4. Approximation of continuous time systems with discrete time systems

Transcription

1 4. Approximtion of continuous time systems with iscrete time systems A. heory (it helps if you re the course too) he continuous-time systems re replce by iscrete-time systems even for the processing of continuous-time signls. ) Impulse Invrince Metho: [ ] h n h n he reltion between the Z trnsform n the Lplce trnsform: kπ ( ) s e s j k he frequency response of the igitl system is the sme with the frequency response of the nlog system of limite bn for frequency less thn hlf of smpling frequency π π ( ω) ( Ω) Ω ω; ω n ω M

2 he reltion between the s n plnes σ jω s jω σ jω r e s σ + jω ; re x+ jy ; e re e e. ω Ω+ k π ) Step invrince metho: [ ] s n s n 3) he metho of the ifferentil eqution pproximtion Differentil eq: N k M k y( t) x( t) k b k k k t t k 0 k 0 Use pproximtion: y t y n y n y n y n t n t () [ ] [ ] Finite ifference eq: N k M k p p p p k k k k C y n p b C k k x n p [ ] [ ] k 0 p 0 k 0 p 0 he reltion between the Z trnsform n the Lplce trnsform: s s he reltion between the s n plnes s

3 4) Biliner rnsformtion Metho + s s + B. Problems Problem. We hve liner time-invrint system with the trnsfer function: s +. s+ ( s+ 3) ) Determine its impulse response, h ( t ) ; b) Fin the impulse response of the igitl system equivlent to the nlog one, using the impulse response invrince metho [ ], the Z trnsform of h [ n ] ; c) Fin h n ; 3

4 Solution. ) o etermine the impulse response from the trnsfer function, we ecompose it into simple frctions: A B C + + s+ s+ ( s + ) 3 s + B ( s+ ) B s s + 3 s s + C ( s+ 3) s 3 s3 4 ( s + ) s+ s+ 3s A ( s+ ) s s s s 3 s s + s + 4 ( 3) We ientify ech term using the tbles of Lplce trnsforms: ( s + ) + s+ ( s + ) s etσ 3 4 ( t) e t 4 s s + tσ X s x() t X x t t e t tx t e tx t s s ( s ) st st () () L{ ()} t t L { te σ () t } te σ t + + ( s ) () h t e t te t e t 4 4 t t 3t () σ () + σ () σ () n n 3n b) h [ n] h ( n ) e σ[ n] + ne σ[ n] e σ [ n] 4 4 c) We ientify ech term using the tbles of Z trnsforms: n n /4 e σ[ n] ( e ) σ [ n] 4 4 e n 3n 3 /4 e σ[ n] ( e ) σ [ n] e n n e ne σ[ n] n( e ) σ [ n] ( e ) 4

5 /4 e /4 ( ) + 3 e ( e ) e Problem. We hve s the trnsfer function of n nlog system, liner n timeinvrint, ( s+ )( s+ ) ) Determine its impulse response, h ( t ) ; b) Compute the impulse response for the igitl system in the cse of the impulse invrince metho. c) he step response is () s t. For the igitl system, its impulse response is h [ n ] n its step response is s [ n ]. Assume we use the impulse response invrince metho to pproximte the nlog system with igitl one: h [ n] h ( n), Verify tht the reltion: n [ ] s n h k k is true. ) Compute the step response of the igitl system, using the step response invrince metho: [ ] s n s n. Solution. ) o etermine the impulse response from the trnsfer function, we ecompose it into simple frctions n we ientify ech term using the tbles of Lplce trnsforms: A B C s+ ( s+ )( s+ ) s ( s+ ) A s+ s ( s + ) s s B ( s+ ) s s + s C s+ s s s s + s s s s ( + ) > h () t e t σ ( t) te t σ ( t) e t σ ( t) b) Impulse invrince metho he impulse response of the equivlent igitl system is:. 5

6 [ ] n [ ] n σ σ[ ] n σ [ ] h n h nt e n n e n e n c) Impulse invrince metho he step response in iscrete-time is the response to the unit step: [ ] σ[ ] [ ] [ ] σ [ ] [ ] s n n h n h k n k h k h k k k k s n h k k Which mens tht: [ ] n. he reltion is true in the generl cse. ) Step invrince metho S s be the Lplce trnsform of the step response for the nlog system: Let S s A B C D s s( s+ )( s+ ) s s+ ( s + ) s+ We compute the step response of the nlog system using the ecomposition in simple frctions n tbles of Lplce trnsforms: A ss s 0 s 0 4 ( s+ )( s+ ) B ( s+ ) S s s s s+ C ( s+ ) S s s s+ s s+ 3 D ( s+ ) S 3/4 s s s s s s s s s t t 3 t > s () t σ () t e σ () t + e σ () t + e σ () t 4 4 n n 3 n s n s n σ n e σ n + e σ n + e σ n 4 4 [ ] [ ] [ ] [ ] [ ] n Problem 3. We hve n nlog, liner time-invrint system with the impulse response h () t n igitl liner time-invrint system h [ n ] equivlent to the nlog system with the impulse response invrince metho. ) If h () t e t σ () t fin the trnsfer function of the nlog system; b) Fin the trnsfer function of the igitl system; c) Sketch the mplitue-frequency chrcteristic of the nlog system n the iscrete-time system. Solution. 6

7 { } s + ) ( s t ) L e σ ( t) n b) h [ n] h ( n) e σ [ n] ( ) c) e Ω e j e Ω + jω + ω Ω e cosω+ je sin Ω ) ( ω) ( ω) + e e cosω e cos e sin Ω + Ω For 0. the mplitue-chrcteristic re lmost the sme for frequency less thn hlf of smpling frequency π/0π3.4. (ω) (Ω) Ωω s Plot it using Mtlb. Wht o you notice when the frequency is higher thn hlf of the smpling frequency? 7

8 >> w-00*pi:0.0*pi:00*pi; >>./sqrt(+w.^); >> 0../sqrt(+exp(-0.)-*exp(-0.).*cos(0.*w)); >> plot(w,,w,); gri on Problem 4 Consier the nlog, liner time-invrint system with the ifferentil eqution: y y () t x() t t + ) Fin the impulse response n the trnsfer function of iscrete-time system tht pproximtes the nlog system using the impulse invrince metho; b) Fin gin the impulse response n the trnsfer function if we use the ifferentil eqution pproximtion; c) he sme for the biliner trnsformtion metho. ) Give implementtion forms for the systems from ), b) n c) ; e) For x() t sin t n smpling perio of 0, fin the mplitue of the output signl for the nlog system n for the igitl system, for the three cses, respectively. Solution. ) impulse invrince metho y Y + y( t) x( t) sy + Y X t X s + t n h () t e σ () t h [ n] e σ [ n] ( ) n e b) ifferentil eqution pproximtion s s n he impulse response of the igitl system is: h [ n] σ [ n] + + c) biliner trnsformtion metho s ( ) s + + s + s ( ) 8

9 ( ) + + ( + ) ( ) + ( ) + ( ) + ( ) + ( ) ( )( + ) + he impulse response of the igitl system using the biliner trnsformtion: ) [ ] [ ] h n δ n + σ n + ( ) n [ ] x - y For the nlog system y + y x x[n] e - D y[n] For the igitl system from ) b ; ; e 0 0 [ ] [ ] [ ] y n e y n x n x[n] (+) - D y[n] For the igitl system from b) b ; + ; 0 0 ( + ) y[ n] y[ n ] x[ n] y[n-] 9

10 x[n] ( + ) D D -(-) y[n] For the igitl system from c) b ; b ; 0 0 ( + ); ( ) y[ n] y[ n ] xn [ ] xn [ ] e) For x() t sin t n smpling perio of 0, we hve to fin the mplitue of the output signl, in the nlog n in the igitl cse. () sin ; () () sin + rg{ } x t t y t t ( ω) () s jω + jω + j () ; rg{ () } rg{ + j} rctg 5 y() t sin trctg, mplitue 5 5 For the igitl systems we hve: n n xn [ ] sin ( 0,n) sin yn [ ] sin rg For the igitl system from ) ( ) e Ω j cos e Ω e Ω+ j e sin Ω ( Ω ) 4 4 e cosω + e sin Ω + e e cosω 0 0,4 0, + e e cos 0 0, ( e ) 0

11 For the igitl system from b) ( ) jω + Ω + e + cos Ω jsin Ω ( Ω ) + cosω + sinω cosΩ 0, 0, 0,083 0, 44, For the igitl system from c) + + e ( ) ( Ω ) +, 0, 9e jω ( cos ) sin (,0,9cosΩ ) + ( 0,9sinΩ) jω + Ω + Ω + cosω ( Ω ), + 0,8,98cosΩ Set the vlue of Ω to 0, (homework). Problem 5. Consier continuous-time system with the impulse response h () t n the s. he ifferentil eqution is: trnsfer function N k y N k k k 0 t k 0 k x bk k t We pproximte this system using iscrete-time system using metho similr to the metho of the ifferentil eqution pproximtion: x x[ n+ ] x[ n] t We efine the ifference of t n ( 0) { [ ]} [ ] [ ] [ ] orer 0: x n x n x n+ x n orer : { xn [ ]} ( k) { [ ]} orer k: x n ) Let s ( s + ) ( k) { { x[ n] }} ( ) ; b) Wht is the connection between. Fin the trnsfer function of the igitl equivlent system? s n

12 Solution We notice tht: { xn [ ]} [ + ] xn [ ] xn { [ ] xn [ + ] xn [ ] xn} { xn [ ] } + { xn [ ]} [ + ] [ + ] [ + ] [ ] x n x n x n x n xn [ + ] xn [ + ] + xn [ ] k + k { xn l [ ]} Cxn [ l ] k l 0 k l ) We hve to fin the trnsfer function of the igitl equivlent system metho similr to the ifferentil eqution pproximtion. s bs ( s ) s s s s b 0; b ; ; ; 0 0 he liner ifferentil eqution is: y t y t x t y() t + t + t t he finite ifference eqution is: mening tht: or: [ ] { [ ]} { [ ]} { [ ]} y n + y n + y n x n [ ] [ ] [ ] [ ] y n+ y n x n+ x n yn [ ] + + yn [ ] yn [ ] yn [ ] using y[ ] + ( y[ + ] y[ ]) + y[ + ] y[ + ] + y[ ] x[ + ] x[ ] n n n n n n n n

13 Applying in both sies the Z trnsform, we obtin: Y + ( ) Y + Y Y + Y X X or: ( ) Y X ( ) + + +? b) Wht is the connection between ( s ) n ( ) mening tht: ( ) ( ) ( + ) + ( ) We cn see tht: ( ) ( s ) s 3