Geeraliig the DTFT The Trasform M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 1 The forward DTFT is defied by X e jω = x e jω i which = Ω is discrete-time radia frequecy, a real variable. The quatity e jω is the a complex siusoid whose magitude is always oe ad whose phase ca rage over all agles. It always lies o the uit circle i the complex plae. If we ow replace e jω with a variable that ca have ay complex value we defie the trasform X. = x The DTFT expresses sigals as liear combiatios of complex siusoids. The trasform expresses sigals as liear combiatios of complex expoetials. = M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 2 Complex Expoetial Excitatio The Trasfer Fuctio M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 3 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 4 Systems Described by Differece Equatios Direct Form II Realiatio Direct Form II realiatio of a discrete-time system is similar i form to Direct Form II realiatio of cotiuous-time systems A cotiuous-time system ca be realied with itegrators, summig juctios ad multipliers A discrete-time system ca be realied with delays, summig juctios ad multipliers M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 5 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 6 1
Direct Form II Realiatio The Iverse Trasform M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 7 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 8 Existece of the Trasform Time Limited Sigals If a discrete-time sigal x[ ] is time limited ad bouded, the trasformatio summatio = x[ ] is fiite ad the trasform of exists for ay o-ero x value of. Existece of the Trasform Right- ad Left-Sided Sigals A right-sided sigal x r < 0 ad a left-sided sigal x l for ay > 0. is oe for which x r [ ] = 0 for ay [ ] is oe for which x l [ ] = 0 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 9 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 10 Existece of the Trasform Right- ad Left-Sided Expoetials x[ ] = α u[ 0 ], α x[ ] = β u[ 0 ], β Existece of the Trasform The trasform of x[ ] = α u[ 0 ], α is = α u 0 X = α 1 = if the series coverges ad it coverges if > α. The path of itegratio of the iverse trasform must lie i the regio of the plae outside a circle of radius α. = 0 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 11 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 12 2
Existece of the Trasform Existece of the Trasform The trasform of x[ ] = β u[ 0 ], β is = β X 0 = β 1 = β 1 = 0 = = 0 if the series coverges ad it coverges if < β. The path of itegratio of the iverse trasform must lie i the regio of the plae iside a circle of radius β. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 13 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 14 Some Commo Trasform Pairs δ [ ] 1, All u[ ] 1 = 1, > 1, u[ 1] 1 1 1, < 1 α u[ ] α = 1, > α, α u[ 1] 1 α 1 α = 1, < α 1 α 1 u[ ] ( 1) = 1 2 1 1 α u[ ] α ( α ) = α 1 2 1 α 1 si( Ω 0 )u[ ] 2, > 1, u 1 ( 1) = 1, < 1 2 ( 1 1 ) 2, > α, α u[ 1] 2 ( α ) = α 1, < α 2 ( 1 α 1 ) 2 + 1, > 1, si ( Ω 0 )u [ 1 ] si( Ω 0 ) 2 2 cos( Ω 0 ) + 1, < 1 + 1, > 1, cos Ω ( 0)u[ 1] = cos ( Ω 0 ) 2 2 cos( Ω 0 ) + 1, < 1, > α, α α si( Ω si( Ω 0)u[ 1] 0 ), < α + α 2 2 2α cos( Ω 0 ) + α 2, > α, α cos( Ω 0)u[ 1] α cos ( Ω 0 ), < α + α 2 2 2α cos( Ω 0 ) + α 2 si Ω 0 2 2 cos Ω 0 cos( Ω 0)u[ ] cos Ω 0 2 2 cos Ω 0 α si( Ω 0)u[ ] α si Ω 0 2 2α cos Ω 0 α cos( Ω 0)u[ ] α cos Ω 0 2 2α cos Ω 0 α u[ 0 ] u[ 1 ] α α 1 1 0 1, α < < α 1 = 1 0 1 + 1 0 2 + + + 1 α 1 1, > 0 Give the -trasform pairs g[ ] G ad h[ ] H with ROC's of ROC G ad ROC H respectively the followig properties apply to the trasform. Liearity + β h[ ] α g ROC H α G + β H Time Shiftig g 0 0 G except perhaps = 0 or Chage of Scale i -Trasform Properties α g[ ] G( / α ) ROC = α ROC G M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 15 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 16 Time Reversal -Trasform Properties g G 1 Covolutio -Trasform Properties g[ ] h[ ] HG Time Expasio ROC = 1 / ROC G, / k ad iteger g / k 0, otherwise 1/k Cojugatio g * G * * G( k ) First Backward Differece g Accumulatio g[ 1] ROC ROC G > 0 m= g[ m] 1 G ROC ROC G > 1 ( 1 1 )G -Domai Differetiatio g[ ] d d G Iitial Value Theorem Fial Value Theorem If g[ ] = 0, < 0 the g[ 0] = lim G If g[ ] = 0, < 0, lim g = lim( 1)G if lim g[ ] exists. 1 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 17 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 18 3
-Trasform Properties all the For the fial-value theorem to apply to a fuctio G fiite poles of the fuctio ( 1)G must lie i the ope iterior of the uit circle of the plae. Notice this does ot say that all the poles of G the uit circle. G must lie i the ope iterior of could have a sigle pole at = 1 ad the fial-value theorem could still apply. The Iverse Trasform Sythetic Divisio For ratioal trasforms of the form H = b M M + b M 1 M 1 + + b 1 + b 0 a N N + a N 1 N 1 + + a 1 + a 0 we ca always fid the iverse trasform by sythetic divisio. For example, = ( 1.2) ( + 0.7) ( + 0.4) ( 0.2) ( 0.8) ( + 0.5) H, > 0.8 H = 3 0.1 2 1.04 0.336 3 0.5 2 0.34 + 0.08, > 0.8 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 19 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 20 The Iverse Trasform Sythetic Divisio The Iverse Trasform Sythetic Divisio M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 21 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 22 The Iverse Trasform Sythetic Divisio Partial Fractio Expasio We ca always fid the iverse trasform of a ratioal fuctio with sythetic divisio but the result is ot i closed form. I most practical cases a closed-form solutio is preferred. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 23 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 24 4
Partial Fractio Expasio -Trasform Properties A LTI system has a trasfer fuctio = Y X = 1 / 2 H 2 + 2 / 9, > 2 / 3 Usig the time-shiftig property of the trasform draw a block diagram realiatio of the system. = X Y 2 + 2 / 9 ( 1 / 2) 2 Y = X ( 1 / 2) X + Y 2 / 9 = 1 X ( 1 / 2) 2 X + 1 Y 2 / 9 Y Y 2 Y M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 25 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 26 -Trasform Properties Y = 1 X ( 1 / 2) 2 X Usig the time-shiftig property y x 2 = x[ 1] 1 / 2 + 1 Y 2 / 9 + y[ 1] 2 / 9 2 Y y 2 Let g G -Trasform Properties = pole-ero diagram for G The poles of G 1 jπ /4 0.8e + jπ /4 ( 0.8e ). Draw a. ad for the trasform of e jπ/8 g are at = 0.8e ± jπ /4 ad its sigle fiite ero is at = 1. Usig the chage of scale property e jπ/8 g G e = jπ /8 G e jπ /8 G e jπ /8 = e jπ /8 jπ /8 0.8e e jπ /8 1 e jπ /8 jπ /4 0.8e e jπ /8 + jπ /4 ( 0.8e ) e jπ /8 jπ /8 ( e ) + j 3π /8 e jπ /8 0.8e e jπ /8 = e jπ /8 jπ /8 + j 3π /8 ( 0.8e )( 0.8e ) M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 27 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 28 -Trasform Properties jπ /8 G( e ) has poles at = 0.8e jπ /8 ad 0.8e + j 3π /8 ad a ero at = e jπ /8. All the fiite ero ad pole locatios have bee rotated i the plae by π /8 radias. -Trasform Properties Usig the accumulatio property ad u[ ] 1, > 1 show that the trasform of u[ ] is u = u[ m 1] u m=0 ( 1), > 1. 2 u[ 1] 1 1 = 1 1, > 1 = u[ m 1] m=0 1 1 1 = 1 2, > 1 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 29 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 30 5
Iverse Trasform Example Fid the iverse trasform of X = 0.5 + 2, 0.5 < < 2 Right-sided sigals have ROC s that are outside a circle ad left-sided sigals have ROC s that are iside a circle. Usig α u[ ] α = 1 1 α 1, > α α u[ 1] α = 1 1 α 1, < α We get ( 0.5) u[ ] + ( 2) u[ 1] X = 0.5 + 2, 0.5 < < 2 Iverse Trasform Example Fid the iverse trasform of X = 0.5 + 2, > 2 I this case, both sigals are right sided. The usig α u[ ] α = 1, > α 1 α 1 We get ( 0.5) ( 2) u X = 0.5 + 2, > 2 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 31 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 32 Iverse Trasform Example The Uilateral Trasform Fid the iverse trasform of X = 0.5 + 2, < 0.5 I this case, both sigals are left sided. The usig α u[ 1] α = 1, < α 1 α 1 We get ( 2) 0.5 = u 1 X 0.5 + 2, < 0.5 Just as it was coveiet to defie a uilateral Laplace trasform it is coveiet for aalogous reasos to defie a uilateral trasform = x X =0 6/19/12 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 33 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 34 Properties of the Uilateral Trasform If two causal discrete-time sigals form these trasform pairs, g G ad h[ ] hold for the uilateral trasform. Time Shiftig Delay: g 0 H the the followig properties 0 G, 0 0 Advace: g[ + 0 ] 0 G Accumulatio: m=0 g[ m] 1 G 0 1 g m m=0 m, > 0 0 Solvig Differece Equatios The uilateral trasform is well suited to solvig differece equatios with iitial coditios. For example, y[ + 2] 3 2 y[ +1] + 1 2 y[ ] = ( 1/ 4), for 0 = 10 ad y[ 1] = 4 y 0 trasformig both sides, 2 Y y 0 1 y[ 1] 3 2 Y y[ 0] the iitial coditios are called for systematically. + 1 2 Y = 1/ 4 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 35 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 36 6
Solvig Differece Equatios Applyig iitial coditios ad solvig, 16 / 3 Y = 1/ 4 + 4 1/ 2 + 2 / 3 1 ad y[ ] = 16 1 3 4 + 4 1 2 + 2 u[ ] 3 This solutio satisfies the differece equatio ad the iitial coditios. Pole-ero Diagrams ad Frequecy Respose For a stable system, the respose to a siusoid applied at time t = 0 approaches the respose to a true siusoid (applied for all time). M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 37 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 38 Pole-ero Diagrams ad Frequecy Respose Let the trasfer fuctio of a system be H = 1+ j2 p 1 = 4 H( e jω ) = 2 / 2 + 5 /16 = p 1 1 j2, p 2 = 4 e jω e jω p 1 e jω p 2 ( p 2 ) Pole-ero Diagrams ad Frequecy Respose M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 39 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 40 Trasform Method Compariso A system with trasfer fuctio H = ( 0.3) ( + 0.8), > 0.8 is excited by a uit sequece. Fid the total respose. Usig -trasform methods, Y Y = H X = ( 0.3) ( + 0.8) 1, > 1 2 = ( 0.3) ( + 0.8) ( 1) = 0.1169 0.3 + 0.3232 y[ ] = 0.1169( 0.3) 1 + 0.3232( 0.8) 1 + 0.7937 + 0.8 + 0.7937, > 1 1 u 1 Trasform Method Compariso M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 41 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 42 7
Trasform Method Compariso Trasform Method Compariso = 0.1169e jω Y e jω 1 0.3e jω Fidig the iverse DTFT, + 0.3232e jω 1+ 0.8e + 0.7937 e jω jω 1 e + πδ ( Ω) jω 2π = 0.1169( 0.3) 1 + 0.3232( 0.8) 1 + 0.7937 y u 1 The result is the same as the result usig the trasform, but the effort ad the probability of error are cosiderably greater. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 43 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 44 System Respose to a Siusoid A system with trasfer fuctio H = 0.9, > 0.9 is excited by the siusoid x. Fid the respose. = cos 2π /12 The trasform of a true siusoid does ot appear i the table of trasforms. The trasform of a causal siusoid of the form x[ ] = cos( 2π /12)u fid the respose to the true siusoid ad the result is y. = 1.995cos 2π /12 1.115 does appear. We ca use the DTFT to System Respose to a Siusoid Usig the trasform we ca fid the respose of the system to a causal siusoid x[ ] = cos( 2π /12)u[ ] ad the respose is y[ ] = 0.1217( 0.9) u[ ]+1.995cos( 2π /12 1.115)u[ ] Notice that the respose cosists of two parts, a trasiet respose 0.1217( 0.9) u[ ] ad a forced respose 1.995cos( 2π /12 1.115)u that, except for the uit sequece factor, is exactly the same as the forced respose we foud usig the DTFT. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 45 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 46 System Respose to a Siusoid M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 47 8