Module 4. Signal Representation and Baseband Processing. Version 2 ECE IIT, Kharagpur

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1 Module 4 Sigl Represettio d Bsed Processig Versio ECE IIT, Khrgpur

2 Lesso 5 Orthogolity Versio ECE IIT, Khrgpur

3 Ater redig this lesso, you will ler out Bsic cocept o orthogolity d orthoormlity; Strum - Lio; Slope overlod distortio; Grulr Noise; Coditio or voidig slope overlodig; The Issue o Orthogolity Let m (x) d (x) e two rel vlued uctios deied over the itervl x. I the product [ m (x) (x)] exists over the itervl, the two uctios re clled orthogol to ech other i the itervl x whe the ollowig coditio holds: m( x) ( x) dx= 0, m 4.5. A set o rel vlued uctios (x), (x) N (x) is clled orthogol set over itervl x i (i) ll the uctios exist i tht itervl d (ii) ll distict pirs o the uctios re orthogol to ech other over the itervl, i.e. ( x) ( x) dx = 0 i j, i =,, ; j =,, d i j 4.5. The orm m( x ) o the uctio m (x) is deied s, m( ) m( ) x = x dx A orthogol set o uctios (x), (x) N (x) is clled orthoorml set i, 0, m m( x). ( x) = 4.5.4, m= A orthoorml set c e otied rom correspodig orthogol set o uctios y dividig ech uctio y its orm. Now, let us cosider set o rel uctios (x), (x) N (x) such tht, or some o-egtive weight uctio w(x) over the itervl x m( x). ( x). w( x) dx= 0, m Do i -s orm orthogol set? We sy tht the i -s orm orthogol set with respect to the weight uctio w(x) over the itervl x y deiig the orm s, Versio ECE IIT, Khrgpur

4 m( ) m( ). ( ) x = x w x dx The set o i -s is orthoorml with respect to w(x) i the orm o ech uctio is. The ove extesio o the ide o orthogol set mkes perect sese. To see this, let m ( ) ( ) ( ) It is ow esy to veriy tht, g x = w x m x, where w(x) is o-egtive uctio m( x). ( x). w( x) dx = gm( x). g( x) dx = This implies tht i we hve orthogol i -s over x, with respect to oegtive weight uctio w(x), the we c orm usul orthogol set o i s over the sme itervl x y usig the sustitutio, g = w( x) x m m ( ) Altertively, orthogol set o g i -s c e used to get orthogol set o i -s with respect to speciic o-egtive weight uctio w(x) over x y the ollowig sustitutio (provided wx ( ) 0, x ): gm ( x) m ( x) = wx ( ) A rel orthogol set c e geerted y usig the cocepts o Strum-Liouville (S-L) equtio. The S-L prolem is oudry vlue prolem i the orm o secod order dieretil equtio with oudry coditios. The dieretil equtio is o the ollowig orm: d dy p( x) q( x) λω. ( x) y dx + + = dx 0, or x ; It stisies the ollowig oudry coditios: dy i) c + cy = 0 ; t x = ; dx dy ii) d + dy = 0 ; t x = ; dx Here c, c, d d d re rel costts such tht t lest oe o c d c is o zero d t lest oe o d d d is o zero. The solutio y = 0 is trivil solutio. All other solutios o the ove equtio suject to speciic oudry coditios re kow s chrcteristic uctios or eigeuctios o the S-L prolem. The vlues o the prmeter λ or the o trivil solutios re kow s chrcteristic vlues or eige vlues. A very importt property o the eige-uctios is tht they re orthogol. Versio ECE IIT, Khrgpur

5 Orthogolity Theorem: Let the uctios p(x), q(x) d ω(x) i the S-L equtio (4.5.0) e rel vlued d cotiuous i the itervl x. Let y m (x) d y (x) e eige uctios o the S-L prolem correspodig to distict eigevlues λ m d λ respectively. The, y m (x) d y (x) re orthogol over x with respect to the weight uctio w(x). Further, i p(x = ) = 0, the the oudry coditio (i) my e omitted d i p(x = ) = 0, the oudry coditio (ii) my e omitted rom the prolem. I p(x = ) = p(x = ), the the oudry coditio c e simpliied s, dy ' ' dy y( ) = y( ) d x= = y ( ) = y ( ) = x= dx dx Aother useul eture is tht, the eigevlues i the S-L prolem, which i geerl my e complex sed o the orms o p(x), q(x) d w(x), re rel vlued whe the weight uctio ω(x) is positive i the itervl x or lwys egtive i the itervl x Exmples o orthogol sets: Ex#: We kow tht, or iteger m d, 0, m cos mx.cos xdx = E4.5., m= 0, m si mx.si xdx = E4.5., m= d cos mx.si xdx = 0 E4.5.3 Let us cosider equtio E4.5. d rewrite it s: 0, m (cos mt).(cos t) dt = E4.5.4, m= y sustitutig x = t = ω t d dx = dt = ω dt Note tht the uctios cosmx d cosx re orthogol over the rge o the idepedet vrile x d its itegrl multiple, i.e. M., i geerl, where M is iteger. This implies tht equtio (E4.5.4) is orthogol i terms o the idepedet Versio ECE IIT, Khrgpur

6 vrile t over the udmetl rge d, i geerl, over M = M T 0, where T 0 idictes the udmetl time itervl over which cosmt d cost re orthogol to ech other. Now m d c hve miimum dierece i (cos mt).(cos t) dt = 0 E4.5.5 i.e., m = = So, i two cosie sigls hve requecy dierece, the we my sy, cos ( c + ) t.cos ( c ) tdt. = 0 E4.5.6 Re-writig equtio (E4.5.6) cos ( c + ) t.cos ( c ) tdt. = 0 where, = Lookig ck t equtio E4.5.5, we my write geerl orm or equtio (E4.5.6): cos ( c + p ) t.cos ( c p ) tdt. = 0 E where m = (+p) d p is iteger. Followig similr oservtios o equtio E4.5., oe c sy, si ( c + p ) t.si ( c p ) tdt. = 0 E4.5.8 Equtio E4.5.3 my lso e expressed s, cos ( c + p ) t.si ( c p ) td. t = si ( c + p ) t.cos ( c p ) td. t =0 E4.5.9 Let us deie s = cos c + p t, s = cos c p t, s3 = si c + p t d s4 si = c p t. C we use the ove oservtios o orthogolity to distiguish mog s i -s over decisio itervl o T 5 = T 0 =? Ex#: x (t) =.0 or 0 t T d zero elsewhere, x (t) =.0 or T t T d zero elsewhere, Versio ECE IIT, Khrgpur

7 Ex#3: x (t) =.0 or 0 t T d x (t) = -.0 or T < t T, while x (t) = -.0 or 0 t T Importce o the cocepts o Orthogolity i Digitl Commuictios. I the desig d selectio o iormtio erig pulses, orthogolity over symol durtio my e used to dvtge or derivig eiciet symol-ysymol demodultio scheme.. Perormce lysis o severl modultio demodultio schemes c e crried out i the iormtio-erig sigl wveorms re kow to e orthogol to ech other. c. The cocepts o orthogolity c e used to dvtge i the desig d selectio o sigle d multiple crriers or modultio, trsmissio d receptio. Orthogolity i complex domi Let, z() t = x() t + jy() t d z( t) = x( t) + jy( t) z () t + z () t z ( t) + z ( t) Now, x () t = d x () t = I x d x re orthogol to ech other over t, () () x t. x t dt = 0 z t z t z t z t dt = i.e., () + () () + () 0 or, z() t. z( t) + z( t). z ( t) + z ( t). z( t) + z ( t). z ( t) dt = 0 Let us cosider complex uctio z t = x t + jy t, t () () () = r() t cos Φ () t + jsi Φ( t) where, r() t = z () t, o egtive uctio o t. x() t = r() t cos Φ () t d y( t) = r( t) si Φ ( t) Now, () (). ( ).cos ( ).si x t y t dt = r t Φ t Φ( t) dt We kow tht cosθ & siθ re orthogol to ech other over - θ <, i.e., Versio ECE IIT, Khrgpur

8 cos θ.siθdθ = 0 So, usig costt weight uctio w = r, which is o-egtive, we my sy r cos θ.siθdθ = 0 Now, x = r cos θ d y = r si θ re lso orthogol over - θ <. Now, let θ e cotiuous uctio o t over - θ <. Ad, θ t = = θ = d θ t = = θ = Assumig lier reltioship, let, θ () t = t dθ () t = dt Uder these coditios, we see, r ().cos t Φ().si t Φ() t dt = r ().cos t Φ().si t () t d Φ Φ r.cos ().si t () t d = Φ Φ Φ = 0 i.e., x(t) d y(t) re orthogol over the itervl t or T T t So, i zt () = xt () + jyt () represets phsor i the complex ple rottig t uiorm requecy o, the x(t) d y(t) re orthogol to ech other over the itervl t or, equivletly T t T T where T =. i.e., xt (). ytdt () = 0 Now, let us cosider two complex uctios: z j () t = x () t + jy () t ( ) () t = z t e Φ d z j () t = x() t + jy() t ( ) () t = z t e Φ T T [x (t),y (t)] d [x (t),y (t)] re orthogol pirs over the itervl t. So, d z ( t) my e viewed s two phsors rottig with equl speed. T z ( t) Now, two sttic phsors re orthogol to ech other i their dot or sclr product is zero, i.e., AB. = A Bcosγ =. B + A. B = 0, where γ is the gle etwee A d B Ax x y y I geerl, two complex uctios z () t d z () t with iite eergy re sid to e orthogol to ech other over itervl t, i Versio ECE IIT, Khrgpur

9 * z (). t z () t dt = 0 Prolems Q4.5.) Veriy whether two sigls re orthogol over oe time period o the sigl with smllest requecy sigl. i) X (t) = Cos t d X (t) = Si t ii) X (t) = Cos t d X (t) = Cos (t + 3 ) iii) X (t) = Cos t d X (t) = Cos (4t + 4 ) iv) X (t) = Si 4t d X (t) = - Cos (t - 6 ) Versio ECE IIT, Khrgpur

Notes 17 Sturm-Liouville Theory

ECE 638 Fll 017 Dvid R. Jckso Notes 17 Sturm-Liouville Theory Notes re from D. R. Wilto, Dept. of ECE 1 Secod-Order Lier Differetil Equtios (SOLDE) A SOLDE hs the form d y dy 0 1 p ( x) + p ( x) + p (