Chapter 3 -Trasform
3.0 Itroductio The -Trasform has the same role as that played by the Laplace Trasform i the cotiuous-time theorem. It is a liear operator that is useful for aalyig LTI systems such as for solvig LCCDE s. Types of -Trasform Oe-sided -trasform (Uilateral -T Two-sided -trasform (Bilateral -T I our textbook, -trasform refers to bilateral -trasform uless specified otherwise.
Motivatio : Limitatio of DTFT For some importat sigals, DTFTs defied above do ot coverge Example x[] a u[] for a > does ot have a DTFT impulse respose of a ustable system For certai cases for which DTFT does ot coverge, we ca relax the covergece requiremets, or allow impulses i DTFT 3
3. -Trasform The Fourier trasform of a sequece x[] is defied as jω ( X e x[ ] e jω The -trasform of a sequece x[] is defied as The -trasform pair is deoted by Z x[ ] X (. Relatioship with Fourier Trasform (if that exists. jω Z{ x[ ] } X( x[ ] ( re is a complex variable X e jω ( e X ( jω. r 4
Sice the -trasform is a fuctio of a complex variable, it ca be iterpreted ad described i the complex -pla. re jω X jω ( e r 0 π uit circle -trasform o uit circle <-> Fourier trasform Liear frequecy axis i Fourier trasform Uit circle i -trasform (periodicity i freq. of Fourier trasform If we plot e jω for ω0 to π we get the uit circle 5
Existece of -trasform ad DTFT jω Sice re, the -trasform becomes jω jω jω X( re x[ ]( re x[ ] r e Therefore, -trasform is the DTFT of [x[] multiplied with expoetial sequece r ] For certai sigals for which DTFT does ot coverge, i.e. there may exit a value r which makes the -trasform coverge, i.e. the sum i -trasform becomes fiite I { xr [ ] } x [ ] x [ ] r x[ ] < 6
Not all sigals have -trasforms Examples havig o -trasform Example: siω [ ] c x π Does't coverge for ay r. It has fiite eergy. DTFT exists oly i a mea square sese. Example: x [ ] cos( ω o Does't coverge for ay r. It does t have eve fiite eergy. But we defie a useful DTFT usig impulse fuctio.
Regio of Covergece (ROC The set of values of for which the -trasform coverges is called regio of covergece (ROC, ROC : x[ ] r x[ ] < Sice the covergece of -trasform depeds oly o r, the regio of covergece will cosist of a rig i the -plae cetered about the origi. Outer boudary, R +, is a circle (may exted to ifiity Ier boudary, R -, is a circle (may exted to iclude the origi If ROC icludes the uit circle, Fourier trasform coverges 8
Uilateral -Trasform Uilateral -trasform cosiders oly causal part of the sigal, i.e. 0 { } Z x [ ] X ( x [ ] + + For a causal sigal (bilateral -trasfrom (uilateral -trasfrom ROC types should be the etire -plae or outside of a disk, > r, for some r 0, so the iverse trasform is uique Ay causal sigal that grows o faster tha a expoetial sigal r 0, for some r 0 has uilateral -trasform Iformatio about x[] for < 0 is lost Used to solve LCCDE with arbitrary iitial coditios a.k.a. oe-sided -trasform 0 9
Zeros ad Poles (Defiitio oly 0
Example 3. Right-sided Expoetial Sequece Cosider the sigal x[ ] a u[ ] ROC ( ( X a a 0 0 0 Iside the ROC series coverges to By settig a a < a < > a X( Z{ a u[ ] } a Z { a } Pole-ero plot. Regio outside the circle of radius a is the ROC Foe a right-sided sequece, its ROC exteds outside a circle
Example 3. Left-sided Expoetial Sequece Cosider the sigal x[ ] a u[ ] X( a u[ ] ROC a a ( a 0 0 a < a < < a Iside the ROC series coverges to a a a ( X Pole-ero plot Regio iside the circle of radius a is the ROC Foe a left-sided sequece, its ROC exteds iside a circle
Example 3.5 Sum of Two Expoetials Cosider the sequece x [ ] u [ ] u [ ]. 3 3 Z u[ ] + 3 Z u[ ],, > ; 3 X( +, < < + 3 ( / ( /( + /3 < 3 Pole-ero plot 3
Example 3.6 Fiite-Legth Sequece Cosider the sigal a, 0 N x [ ] 0, otherwise X ( N 0 a N -( a a (ROC : 0 N 0 N ( a N -a a N Pole-ero plot (N6 Specifically, the N roots of the umerator polyomial (eros are at k j(πk / N ae, k 0,,..., N. 4
Some Commo -trasform Pairs
3. Properties of The ROC of -Trasform The -trasform X( (a power series does ot coverge for all sequece or for all values of. For ay give sequece, the set of values of for which the - trasform coverges is called the regio of covergece (ROC The ROC is a rig or disk cetered at the origi 0 rr < < rl Im Im Im a a Re Re a b Re Right-sided sigal (e.g. causal Left-sided sigal (e.g. ati-causal Two-sided sigal (o-causal 6
Importace of ROC The importace of ROC caot be overemphasied. It is part of the - trasform. I specifyig the -trasform X( of a sigal x[], the ROC must be give - otherwise, the iverse -trasform x[] caot be computed uiquely from X(. Example - Cosider the two sequeces a ; < 0 x[ ] 0; 0 X ( for < a a 0; < 0 y [ ] a ; 0 Y( for > a a Here it is importat to uderstad that X( Y(. The ROC s of these two trasforms do ot eve itersect. Not equal! X ( + ROC uique x[ ] 7
Some importat properties of ROC iclude: DTFT of a sequece x[] coverges uiformly if ad oly if the ROC of the -trasform of x[] icludes the uit circle The ROC caot cotai ay poles : o poles i ROC The ROC must be a coected regio
Example of ROC Cosider the discrete sigal It has a two-sided -trasform provided that both sums coverge. Thus covergece of this Z- trasform requires that, < 0 ; 0 ; ( b a x 0 0 ( + + + a b a b a b Z X ad < < a b }. : { ROC so b a < < Im Re R + R 9 Two-sided sigal (o-causal
Properties of ROC ROC of Differet Types of Sequeces Fiite legth: 0 The ROC for a fiite-legth sequece is the etire -plae ROC { :0 < < } except possibly 0 ad Righted-sided: The ROC for a right-haded sequece exteds outward from the outermost pole possibly icludig R Left-sided: The ROC for a left-haded sequece exteds iward from the iermost pole possibly icludig 0 ROC { : 0 < < R + } Two-sided (ifiite x [ ] 0 for < ad > x [ ] 0 for < x [ ] 0 for > The ROC of a two-sided sequece is a rig bouded by poles ROC { : R < < R + } 0 ROC { : < < } 0
3.4 -Trasform Properties
Liearity otatio Example:
Note that the ROC of combied sequece may be larger tha either ROC This would happe if some pole/ero cacellatio occurs Example: x a u - a u-n [ ] [ ] [ ] Both sequeces are right-sided Both sequeces have a pole a Both have a ROC defied as > a I the combied sequece the pole at a cacels with a ero at a The combied ROC is the etire plae except 0 3
Time Shiftig (Bilateral Trasform o > 0 ; right shift o < 0 ; left shift 3.4 The ew trasform may add or remove poles at 0 or Uit delay Z{ x[ ]} x[ ] m x[ ] x[ m] m ( Z{ x[ ]} 4
Time Shiftig (Uilateral Right Shift (Delay For a ocausal sigal x[], fid the uilateral -trasform of x(-m i terms of X(, the uilateral -trasform of x[] Let X( Z{ x( u( } : Uilateral -trasform of x[ ] The the uilateral -trasform of x( - m, m> 0 is { } Z x( m u( x( m Let r -m r+ m { } 0 Z x( m u( x( r rm ( r+ m -m We ca pull out i fro -m r -m r -m -m r { } { } { } t of the summatio: Z x ( m xr ( + xr ( Z x ( + xr ( r 0 r m rm -m -m Z x( m X( + x( m 5
Left Shift (Advace If xu [ ] [ ] X( m m m the x [ + mu ] [ ] X( x [ ] m > 0 0 6
Example: Time Shiftig (Uilateral Example: Fid the -trasform of the followig differece equatio: y ( y ( x ( for 0 ad y( 3 Trasform the differece equatio usig the uilateral Z-trasform: { ( } Y( { } Z y Z y( Y( + y( Y( + 3 3 Y( X( 3 Y( X( + X( 3 Y( + X( 3 Y( + 7
Multiplicatio by Expoetial Sequece 3.5. ROC is scaled by o. All pole/ero locatios are scaled 3. If o is a positive real umber: -plae shriks or expads 4. If o is a complex umber with uit magitude -plae rotates 8
Differetiatio of X( 3.6 9
Proof Fid the trasform of x( : First,write the Z trasform of x ( : { } X( Z x( x( dx ( d dx ( d x( x( dx ( Z{ x( } d Example: x ( au ( X( Z{ a u( } a dx ( d a a d d a a ( a ( a ( a 30
Cojugatio of a Complex Sequece Time Reversal 3.8 ROC is iverted 3
Proof Fid the -trasform of x[ ]: Z x[ ] x[ ] { } Let m - Z x[ ] x[ m] m { } m m m m Z x x m X { [ ]} [ ] X m ( Poles i X( become eros i X. Zeros i X( move to poles The ROC for X is: where R is the ROC for X(. R i X. 3
Covolutio of Sequeces Covolutio i time domai is multiplicatio i -domai 33
Proof Z x k x k Z x m x k m m { [ ] [ ]} [ ] [ ] x[ m] x[ km] k m m m [ ] [ ] x m x km k ( r+ m [ ] [ ] x m x r k [ ] [ ] k k m r x[ m ] x[ r] m k X X Covolutio defiitio Take -trasform Z-trasform defiitio Iterchage summatio r k - m Z-trasform defiitio 34
Example 3.9 : Evaluatig a Covolutio Let The correspodig -trasform are If a <, the -trasform of the covolutio is the Correspodig sequece is ] [ ] [ ad ] [ ] [ u x u a x,, ( 0 a a a X >,, ( 0 > X. ( ( ( ( ( > a a a a Y ]. [ ] [ ( ] [ u a u a y + 35
Iitial Value Theorem x[ ] X ( If the iitial value of x[] ca be computed directly from X(: x[0] lim X ( x[] x[ q] lim lim [ X ( x[0] ] q q q [ X ( x[0] x[]... x[ q ] ] To prove this: 0 for 0 x[ ] 0 for 0 ad applyig the limit left ad right of X ( x[ ]... 0 36
Example : Iitial Value Theorem ( Cosider X ( x[ ] ( u[ ] For the iitial value theorem x[0] lim X ( x[] lim x[ q] lim [ X ( x[0] ] q q q [ X ( x[0] x[]... x[ q ] ] So for the particular case x[0] lim x[] lim lim ( lim 37
Fial Value Theorem Suppose that x[] has a limit for approachig ifiity, the lim x[ ] lim( X ( The fial value theorem oly holds if x[] has a limit for approachig ifiity, this coditio holds if the -trasform X( is a ratioal fuctio i ad all its poles are, i magitude, strictly less tha, except oe that may be equal to (ad the is caceled by (- A pole equal to meas a costat term i the discrete-time domai, so a limited term A pole larger tha (e.g. meas a ulimited term i the discrete-time domai ( u[] If these coditios are verified, the [( X ( ] lim x[ ] 38
Example : Fial Value Theorem Cosider ( x[ ] ( 0.5 u[ ] X ( + 0.5 lim x[ ] Check if the limit exists if X( is a ratioal fuctio the magitude of all the poles must be strictly less tha, except oe pole oly that may be equal to this is the same as sayig that the locus must belog to the regio of covergece of (-X( the pole i this case is p-0.5 so p -0.5 < there are o other poles The limit exists ad ca be computed as lim x[ ] + 0.5 [( X ( ] ( 0 39
40
3.3 The Iverse -Trasform X ( x[ ] -Trasform: aalysis equatio time domai x [ ] X( -domai Iverse -Trasform: sythesis equatio x[ ] X( d π j C 4
Iverse Z-Trasform X ( + ROC uique x[ ] Formal iverse -trasform is based o a Cauchy itegral I most applicatios, we may take less formal ways for iversio Ispectio method Partial fractio expasio Power series expasio
Cauchy Itegral Theorem Augusti Louis Cauchy (789-857 Frech mathematicia a early pioeer of aalysis gave several importat theorems i complex aalysis 43
Cauchy Itegral Theorem says that if two differet paths coect the same two poits, ad a fuctio is holomorphic everywhere "i betwee" the two paths, the the two path itegrals of the fuctio will be the same. A complex fuctio is holomorphic if ad oly if it satisfies the Cauchy-Riema equatios. The theorem is usually formulated for closed paths as follows: Let f( be a complex fuctio ad aalytic (holomorphic o a simply coected domai D. The for ay simple closed cotour C i D, f ( d 0 C Cauchy s Itegral Formula Let f( be aalytic o ad iside a simple closed cotour C. The for ay iside C, f ( f ( w dw πi w C 44
Cauchy Residue Theorem Let f( be aalytic o ad iside a simple closed cotour C except for a fiite umber of isolated sigularities at,,, N. The i N f ( d Res( f(, π C 45
Cauchy-Lauret Series The -trasform is a form of the Cauchy-Lauret series ad is a aalytic fuctio at every poit i the ROC Let f( deote a aalytic (or holomorphic fuctio over a aular regio Ω cetered at o Im( o Ω Re( 46
The f ( ca be expressed as the bilateral series f( α( o where ( + α f ( ( o d π j γ γ beig a closed ad couterclockwise itegratio cotour cotaied i Ω 47
Iverse Trasform Geeral Expressio: Recall that, for r e, the -trasform X( give by is merely the DTFT of the modified sequece jω j X ( x [ ] x [ ] r e ω x[ r ] Accordigly, the iverse DTFT is thus give by j j xr [ ] π X( re ω e ω dω π π By makig a chage of variable jω r e, the previous equatio ca be coverted ito a cotour itegral give by C [ ] ( π j x X d C where is a couterclockwise cotour of itegratio defied by r 48
But the itegral remais uchaged whe it is replaced with ay cotour C ecirclig the poit 0 i the ROC of X( The cotour itegral ca be evaluated usig the Cauchy s residue theorem resultig i x [ ] X( d π i c [residues of X( at all poles iside C] X ( i ( i i Calculatio of residues a simple pole at 0 a s th order pole at 0 Res[ X ( at 0] ψ ( 0 ψ ( where X( 0 s d ψ ( Res[ X( at 0] s ψ ( where X( s ( ( s! d 0 0 49
Example of Iverse -Trasform X ( > a a x[ ] d d πi c a πi c a Ψ ( ad Ψ ( Ψ ( a a 0 x [ ] a, 0 50
Practical Ways to fid Iverse Trasform Iverse -Trasform is defied: x [ ] X( d π j Yuk! Usig the defiitio requires a cotour itegratio i the complex -plae. Fortuately, we ted to be iterested i oly a few basic sigals (pulse, step, etc. Virtually all of the sigals we ll see ca be built up from these basic sigals. For these commo sigals, the -trasform pairs have bee tabulated, ad we ca fid iverse -trasform by ispectio Two other methods to fid the iverse of -Trasform for more geer al cases: Partial fractio expasio method Power-series method 5
Partial Fractio Expasio Method A -trasform X( is said to be ratioal if it is a ratio of polyomials i. For a ratioal fuctio X(, we ca write X( as a sum of simpler terms by partial fractio expasio (PFE After PFE, we ca use kow trasform pairs to get the iverse trasform of each term. Geeral form of a ratioal X( is X ( M M k bk k 0 b0 k N N k a0 ak k 0 k ( c ( d k k eros poles 5
Partial fractio expasio If M < N (we say X( is proper If M N (ot proper N k k M k k k N k k k M k k d c a b a b X 0 0 0 0 ( ( (. ( ( where ( 0 k d k k N k k k X d A d A X ( 0 0 + K k k k N M r r r d A B X ca be obtaied by log divisio. B k 53
If X( has multiple-order poles of order s at d i, the the formula becomes ( X exists oly if M N MN s A C + + + r k m Br m r 0 d k m ( d i where B r all first order poles A k ( dk X ( ca be obtaied by log divisio. a order s pole sm C d s l ( d sm sm i X ( ( s m!( di d d There will be a similar term for every high-order pole p k. i Third term represets a order s pole There will be a similar term for every high-order pole Each term ca be iverse trasformed by ispectio 54
Iversio of each term Simple poles γ u [ ] if ROC > γ γ γ u[ ] if ROC < γ γ Multiple poles + m m ( + ( + m u [ ] if ROC m γ > γ ( m! m γ γ ( + ( + m γ u[ ] if ROC < γ ( m! m ( ( + m m 55
Example : Secod-order -Trasform X ( ( a ( b A B + ( a ( b a b ( + ( a b ( a b a ( b Give a > b. > a > b a b x [ ] ( au [ ] + ( bu [ ] (right-sided ab ba a > > b b a ROC x [ ] ( bu [ ] ( au[ ] (double-sided ba ab a > b > a b x [ ] ( au[ ] ( bu[ ] (left-sided ab ba 56
Example 3.8 ROC exteds to ifiity Idicates a right-sided sequece 57
Note: Sometimes the -trasform X( is give i positive powers of I this case, i order to achieve the PFE form give above, we eed to factor a out of the umerator term F( F( X( G( G( The term i parethesis is used i the partial fractio expasio: X( F( A A A3 A + + + + G( p p p3 p F ( A A A 3 X( + + + + G ( p p p 3 p A A A A + + + + p p p p 3 3 A 58
Example Suppose the system trasfer fuctio is: H( ad we wated the fid the step respose of the system: x( u( X( Z{ x( } Y( X( H( F( Y( G ( ( F( Y ( G( ( 59
F( Now we do a partial fractio expasio of G( A B + ( ( A B( + ( ( A ( A+ B + B A+ B A B 0 A B Y( ( y [ ] Z Y( { } y [ ] u [ ] 60
Example X( F( G ( F( We ca multiply ad divide by without chagig the expressio: G ( F( G( ( The expressio i brackets is the fuctio we wat to write i a partial fractio expasio: F( G( ( Factorig the deomiator: ( ( ( + The partial fractio expasio is A B C + + ( ( + ( ( + 6
A ( ( + 0 B ( + 4 C ( 4 X( + + 4 4 + x [ ] Z { X( } Z { } + Z + Z 4 4 + x [ ] δ[ ] + ( u [ ] + ( u [ ] 4 4 6
What about M N?, i.e. X( is ot proper Example 3.9 Cosider a sequece. ( ( ( 3 ( > + + + + X 63
Sice M N, we have We fid Sice the ROC is >, 0 ( + + A A B X 8 9 ( + + X ]. [ 8 ] [ 9 ] [ ] [ u u x + δ 64 5 3 3 + + + + ROC exteds to ifiity Idicates a right-sided sequece
X( + + 3 + + + X( 3 + + + X( ( A A X( B + + 0 Example: We ca fid the iverse ot proper usig polyomials i - Ratio of polyomials i terms of Divide through by the highest power of Factor deomiator ito firstorder factors Use partial fractio decompositio to get firstorder terms 65
Fid B 0 by polyomial divisio Express i terms of B 0 Solve for A ad A 5 3 3 + + + + ( 5 ( X + + 8 9 4 4 + + + + + + + + A A proper 66
Express X( i terms of B 0, A, ad A X 9 ( + Use table to obtai iverse - trasform 8 With the uilateral -trasform, or the bilateral -trasform with regio of covergece, the iverse -trasform is uique. γ u [ ] γ γ x [] k δ [] k 9 u[ k] + 8 u[ k] k 67
Complex Poles We cosider oly distict complex poles If x[] is real, ad if its -trasform has a complex pole i, the its complex cojugate i* is also a pole It ca be show that their partial fractio coefficiets are also complex cojugates Therefore assumig causal, the cotributio of two complex cojugate poles is of the form * * x [ ] [ A ( p + A ( p ] u[ ] k k k k k If we express the poles ad coefficiets as j k A A e α, p re k k k k jβ k The we ca show that the iverse trasform is x [ ] A r cos( β + α u[ ] k k k k k 68
Example 69
70
7
Power Series Expasio Method The -trasform is a power series I expaded form X X ( x[] -trasforms of this form ca geerally be iversed easily Especially useful for fiite-legth series ( + x[ ] + x[ ] + x[ 0] + x[ ] + x[ ] +
Example 3.0 : Suppose X( is give i the form ( ( ( ( + X ( + X Otherwise. 0, 0,,, ] [ x 73 [] [ ] [ ] [ ] [ ] + + + δ δ δ δ x
Example 3. 74
Example Assume a causal x[] X( 0.4 Dividig by ( -0.4: 3 3 ( 0.4 + 0.4 + (0.4 + (0.4 + + (0.4 X 3 3 ( + 0.4 + (0.4 + (0.4 + + (0.4 3 { 04 04 04 04 } x(,.,(.,(., (. Which we recogie as: x( (0.4 u ( 75
Example Assume a causal x[] 0.5 0.5 X( + ( 0.5( 0.7. 0.35 Perform the idicated divisio: 0.5 + 0.6 + 0.545 +. 0.35 0.5 3 0.5 0.6 + 0.75 0.6 0.75 + 0.6 070 0. 0.545 0. x [ ] {0,0.5,0.6,0.54, } Note that: the results should be exactly the same as the results obtaied via partial fractio expasio 76
NB: For bilateral trasforms, we eed to get a series i or i - depedig o the ROC!!! 77
DE Solutio via (oe-sided -Trasform Cosider the LCCDE Assume the iitial coditio y[0] Apply the oe-sided -trasform to both sides: Solve: (by partial fractio expasio Iverse -trasform [ ] ( ( (0 ( U Y y Y + 0 coverget (if ( U [ ] [ ] Y Y Y Y y Y + + + ( ( ( ( ( (0 ( ( ( ( ( ( ( + + + + + + Y 3 3 ( + + Y ] [ ( 3 ] [ 3 ] [ u u y + 78 [ ] [ ] [ ]. y y u + +
Example : Cosider a LTI system with iput x[] ad output y[] that satisfies the differece equatio Determie all possible values for the system s impulse respose h[] at 0. ] [ ] [ ] [ ] [ 5 ] [ + x x y y y ( ( 5 ( ( + X Y ( 3 / ( 3 / ( ( ( 5 ( ( ( ( ( + + X Y H 79
+ > + < < < 3 [0] ] [ ( 3 ] [ 3 ( [0] ] [ ( 3 ] [ 3 ( 3 [0] ] [ ( 3 ] [ 3 ( / 0 [0] ] [ ( 3 ] [ 3 ( / ROC h u u h No h u u h h u u h h u u h ( 3 / ( 3 / ( ( ( 5 ( ( ( ( ( + + X Y H 80
3.x. DT LTI System Aalysis usig the - Trasform Liear Costat-Coefficiet Differetial Equatios for CT systems N k M r d y( t d x( t α βr r dt k k k 0 dt r 0 Liear Costat-Coefficiet Differece Equatios for DT systems N a y( k b x( r M k k 0 r 0 N M ak br y ( y ( k + x ( r a a k 0 r 0 0 r 8
For a causal system, we require M N (most geeral case MN Without loss of geerality, we set a 0 [ ] [ ] N [ + ] N [ ] + b x[ ] + bx[ ] + + b x[ M + ] + b x[ M] y ay a y N a y N 0 M M Fiite impulsive respose system (FIR: whe all a i 0 except a 0 Ifiite impulsive respose system (IIR: at least oe a i 0 A DT LTI system is defied by Differece equatio + Iitial coditios 8
Advaces ad Delays Sometimes differece equatios may be preseted i terms of uit advace operators rather tha uit delay operators y[k+] 5 y[k+] + 6 y[k] 3 x[k+] + 5 x[k] Oe ca make a substitutio that reidexes the equatio so that it is i terms of delays For example, i above equatio, substitute k with k- to yield y[k] 5 y[k-] + 6 y[k-] 3 x[k-] + 5 x[k-] Before takig the -trasform, recogie that we work with time k 0 so u[k] is ofte implied NB: y[k-] y[k-] u[k] y[k-] u[k-] 83
- as a Delay Operator Let x[ ] y[ ] x[ ] x[ ] - is somethig like a uit delay operator! Geerally, Let X( x[ ] ad Y( X( The ( [ ] Y x x [ ] ( + xm [ ] m m Z{ x[ ]} Therefore y [ ] x [ ] X( - X( - x[] x[-] 84
Liear Differece Equatios Discrete-time LTI systems ca be characteried by differece equatios, e.g. y[k] (/ y[k-] + (/8 y[k-] + x[k] By usig the uit delay operators -, we ca represet the equatio i a sigal flow graph x[k] + + Σ + - y[k] / y[k-] - /8 y[k-] 85
Trasform Solutio of Liear Differece Equatios A DT LTI system is defied by Differece equatio + Iitial coditios The uilateral -trasform is well suited to solvig differece equatios with iitial coditios. As i the case of the Laplace trasform with differetial equatios, the -trasform coverts differece equatios ito algebraic equatios 86
First-order Differece Equatios Cosider a liear, time-ivariat discrete-time system y [ ] + ay[ ] bx[ ] Applyig the -trasform to the equatio it follows: ( Y ( + y[ ] bx ( Y ( + a Solvig for Y( ad rearragig the terms: ay[ ] b Y ( + X ( + a + a Notice the presece of a term related to the iitial coditio ad a term related to the iput For ero iitial coditio Y ( b + a X ( y [ ] Y ( + y[ ] 87
Example As a specific example, cosider y [ ] 0.8 y [ ] x [ ] (0.5 u [ ] y[ ] 0 : ero iitial coditio 0.8 y[ ] Y( + X( 0.8 0.8 but 0.5 Z u[ ] X ( 0.5 The (-trasform of the output is therefore: Y ( y[ ] ( 0.5( 0.8 0.3 (0.8* 0.5 0.8 0.5 ( 0.5 ( 0.8 u[ ] 0.5* 0.8 u[ ] / 0.3 ROC >0.8 ero-state respose 88
Secod-order Differece Equatios Cosider a liear, time-ivariat discrete-time system y [ ] + a y[ ] + a y[ ] b0 x[ ] + b x[ ] Applyig the -trasform to the equatio it follows: ( ( ( ( ( [ ] Y( + a Y( + y[ ] + a Y( + y[ ] + y[ ] bx + b X + x 0 Solvig for Y( ad rearragig the terms: ( ay[ ] ay[ ] ay[ ] bx[ ] + + Y( + a+ a b + b + a a 0 + + X( y y [ ] Y ( + y[ ] + y[ ] [ ] Y ( + y[ ] 89
Example : As a specific example, cosider the system described by a differece equatio y[k] 5 y[k-] + 6 y[k-] 3 x[k-] + 5 x[k-] y[-] /6, y[-] 37/36 x[k] -k u[k] Y[ ] 5 Y[ ] + y[ ] + 6 Y [ ] + y[ ] + y[ ] 3 X[ ] + x[ ] + 5 X [ ] + x[ ] + x[ ] 5 6 3 5 + Y [ ] 3 + 0.5 0.5 [ ] Y 6 7 8 + 5 0.5 3 5 3 6 7 8 5 3 5 k k k [ ] ( 0.5 ( + ( 3 uk [ ] yk ( 90
Example : Cosider the discrete time step iput sigal u Z [ ] X ( to the secod order differece equatio with ero iitial coditio 6 y [ ] 5 y [ ] + y [ ] x [ ] To calculate the solutio, multiply ad express as partial fractios Y ( (3 ( ( 0.5 + 0.5 (3 ( ( 0.67 ( / 3 y[ ] (0.67(/ 3 0.5 ( / 0.5(/ + 0.5 u[ ] + 0.5 ( ero-state respose 9
Zero-State Respose Cosider the ero-state respose Zero iitial coditios: y[-k] 0 for all k > 0 Oly causal iputs: x[-k] 0 for all k > 0 Write geeral th order differece equatio [ ] + [ ] + + [ ] [ ] + [ ] + + [ ] yk ayk a yk N bxk bxk b xk M N 0 yk [ m] uk [ ] Y[ ] + 0 xk [ m] uk [ ] X[ ] + 0 m m N+ N M+ M ( + a + + an + an Y[ ] ( b0 + b + + bm + bm X[ ] [ ] { ero state respose} [ ] X[ ] Z{ iput} [ ] H[ ] X[ ] H Y Y Z b + b + + b + b yk [ ] hk [ ] xk [ ] M+ M 0 M N N+ N + a + + an + an Zero state respose covolutio of iput with impulse respose M 9
Zero-Iput ad Zero-State Compoets 93
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Example 97
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Trasfer Fuctios We ca separate the respose of a DT LTI system Zero state respose of the system to a give iput with ero iitial coditios Zero iput respose to iitial coditios oly Cosider the ero-state respose Zero iitial coditios: y[-k] 0 for all k > 0 Oly causal iputs: x[-k] 0 for all k > 0 00
Write geeral th order differece equatio [ ] + [ ] + + [ ] [ ] + [ ] + + [ ] yk ayk a yk N bxk bxk b xk M N 0 yk [ m] uk [ ] Y[ ] + 0 xk [ m] uk [ ] X[ ] + 0 m m N N M M ( + a + + an + an Y[ ] ( b0 + b + + bm + bm X[ ] [ ] { ero state respose} [ ] X[ ] Z{ iput} [ ] H[ ] X[ ] H Y Y Z b + b + + b + b ( M M 0 M M ( N N + a + + an + an M 0
The trasfer fuctio of the system is defied as H( Y( X( + M k 0 N k b k k a k k a polyomial i (or - H(: may be viewed as a ratioal fuctio of a complex variable ( or -. ero iitial coditios!! 0
Relatio betwee h[] ad H [] I the discrete time domai, a system ca be described by the covolutio represetatio, i particular, h[] beig the uit-impulse respose with ero iitial coditio [ ] [ ] [ ] y h x h[ i] x[ i] 0,,,... i 0 Applyig the -trasform: Y ( H ( X ( Notice that H( represets the output whe the iput X(, beig the -trasform of δ[] h [ ] H( 03
Either ca be used to describe the system Havig oe is equivalet to havig the other sice they are a - trasform pair By defiitio, the impulse respose, h[k], is y[k] h[k] whe x[k] δ[k] Z{h[k]} H[] Z{δ[k]} H[] H[] h[k] H[] Sice discrete-time sigals ca be built up from uit impulses, kowig the impulse respose completely characteries the LTI system 04
Example : Trasfer Fuctio Cosider the system defied by the differece equatio y [ ] y[ ] x[ ] The same system ca be represeted by its uit-pulse respose h[] h [ ] u[ ] H( By takig -trasform of the differece equatio for IC0, we get Y ( X ( We get the trasfer fuctio which is idetical to the -trasform of impulse respose ( H 05
Trasfer Fuctio iput sigal x( [ ] X ( Z x( LTI System h ( H ( [ ] H ( Z h( output sigal y ( [ ] Y( Z y( Trasfer fuctio: the ratio of the -trasform of the output sigal ad the -trasform of the iput sigal of the LTI system H( Y( Z[ y( ] X ( Z[ x( ] 06
A discrete-time system that has a descriptio i terms of differece equatio ca be described i terms of a ratioal trasfer fuctio The trasfer fuctio carries the iformatio about the dyamics of the system ad it s the -trasform of the uit-impulse respose The poles of the trasfer fuctio tell you the if the step respose is limited (stability Stable: Covergece regio must iclude the uit circle, ad vise versa. Causal: Covergece regio is the exterior of a circle. Stable ad Causal Covergece regio will iclude the uit circle ad the etire -plae outside the uit circle. The output of a system ca be computed (for a give iput ad IC0 i the domai with a simple multiplicatio of the trasfer fuctio times the -trasform of the iput 07
Ratioal Trasfer Fuctio Ratioal fuctio is defied as the ratio of polyomials i (or - P( X ( where P( ad Q( are polyomials i Q( Most importat ad useful -trasforms are those for which X( is a ratioal fuctio iside the regio of covergece The trasfer fuctio of a DT LTI system is a ratioal fuctio H( M M k b k k 0 b 0 NM k N N k a0 k k k ( k + a ( p k irreducible or factoried form 08
Zeros ad Poles Zeros of H(: values of for which H(0 Poles of H(: values of for which H( For a ratioal trasfer fuctio: The roots of the umerator polyomial (the set { k } are those values of for which H( 0, ad therefore are the eros of H(. The roots of the deomiator polyomial (the set {p k } are those values of for which H(, ad therefore are the poles of H(. Poles may also occur at 0, ad Poles ad regio of covergece No poles of H( ca occur withi the regio of covergece The regio of covergece is bouded by poles 09
0 Graphic Represetatio Which coverges Zeros: 0 Poles: a + + 0 ( ( ( ( ( a u a X u a x a whe a a X > (
Discrete Time Eigefuctios Cosider a discrete-time iput sequece ( is a complex umber: x[] The usig discrete-time covolutio for a LTI system: y[ ] k k h[ k] x[ k] h[ k] k H ( h[ k] k k H ( x[ ] Z-trasform of the impulse respose H( h[ k] k But this is just the iput sigal multiplied by H(, the -trasform of the impulse respose, which is a complex fuctio of. Therefore, is a eigefuctio of a DT LTI system, ad H( is correspodig eigevalue k
BIBO Stability BIBO stability is exteral stability, ad it depeds o iput-output relatioship The iput-output relatioship is characteried by the impulse respose h[] or trasfer fuctio H[] BIBO stability coditios we have show this before h[] is absolutely summable h [ ] < All the terms i h[] are decayig expoetials All the poles of H[] are withi the uit circle we will show this ow Note that the absolute summability coditio guaratees that the DTFT of h[] will exist. This meas that a discrete-time LTI system is stable if ad oly if the trasfer fuctio has a regio of covergece that icludes the uit circle.
Example: a a This is stable if a < k u Z [ k] for > a 3
Theorem: A causal LTI system with ratioal trasfer fuctio is BIBO stable if ad oly if all the poles of H( lie iside (ot o the uit circle. Im p max Re ROC uit circle 4
Proof Assume all the poles are simple (for multiple poles, we ca make a similar argumet H( ca be expaded i PFE form The (we will show this later whe we discuss iverse trasform Clearly, the -trasform of coverges if ad oly if Therefore, H( coverges for all such that K k k k p A H ( K k K k k k k h u p A h ] [ ] [ ( ] [ <. p k. max max k K k p p > h k [] 5
Fact: The ROC of causal H( icludes the uit circle if ad oly if all the poles of H( lie iside the uit circle. Sufficiecy Assume Sice the p k < for k,..., K. h[ ] h[ ] K k K k A ( p k A k p k k u[ ] ad so 0 K K h( Ak pk 0 k A k p k k 0 < sice p k < Hece H( is the trasfer fuctio of a BIBO stable system. 6
Necessity: (by cotradictio Suppose there exists a pole p i such that p i >. Clearly, from the Fact the uit circle is ot i the ROC of H(. Therefore, there exists 0 with 0 such that H( 0 does ot absolutely coverge: H ( 0 h( ( 0 0 If there is o such 0 the the uit circle is i the ROC! But the 0 h[ ] 0 0 0 h[ ]( h[ ] 0 so H( is ot BIBO stable. H ( 0 0 h[ ] 0 7
Commets o the Theorem We say that a system with its impulse respose g[] is ati-causal if g[-] is a causal sequece. Corollary: A ati-causal liear system with ratioal trasfer fuctio G ( Z{ g( } is BIBO stable if ad oly if all the poles of G( lie outside (ot o the uit circle. Theorem (most geeral: A liear system with ratioal trasfer fuctio is BIBO stable if ad oly if the ROC cotais the uit circle. This applies to causal, ati-causal, or otherwise. 8
Causal Systems For causal systems, h[] 0 for egative. I this case, the trasfer fuctio sum oly has terms for positive : H( h[ ] 0 So if the ROC of the trasfer fuctio of a causal system cotais a circle of radius r i the complex plae, it must also cotai all circles of radius greater tha r. The ROC of the trasfer fuctio of a causal system always looks like this: Im{} RoC Re{} 9
Stability of Causal Systems The places where the trasfer fuctio is ifiite (the poles determie the regio of covergece. As log as all the poles lie strictly iside the uit circle, the regio of covergece will iclude the uit circle ad the system will be stable. This is true oly for causal systems. Im{} x x x x RoC Re{} I the case of a causal LTI system, the stability coditio is equivalet to the requiremet that the trasfer fuctio H( has all of its poles iside the uit circle. 0
Example 3.7 : Stability, Causality, ad the ROC Cosider a system with impulse respose h[] The -trasform H( ad the pole-ero plot show below Without ay other iformatio h[] is ot uiquely determied Three possible ROCs are >; or <½; or ½< < (why? If we kow some other iformatio about the system, we may be able to determie the ROC If system stable, the the ROC must iclude uit-circle ROC: ½< < If system is causal, h[] is right sided ROC : > (ot stable
LTI System Descriptio: Summary N Time Domai impulse respose y ( bkx ( ( k ak ( y ( k h [ ] M k 0 k Z Domai trasfer fuctio Z h( FT Frequecy Domai: frequecy respose H( N k 0 M bk ( + ak ( k k k Z - FT- j j e ω ω e jω He ( N k 0 M bke ( + ake ( k jω k jω k
Appedix I. Geometric Series Formula A basic aalysis tool used i -trasform evaluatio ad iversio is the geometric series formula. If N s[ N ] a[ ] a[0] + a[] +... + a[ N ] 0 where a[ i + ] a[ i] r commo ratio the s[ N ] a[0] N r a[0] 0 r r N ad if r < (covergece criteria s[ ] a[0] r 0 a[0] r 3
Appedix II. Iteral Stability All trasiet respose vaishes as t Trasiet respose cosists of atural modes That meas the magitudes of all the atural frequecies are less tha Stable: All the poles of H[] is withi the uit circle Ustable: Oe or both of the followig coditios are met At least oe pole of H[] is outside of the uit circle There are repeated poles o the uit circle Margially stable: No poles outside of the uit circle, ad there are some simple poles o the uit circle Iteral Stability BIBO Stability, but ot ecessarily the other directio 4
Examples Example : stable system H( 0.9 + 0.8 0.8 + 0.64 0.3 p 0.4000 + 0.698 j p 0.8 < 0.6 p 0.4000 0.698 j p 0.8 < Example : ustable system 0.6 H(. +. 0.4 p 0.5500 + 0.956 j p. > 0.4 p 0.5500 0.956 j p. > 5
Appedix III. -Trasform ad Laplace Trasform Let x[] be a discrete-time sigal geerated by samplig a aalog sigal x(t We ca cosider a aalog sampled sigal where f x[ ] x( T s ( ( ( x t x t f comb f t x( kt δ ( tkt s δ T s k 0 s s s s k 0 xk [ ] δ ( tkt These two sigals are equivalet i the sese that their stregths, i.e., the heights, are same at correspodig poits. s 6
Let s cosider a DT system with its impulse respose, h[], ad a CT system with its impulse respose,. ( [ ] δ ( hδ t h t T If x[] is applied to the DT system ad δ is applied to the CT system, the their resposes will be equivalet i the sese that the stregths (heights are the same. x 0 ( t - s 7
The trasfer fuctio of the DT system is H ( [ ] h 0 ad the trasfer fuctio of the CT system is The equivalece betwee them ca be obtaied by, The relatioship betwee s-plae ad -plae 0 Hδ ( s h[ ] e T s ( ( st s H s H δ e s st e s, s l T s 8
Mappig of the s-plae to the -plae 9
s σ + jω Let, the σts e e jωt s The above relatios show the followig: The imagiary axis of the s-plae betwee mius half the samplig ad plus half the samplig frequecy maps oto the uit circle i the -plae. The portio of the s-plae to the left of the red lie maps to the iterior of the uit circle i the plae. The portio of the s-plae to the right of the red lie maps to the exterior of the uit circle i the plae. The gree lie (lie of costat sigma maps to a circle iside the uit circle i the -plae. Lies of costat frequecy i the s-plae maps to radial lies i the -plae. The origi of the s-plae maps to i the -plae. The egative real axis i the s-plae maps to the uit iterval 0 to i the -plae. 30
Differet lie segmets i the s-plae may map ito the same cotour i the -plae. e st s 3
The s-plae ca be divided ito horiotal strips of width equal to the samplig frequecy. Each strip maps oto a differet Riema surface of the "plae". Mappig of differet areas of the s plae oto the -plae is show below. 3
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