Finite Fourier Decomposition of Signals Using Generalized Difference Operator

Transcription

1 SCIENTIFIC PUBLICATIONS OF THE STATE UNIVERSITY OF NOVI PAZAR SER. A: APPL. ATH. INFOR. AND ECH. vo. 9, (27, Finite Fourier Decoosition of Signas Using Generaized Difference Oerator G. B. A. Xavier, B. Govindan, S. J. Borg,. eganathan Abstract: In this aer, we introduce discrete inner roduct of two functions, discrete orthogona orthonora syste of functions deveo finite Fourier series for oynoia factoria, oynoia, exonentia, rationa ogarith functions using the inverse of generaized difference oerator Keywords: Discrete inner roduct, Discrete orthonora syste, Finite Fourier series Generaized difference oerator. Introduction In Fourier anaysis, a signa is decoosed into its constituent sinusoids. In the reverse by oerating the inverse Fourier transfor, the signa can be synthesized by adding u its constituent frequencies. any signas that we encounter in daiy ife such as seech, autoobie noise, chirs of birds, usic etc. have a eriodic or quasi-eriodic structure, that the cochea in the huan hearing syste erfors a kind of haronic anaysis of the inut audio signas in bioogica hysica systes []. The Fourier series decooses the given inut signas into a su of sinusoids. By reoving the high frequency ters(noise of Fourier series then adding the reaining ters can yied better signas [4]. Finite Fourier series is a owerfu too for attacking any robes in the theory of nubers. It is reated to certain tyes of exonentia trigonoetric sus. It ay therefore be exed into a finite Fourier series of the for f (α µ = j= α a j α b j = j= g( jα µ j (µ =,,,. The orthogonaity reation (a b(od, (a b(od, enabes us to deterine the finite Fourier coef- anuscrit received arch 2, 26; acceted August 29, 26. G. B. A. Xavier, B. Govindan, S.J. Borg,. eganathan are with the Deartent of atheatics,sacred Heart Coege, Tiruattur , Veore District, Tai Nadu, S. India 47

2 4 G. B. A. Xavier, B. Govindan, S.J. Borg,. eganathan f (α µ α µk [2]. If we are µ= ficients g(k exicity by eans of the forua g(k = given k distinct coex nubers z,z,,z k, then there is one ony one oynoia P(x = ζ + ζ x + + ζ k x k satisfying the equations P(ω ν = z ν (ν =,,,k []. A finite Fourier series: η(t = A + N/2 q= A q cos(qσ t + N/2 B q sin(qσ t, where q= η = sea surface eevation (, t = tie (s, A = recond ean (, N = tota nuber of saing oints, A q B q = Fourier coefficients (, q = haronic coonent index (in the frequency doain σ = fundaenta radian frequency, is used in [6]. The su of N sine waves defined over the tie interva, t T : y = N a n cos(ω n t +ϕ n, t n T, a n, ϕ n < 2π, where a n is aitude t is tie, is aso a finite Fourier series[3]. In [7], the authors describe an efficient foruation, based on a discrete Fourier series exansion, of the anaysis of axi-syetric soids subjected to non-syetric oading. They have discussed the Fourier series aroach, the discrete Fourier series reresentation robes such as the resence of Gibb s henoenon the ack of confority of eeents. Here we arrive a new tye of finite Fourier series for functions (signas by defining discrete orthonora faiy of functions using inverse generaized difference oerator. This finite Fourier series becoes Fourier series as tends to zero. Suitabe exaes verified by ATLAB are inserted to iustrate the findings. 2 Preiinaries The riitive N th roots of unity (z N = but z r ; < r < N z n = e i(2π/nn, n =,2,3,...,N, ( where n N are co-rie, satisfies the geoetric series exressed as i f N = N k= z k n = z k n N = zn n k= z n = i f N >. (2 Fro ( (2, the coex discrete-tie sequence e r (k is defined as e n (k = (z n k = e i(2π/nnk ; n, k =,,2,...,N. (3 For the ositive integers n, r N, the e n (k defined in (3 satisfies the identity N i f n = rn N k= e n (k = e n (k N = k= e i(2πn/nk N k= = i f n rn. (4

3 Finite Fourier Decoosition of Signas Using Generaized Difference Oerator 49 This atheatica roerty is utiized with the factorization into two orthogona exonentia functions, e n (k} satisfying e n (ke (k N = 2π(n k k= i( e N N = N i f n = rn (5 k= i f n rn, where,n r are integers, the notation ( reresents the coex conjugate. The equation (5 induces us to define a generaized discrete orthonora syste a finite Fourier series by reacing by e n (k by u n (k. Nonexistence of soutions of certain tye of second order generaized α-difference equation with the oerator α( has been discussed in [9]. When α = the oerator α( becoes the generaized difference oerator. Definition [] Let u(k, k [,, be a rea or coex vaued function > be fixed. Then, the generaized difference oerator on u(k is defined as u(k = u(k + u(k, (6 its inverse is defined as if there is a function v(k such that v(k = u(k, then v(k = u(k + c j, f or k where c j is constant, j = k [k/] [k/] is the integer art of k/. } j + r, (7 Lea [] Let s r Sr are the Stiring nubers of first second kinds, k ( =, k ( = k k ( = k(k (k 2 (k (. Then we have k ( = s r r k r, k = k ( = k(+ ( +, k = Sr r k (r, k ( = (k ( ( S r r k (r (r +. (9 Lea 2 [5] Let be rea, >, k (, 2π. Then, we have sin (k sin k sin k = + c j, ( 2( cos cos (k cos k cos k = + c j ( 2( cos (u(kw(k = v(k w(k ( w(k + u(k. (2

4 5 G. B. A. Xavier, B. Govindan, S.J. Borg,. eganathan Reark Fro ( (, we have sin k 2π = = cos k 2π. (3 Lea 3 [] If u(k is a bounded function on [a,b] = b a, then we have In genera, when k (,, we exress u(k k k [ ] = j u(k b = a r = u(b [ k ] u(k r = u(a + r. (4 u( j + r, j = k [k/]. (5 3 Discrete Orthogona Syste Finite Fourier Series Since a finite Fourier series is described by a faiy of discrete orthonora functions, we introduce a orthogona further a orthonora syste of functions by defining the discrete inner roduct. Definition 2 Let u(k v(k be bounded functions defined on [a,b] = b a. The discrete inner roduct of u v with resect to is defined as (u,v = u(kv (k b = u(a + rv (a + r (6 a the discrete nor of u, denoted by u ( is defined as u ( = (u,u /2 = u(k 2 b} /2 = a } /2 u(a + r 2. (7 Definition 3 Let I = [a, b], = b a S = ϕ,ϕ,ϕ 2,...ϕ } be a coection of bounded coex vaued functions defined on I. If (ϕ n,ϕ = whenever n, the coection S is said to be a discrete orthogona syste, if in addition ϕ n = for each n, then S is said to be discrete orthonora.

5 Finite Fourier Decoosition of Signas Using Generaized Difference Oerator 5 Definition 4 Let S = ϕ,ϕ,ϕ 2,...ϕ } be an orthonora on I = [a,b], = b a, u(k is a bounded function on I c n = (u,ϕ n, (say finite Fourier coefficients. Then the finite Fourier series of u(k reated to S is defined as u(k = n= } c n ϕ n (k, k a + r. ( Exae Let I = [a,a + 2π], = π N (here = 2N By (3 ϕ (k = 2π, ϕ 2n (k = sinnk π, ϕ 2n (k = cosnk π, n =,2,, N. (9 k ( a+2π a = 2π, we find that S = ϕ,ϕ,ϕ 2,...ϕ 2N } is a syste of discrete orthonora functions on I. Fro (, the finite Fourier series reated to (9 is given by where a n = π u(k = a N 2 + (a n cosnk + b n sinnk + a } N 2N 2 cosnk, k a + r, (2 (u(k cos nk a+2π a Theore 3. Let u(k = c n ϕ n (k, k n=, b n = π u(ksinnk reative to a discrete orthonora set S. Then we have a+2π a are got by (4. } a + r be the finite Fourier series of u(k c n 2 = u 2 ( (Discrete Parseva s Forua (2 n= Proof Since S is orthonora is inear, (2 foows fro (6, (7, ( the Definition 3. Theore 3.2 Let k (, >. If n 2π, then we have k ( cosnk = k ( sinnk = t= t= ( (t t + ( t k ( t cosn(k + r r ( r ( 2(cosn (22 ( (t t + ( t k ( t sinn(k + r r ( r ( 2(cosn. (23

6 52 G. B. A. Xavier, B. Govindan, S.J. Borg,. eganathan Proof Taking u(k = k (, w(k = cosnk in (2 using (9 (, we get ( ( k ( cos nk = k ( cosn(k cosnk ( cosnk cosn(k +. 2( cosn 2( cosn Since is inear, aying ( for both cosnk cosn(k +, we get ( ( k ( cos nk = k ( cosn(k cosnk 2( cosn ( cosn(k 2cosnk + cosn(k + (2( cosn 2. (24 Taking u(k = k (2, w(k = cosnk in (2, using (9, ( (24, we get ( k (2 cos nk = k(2 (cosn(k cosnk 2( ( cosn 2k ( cosn(k 2cosnk + cosn(k + (2( cosn 2 ( ((2 cosn(k 3cosnk + 3cosn(k + cosn(k (2( cosn 3 (25 which can be exressed as k (2 cosnk = 2 ( (2 (t t k (2 t cosn(k + r r t= ( r ( 2(cosn. Continuing the above rocess, we get the reation (22. Now, (23 foows by reacing cosnk by sinnk in (22. Coroary When I = [, 2π], = π N, k r b n for the oynoia factoria k ( a n = π b n = π k ( k ( cos nk sin nk a = π 2π = t= 2π = t= are given by } 2N k ( 2π = (2π(+, the finite Fourier coefficients a n π( +, (26 ( (t t + ( t (2π ( t cosn(r r N( r ( 2(cosn (27 ( (t t + ( r t (2π ( t sinn(r N( r ( 2(cosn. (2

7 Finite Fourier Decoosition of Signas Using Generaized Difference Oerator 53 Proof The roof foows by aying the iit to 2π in (22 (23, then utiying by /π. Exae 2 Fro (2, using (26, (27 (2 for a, a n b n resectivey, we get the finite Fourier series for the oynoia factoria k ( as k ( = a N 2 + (a n cosnk + b n sinnk + a N 2 cosnk, k } 2N r. (29 In articuar, when N = 2, = π = 5, (29 becoes, 2 (k (5 π/2 = (2π6 π/2 + 9 (a n cosnk + b n sinnk + a 2 rπ } 39 32π 2 cos2k, k 2. Theore 3.3 Let k (,, > n 2π, then we have k cosnk = k sinnk = = = t= t= ( t + S ( (t k( t cosn(k + r r ( r t ( 2(cosn (3 ( t + S ( (t k( t sinn(k + r r ( r t ( 2(cosn. (3 Proof The roof foows by second ter of ( aying (22. Coroary 2 When I = [, 2π], = π N, the finite Fourier coefficients a n b n for n =,,2,,N for oynoia k are given by a n = π b n = π k cosnk k sinnk 2π 2π = = = = t= t= ( t + S ( (t (2π( t cosn(r r ( r N t ( 2(cosn (32 ( t + S ( (t (2π( t sinn(r r ( r N t ( 2(cosn. (33 Proof The roof foows by aying the iits to 2π in (3 (3, then utiying by /π.

8 54 G. B. A. Xavier, B. Govindan, S.J. Borg,. eganathan Coroary 3 Fro (2, using (32 (33 for a n b n resectivey, we get the finite Fourier series for the oynoia k as k = a N 2 + (a n cosnk + b n sinnk + a } N 2N 2 cosnk, k r. (34 Coroary 4 Let I = [,2π], = π } 2N r N, c > be a constant. Then, for k, the finite Fourier series of the geoetric function c k is given by c k = 2π ( c 2π c + π N [2π/] sinnk (cosnk [2π/] c (2π r cosnr c (2π r sinnr + [2π/] 2π cosnk c (2π r cosnr. (35 Proof The roof foows by taking u(k = c k in (2, then aying (5. The foowing exae is a verification of Coroary 4 Exae 3 Taking c =, N = = k = π in (35, we have ( 2 (2π r(π/ cos nrπ 2π 2( π/ (2π r(π/ sin nrπ sinnk cosnk cosrπ cosk, k rπ } 99. Theore 3.4 Let I = [2π,4π], (,, = π. Then the finite Fourier series of N rationa function is given by k k = [2π/] 2π (4π r + N ( [2π/] cos nr π (4π r cosnk [2π/] sin nr (4π r sinnk + [2π/] 2π cos Nr } 2N (4π r cosnk, k 2π + r. (36 Proof Taking a = 2π, u(k = in (2 aying (5, we get (36. k

9 Finite Fourier Decoosition of Signas Using Generaized Difference Oerator 55 The foowing exae is a verification of the Coroary 3.4. Exae 4 Taking = 5, N = 24, = π in (36, then we have 24 k 5 = 4 4 (4π r(π/ ( 4 cos nr(π/24 24 (4π r(π/24 5 cosnk 4 sin nr(π/24 (4π r(π/24 5 sinnk + 4 cosrπ 4 (4π r(π/24 5 cos24k, k 2π + rπ } Theore 3.5 Let I = [π/,7π/], = π. Then, the finite Fourier series of the ogarithic function ogk is given N by ogk = 2π [2π/] og( 7π r + π + [2π/] 2π + [2π/] N ( [2π/] og( 7π 7π rcosn( rcosnk og( 7π 7π rsinn( rsinnk og ( 7π r cosn ( 7π π } 2N r cosnk, k + r. (37 Proof The roof foows by taking a = π, u(k = ogk in (2 aying (5. The foowing exae is a verification of the Coroary 3.5 Exae 5 Taking N =, = π } 5 in (37, for k π+rπ 7 ( 6 og( 7π rπ cosn( 7π rπ cosnk + 6 og( 7π rπ sinn( 7π rπ sinnk 2, N = = π in (34, we have k 2 = 3π ( 2π 2 2 ( + 6 cosn(π/ 6, we have ogk = 6 6 og( 7π rπ + og( 7π rπ cos( 7π rπ cosk. Taking = cosnk + 22π2 sinn(2π/ 2( cosn(π/ sinnk 2π2 rπ 242 cosk, k } 2. Here we rovide ATLAB coding for verification FS: sys n i. 2 = ((3 i. 2./242+sysu((( 2 i. 2./2 (./( cos(n i./ cos(n i + ((22 i. 2./2 (sin(n 2 i././( cos(n i./ sin(n i,n,, (2 i. 2./242 (./( cos(i cos( i

10 56 G. B. A. Xavier, B. Govindan, S.J. Borg,. eganathan Discussion: The diagras 3.2 to 3.4 give 9 coonents of the decoosition of the function u(k = k 2 (inut signa. One can get the reaining 3 coonents easiy. 4 Concusion: The Fourier series its transfors have wide range of aications seciay in the fied of digita signa rocess. For functions which have no usua Fourier series exression, we are abe to find finite Fourier series exression (decoosition using suation soution of generaized difference equation. The ethod discussed in this aer eads to severa aications in signa rocess. References [] R.P. AGARWAL, Difference Equations Inequaities, arce Dekker, New York, 2. [2] ALBERT LEON WHITEAN, Finite Fourier Series Cyctoy, Nationa Acadey of Sciences, Vo. 37, No. 6 (Jun. 5, 95,

11 Finite Fourier Decoosition of Signas Using Generaized Difference Oerator 57 [3] R.D.BLEVINS, Probabiity Density of Finite Fourier Series with Ro Phases, Journa of Sound Vibration (997 2(4, [4] V. Britanak K. R. Rao, The Fast Generaized discrete Fourier Transfors: A unified Aroach to the Discrete Sinusoida Transfors Coutation, Sigana Processing, vo. 79,. 35-5, Dec 999.c [5] G.BRITTO ANTONY XAVIER, S.SATHYA, S.U.VASANTHAKUAR, -Series of the Generaized Difference Equation to Circuar Functions,Internationa Journa of atheatica Archive- 4(7, (23, [6] DENNIS J.WHITFORD AND ARIO E.C. VIEIRA, Teaching tie series anaysis.i.finite Fourier anaysis of ocean waves, Aerican Journa of Physis.69(4, Ari 2. [7] J.Y. LAI AND J.R. BOOKER, Aication of Discrete Fourier Series to the Finite Eeent Stress Anaysis of Axi-Syetric Soids, Internationa Journa for Nuerica ethods in Engineering, Vo.3,69-547, 99. [].ARIA SUSAI ANUEL, G.BRITTO ANTONY XAVIER AND E.THANDAPANI, Theory of Generaized Difference Oerator Its Aications, Far East Journa of atheatica Sciences, 2(2 (26, [9].ARIA SUSAI ANUEL, G.BRITTO ANTONY XAVIER, D.S.DILIP AND G.DOINIC BABU, Nonexistence of Soutions of Certain Tye of Second Order Generaized α-difference Equation in 2 (α( c (α( Saces, Southeast Asian Buetin of atheatics (26 4: [] I.J.SCHOENBERG, The Finite Fourier Series Eeentary Geoetry, atheatica Association of Aerica, Vo. 57, No. 6 (Jun. - Ju., 95, [] STEVEN W. SITH, The Scientist Engineer s Guide to Digita Signa Processing, Second Edition, Caifornia Technica Pubishing San Diego, Caifornia, 999.