Extended Abstract Legendre Polynomials

8/8/2019 Extended Abstract Legendre Polynomials

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Polynomial Based Recursive and Non Recursive FilterDesign

Vinay Kumar

Reviewers for thesis:Prof. Dr. rer. nat. Manfred Thumm (Institut fur Hochstfrequenztechnik und Elektronik,

Karlsruhe, Germany)and

Dr. M V Kartikeyan (Indian Institute of Technology, Roorkee, India)

1 Overview

1.1 Digital Filters: An Introduction

The digital filter is a digital system that can be used to filter discrete-time signals. The realizationof a digital filter is accomplished by burning a program on a reprogrammable device, like FPGA,or by using a software program. Therefore, if we want to change the characteristics of a digitalfilter we only have to change the program which defines the circuit. Figure (1) shows a typicaldiscrete time filtering system.

xa(t) ya(t)

H(ej)x[n] y[n]

C/D D/CAnalog SignalAnalog Signal

Figure 1: Typical system to process analog signals via a discrete time filter

In Figure (1), xa(t) is the analog input signal, ya(t) is the output analog signal, x[n] is the digitizedxa(t), y[n] is the digitized ya(t),H(exp(j)) is the transfer function of the filter, C/D is the continuousto digital signal converter and D/C is the digital to continuous signal converter [1].

One can design a digital filter having linear or non-linear phase. With respect to the previousstatement consider the Fourier transform pair

X()ej

x[n

] (1)

The Equation (1) shows that applying a time delay (in the time domain) is equivalent to amultiplication of the Fourier transform by a factor of exp

j which implies the introductionof a linear phase. Thus when we transmit a message, a time delay is naturally introduced and no

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We propose a polynomial based approach to design the digital filters. Although polynomialbased approaches have been discussed by other authors earlier, yet they are completely differentfrom thepresent one. Oneprevailing method for optimized design forFIR filters is Parks-McClellan[16, 17] algorithm. The Parks-McClellan algorithm is based on considering filter design problem as

a problem in polynomial approximation. In [16, 17] approach an approximation error function iscalculated between desired frequency response and Lth order polynomial, in cos , approximationof frequency response, various techniques have been proposed to calculate the optimal solutionfor this error function [18]. Another method was proposed by Hofstetter, Oppenheim, and Siegel[19] for designing maximal ripple filters. This algorithm is an iterative procedure for producinga polynomial H(ej) that has extrema of desired values. But the number of iterations dependson the initial guess. Whereas the method presented in the present thesis neither requires anyoptimization technique to get the result nor any guess which has to be minimized iteratively.

The digital filters proposed in the present thesis are easy to implement and produce excellentmagnitude response with absolute linear phase without any approximation. To date there is nooptimal algorithmto design absolute zero group delay IIRfilters, thepresent dissertation proposes atechnique which results in an absolute zero group delay IIR filter. By the present design approach

we can design a filter having user defined type of pass band and stop band. The technique isextended to design filters to process images also; that is, 2-dimensional digital recursive and nonrecursive filters.

The outline of the procedure to design a digital filter with desired filter characteristics is asfollows

1. First transform the filter characteristics to a function, say object function, by using a special transfor-mation,

2. This object function is thereafter realized by using a previously defined set of polynomials, and then

3. This object function is converted back to filter characteristics using inverse of the transform used instep 1.

Therefore, to design a filter we need to understand the transformation which will be used to framethe object function and later its inverse to realize the filter characteristics itself. In the succeedingsection we discuss the transformation.

1.3.1 Transformation

In general transformations, like bilinear transformation, are used to convert analog filters to digitalfilters [20] or transformations are used to convert one type of filter to other type of filters; forexample, low pass filter to band pass filter etc. Whereas the transformation, we are going to useworks in a completely different way. This transformation is used in antenna theory [15] and weapply this to design signal processing filters. The transformation used in the procedure is

x = x0cos(/2) (2)where, x0 is the maximum value ofx and it will be clear how x0 is chosen.

From the properties of Fourier transform, we know that the signal spectrum occupies a rangeof 2. Therefore, the unique value of [, ] or [0, 2] or any other such interval. Weassume that [, ] in the present work, wherever it is not explicitly mentioned.

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Suppose we need to design filter characteristics shown in Figure (2). First we apply thistransformation on the filter characteristics, defined in terms of , it results in a new function,defined in terms ofx, which we call as object function, as shown in Figure (3). This object functionis an inverse of the filter definition, or in other words, the maximum value of filter characteristics,

or pass band, will be mapped to minimum values in object function and vice versa. Similarly,transition region with positive slope becomes a ramp having negative slope and vice versa in objectfunction, this becomes clear if we look at Figures (2) and (3) which show desiredfilter characteristicsand corresponding object function, respectively. The transformation defined, Equation (2), is non-linear. Therefore, linearly increasing values of are mapped onto non linearly increasing valuesofx.

Amin

Amax

0L 0L

0H 0H

3 dB ofAmax

|H()|

Figure 2: Desired filter characteristics

Amin

Amax

x0x0 x0 cos(0L/2) x0 cos(0L/2)

x0 cos(0H/2) x0 cos(0H/2)f(x)

x

Figure 3: Object function for filter characteristics of Figure (2), showing the slight non linearity inthe transition region.

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We know that the signal spectrum which is spread in a range of 2 is symmetric. Therefore,in the transformation, Equation (2), is divided by 2, so that we only have to deal with the onehalf of the spectrum portion, other half being mirror image. Digital filters are defined in terms ofdelay elements, or z, as they have to be realized using digital storage devices. The transformation

of Equation (2) gives us filter characteristics directly in terms of z, as shown below

x = x0cos(/2)x

x0= cos(/2)

=ej/2 + ej/2

2

=z1/2 + z1/2

2(replacing ej with z) (3)

Therefore the object function, defined in terms ofx, can directly be converted to delay elements,

in terms ofz, and can be realized. But in the present form they can not be realized.But as we know that the filter characteristics always have even symmetry, therefore, whileapproximating the object function we use only even power terms of variables which constitutesthe polynomial, like x0, x2, x4, . . . (this point will be clear in the next section where we design anactual filter). Since we shall be using only even power terms, Equation (3) will always appear as:

z1/2 + z1/2

2

2n

or,

z

4+

z1

4+

1

2

n

Hence, we always have filter response in terms of z and z1. This transformation is used in thesuccessive sections as the fundamental transformation. Let us have a look on the design techniquesto generate a new type of digital filters.

2 Design of 1 Dimensional Linear Phase FIR Filters with Simple

Polynomials

2.1 Procedure

In this section, we present an approach to realize FIR filters using simple polynomials. A stepwisedescription to design the FIR filter using simple polynomials is discussed below

Step 1. Choose a polynomial, f(x), which has the following properties:

1. The chosen polynomial must have real roots,

2. The polynomial should have zeros near the origin; that is, 0 |x| 1, and

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3. The polynomial should increase sharply for real values ofx, which are far away from x = 1;that is, x > 1.

Step 2. We use the transformation discussed in section 1 for mapping the polynomial variable

x to frequency variable of filter characteristics.

Step 3. The zeros of the transfer function of the FIR filter are calculated next. Let these zerosbe represented by zi = exp(ji), i = 1, ..n, where the is are calculated using the inverse of thetransformation equation; that is,

i = 2 cos1

xix0

(4)

Note that the xis are the zeros of the original polynomial f(x). The FIR filter transfer functionis then given by

H(z) =n

i=1

(z zi) (5)

or,

H() =n

i=1

(ej eji ) (6)

We consider a specific case of filter design in the next section.

2.1.1 Design Example

Let us consider a polynomial where the values ofxis are all equal to 0; that is, all the zeros of theobject function lie on the origin, in this case the polynomial is

f(x) = xn (7)

where, n is an integer. This polynomial satisfies all the conditions specified in Step 1.The value of the function, where we want our stop band to start, can be taken to be x = 1. This

value is arbitrary, but x = 1 gives good results. Therefore, at the start of the stop band

xn = 1 (8)

Looking at the transformation x = x0 cos(/2), we observe that x = 0 transforms to = , x = 1transforms to = 2 cos1(1/x0) which is the frequency where the stop-band starts.

The values ofs and p are given by [15]

s = 2cos1(1/b(1/n)) (9)

p = 2 cos1(1/21/2n) (10)

At x = x0, the polynomial has the value b. If we want our stop band to be, let us say p dB downthan that of the maximum value of pass band, consequently the value of b will be absolute valueofp dB; that is,

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b = 10p/20 (11)

Suppose in the polynomial discussed above, the value ofn is 6; that is, our filter is of the order

of 6, and we want stop band to be 40 dB down; that is, b = 1040/20

= 100 (Equation (11)). The valuesof s and p are 2.17622 and 0.6733 respectively as calculated using the Equations (9) and (10) .The transfer function, as discussed in Step 3, for this FIR filter, where i = with i = 1, 2,...6, is

H() = {x0 cos(/2)}6 (12)The magnitude response of this filter in dB, and phase response for this low pass FIR filter is

shown in Figures (4) and (5) respectively.

0 0.5 1 1.5 2 2.5 3 3.5200

150

100

50

0

50

Frequency in Radians

Amplitudein

dB

Frequency Response in dB

Figure 4: Magnitude response in dB ofH(z) of Equation (12)

0 0.5 1 1.5 2 2.5 3 3.51

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1


PhaseinRadiance

Phase Response

Figure 5: Phase response of Equation (12)

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3 Design of 1 Dimensional Linear Phase FIR Filter With Cheby-

shev Polynomials

3.1 ProcedureIn this section, we discuss a new approach to realize FIR filters using Chebyshev polynomials [15].Chebyshev polynomials play a vital role in antenna as well as in signal processing theory. TheDolph-Chebyshev distribution of currents feeding the elements of a linear array comprising anantenna gives a sharp main lobe and small side lobes all of which have the same power level [15],we use this concept in the present discussion. The Chebyshev polynomials are given by

Tm(x) =

cos(mcos1x) 0 < |x| < 1cosh(m cosh1x) 1 < |x| (13)

We proceed to design an FIR filter based on the antenna design [15]. Let us assume that theorder of the filter, which we intend to design, is m. The step vise procedure to design the required

FIR filter is as followsStep 1 : First we calculate the absolute value of attenuation in the stop band, b,

b = 10(attenuation in dB)/20 (14)

Step 2 : We find the stop band, s, and pass band, p, frequencies following the steps discussedin [15]

s = 2cos1

1cosh(1/mcosh1b)

(15)

p = 2cos1

cosh(1/m) cosh1(b/

2)

cosh(1/mcosh1b)

(16)

where, m is the order of the filter.Step 3 : The location of zeros, m, on unit circle can be calculated by the following equation,

which is discussed in [15]

m = 2cos1

cos(k)cosh(1/mcosh1b)

(17)where, k = (2k 1)/2m, and k = 1 . . . m.Step 4 : Using the relation zm = ejwm , we can write the frequency response, H(z) in z-transform

domain as follows

H(z) = (z z1)(z z2) . . . (z zm) (18)

where, z1, z2,. . . are location of zeros. Replacing z by ej and zms by ejm s in Equation (18) weget

H() = (ej ej1 )(ej ej2 ) . . . (ej ejm ) (19)We apply the procedure discussed above to design a filter in the following section.

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3.2 Design Example

Let us design a Chebyshev low pass FIR filter with side band 40 dB down from the pass band; thatis, b = 1040/20 = 100 and with order 6; that is, m = 6. From Equations (15) and (16) we can calculate

s = 1.5732, p = 0.5622.The values of the ms can be calculated by using Equation (17). We can write H() as

H() = (ej ej1.64)(ej ej2.0958)(ej ej2.7739)(ej ej3.5093)(ej ej4.1874)(ej ej4.6431)

0 0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency

Magnitude

alpha= 1

alpha= 1.33

alpha= 0.9

Figure 6: Magnitude response of 6th order Chebyshev low pass FIR filter

0 0.5 1 1.5 2 2.5 3 3.5120

100

80

60

40

20

0

Frequency

Mag

nituindB

alpha=1

alpha=1..33

alpha=0..9

Figure 7: Magnitude response of 6th order Chebyshev low pass FIR filter in dB

If we plot the filter characteristics; that is, |H()| verses frequency , it will be much clear.The dark continuous lines in Figures (6), (7), and (8) show the magnitude response, magnituderesponse in dB, and phase response of above mentioned FIR filter respectively. From Figure (8) itis clear that the phase is linear.

3.3 The Modified Chebyshev Filter

By using the procedure discussed above we can not control the bandwidth of the resulting filter.Therefore, we introduce a new parameter in the Chebyshev polynomial. In the original Chebyshev

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0 0.5 1 1.5 2 2.5 3 3.54

3

2

1

0

1

2

3

Frequency

Angleinradian

s

alpha=1..33

alpha=0..9

alpha=1

Figure 8: Phase response of 6th order Chebyshev low-pass FIR filter

polynomial (Equation (13)) we multiply a new parameter with parameter x. controls the

bandwidth of the filter. Thus, Equation (13) becomes

Tm(x) =

cos(mcos1x) 0 < |x| < 1cosh(m cosh1x) 1 < |x| (20)

When we multiply x with , it results in a change in s only, where p remains the same, as will be present in both numerator as well as in denominator term (see Equations (15) and (16)).Therefore, new stop band frequency, snew, will be

snew = 2 cos11/

cosh(1/mcosh1b)

(21)

and

pnew = p (22)We can calculate the location of zeros for modified Chebyshev polynomials by

mnew = 2 cos1cos(k)/

(cosh(1/mcosh1b))

(23)

where, k=(2k 1)/2m, and k=0 . . . m. We can write H() as

H() = (ej ej1 )(ej ej2 ) . . . (ej ejm ) (24)with new values ofms calculated using Equation (23).

3.4 Design Example

Plotting the magnitude response of design; that is, for 6 th order filter, with the new parameter taken into consideration, we get the magnitude response and magnitude response in dB shownin Figures (6) and (7) respectively, dash followed by dot for values > 1 and dots for values < 1 respectively. It is evident from figures that the bandwidth of our filter is increased in case of > 1 and the stop band is further down. When < 1 we can easily conclude from Figure (7) that

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the bandwidth is reduced with an increase in the stop band level and the magnitude of the stopband is equal to the pass band. Thus we should not use filters designed using value of < 1.

Figure (8) shows that our modified Chebyshev FIR filter has linear phase characteristics, furtherit confirms that the filter retains its linear phase even after introducing the new parameter .

4 Design of 2 Dimensional Linear Phase FIR Filters with Cheby-

shev Polynomials

4.1 Procedure

The 1-D Chebyshev polynomials are given by Equation (13). But we desire to design a 2D linearphase FIR filter and for that we require 2D polynomials. To convert the 1-D Chebyshev polynomial,defined in Equation (13), to 2D Chebyshev polynomial we changethe variable x with a new variable, which will represent the Chebyshev polynomial in cylindrical coordinate system.

Thus Equation (13) with new variable can be represented as

Tm() =

cos(m cos1) 1 1 (25)The procedure to design the filter is discussed below: Step 1 : Calculate the absolute value of

attenuation in the stop band, b

b = 10attenuationindB/20 (26)

Step 2 : Calculate the stop band, s, and pass band, p, frequencies using

s = 2cos1

1cosh(1/m cosh1 b)

(27)

p = 2cos1

cosh (1/m)cosh1(b/

2)cosh(1/mcosh1b)

(28)Step 3 : The location of zeros, m, on unit circle can be calculated by the following equation [15]

m = 2cos1

cos(k)cosh(1/mcosh1b)

(29)where, k = (2k 1)/2m, and k = 1 . . . m.Step 4 : Calculate the value ofHm(u, v), where the relationship between and u, v is given by

2 = u2 + v2.Let us design a 2D Chebyshev filter with the present approach.

4.2 Design ExampleSuppose we need to design a low pass filter of order 6; that is, according to the earlier discussionm = 6. The values ofs and p are calculated by using Equations (27) and (28) and result in 1 .5732and 0.5622 respectively. When we realize the filter characteristics, H(u, v), they result as shown inFigure (9).

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4

2

0

2

4

4

2

0

2

480

60

40

20

0

20

40

u

20th

order Low Pass Filter

v

H(u,v

)indB

Figure 9: Magnitude response in dB of 6th order low pass FIR filter

5 Design of 1 Dimensional Linear Phase FIR Filter With Orthog-

onal Polynomials

5.1 Procedure

A sequence of orthogonal polynomials with weight W(x) satisfies the relation

fi(x), fj(x)

=

x1x2

fi(x)fj(x)W(x)dx = 0 i j (30)

where, W(x) is the weight function, fi(x) and fj(x) are any two members of the orthogonal set, anda, b represents the inner product.

In the present discussion we consider Legendre polynomials only as an example of orthogonalpolynomials because they are the simplest orthogonal polynomials among all the generally used,

yet the procedure is general and user can use any of the other orthogonal polynomials and considertheir interval of orthogonality.

For Legendre polynomials the interval of orthogonality is [1, 1] and weight function is 1. Wecan generate the Legendre polynomials by using Rodrigues Formula [21] given by

Pn(x) =1

2nn!

dn

dxn(x2 1)n (31)

where, n=1, 2 . . .. Suppose user needs to design a low pass filter (LPF) whose magnituderesponse is shown in Figure (2)1, whereAmax represents the amplitude of the pass band of the filter,Amin represents the amplitude of the stop band of the filter, OH is the end of pass band and thestart of the transition band, and OL is the end of transition band and start of the stop band.

This ideal characteristic is first converted to an object function (independent variable x), shownin Figure (3). The mapping from Figure (2) to (3) is done by previously defined transformation,Equation (2).

1Though here an LPF is considered, any other ideal magnitude characteristic may be used. The procedure is general innature.

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The object function is then approximated using a linear combination of several Legendre poly-nomials. Note that we approximate the characteristics only with the even Legendre polynomials;that is, P0, P2, P4, . . . etc., since the characteristics of Figure (3) are symmetric [22]. The transforma-tion from Figure (2) to Figure (3) is accomplished by using Equation (2).

Let us call the object function of Figure (3) as f(x), which can be written as

f(x) =

n=0

a2nP2n(x) (32)

where, a0, a2, a4,... arecoefficientsof thecorresponding Legendre polynomials to getthe requiredcharacteristics, shown in Figure (3).

Let us discuss the procedure to design the filter step by step.Step 1 : To calculate the coefficients we obtain a set of equations by multiplying Equation (32)

by P0(x), P2(x), . . . one by one and integrating over the interval [1, 1] [22]. With this procedure weget the following array of equations.

21

0f(x)P0(x)dx = 2

10

n=0a2nP2n(x)

P0dx (33)

2

10

f(x)P2(x)dx = 2

10

n=0

a2nP2n(x)

P2dx (34)

2

10

f(x)P4(x)dx = 2

10

n=0

a2nP2n(x)

P4dx (35)...

The number ofequations which we have tosolve depends onhow many coefficients are required

or in other words how many Legendre polynomials we are going to use for the approximation ofour object function. By using the orthogonality property Equations (33), (34), . . . reduce to

10

f(x)P0(x)dx =

10

a0(P0(x)P0(x))2dx (36)

10

f(x)P2(x)dx =

10

a2(P2(x)P2(x))2dx (37)

...

From the above equations it is clear that, the coefficients a0, a2, . . . may be calculated directly byusing

ai =

10

f(x)Pi(x)dx10

Pi(x)Pi(x)dx(38)

where, i = 0, 2, 4, . . .

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Step2 : The coefficients arereplacedin Equation (32), andthe resulting approximatepolynomialfa(x) is calculated. Note that fa(x) has a limited number of terms, up to N, when compared withf(x); that is,

fa(x) =Nn=0

a2nP2n(x) (39)

Step 3 : To transform from polynomial or x domain to frequency or domain we use thetransformation, Equation (2). We choose x0 to be 1, albeit one can choose any value.

Step 4 : The zeros of fa(x) are calculated using any standard routine and the zeros of fa() areobtained by using

xiT i

here xis are the zeros of fa(x). The zeros ofH(z) are zi = exp(ji), where i = 1, 2, 3 . . ..Step 5 : Using these zeros we calculate the transfer function in z domain; that is,

H(z) = (z

z1)(z

z2)(z

z3) . . . (40)

5.2 Design Example and Discussion

Let us design a filter with following values (refer Figure (2))

Amax = 1000,Amin = 0,OL = 2.3186,OH= 2.0007,3dB = 2.0944.

Then, the object function characteristics are calculated to be

f(x) =

0 if 0 x < 0.47142.86(x 0.4) if 0.4 x < 0.541000 if 0.54 x < 1.0

(41)

Let us take some specific cases with above mentioned values. As we keep increasing the numberof terms in our approximation we get better and better results. It is up to the designer, where hewould like to stop. For example, when ten even Legendre polynomial terms are used then

fa(x) = a0P0 + a2P2 + a4P4 + a6P6 + . . . + a18P18 (42)

where fa(x) is the approximation to f(x), Figure (10).Following the procedure of previous section we get the magnitude response shown in Figure

(11) and the magnitude response in dB which is shown in Figure (12).

6 Design of 2 Dimensional Linear Phase FIR Filter With Orthog-

onal Polynomials6.1 Procedure

We propose to design a circularly symmetric 2DFIR filter. Therefore half of the symmetric mag-nitude response of such a filter is shown in Figure (13). As previously, the filter will be designed

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0 0.2 0.4 0.6 0.8 1200

0

200

400

600

800

1000

1200Polynomial

x

f(x)

Object Function

Approximationusing

Polynomials

Figure 10: Approximation of object function using first 10 Legendre polynomials terms, P0 to P18(the object function here is not showing the non linearities for simplicity).

0 0.5 1 1.5 2 2.5 3 3.5200

0

200

400

600

800

1000

1200Frequency Response


Amplitude

ApproximationusingPolynomials

Object Function

Figure 11: Magnitude response of the low pass FIR filter corresponding to the object function

shown in Figure (10).

from the object function. The 2D object function for the filter characteristics of Figure (13) is shownin Figure (14).

Since the filter is circularly symmetric therefore, if we take the cross section of the 2D FIR filterperpendicular to u v plane passing through the origin, it will look like a 1D filter. Figure (2)shows the 1D equivalent magnitude response for the filter shown in Figure (13). Similarly the 1Dobject function will look as shown in Figure (3). In Figures (13) and (14)

Amax represents the amplitude of the pass band of the filter, Amin represents the amplitude ofthe stop band of the filter, OH is the end of pass band and the start of the transition band, and OLis the end of transition band and start of the stop band.

The relationship between the 1D frequency axis, -axis, of Figure (2) and 2D frequency axis,

u, v-axis, of Figure (13) is given by

2 = u2 + v2 (43)

The variable is related to x,y by the relationship

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0 0.5 1 1.5 2 2.5 3 3.520

10

0

10

20

30

40

50

60

70Frequency Response in dB


AmplitudeindB

Figure 12: Magnitude response in dB of the low pass FIR filter corresponding to the object functionshown in Figure (10).

1 01 2

3 45 6

5

0

5

5

4.5

4

3.5

3

2.5

2

1.5

1

uv

H(u,v

)

Figure 13: 3D desired filter response

0 12 3

4 56

5

0

5

0

1

2

3

4

5

xy

f(x,y

)

Figure 14: 3D object function for filter characteristics of Figure (13), showing slight non linearityin the transition region

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2 = x2 + y2 (44)

The 1D filter, shown in Figure (2), and then the 1D filter function is transformed to the object

function, shown in Figure (3)

= 2cos1(x/x0) (45)

where, x0 is the maximum value ofx. The 1D object function is shown in Figure (3).From Equations (43), (44) and (45) we can conclude that to transform the 2D filter function to

2D object function we use

u = 2cos1(x/0) |u| , 0 < |x|

v = 2cos1(y/0) |v| , 0

Extended Abstract Legendre Polynomials

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