Canonical Cosine Transform Novel Stools in Signal processing

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1 nternationa Journa of Engineering Research and Genera Science Voume 2, ssue 5, August-Septemer, 204 SSN Canonica Cosine Transform Nove Stoos in Signa processing S.B.Chavhan Yeshwant Mahavidyaaya, Nanded-43602, ndia E-mai- Astract: n this paper a theory of distriutiona two-dimensiona (2-D) canonica cosine is deveoped using Gefand-Shiov technique and defined some operators on these spaces aso the topoogica structure of some of the S-type spaces of distriutiona two dimensiona canonica cosine transform. Keywords: 2-D canonica transforms, generaized function, testing function space, s-type spaces, canonica cosine transform.. NTRODUCTON: Linear canonica transform is usefu toos for optica anaysis and signa processing.the Fourier Anaysis is undoutedy the one of the most vauae and powerfu toos in signa processing, image processing and many other ranches of engineering.the fractiona Fourier transform, a specia case of inear canonica transform is studied through different anges. Ameida [], [2] Had introduced it and proved many of its properties Namias [5].Opened the way of defining the fractiona transform through the Eigen vaue as in case of fractiona Fourier transform. The conversiona canonica cosine transform is defined as. i d 2 i a 2 s t 2 s 2 { C CTf ( t)}( s) e cos t e f ( t) dt, 2i t is easiy seen that for each s n R and the function K, c t s eongs to E(R n ) as a function of t, where K ( t, s) c i d 2 i a 2 s t 2 2 s e e cos t 2 i Hence the canonica cosine transform of f E ( R n ) can e defined y { C CTf ( t)}( s) f t, K t, s, c where right hand side has a meaning as the appication of f E to Kc ( t, s) E. As compared to one dimensiona, canonica cosine transform has a consideray richer structure in two dimensiona. The definition of distriutiona two dimensiona canonica cosine transform is given in section 2. S-type spaces using Gefand-shiov technique are deveoped in section 3.Section 4 is devoted for the operators on the aove spaces. n section 232

2 nternationa Journa of Engineering Research and Genera Science Voume 2, ssue 5, August-Septemer, 204 SSN , discuss the resut on the topoogica structures of some spaces. The notation and terminoogy as per Zemanian[6],[7]. Gefand-Shiove[3],[4]. 2. DEFNTON OF TWO DMENSONAL (2D) CANONCAL COSNES TRANSFORMS: Let E ( R R ) denote the dua of E( R R ). Therefore the generaized canonica cosine-cosine transform of ' f ( t, ) E ( R R) is defined as C C2 2 DCCCT f ( t, ) ( s, w) f ( t, ), K ( t, s) K (, w) 2 DCCCT f ( t, ) ( s, w) i d 2 i d 2 i a 2 i a 2 s w t 2 2 s w 2 2 e e cos t cos e. e f ( t, ) ddt 2i 2i where, i d 2 i a 2 s t 2 2 s KC ( t, ) e..cos t 2i when 0 i 2 cds 2 de ( tds) when =0 & i d 2 i a 2 w 2 2 w KC (, w) e..cos 2 2i when 0 i 2 ( cdw ) 2 d e ( dw) where 0 where, K ( t, s) K (, w) sup D D K ( t, s) K (, w). k E k C C2 t t C C2 3. VAROUS TESTNG FUNCTON SPACES: n this section severa spaces consisting of infinitey differentiae function are defined on the first and second quadrants of coordinate pane. 3. The space CC : t is given y k q sup a, t. Dt D t, CC : E /, k, q t, Ck, q A. (3.) 233

3 nternationa Journa of Engineering Research and Genera Science Voume 2, ssue 5, August-Septemer, 204 SSN The constant C k,q and A depend on. 3.2 The space CC : a, K q k k CC. E /, k, q t, sup t Dt D t, C, qb k (3.2) The constants C q, and B depend on. 3.3 The space CC, : This space is formed y comining the condition (3.) and (3.2) CC : E / t, t D D t, C A B k (3.3) a,, sup q k k k, a,, q,, k t, k, q 0,,2... Where A,B,C depend on. n net we have introduced suspaces of each of the aove space that are used in defining the inductive imits of these spaces. 3.4 The space CC, m :t is defined as, CC : E / t, t D D t, C m (3.4) a, sup k q, m a,, q,, k t k, q For any 0 where m is the constant, depending on the function. 3.5 The space CC :This space is given y, n a, sup k q k k, n a,, q,, k t,, q CC : E / t, t D D t, C n k (3.5) For any 0 where n the constant is depends on the function. 3.6 The space CC a,,, n, m : This space is defined y comining the conditions in (3.4) and (3.5)

4 nternationa Journa of Engineering Research and Genera Science Voume 2, ssue 5, August-Septemer, 204 SSN k k C m n k C : E / t, t D D t, a,,, n sup k q, m, k, q t. (3.6) For any 0, 0 and for given m > 0, n > 0 uness specified otherwise the space introduced in (3.) through (3.6) wi henceforth e consider equipped with their natura, Hausdoff, ocay conve topoogies to e denoted respectivey y, T, T, T, T, T, T m a, a, a,,, a, a, a,,, n, m, n, These topoogies are respectivey generaized y the tota famiies of seminorms. a,, q,, k,,,, and a,, q,, k a,, q,, k a,, q,, k a,, q,, k a,, q,, k 4 SOME BOUNDED OPERATORS N S-TYPE SPACES: This section is devoted to the study of different types of inear operators, namey, shifting operator, differentiation operator, scaing operator, in the, CC space. These operators are found to e ounded (continuous aso) in the CC,., Proposition 4.: f t, CC, and is fied rea numer then t, C, t 0 Proof: Consider, t, sup t D q D k t, k t ' k q ' t, sup t D D t, ' where t t k t CA B k k k, thus t, C, for t 0. Proposition 4.2: The transation (shifting) operator T : t, t, is a topoogica automorphism on C, for t 0. t, C and 0, Proposition 4.3: f t, C, stricty positive numer then 235

5 nternationa Journa of Engineering Research and Genera Science Voume 2, ssue 5, August-Septemer, 204 SSN sup k q Proof: Consider t t D D t k t,,, sup T k t sup k q T q D D T, C T D D T, Where C is constant depending on k k 2. C C A B k. y k k CA.. B K, where C = C C 2, Thus t, C for 0 Proposition 4.4: f 0, and, t C then the scaing operator. R : C a,, a,, C defined R Where t, t, is a topoogica automorphism. Proposition 4.5:The operator t, Dt, is defined on the space t, C and transform this space into itsef, Proof: Let t, C,if Dt, t, t we have, sup k q sup k q,, k t Dt D t, sup q k t t D D t, CA B k k k t Dt D D t t 236

6 nternationa Journa of Engineering Research and Genera Science Voume 2, ssue 5, August-Septemer, 204 SSN ,, t C 5 TOPOLOGCAL PROPERTES OF CC - SPACE: This section is devoted to discuss the resut on the topoogica structures of some of the spaces and the resuts ehiiting their reationship. Then attention is aso paid to e strict inductive imits of some of these spaces. a, a, Theorem 5.: CC, T is a Frechet space Proof: As the famiy competeness of the space CC A of seminorms, k, q, a,, k, q 0 a, a,, T. generating T is countae it, suffices to prove the n Let us consider a Cauchy sequence in CC. Hence for a given 0 there eist an N N, k, q such that for m, n N sup q k t D D (5.) a,,, k, q m n t m n n particuar for k q 0, m, n N,, sup m t n t (5.2) Consequenty for fied t in t, is a numerica Cauchy sequence. et t, e the point wise imit of t, using (5.2) we can easiy deduce that t, m m converges to uniformy on. Thus is continuous moreover, repeated use of (5.) for different vaues,k,q, yieds that is smooth i.e. E further from,(5.) We get, mn a,, q, k m a,,, k, q N, y Ckq, A E 237

7 nternationa Journa of Engineering Research and Genera Science Voume 2, ssue 5, August-Septemer, 204 SSN taking m and is aritrary we get, sup q k a,,, k t D Dt t, C A k Hence CC and it is the T imit of m y (5.). This proves the competeness of a, a, CC and CC, T is a Frechet space. Proposition 5.2: f m m 2 then C a, a,, m C, m. 2 The topoogy of, m C is equivaent to the topoogy induced on C, m y C, m 2 a, a, a,, m, m, m i. e T ~ T / C 2 Proof: For C, m and, a,,, k Ck, m Ck, m2 thus, C a, a, m C m,, 2 The second part is ceary from the definition of topoogies of these spaces. The space, C can e epressed as union of countay normed spaces. 6. CONCLUSON: n this paper two-dimensiona canonica cosine is generaized in the form the distriutiona sense, and proved some operators on these spaces aso discussed the topoogica structure of some of the S-type spaces

8 nternationa Journa of Engineering Research and Genera Science Voume 2, ssue 5, August-Septemer, 204 SSN REFERENCES: [] Ameida Lufs B., The fractiona Fourier transform and time frequency, representation EEE trans. On Signa Processing, Vo. 42, No., Nov [2] Ameida Lufs B., An introduction to the Anguar Fourier transform, EEE, 993. [3] Gefand.M. and Shiov G.E., Generaized Functions, Voume Academic Press, New York, 964. [4] Gefand.M. and Shiov G.E., Generaized Functions, Voume, Academic Press, New York, 967. [5] Namias Victor., The fractiona order Fourier transform and its appication to quantum mechanics, J. nst Math. Apptics, Vo. 25, P , 980. [6] Zemanian A.H., Distriution theory and transform anaysis, McGraw Hi, New York, 965. [7] Zemanian A.H., Generaized integra transform, nter Science Puisher s New York,