Analysis clonal algorithm based window Using Fractional Fourier Transform

Analysis clonal algorithm based window Using Fractional Fourier Transform

K.Prasada Rao 1, A.S.Srinivasa Rao 2, P.V.Muralidhar 3

1. Aditya Institute of Technology and Management, Tekkali 532001 (A. P., India)

2. Professor & HOD, Dept. of ECE. College of Engineering, AITAM,Tekkali 532001

   (A. P., India)

3. Associate Professor, Dept. of ECE.College of Engineering, AITAM,Tekkali 532001 (A. P., India),

Abstract

An Approximated Exponential Fractional Fourier Transforms(FrFT) Mathematical derivation for clonal algorithm based window is proposed. By control the parameter of FrFT,it is possible to control the Spectral parameters of above windows like Half Bandwidth (HBW), Maximum Side Lobe Attenuation(MSLA) and Side Lobe Fall Of Ratio(SLFOR).This proposed derivation is also holds good for generalization of FrFT with Fourier Transform(FT).

Index TermsFractional Fourier transform, clonal algorithm based window.

NOMENCLATURE

FT: Fourier Transform

FrFT: Fractional Fourier Transform

1. INTRODUCTION

   In order to reduce the effects of spectral leakages in Harmonic analysis, windows are used [1]. window functions successfully used in the areas like interpolation factors to design Anti-Imaging filters, speech processing systems, digital filter design and beam forming [2]-[3].windows are also useful to solve reconstructive errors which are objective functions to

2. Fractional Fourier Transform

Fractional Fourier Transform widely used in quantum mechanics and quantum optics [13].Fractional Fourier Analysis can obtain the mixed time and frequency components of signals[14].it finds various applications like pattern recognisition with some spatial distortion, Image representation , compression and noise removal in signal processing [15]-[17].FrFT used for Interpretation of sinusoidal signals and design of Digital FIR Filters[18]-[19].

The continuous time Fractional Fourier Transform of a signal

()is defined through an interval [3]

() = ()(, ) (1)

Where the transform kernel (, ) of the FRFT is Given by

(, )

design the prototype filters [4].windows are essentially Applicable in spectral analysis of signals[5]- [6].According to [3],as the parameter of FrFTi.e =

2

1 ()

= [ (

2

2 + 2 2

) cot()

which could not holds good for generalization of FrFT to FT [7].In this proposed Derivation of FrFT, An attempt is made to study the variations of window parameters like HBW,MSLA and MSLFOR by different values of fluid parameter of FrFT to FT at =

This paper is organized as follows. ::section-II gives

2

an overview of FT, and mathematical model of

windows by using FT. section-III gives an overview of FrFT, and mathematical model of windows by using FRFT. Later conclusive remarks are discussed in section-IV.

()]

= ( ) 2

= ( + ) + 2 (2)

Where indicates rotation of angle of the Transformed signal for FrFT.

2.1) window function based on Clonal Algorithm

The Expression for clonal Algorithm based window is [16]

() = 0.5154 0.4711 cos( 2)

+ 0.08cos(4) || < 1 (1)

= 0 otherwise.

() =

2 ()1() [ (2+2) cot()

Now solving for I4

4 2

4 2

I =2(0.4711)cos(2) exp ( 2 cot()

1

()) – – – – (10) According to [17]

exp(2) + exp(2)

1

2 2

cos(2) =

2 (11)

()] (2)

Substitute equation-(11) in equation-(10) then

=

1()

2

2cot()

2 (3)

I4=2((0.4711)

1

Then equation-(21) becomes

(exp(2)+exp(2)) )exp ( 2 cot()

2

2 2

2

2

2

() = () exp ( cot()

1

()) —————————————- (12)

= 0.23555((2 exp(2) exp ( 2 cot()

()) (4) 4

1 2

Substitute equation-(20) in equation-(23) then

()) ) +

(2 exp(2) exp ( 2 cot() ())))

2

() = (0.5154

1

2

– – – (13)

2

2

2

1

0.4711 cos( 2)

2

Now multiply both sides with

results to

cot()

which

+ 0.0135 cos(4)) exp ( 2 cot()

()) (5)

2

2 cot()4 =

2 cot()

2 ( )

2 ( )

2

(0.23555((1 exp 2

exp ( cot

2

Equation-(5) divided into four parts like I3,I4,I5,I6where

3 2

3 2

I =2(0.5154)exp ( 2 cot()

1

()) ) +

2

2

(2 exp(2) exp ( 2 cot()

1

())))) ———- (14)

()) (6)

I =2(0.4711)cos(2) exp ( 2 cot()

2cot()

2

4 = 0.23555(( (exp(2

1

2

2

())))

4 1 2

2

+ ( (exp(2

()) – – – (7)

5 2

5 2

I =2(0.0135)cos(4) exp ( 2 cot()

1

1

())))) (15)

Integration above equation and applying limits, you

()) – – – (8)

Now solving for I3

will get

2cot()

exp(2 t ())

I3=2

(0.5154)exp ( 2 cot() ())

2

2

= 0.23555 ( 2 2

4

4

2()

1

According to that

2()

1()

exp(211())) – – -(16)

2()

Now Integrate both sides with limits t1 and t2

I3= 0.5154

( () () )

( )+() 33) – – – – – (9)

2 1 6 ( 2 1

2

2

2

(

1

2

2cot()

4)

exp(22 t2())

Finally I4

4 =

exp(22t2()) exp(211())

= (0.23555 (

2 ()

0.23555(

2()

2() )(21)

1

(2 1) () 3 3

exp(2 ())

6 2 1

( )

( )

1 1 )) (17)

2 ()

Now solving for I5

– – – (25)

The above equation can be written as

I4=

2(0.23555(exp(22t2())exp(211())))

I =2(0.01 35)cos(4) exp ( 2 cot()

5 2

5 2

1

()) – – – – (26)

1

2()

2()

(18)

2(

1

2cot()

)

According to [17]

2

2

Now equation-(18) can be divided into two parts x1 and x2

Where

1

cos(4) =

exp(4) + exp(4) 2

(27)

2

= (0.23555 (

1

exp(22 t2())

2 ()

Substitute equation-(27) in equation-(26) then

I5=2((0.4711)

exp(2 ())

1

1 1 ))

(exp(4)+exp(4)) )exp ( 2 cot()

2 () 2 2

And

2

2 = (

2 cot()

) /p>

()) —————————————- (28)

2

2

1

= 0.23555((2 exp(4) exp ( 2 cot()

5 1 2

Now solving for 1

Integrating 1 and applying lilits

()) ) +

(2 exp(4) exp ( 2 cot() ())))-

1

= 0.23555 (exp(22t2())

2()

1

2

——— (29)

exp(211())) (

) – – – (19)

2() 2 1

Now solving for 2

2

2

2

Now multiply both sides with 2 cot() which results to

2

2 = (

1

2 cot()

) (20)

2

2 cot()5 =

2 cot()

2

2

According to [17]

2

(0.23555((1 exp(4) exp ( 2 cot()

2 ( )

2

()) ) +

2 cot

= 1

cot() (21)

(2 exp(4) exp ( 2 cot()

2

Substitute equation-(21) in

equation-(20) to get

1 2

= 2 (1 2 cot())

– – – (22)

())))) ———- (30)

2 1 2

2

2

Integrating and applying limits to get

2 cot()5 = 0.23555(( (exp(4

1

())))

2

2 =

1

() 2

2

2 (23)

1

2

+ ( (exp(4

1

())))) (31)

= ( ) () (3 3) – – – (24)

Integration above equation and applying limits, you

2 2 1

6 2 1

will get

2

2

2cot()

5

Substitute equation-(37) in equation-(20) to get

= 0.23555 (exp(42 t2())

= 2 (1 2 cot()) (36)

4 () 2 1 2

exp(41 1())) (32)

Integrating and applying limits to get

4 ()

2

() 2

Now Integrate both sides with limits t1 and t2

2 =

1

2 (38)

2

2

1

2

(

2cot()

5)

= ( ) () (3 3) (39)

2

2

1

2

exp(42 t2())

2 2 1

6 2 1

= (0.23555 (

1

4 ()

Finally I5

exp(41 1()))) (33)

4 ()

5 =

0.23555(exp(42t2())exp(411()))(21)

4() 4() (40)

( )

( )

(2 1) () 3 3

The above equation can be written as

I5=

2(0.23555(exp(42t2())exp(411())))

Finally

6 2 1

1

4()

4()

(34)

() = 3 4 + 5 (41)

2(

1

2cot()

)

Thus equation-(41) is the FRFT based clonal

2

2

Now equation-(34) can be divided into two parts x1 and x2

Where

1

algorithm based window.

When substitute = = 1 in equation-

2

(41) results to generalized Fourier Transform based

2

= (0.23555 (

1

exp(42 t2())

4 ()

clonal

algorithm based window.

exp(41 1())))

4 ()

And

The spectral parameters of above window is shown in Table and its spectral responses are shown from Fig1 to Fig6.

2

2

2

2 = (

1

Now solving for 1

2 cot()

)

Table:

Spectral Parameters of FrFT Based clonal algorithm based window for variations in a

A	SLA in dB	HBW in dB	SLFOR in dB
0.93	-57	0.0195	-73.94
0.94	-60.8	0.0195	-71.85
0.95	-65.5	0.0185	-71.87
0.96	-63.9	0.0185	-72.55
0.97	-62.2	0.0185	-72.09
0.98	-61.0	0.0185	-70.76
0.99	-60.3	0.0185	-69.49
1	-60	0.0185	-69.1

A	SLA in dB	HBW in dB	SLFOR in dB
0.93	-57	0.0195	-73.94
0.94	-60.8	0.0195	-71.85
0.95	-65.5	0.0185	-71.87
0.96	-63.9	0.0185	-72.55
0.97	-62.2	0.0185	-72.09
0.98	-61.0	0.0185	-70.76
0.99	-60.3	0.0185	-69.49
1	-60	0.0185	-69.1

Integrating 1 and applying lilits

1

= 0.23555 (exp(42t2())

4()

exp(411())) (

) – – – (35)

4()

Now solving for 2

2

2 1

2

2 = ( 2 cot()) (36)

1

According to [17]

2

2

2 cot()

2

= 1 2 cot() (37)

Fig1:Spectral Response of window for a=0.93

Fig2:Spectral Response of window for a=0.94

Fig3:Spectral Response of window for a=0.95

Fig4:Spectral Response of window for a=0.96

Fig5:Spectral Response of window for a=0.97

Fig6:Spectral Response of window for a=0.98

Fig7:Spectral Response of window for a=0.99

Fig8:Spectral Response of window for a=1

3 Conclusion:

From the study of Exponential derivation of FrFT for clonal algorithm based window, The controllability of window parameters like HBW,MSLA and SLFOR is possible..i.e. The MSLA increases from -60 dB to –

65.5 dB for a= 1 to a=0.95 decreases to -57 dB for a=0.93[Table].so that from the table it is observed that the spectral parameters of clonal based window is improved by controlling the a parameter of FrFT. and

also one of the property of FrFT is Generalization of

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FrFTto FT i.e. when =

2

where a=1;then The FrFT

representation-

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should equals to FT .This proposed Mathematical derivation of FrFT fulfills the Property of FT.

4:References

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Prasada rao Koppala obtained bachelor degree under JNTU Hyderabad, pursuing M.Tech Under JNTU Kakinada His areas of interest are Automation of Industrial process, Tuning of Controllers, Signal Processing and Designing of Digital

Filters

Adari Satya Srinivasa Rao received his M Tech. degree from Andhra University, Vishakapatnam in 2004 and he is a PhD student in

Andhra University. He is having 15 years of teaching experience in various engineering colleges. Presently, he is

working as Professor, Department of ECE, Aditya Institute of Technology and Management,Tekkali. His interests include signal processing,adaptive antenna arrays and communication systems.

First Other:P.V.Muralidhaar obtained

M. Tech from JNTU, Hyderabad, pursuing PhD form Berhampur University. He is having an experience more than 10 years and also having more number of both national and international journals, conferences. His area of interest is signal processing,presently working with AITAM , Tekkali, Srikakulam, A.P