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1 Journal of Science and Arts Year 14 No. (7) pp ORIGINAL PAPER WEIGHTED CORRELATION ON EXTENDED FRACTIONAL FOURIER TRANSFORM DOMAIN ALKA S. GUDADHE 1 NAVEEN REDDY A. Manuscript received: ; Accepted paper: ; Published online: Abstract. In this paper we defined the correlations based on the extended fractional Fourier transform domain. The correlation uses two functions to produce a third function in its own form. This third function is called the correlation of the two functions. Properties of correlations on the extended fractional Fourier transform domain are also presented. Keywords: Extended fractional Fourier transform fractional Fourier transform cross-correlation and auto-correlation. 1. INTRODUCTION The fractional Fourier transform (FrFT) is a generalization of the classical Fourier transform (FT) into fractional domains is introduced way back in 190 s but remained largely unknown until the work of Namias [5] in It is used in many areas of communication systems signal processing and optics [7 8]. Number of applications of his FrFT can be found in [1 8 9]. Extended fractional Fourier transform (EFrFT) is the generalization of FrFT can be seen in Jianwen Hua et. al. [4] with two more parameters as where ( ) ( ) ( ) = ( ) = F f t u F u f t K t u d t (1.1) i ( a t b u ) cot abtu csc K tu e π + =. In [3] we are introduced the properties on EFrFT domain and in [] we have enhanced the concept of convolution to the domain of EFrFT. In this paper we have defined the correlations of EFrFT and proved some properties on it. 1 Govt. Vidarbha Institute of Science and Humanities Amravati. (M. S.) India. alka.gudadhe@gmail.com. Vidyabharti Mahavidyalaya Amravati. (M. S.) India. anreddyssv@gmail.com. ISSN:

2 106 Weighted correlation on extended Alka S. Gudadhe Naveen Reddy A.. CROSS-CORRELATION THEOREM FOR EFrFT: Definition.1: For two functions f(t) and g(t) of extended fractional Fourier transform the weighted correlation operation can be defined as ( ) = ( )( ) = ( ) ( + ) i cot (.1) ht f g t f gt e d where is the cross correlation operation for the extended fractional Fourier transform. H = be the weighted correlation and F G and transforms of functions f g and h respectively Theorem.1. Let ht ( ) ( f g)( t) cot H u = e F u G u (.) π Proof: By (1.1) and (.1) i a ( t ) cot π ( ) cot csc π τ + τ + H u f τ g t τ e dτ e dt Replacing ( ) = ( ) ( + ). i a t b u abtu ab Hence proved. t + τ = v t = v τ and dt = dv ( ) = cot ( ) ( ) H u e F u G u π 3. PROPERTIES OF CROSS CORRELATION OF EFrFT 3.1. Commutative Law: If F G and H transform of functions f g and h respectively and ( ) cot ( ) i π a τ t+ τ ht = f g t = f gt+ e d weighted cross correlation of EFrFT of f and g f g t g f t ( )( ) Proof: Let z( t) = ( g f )( t) ( ) = ( ) ( + ) By (1.1) i a ( t ) cot iπ ( a t b u ) cot abtu csc π τ + τ + Z u g τ f t τ e e dτdt Replace t + τ = v t = v τ and dt = dv. Therefore iπbu Z u e cot = G u F u (3.1) Therefore from (.) and (3.1) ( )

3 Weighted correlation on extended Alka S. Gudadhe Naveen Reddy A. 107 That is ( f g)( t) ( g f )( t) ( ) ( ) H u Z u 3.. Associative Law: If F G and H transform of functions f g and h respectively and i cot ht f g t f gt e d ( ) = ( )( ) = ( ) ( + ) ( f g) y ( t) f ( g y) ( t) Proof: The proof is simple and hence omitted Distributive Law: If F G and H transform of functions f g and h respectively and ( ) cot ( ) i π a τ t+ τ ht = f g t = f τ gt+ τ e dτ f ( g + y) ( t) = ( f g) + ( f y) ( t) Proof: Let z1 ( t) = f ( g + y) ( t) By (1.1) EFrFT of z1 ( t ) will be ( ) = ( ) ( + ) + ( + ) i a ( t ) cot i ( a t b u ) cot i abtu Z csc 1 u π τ + τ π π f τ g t τ y t τ e e + dτdt Substituting t + τ = w t = w τ and dt = dw we obtain ( ) = cot ( ) ( ) + ( ) π Z1 u e F u G u Yab u (3.) Let z ( t) = ( f g) + ( f y) ( t) = ( f g)( t) + ( f y)( t) By (1.1) EFrFT of z ( t ) will be ( ) iπa τ( t+ cot iπa τ( t+ cot { ( ( τ} Z u = f g t + e d + f y t + e d iπ( a t + b u ) cot iπabtu csc e dt Substituting t + τ = w t = w τ and dt = dw we obtain ( ) = cot ( ) ( ) + ( ) π Z u e F u G u Yab u (3.3) Therefore from (3.) and (3.3) f g + y t = f g + f y t Hence proved. ( ) 3.4. Evenness: If F G and H transform of f g and h respectively and ISSN:

4 108 Weighted correlation on extended Alka S. Gudadhe Naveen Reddy A. ( ) = ( )( ) = ( ) ( + ) i cot ht f g t f gt e d ( f g)( t) = ( g f )( t) ( f g)( t) Proof: Let h ( t) = ( f g)( t) By (.) 1 cot H u = e F u G u (3.4) π 1 Let ( ) = ( )( ) = ( ) ( + ) iπa τ( + t cot h t f g t f g t e d By (1.1) ( ) = ( ) ( + ) i a ( t ) cot π ( ) cot csc π τ + τ + H u f τ g t τ e dτ e dt. i a t b u abtu ab Replacing τ t = v t = τ + v and dt = dv Let h ( t) = ( g f )( t) 3 Therefore by (1.1) cot H u = e F u G u (3.5) ( ) = ( ) ( ) π i a ( t) cot iπ ( a t b u ) cot abtu csc π τ τ + H u g τ f τ t e e dτdt 3 Replace τ t = v t = v + τ and dt = dv cot H u = e G u F u (3.6) π 3 By (3.4) (3.5) and (3.6) f g t = g f t f g t ( )( ) 4. AUTO-CORRELATION THEOREM FOR EFrFT: Definition 4.1. For any functions f ( t ) and ht ( ) the weighted correlation operation can be defined as ( ) = ( )( ) = ( ) ( + ) i (4.1) cot where is the auto-correlation operation for the extended fractional Fourier transform. = is the weighted correlation and F and H transforms of functions f and h respectively Theorem 4.1. Let ht ( ) ( f f)( t) Proof: By (1.1) and (4.1) cot H u = e F u F u (4.) π

5 Weighted correlation on extended Alka S. Gudadhe Naveen Reddy A. 109 i a ( t ) cot π ( ) cot csc π τ + τ + H u f τ f t τ e dτ e dt Replacing ( ) = ( ) ( + ). i a t b u abtu ab Hence proved. t + τ = v t = v τ and dt = dv ( ) = cot ( ) ( ) H u e F u F u π 5. PROPERTIES OF AUTO-CORRELATION OF EFrFT: 5.1. Commutative Law: If F and H transform of functions f and h respectively where h is auto correlation Proof: It is obvious. i cot ( f f )( t) = ( f f )( t) 5.. Associative Law: If F and H transform of functions f and h respectively and i cot ( f f ) f ( t) f ( f f ) ( t) Proof: By associative law of cross-correlation of EFrFT. ( f f ) f ( t) f ( f f ) ( t) 5.3. Distributive Law: If F and H transform of functions f and h respectively and i cot ( + ) ( ) = ( ) + f f f t f f f f t Proof: By distributive law of cross-correlation of EFrFT. f f + f t = f f + f f t 5.4. Evenness: If F and H functions f and h respectively and ( ) transform of i cot ( f f )( t) = ( f f )( t) ISSN:

6 110 Weighted correlation on extended Alka S. Gudadhe Naveen Reddy A. Proof: By evenness of cross-correlation of EFrFT. f f t = f f t ( )( ) 6. CONCLUSIONS In this paper we have defined the weighted cross-correlation and auto correlation in the extended fractional Fourier transform domain. We have also established the cross correlation theorem and proved some related properties. It should also be noted that the theorem and properties are converted to the case of FrFt if we put a=b=1 and moreover they are transformed to the case of conventional FT if further is taken as π/. All these properties and theorem can be utilized in the theory of signal processing. REFERENCES [1] Almeida L.B. IEEE Trans. Signal Process 4(11) [] Gudadhe A.S. Naveen A. Journal of Science and Arts 4(5) [3] Gudadhe A.S. Naveen A. International Journal of Engineering Research and Technology (1) [4] Hua J. Liren L. Guoqiang L. J. Opt. Soc. Am. A 14(1) [5] Namias V. J. Inst. Math. Appl [6] Olcay A. Fractional convolution and correlation: simulation examples and an application to radar signal detection Norsig Proceedings 00. [7] Ozaktas H.M. Mendlovic D. J. Opt. Soc. Am. A [8] Ozaktas H.M. Mendlovic D. J. Opt. Soc. Am. A. 10(1) [9] Ozaktas H.M. Zulevsky Z. Kutay M.A. The fractional Fourier transform with Applications in Optics and Signal Process John Wiley and Sons Chichester 001. [10] Pei. S.-C. Liu. C.-L. Lai. Y.-C. The generalized fractional Fourier transform Acoustics Speech and Signal Processing (ICASSP) IEEE