Comlex exoetial A discrete time sigal may be comlex valued I digital commuicatios comlex sigals arise aturally A comlex sigal may be rereseted i two forms: jarg { z( ) } { } z ( ) = Re { z ( )} + jim { z ( )} = z ( ) e ; arg { ( )} ta Im ( ) z z = Re z ( ) Classificatio of D sigals Periodic ad aeriodic sigals: A sigal x( ) is said to be eriodic if, for some ositive real iteger, x( ) = x( + ) Eergy sigals ad ower sigals: A sigal x( ) is said to be a eergy sigal if its eergy is fiite E = lim x ( ) [ P = ] If the average ower of a sigal is fiite the the sigal is called ower sigal P = lim x( ) Simle maiulatio of D sigals [ E = ] + { } he time shiftig ad time reversal oeratio is show i Figure (b) ad (c) above Figure (d) shows dow samlig oeratio ad Figure (e) shows u samlig oeratio Mathematically, Dow samlig: f ( ) = x( ), where is a iteger f ( ) is formed by taig every th samle of x( ) x ( / M) =, ± M, ± M, U samlig: f( ) = Here there will be M cosecutive zeros of f ( ) otherwise betwee cosecutive samles of x( ) Shiftig, reversal ad time scalig oeratios are order deedet herefore, oe eeds to be careful i evaluatig comositios of these fuctios
Aalysis of D systems x( ) y( ) [ ] is used to rereset a geeral system he iut sigal x( ) is trasformed ito a outut sigal y( ) through the trasformatio [ ] Classificatio of systems Liear system: If x( ) y( ) ad x( ) y( ) the for a liear system, ax ( ) + ax ( ) ay ( ) + ay( ) ime ivariat system: If x( ) y( ) the for a time ivariat system, x( ) y ( ) 3 BIBO stable system: A system is BIBO stable if ad oly if wheever the iut sigal is bouded as x( ) Afor all, the outut is also bouded as y ( ) Bfor all If this is violated by the iut outut air the system is ot stable 4 Causal system: A system is causal if the outut y ( ) at ay time is ideedet o ay value of x( + m) for m > Mamoryless system: A system is said to be memoryless if the outut at ay time = deeds oly o the iut at time = 6 Ivertible system: A system is said to be ivertible if the iut to the system may be uiquely determied from the outut Examle π j π 6 A Determie the eriodicity of the sigals, i) x( ) = e cos( ) ad ii) x( ) = cos( π ) 7 π π j j 6 6 π π i) x ( + ) = e e cos + 7 7 ; For eriodicity, π π = π ad = π 6 7 7 = 3 = 34 = ; For = 7, = 3 7 = 44 [As] 6 So, the eriod of the sigal is, 44 ii) = π = 6 herefore, = 6 [As] π B Fid the eve ad odd art of the sigal x( ) = α u( ) Hits: xe ( ) = [ x( ) + x( ) ], etc
π C Is the system y ( ) = x ( )si liear? Let, x( ) y( ) ad x( ) y( ) he for x( ) = ax( ) + ax( ), π π π y( ) = [ ax ( ) + ax ( ) ] si = ax ( )si + ax ( )si = ay ( ) + ay ( ) As the liearity roerty holds, the system is liear D Is the system y ( ) = x ( ) x ( ) causal? For =, y( ) = x( ) x() As ydeeds ( ) o the future value of x( ), the system is ocausal x ( ) E Is the system y ( ) = e / x ( ) stable? Let x( ) = ( ) hus, y ( ) = hus the system is ot stable F Is the system y ( ) = x ( ) x ( ) ivertible? Let, x( ) = x ( ) he, y( ) = x( ) x( ) Agai let, x( ) = x ( ) + C he, y ( ) = x( ) x( ) I both cases the outut is the same Hece, the system is ot ivertible G Is the system y ( ) = x ( ) time ivariat? Let, x( ) y( ) ad x( ) = x( ) he, y( ) = x( ) = x( ) If the system is time ivariat, y ( ) = ( ) x( ) Here, y( ) y( ) Hece the system is time variat [As] Resose of LI systems to arbitrary iuts: he covolutio sum We deote the resose of a system to uit samle sequece as imulse resose, h() h ( ) = [ ( )] For a time ivariat system, h ( ) = [ ( )] + + ow, y( ) = [ x( )] = = [ x ( ) ( )] = x ( ) [ ( )] = + + y ( ) = xh ( ) ( ) = x( ) h( ) = h( ) x( ) = h( ) x( ) = = he above oeratio is called covolutio oeratio Stes to erform covolutio: Chage the variable from to : x( ), h( ) x( ), h( ) Perform the foldig oeratio: h ( ) h( ) 3 Perform the shiftig oeratio: h( ) h( ) y ( ) = xh ( ) ( ) = 4 Multilicatio: x( h ) ( ) = v ( ) Summatio: y ( ) = v ( ) = Examle If x()=u() ad h() = 8 u(), y()=? + 8 y ( ) = 8 u ( ) u ( ) = 8 = ; = = 8 x ( ) = {,,3,}, h ( ) = {,,, }, y ( ) =? x ( ) = {,,3,}, h ( ) = {,,, } ; h( ) = {,,,} Lower limit of yis ( ) = ad the uer limit is 3+= ow, y( ) = = ; 3 y() = = + = 4 ; 3 [As] y() = = + 4 + 3 = 8 3
y() = = + + 6+ = 8; y(3) = = 3; 3 3 Similarly, y () = ; y (6) = hus, y ( ) = {,4,8,8,3,, } [As] 3 Setch y ( ) = x( 3), ad y ( ) = x(8 3 ) for x ( ) = (6 )[ u ( ) u ( 6)] y(4) = = 3 Proerties of covolutio Commutative roerty: x( ) h( ) = h( ) x( ) Associative roerty: {() x h()} h() = x() { h() h()} 3 Distributive roerty: x( ) { h( )} + h( )} = x( ) h( ) + x( ) h( ) # x( ) ( ) = x( ) ad x( ) ( ) = x( ) # if x( ), h( ), the, y ( ) + For causal system ad causal iut the covolutio sum becomes, Some commo series y ( ) = xh ( ) ( ) = hx ( ) ( ) = =
Stability of LI systems y ( ) = hx ( ) ( ) ; Let, x( ) < A, the, y ( ) h ( ) x ( ) A = = herefore, the outut y() is always bouded if h ( ) is always bouded Examle: h ( ) = au ( ) = = = S = h( ) = a = ; a < a he sum is fiite for a < Hece, the system is stable System described by differece equatios Let, a system has h ( ) = α u ( ) herefore the resose of the system would be, y ( ) = hx ( ) ( ) = α x ( ) = = he first equatio may be writte as, y ( ) = x ( ) + α x ( ) Puttig =+, = y ( ) = α x ( ) = + ( ) = ( ) + α ( ) = ( ) + α α ( ) = = = x ( ) + α y ( ) y x x x x ie, y ( ) = x ( ) + α y ( ) he above equatio is called liear costat coefficiet differece equatio he geeral form of LCCDE is, M y ( ) = bx ( ) ( ) a ( ) y ( ) = = () where, the coefficiets a() ad b() are costats that defie the system if, all a ( ) s are ot zero the system is recursive If a()= the system is o recursive he iteger is called the order of the differece equatio or the order of the system Differece equatios rovide a way for comutig the resose of a system, y() to a arbitrary iut x() It is ecessary to satisfy a set of iitial coditios to solve the above equatio For examle, if x() begis at =, the solutio at time = deeds o the values of y( ), y( ),, y( ) Whe the iitial coditios are zero, the system is said to be i iitial rest he geeral solutio of a LCCDE system is give as, y( ) = yh( ) + y ( ) () he homogeeous solutio, yh( ) is the resose of the system to the iitial coditios assumig that the iut x ( ) = he articular solutio is the resose of the system to the iut x( ), assumig zero iitial coditios he homogeeous solutio may be foud by assumig a solutio of the form, y ( ) = z (3) Substitutig i the geeral equatio, Or, h = z + a( ) z = = z [ z + a( ) z ] =, z + a() z + a() z + + a( ) = (4) he olyomial i equatio (4) is called the characteristic olyomial It has roots If all the roots are distict the homogeeous solutio will be of the form, yh( ) = Az + Az+ + Az If there are m reeated roots, the solutio would be, m yh( ) = ( A + A + + A m ) z + A z + + Az j If there are two comlex roots, z, z = a+ jb= re ± θ, the solutio would be, y ( ) = r ( B cosθ + B cos θ) + A z + + A z h 3 3
Examle: Fid yh( ) for the system, y ( ) = y ( ) + y ( ) he characteristic olyomial, z z = ± z = + Hece, yh( ) = A + A ; Let the iitial coditios be y() =, y() = Alyig iitial coditios, + A+ A = ; A + A = A = A = + yh( ) = [As] Fid yh( ) for the system, y ( ) 4 y ( ) + 4 y ( ) = he characteristic olyomial, z 4z+ 4= ; z, z =, y ( ) ( ) h = A + A [As] 3 Fid yh( ) for the system, y ( ) y ( ) + y ( ) = ± 3 /4 he characteristic olyomial, z z ; z, z j e j π + = = ± = 3π 3π yh( ) = ( ) [ Acos + Asi ] [As] 4 4 For the articular solutio it is ecessary to fid the sequece y ( ) that satisfies the differece equatio for the give x( ) able below lists articular solutios for some commo iuts Examle: Fid the solutio of the differece equatio, y ( ) y ( ) = x ( ) () for x( ) = u( ) with y( ) = ad y( ) = For x( ) = u( ), y ( ) = C Substitutig this i the differece equatio we get, y ( ) y ( ) =, or, C C = ; C = 4/3 ow, the characteristic olyomial is, y ( ) = A() + A ( ) h z = z =± he total solutio is, y ( ) = A() + A( ) + 4/3; () he total solutio oly alies to, therefore, we have to derive a equivalet set of iitial coditios y () ad y() from the system equatio () y() = y( ) + x() = + = y() = y( ) + x() = + = Usig equatio (), = A+ A + 4 / 3; = A A + 4 / 3 A =, A = / y ( ) = () (/)( ) + 4 / 3; [As]
Fid the resose of the system y ( ) = y ( ) y ( ) + x ( ) + x ( ) to the iut x( ) = u( ) with iitial coditio y( ) = 7ad y( ) = he characteristic equatio of the system is, z z+ = z = ( ± j j /3 3) = e π jπ/3 jπ/3 hus, yh ( ) = Ae + Ae Ad, y( ) = C u( ); Substitutig this i the differece equatio we get, C = C C + + C = C 4C+ + ; C = jπ/3 jπ/3 herefore, y ( ) = Ae + Ae + ; y() = y( ) y( ) + x() + x( ) = A+ A + = Alyig iitial coditios, jπ /3 jπ /3 y() = y() y( ) + x() + x() = Ae + Ae + = jπ /3 e 3/4 A A j = jπ /3 jπ /3 e e A 7 A = 3 jπ /3 e + 3/4 + 3 π π( ) his yields, y ( ) = + si 3 si [As] 3 3 3 3 Fid uit samle resose, h ( ) of the system described as y ( ) = y ( ) y ( ) + x ( ) x ( ) 4 8 he imulse resose is the resose of a system with x( ) = ( ) ad iitial rest coditios 3 he characteristic equatio is, z z+ = ( z )( z ) z =, 4 8 4 4 Hece, y ( ) = A + A + ; 4 With iitial rest coditio, y( ) = y( ) =, it follows that, 3 y() = y( ) y( ) + x() x( ) = = A+ A 4 8 A =, A = 3 3 3 y() = y() y( ) + x() x() = = = A+ A 4 8 4 4 4 Hece, h ( ) = + 3 ;, or, h ( ) = + 3 u ( ) [As] 4 4 4 Fid the resose of the system of roblem 3 for the iut x ( ) = u ( ) u ( ) with zero iitial coditio Let, s( ) be the ste resose of the system he, y ( ) = s ( ) s ( ) ow, s( ) = h( ) u( ) = h ( ) = + 3 = = 4 + + (/ ) (/ 4) Or, s( ) = + 3 u( ) = (/) (/4) u( ) / /4 hus, y ( ) = (/ ) (/ 4) u ( ) (/ ) (/ 4) u ( ) [As] Fid h ( ) of the system described as: y ( ) 3 y ( ) + y ( ) = x ( ) + 3 x ( ) + x ( ) h ( ) 3 h ( ) + h ( ) = ( ) + 3 ( ) + ( ) z 3z+ = ; z, z =, h ( ) = C+ C u ( ) + C3 ( ) () ow, from the differece equatio, h()=; h()=3+3=6, h()=8 +=8
Puttig these i equatio () we get, = C+ C + C3 6= C+ C C = 6, C = 6, C3 = 8 = C+ 4C h ( ) = 6+ 6 u ( ) + ( ) = 6( ) u ( ) + ( ) [As] 6 Determie the resose y, ( ), of the system described by the secod order differece equatio y ( ) 3 y ( ) 4 y ( ) = x ( ) + x ( ) whe the iut sequece is x( ) = 4 u( ) Here, Ch Eq: z 3z 4= ( z+ )( z 4) = z, z =,4 herefore, yh( ) = c( ) + c(4) As the system root cotais oe characteristic root, y ( ) 3 (4) = c Puttig this i system equatio we get, c 3 (4) u ( ) 3 c3( )(4) u ( ) 4 c3( )(4) u ( ) = (4) u ( ) + (4) u ( ) For oe of the uit ste terms vaish We ca solve c3 for ay Use = 6 3c3 c3 = 4 c3 = 6 hus, the total solutio of the differece equatio is, y ( ) = c( ) + c(4) + 4 Assume that the iitial coditios, y( ) = y( ) = y() = c+ c = 3 y( ) + 4 y( ) + = 6 he, 4 c =, ad c = y() = c+ 4c + = 3 y() + 4 y( ) + 6 = 9 6 6 Hece, the zero state resose of the system is, y ( ) = ( ) + (4) + (4) ; [As] Correlatio of D sigals he correlatio of two sequeces is a oeratio defied by the relatio, r () l = x ( ) y ( l) = x ( + ly ) ( ) xy = = r ( l) = y( ) x( l) = y( + l) x( ) yx = = r () l = r ( l) he comutatio of correlatio sequece ivolves the same oeratio as covolutio excet for the foldig oeratio, rxy () l = x() l y( l) If y ( ) = x ( ), we have autocorrelatio: rxx () l = xx ( ) ( l) If x( ) ad ( ) = yare causal sequeces of legth, ( ),the, rxy () l = x( ) y( l), where, for l, i = l & = ad for l, i = & = l = i Alicatios: Correlatio measures the degree to which two sigals are similar his cocet is ofte used i radar, soar ad digital commuicatio Let, x( ) is the trasmitted sigal yis ( ) the received sigal which is the delayed versio of the iut sigal he, y ( ) = α x ( l) + w ( ) Correlatio of x( ) ad ywill ( ) be maximum at lag l from where we ca measure the distace of a target I commuicatio system the cocet of correlatio is used to determie whether the received sigal is zero or oe o trasmit zero we sed x ( ), L ; o trasmit oe we sed x ( ), L Sigal received by the receiver is, y ( ) = xi ( ) + w ( ); i=, he receiver comares ywith ( ) x ( ) ad x ( ) {atter available i receiver} to determie the sigal that better match y ( ) xy yx
Examle: Determie the correlatio betwee the two sequeces x( ) ad ygive ( ) below x ( ) = {,,3,7,,, 3,7,,, 3} y ( ) = {,,,,4,,,} Proerties of correlatio sequeces Let, x( ) ad yare ( ) two fiite eergy sequeces ow, the eergy of the combied sequece, ax( ) + y( l) : E = ax( ) + y( l) = a r () + r () + ar ( l) = { } xx yy xy rxx () rxy ( l) a his equatio ca be rewritte as: [ a ] rxy () l ryy () for ay fiite value of a rxx () rxy ( l) hus, the matrix rxy () l ryy () is ositive semi defiite his imlies, rxx () ryy () rxy ( l), ie, rxy() l rxx() ryy() = ExEy For, y ( ) = x ( ), r () l E xx hus autocorrelatio attais its maximum value at zero lag x Quadratic Forms A quadratic form q i variables x, x,, x is a liear combiatio of terms x, x,, x ad cross terms xx, xx 3, If =3, q = ax + ax + a33x3 + axx + axx+ a3xx3+ a3x3x+ a3xx3+ a3x3x his sum ca be writte comactly as a matrix roduct, q(x) = x Ax, x, A Whe A = A the, q (x) > for all x if ad oly if A is ositive defiite (all eigevalues are ositive) I this case q is called ositive defiite λ x x3 λ x x3 λi A = = λ x3 x ; = x3 λ x λ ( x+ x) λ+ ( xx x3) = ; λ would be ositive if ( xx x3) > ote that, a i the eergy equatio is a atteuatio factor herefore the roots of a must ot be comlex his imlies, rxx () ryy () rxy ( l) Detectio ad estimatio of eriodic sigals i oise he eriod of a sigal is first estimated by autocorrelatig the oisy sigal he oisy sigal is the cross correlated with a eriodic imulse trai of eriod equal to that of the sigal he resultig cross correlatio fuctio is the sigal estimate s( ) = x( ) + q( ) ; Period of x(), ad legth of sigal Let, ( ) be the eriodic imulse trai used for autocorrelatio Let be the umber of imulses used for correlatio rs ( j) = [ x( ) + q( )] ( j) ; =,,, = j For j=, rs () = [ x( ) + q( )] ( ); =,,, = j Or, rs () = { x() + q() + x( ) + q( ) + x( ) + q( ) + x( ) + q( ) } As x( ) is eriodic, x( ) = x( + ); hus, rs () = { x() + q() + q( ) + q( ) + + q( ) } / Or, rs () = x() + q( ), he secod term is zero = Similarly, for the other values of j, rs ( j) x( j), which is the required sigal