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 method would be researched in 2-D window design, such as McClellan transformation.

Chapter 7

Conclusions and Future Works

In this thesis, the design of variable digital filter has been investigated in various fields and applications. In terms of variable fractional-delay, first, we have proposed a new coefficient relationship method to design VFD FIR filters, such can achieve high performance of frequency response in the whole band and less number of the designed coefficients then latest one. Second, contrast to FIR filters, the design of VFD allpass filters is also introduced.

Although the design of allpass filter is difficult to analyze, the latest noniterative WLS design is presented to avoid iterative integrals and simplify the procedure of finding the optimal solution. Comparing FIR filters with allpass filters, FIR cases have the simpler structure to implement designed filters than allpass cases, but in the viewpoint of design accuracy, allpass cases are more accurate. So, how to choose the suitable filter between these two is based on the design requirement.

On the other hand, the design of variable magnitude response digital filters is researched as well. First, two methods (binomial series expansions and Taylor series expansions) have been proposed to design variable fractional-order differintegrators, which can derive the objective WLS error function into closed-form and the other advantage of these two methods is that more accurate frequency response can be achieved at high frequency band than the conventional methods. Second, in terms of variable 2-D filters, the variable 2-D subfilters and variable 1-D prototype filter were designed with the same variable parameter to construct variable 2-D FIR filters by using McClellan transformation. In the transformation, the variable structure was proposed to implement the designed filters. Some design examples were detail derived to illustrate the effectiveness of the proposed method, such as variable 2-D fan-type filters, variable circularly symmetric filters and variable elliptically symmetric filters with arbitrary orientation.

Finally, except the variable delay and magnitude filters, the variable window was also generated which can be applied in designing variable transition bandwidth FIR filters. This window is based on the adjustable Kaiser window, and can be implemented in the Farrow structure. By applying the WLS approach, the optimal impulse response can be obtained to improve the performance of the frequency response.

In the future, the research of variable digital filter will keep up and it will be classified into three aspects. First, some new structures and modified Farrow structures could be investigated to achieve high performance of the designed filters. Second, the new approach techniques and error criterions are worth to study due to the different optimal coefficients would be obtained in the distinct objective error function. Third, the variable impulse response of the designed filters can be expressed in other basis forms except the polynomials of parameter. The new variable basis may achieve more approximate response of the designed filter than polynomial one. All these aspect will be researched and developed into 2-D filters.

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