.36^s

N⁼ A⁻ ω π ^(6.9)

where ω is the transition bandwidth. Assuming ω_p and ω_s be the passband and stopband edge angular frequencies of the lowpass filter respectively, and by means of the identity lowpass filter using Kaiser window function can be obtained and the corresponding magnitude response is shown in Fig. 6-2 when N=13, 25A_s= and ω_c=0.45π.

Fig. 6-2 The magnitude response of the lowpass filter using Kaiser window when N=13, 22A_s= and ω_c=0.45π .

6.3 Design of Variable Transition Bandwidth FIR Filters

By analyzing the Gibbs phenomenon, we can know the relationship between the size of ripple and the width of transition band. The magnitude response with small ripple would have large transition bandwidth and contrariwise. To control this relationship, the variable Kaiser window is developed in next subsection.

6.3.1 Generation of Variable Kaiser Window

The frequency response of the variable window for FIR filter design is characterized as

( )

^{( )}

where the parameter p is setting to be the variable passband edge ω_p in the range

1, 2

p p

⎡ω ω ⎤

⎣ ⎦ , and the coefficients w n are assumed to be symmetric at _p( ) ⁿ⁼⁰ and expressed as the polynomials of the parameter p as

( )

( )

Now, based on Kaiser window function (6.5), we first make Kaiser window be variable in FIR filter design. Defining variable parameter ω_p=p in (6.11) and after some manipulations, the minimum attenuation A can be formulated in the p function as _s

( )

^28.72

(

c

)

_7.95

s

A p ω p N π

= − + (6.15)

and therefore the parameter α is controlled over p in (6.7), then Kaiser window coefficients (6.5) can be expressed in the function of p as

( )

Thus, the problem reduces to find the optimal window coefficients w n m such that

(

,

)

w n _p( ) could approximate w n as well as possible. Here, the objective errors function between _k( )

p( )

Then, the optimal window coefficient vector A_n can be obtain by

1 -1 .

A_n=−2 Q P_{n n} (6.21)

6.3.2 Design of Variable Transition Bandwidth FIR Filters with Simplest Finite Impulse Response

After variable Kaiser window is developed in previous section, we would use (6.13) to multiply by the simplest finite lowpass impulse response (6.4) , then the windowed variable transition bandwidth finite impulse response coefficients can be achieved as

( ) ( ) ( )

and its corresponding zero-phase frequency response is

( ) ( )

( )

Fig. 6-3 The magnitude response of the designed lowpass variable transition bandwidth filter with the simplest finite impulse response when N=13, 4M= , 0.45ω_c= π, ω_p1=0.35π and ω_p2=0.44π .

6.3.3 Design of Variable Transition Bandwidth FIR Filters with WLS Impulse Response

To further improve the performance of the designed filter, the WLS technique [17], [24] is used to find the optimal impulse response coefficients. The desired variable transition bandwidth frequency response is given by

(

^,

)

^{1, 0} expressed as the polynomial of the parameter p as

( )

( )

Then the transfer function can be rewritten as

( ) ( )

and its frequency response

( ) ( )

where

is designed to approach the desired response (6.24). Due to the symmetric assumption of

p( )

w n and h n , the coefficients _p( ) h n m have the following symmetry

(

,

)

(

,

) (

,

)

^{, for 1 .}

h n m =h −n m ≤ ≤n N (6.30)

Then, the WLS method is used to find the optimal coefficients h n m . Thus, the objective

(

^,

)

error function between ^D

(

^ω,p

)

^{and H e}

(

^jω,p

)

can be expressed as

To find the optimal B , (6.31) is differentiated with respective to B and set the result to zero. Finally, B can be achieved by the formula

1 1 . 2 ^{B B}

B=− Q P⁻ (6.33)

Without loss of generality, the elements of P and _B Q are detail derived in the closed- _B forms as

Now, multiplying h n m by

(

,

)

w n m , the transfer function of the designed variable

(

,

)

windowed FIR filter can be expressed as

( ) ( ) ( )

According to chapter 1, (6.35) can also be implemented in the Farrow structure as shown in Fig. 1-1 .To demonstrate the effectiveness of the proposed method, the magnitude response of (6.35) is shown in Fig. 6-4 with N=13, 4M= , 0.45ω_c= π, ω_p1=0.35π and ω_p2=0.44π .

Fig. 6-4 The magnitude response of the designed lowpass variable transition bandwidth filter with the WLS impulse response when

13

6.4 Conclusions

This chapter presents the technique for generating the variable window. By using this window, the lowpass variable transition bandwidth FIR filters can be designed easily. Then, to improve the performance of the designed frequency response, the WLS method is used to find the optimal impulse response coefficients. Moreover, in the procedures of deriving error functions, all elements of the relative vectors and matrices can be calculated in the closed- forms by truncating the series expansion of the Bessel function. In the future work, this

This chapter presents the technique for generating the variable window. By using this window, the lowpass variable transition bandwidth FIR filters can be designed easily. Then, to improve the performance of the designed frequency response, the WLS method is used to find the optimal impulse response coefficients. Moreover, in the procedures of deriving error functions, all elements of the relative vectors and matrices can be calculated in the closed- forms by truncating the series expansion of the Bessel function. In the future work, this

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