( ) ( ) ω = X x t e dt

Transcription

1 The Laplace Tranform The Laplace Tranform generalize the Fourier Traform for the entire complex plane For an ignal x(t) the pectrum, or it Fourier tranform i (if it exit): t X x t e dt ω = For the ame ignal x(t) the bilateral Laplace tranform i t L ( x() t )() = X ( ) = x( t) e dt; = σ+ The integral i computed for σ= ct from to For σ= the Laplace tranform i the pectrum of the ignal or the Fourier tranform σ=ct σ Region of convergence (ROC) = region of the complex plane where the bilateral Laplace tranform X ( ) i convergent Some propertie The ROC i compoed of vertical band of the plane parallel with the imaginar axi The ROC doe not contain an pole of the Laplace tranform 3 For right ided ignal (thi include caual ignal), the ROC i a right ided half plane If additionall, the Laplace tranform i rational, the ROC i the region in the -plane to the right of the rightmot pole Re{ } >σ p max For left ided ignal (thi include anti-caual ignal) the ROC i a left ided half plane If additionall, the Laplace tranform i rational, the ROC i the region in the -plane to the left of the leftmot pole Re{ } <σ p min 5 For ignal with infinite upport (or two-ided ignal), the ROC i a vertical band parallel with the imaginar axi that doen t include pole 6 For ignal of finite duration and abolutel integrable, the ROC i the entire complex plane 7 For a table tem the ROC mut contain the imaginar axi

2 The invere Laplace Tranform t x() t = X ( ) e d π j Integral i computed on a line parallel with the imaginar axi ( σ = ct ) More frequentl, we ue the partial fraction decompoition and inverting the tranform uing the table Problem The Laplace tranform X () of a ignal Find x() t in the following condition a) Re{ } < 3 ; b) 3< Re{ } < ; c) { } Solution + A A B X() = = ( ) ( 3) - +3 ( -) ; + x t i X() = ( ) ( + 3) Re > + => B= ( + 3) X( ) = = /6 3 ( ) + A = X = = 3 3/ + 3 d + 3 A = ( ) X( ) = = /6 d ( + 3) + /6 3/ /6 X() = = + ( ) ( + 3) - +3 ( -) t t 3t a) ROC : Re{ } < 3 : x() t = e σ( t) te σ( t) + e σ( t) t t 3t b) ROC: 3< Re{ } < : x() t = eσ ( t) teσ ( t) e σ () t t t 3t c) ROC: Re{ } > : x() t = eσ () t + teσ () t e σ () t Verif uing Matlab the ignal expreion: >> m >> L=(+)/((-)^*(+3)) >> ilaplace(l)

3 The tranfer function of a linear, time-invariant (LTI) caual tem i + H() = Compute and ketch the repone of the tem () t, if the input + + t ignal i x() t = e Solution () t = x() t () t Y( ) = X ( ) H( ) + + H() = = with complex pole, = ± j σ p max = + + ( )( ) - ROC for H() σ - - The tem i caual: it ROC i Re{ } > t t e, t t t x() t = e = = e σ () t + e σ ( t) t e, t < X() = = with the ROC: < Re { } < + + Y( ) = X ( ) H( ) = = It ROC i the interection between the ROC for H( ) and ROC for < Re{ } < Y A B+ C = = ( )( + + ) => A( ) ( B C)( ) = =>, A = -/5, C = 8/5, B = /5 => Y( ) X : = + = ( + ) + ( + ) + () t t t t = e σ t + e cot+ 3e int σ t 5 5, ROC: < Re{ } < => () 3

4 3 Conider a linear, time-invariant caual tem with the tranfer function H() having the following pole/zero contellation: -3 - σ a) Give all the poible region of convergence; b) The tem fulfill the condition of tabilit and caualit in each cae or not? Solution Re()<-3, the tem i both anti-caual and untable, -3<Re()<-, the tem untable, -<Re(), the tem i both caual and table Prove the following propertie of the Laplace tranform: a) time hifting; b) hifting in the complex domain ; c) time caling x( at ) ); d) convolution in time; e) derivation in the complex domain - homework 5 Conider a tem obtain b interconnecting two LTI tem, whoe tranfer function are H () and H () If the tranfer function of the equivalent tem i, the tem H () i referred to a the invere tem of the tem H () a) Determine the relation between H () and H () b) Below it i repreented the pole/zero contellation of H (), correponding to a caual and table tem Determine the pole/zero contellation of the invere tem c) Determine the impule repone h (t), conidering the tem i table d) Prove that the impule repone of the equivalent tem i identical with the unit impule δ(t) -3 - σ - homework 6 An LTI tem with the input ignal x(t) and the output ignal (t) i decribed b the differential equation: d d + 3 = x dt dt a) Determine the tranfer function of the tem, H() Sketch the pole/zero contellation

5 b) Determine h(t) if the tem i table; if the tem i caual; if the tem i neither table nor caual Solution d d a) x() t X ( ) ; x () t X ( ) dt dt A B => Y+ Y 3Y= X => H( ) = = = ( )( + 3) + 3 A= ( ) H( ) = = + 3 B= ( + 3) H( ) = = 3 H = σ ht t 3t = eσ t e σ t (untable) t 3t b) table tem ROC: -3<Re()<, ht () = eσ ( t) e σ () t caual tem ROC: Re()>, () () () Verif uing Matlab the impule repone expreion: >> m >> L=//(-)-//(+3) >> ilaplace(l) untable & anti-caual tem ROC: Re()<-3, t 3t ht () = eσ( t) + e σ( t) 7 An LTI tem ha the output ignal: ( t t t e e ) ( t) t 3t () x t = e + e σ t a) Find the frequenc repone of the tem, H ( ω ) b) Find the impule repone of the tem, h(t) c) Find the differential equation that characterize the tem, = σ if the input ignal i 5

6 d) Implement the tem uing direct form I or II Solution a) Y( ) = X ( ) = ; Y( ) 6 ( + )( + 3) 3( + 3) H( ) = = = X ( ) ( + )( + ) ( + ) + + tem 3( + 3) H( ω) = H( ) = ( + )( + ) 3( + 3) A B b) H( ) = = ( + ) ( ) A= + H = = + ; 3 3 H h t e e t + + t t = + () = ( + ) σ () Verif uing Matlab the impule repone expreion: >> m >> L=(3/)/(+)+(3/)/(+) >> ilaplace(l) c) Differential equation ; ROC: Re()>-, caual ( + ) B= + H( ) = = + M k k k b N M k d d x Y( ) k = ak = k bk H k ( ) = = N k= dt k= dt X ( ) k a k k = => = 3x + 9x d) We can ue one of the direct form of implementation: a+ a + a = bx+ bx + bx a + a + a = b x+ bx+ b x a + a + a = b x+ b x+ b x b x b x b x a a => = + + a 6

7 Direct form I x b /a Direct form II x /a b b -a -a b b -a -a b x For the caual, linear time-invariant tem with the frequenc repone 7 + H ( ω) = ( + )( ω + ) a) Find h(t), b) Implement the tem uing two tem interconnected in erie, c) Implement the tem uing two tem interconnected in parallel Solution + 7 A B+ C H( ) = H( ω ) = = = A= ( + ) H( ) = = ; for => C=/3 => B=-3/ = = H

8 H H = = ht e t e t t e t t () t t () / t = σ co σ () + / in σ () Verif uing Matlab the impule repone expreion: >> m >> L=(+7)/(+)/(^++) >> ilaplace(l) H = = = H () () a Hb b) ( + )( + + ) For Ha ( ) : a + a = bx + bx or + = x For Hb( ) : a + a + a = bx + bx + bx or + + = x + 7x x - u u c) H( ) = = H ( ) + H ( ) 3 α β homework 8

9 x H α () H β () 9 Conider the tem repreented below: I () R I 3 () R x X() C I () C Y() a) Find the tranfer function of the circuit, for R=k Ω, C=uF b) What tpe of characteritic doe thi circuit have? Find the qualit factor Q, the amplification A and the natural pulation of reonance ω c) Find the repone of the tem to the unit impule and to the harmonic ignal x t = A coω t, repectivel () Solution Y ( I ) CY( ) () 3 = = ; C I( ) = I3( ) R + I( ) = I3( )( + CR) = ( + CR) CY ( ) C C I( ) = I( ) + I3( ) = C( + CR) Y( ) X ( ) = I( ) R+ I( ) = ( CR) 3CR Y( ) C + + Y => H( ) = = =, for RC= (low pa filter) X ( CR) + 3( CR) A b) H ( ω) = = ω + 3 ω ω + jξωω => A=, amplification at low frequenc Q = = 33, qualit factor, ω = natural pulation ξ 9

10 c) H( ) H = = + 3+ A B = + => A ( ) H( ),, 3± 5 =, 5 = = = = = 3+ 5 ( 3 5) 5 5 => B ( ) H( ) 5 = = = = = 3 5 ( 3+ 5) t t => ht () = e σ () t e σ () t 5 Repone of the tem to the unit impule, x( t) = δ ( t) => ( t) = h( t) Repone of the tem to x() t = Acoωt => ( { }) () = ( ω ) co ω + arg ( ω ) t A H t H H ( ω) = H( ω ) = ω + 9ω ω + 9ω { H( ω) } = { ω + } arg arg 3 If ω < then H ( ω ) If ω > then 3ω arg{ } = arctg ω 3ω arg{ H ω } = π + arctg ω For an LTI tem the frequenc repone i H ( ω) a) Find the impule repone of the tem h(t), b) Implement the tem homework a) H( ) = = ( + )( + + ) = + + H =

11 () t t 3 () t 3 ht = e σ t+ e co tσ () t+ e in tσ () t Verif uing Matlab the impule repone expreion: >> m >> L=-/(+)+(+)/(^++) >> ilaplace(l) b) H( ) = = ( + )( + + ) b =5, b =7 a 3 =, a =5, a =5, a = or: = 5x + 7x