Approximtion of continuou-time ytem with icrete-time ytem he continuou-time ytem re replce by icrete-time ytem even for the proceing of continuou-time ignl.. Impule invrince metho 2. Step invrince metho 3. Finite Difference Approximtion (FDA) 4. Biliner rnform Generl metho for qulity evlution of pproximtion for bn-limite ytem AD converter Anlog to Digitl Converter DA converter Digitl to Anlog Converter - mpling tep, ω S mpling frequency 2
H H ( ) ( z) - trnfer function of the nlog ytem to be pproximte - trnfer function of equivlent igitl ytem. he igitl ytem - minimize error e. Suppoe the ytem n input re bn limite: H X ( j ) ( ) ω =, for ω >ω M ω =, for ω >ω M 3 Sytem ientifiction δ ( t) in ( πt ) πt -ientifie ytem, not necerilly bn limite. -ientifie bn - limite π ytem with ω M < For the firt ce: we cnnot mple the Dirc impule, ince δ(t) δ(t) i not efine. 4
Ientifiction of bn-limite ytem uing the crinl ine. Frequency repone of the bnlimite ytem tht mut be ientifie Spectrum of the input ignl (crinl ine) Spectrum of the output ignl 5 6
. Impule Invrince Metho h [ n] = h ( n) πt Anlog ytem: x t inc y t h t y n h n Digitl ytem: x[ n] = δ[ n] y[ n] = δ[ n] h[ n] = h[ n] Minimum error e( n) : h( n) = h ( n) 7 () = ( ) = ( ) ( ) = ( ) ( ) ( ) Now conier the input ignl, x t X, bn-limite t ω = π / = ω / 2 (hlf of mpling frequency) M he repone of the nlog ytem, t moment n, i: - ( ) = L { ( ) ( )}( ) y n X H n he repone of the igitl ytem i: ( ) = [ ] = Z { ( ) ( )} e( n) = y[ n] = y ( n) = y( n), or: { X ( ) ( )} { ( ) ( ) }( ) z H z = L X H n y n y n X z H z For trnformtion without error, : Z he igitl ytem trnfer function epen on the input ignl ue H ( z) = Z L X H n X ( ) ( ) to the preence of function X n X z. ( z) - { { ( ) ( )}( )} 8
H ( z) = Z L X H n X ( z) Chnge the input ignl : chnge the igitl ytem. No perfect pproximtion. Approximtion uing tnr ignl: Unit tep σ(t) Rmp ignl t σ(t). - { { ( ) ( )}( )} 9 Approximtion by um of unit tep ignl, hifte n weighte ( ) σ( ) x t t t
Approximtion by um of rmp ignl, hifte n weighte ( ) ( ) σ( ) x t t t t t Impule invrince metho for igitl ytem equivlent with bn-limite c.t. ytem πt For x t inc ; x n n ; h n h n he Lplce trnform of the mple ignl y i () = [ ] = δ [ ] [ ] = ( ) n= ( ) L{ δ ( )} ( ) = h n t n = h n e n= ( nt) ˆ Y( ) = L{ y() t δ () t } = L h( n) δ ( t n) n= n 2
he Z trnform of the output ignl from the igitl ytem i n ( ) = [ ] = ( ) H z h n z h n z n= n= Yˆ ( ) = H ( z) z= e For ny mple ignl we hve Yˆ j H j j Replcing jω= : n ( ω ) = ( ω ω ) = Yˆ ( ) = H( jω). = 2π H ( z) = H( j ) ; z= e ω ω = = 3 Reltion between n z plne for the Impule Invrince Metho σ jω jω σ jω r = e =σ+ jω ; z = re ; z = e re = e e ω =Ω+ 2 π Left hlf plne σ< interior of unit ic z =r< Imginry unit ic z = xi σ= Right hlf plne σ> exterior of unit ic z =r> egment [-π/, π/) on imginry xi one wrpping on the unit ic 4
In orer to voi liing error in the frequency repone of the igitl ytem obtine, the frequency repone of the nlog ytem houl be completely inclue in frequency bnwith -π/, π/. Bn-limite nlog ytem ωm < π / with mpling frequency ω = 2 π/ 2ω M 5 Impule invrince metho: lin between frequency repone of equivlent ytem ( ) ( ) jω H rnfer z = H jω ; z = e ; = jω, ω = 2 π/ z= e = function jω Ω 2π H ( ) ( ). Freq. e = H ω ω Ω = H ω= = = repone he frequency repone of the igitl ytem i the me with the frequency repone of the nlog ytem of limite bn for frequency le thn hlf of mpling frequency (or F=.5 normlize frequency) H π π ω = Ω ω ω ( ) H ( ) ; n Ω=ω M 6
frequency repone of the igitl ytem ~ comb F=f normlize frequency. he normlize mpling frequency i F = 7 Exmple: RC circuit non bn limite ytem liing error () t y RC + y() t = x() t H ( ) = ; ω = =. t + RC ω ( ) Freq. repone H ω = =. ω ω + jω + j ω t ωt Anlog ytem: h () t =ωe σ () t = e σ() t. n Digitl ytem: h[ n] = h( n) = e σ[ n]. ω 8
he impule repone of the icrete ytem i n h[ n] = h( n) = e σ[ n]. provie tht the mpling tep i mll enough compre to (time contnt of the circuit) Since thi i non-bn limite ytem, liing error occur (the frequency repone of the igitl ytem i ffecte t higher frequencie) 9 he unit tep repone of both nlog n igitl ytem (repone to σ(t), σ[n]) t ω Y( ) = H( ) X( ) = y t = e σ t ( ω + ) Y ( z) = H ( z) X ( z) = e z ( z ) n y [ n] = e e σ[ n] e () () ( ) ( ) ( ) he error: e n = y n y n 2
( ) n For n : y n = e, n y ( n) e, if <<. n e( n) e, <<, n. n For =, e,95, the error i: e( n), e. For mller vlue of / the error i even mller 2 2π For / =ω =ω =. ω he mpling frequency i: ω = 2πω ω At ω= = πω, the mgnitue repone of the nlog ytem 2 H ( πω ) =.38 + jπ π ( πω ) 2log H 3B which men the liing error i negligeble 22
H Frequency repone ( ) H ω = ; + jω ( Ω ) = e e jω 23 24
25 2. Step Invrince Metho [ n] = ( n) Input ignl: unit tep x() t = σ() t X( ) = Step repone of the nlog ytem y () t = h () t σ() t Y () = H () Digitl input ignl: Z x[ n] =σ[ n] X ( z) = z 26
Digitl ytem trnfer function: ( ) H H ( z) = ( z ) Z L ( n) Step repone: () = () σ () n [ ] = [ ] σ[ ] t h t t n h n n Z ( ) = L [ ] ( ) n = n ( ) H z H [ n] z ( n) 27 Exmple: RC circuit Step repone t () t e = σ() t Step repone of the equivlent igitl ytem n n e n with impule repone [ ] = σ[ ] n [ ] [ ] [ ] ( t) ( ) h n = e e σ n δ n h n. n For << : h [ n] e σ[ n] δ[ n]. y () () RC + y t = x t t 28
he igitl equivlent ytem, for << : n h n e n n [ ] σ[ ] δ[ ] If we pply to the ytem Dirc impule δ () t h () t [ n] h [ n] h ( n) δ he error i e( n) = h( n) h[ n] δ [ n], <<.. 29 3. Finite ifference pproximtion ( ) ( t) y RC + y() t = x() t, = RC = time contnt t ω H = + Firt erivtive pproximtion y () t y ( n ) y ( n ) y [ n] y [ n ] t= n = t + yn yn = xn [ ] [ ] [ ] rnfer function of the igitl ytem: H z H + z z + ( ) = = = ( ) z = 3
M b N M y() t x() t = = b H ( ) = N = t = t = () ut t t= n Generl ce p p ( ) Cun [ p] p= p p z ( ) C ( z ) U( z) = U( z) p= N M = p p p p ( ) [ ] ( C y n p b ) C x[ n p] = p= = p= 3 Apply z trnform: N M z z Y z b X z = = ( ) = ( ) M z b = H ( z) = = H( ) N z z =... = trnfer function, originl c.t. ytem equivlent.t. ytem, finite ifference pproximtion. z = z H ( z) = H ( ) ; = 32
Reltion Between n z plne Finite Difference Approximtion jω z = x+ jy = re n =σ+ jω. z = z = r = z = ; σ< r < ( σ ) + ( ω) 2 2 ω x= n y = 2 2 + ( ω) + ( ω) 2 y x nottion: = ω x= or x= x y x + y + 2 x 2 2 or x + y x=,becue generlly x 2 2 2 33 he rel n imginry prt of z ecribe circle: 2 2 2 x + y = 2 2 plne: imginry xi j ω z plne: contour of circle C= /2,, ( ) riu /2 Reltion nlog frequency ω - igitl frequency Ω: y = tg Ω=ω Ω= rctg ω x 34
-plne z-plne 2 2 2 x + y = 2 2 Left hlf plne σ< ic z < / 2,center = / 2, ( ) ( ) ( ) Right hlf plne σ> exterior of ic z > / 2,center = / 2, Imginry xi = jω, σ= ic z = / 2,center = / 2, + jω z = = = x + jy jω + ω ( ) 2 35 When n nlog tble ytem i pproximte, ielly the equivlent igitl ytem houl lo be tble Anlog ytem n igitl ytem hve the me frequency repone if the imginry xi on -plne become the unit circle on the z-plne. Not true for thi metho 36
π Ω π or Ω 36 2 2 2 x + y = n z = very cloe 2 2 π π H ( ω) limite t ωm : Ω ω M = tg ΩM ΩM 36 36 ω 36 2 ω very high vlue!!!!! M Smpling theorem ω 2 ω t limit ω = 36ω M Very high mpling frequency pproprite for moeling low frequency ytem (electromechnicl ytem) 37 Exmple: firt orer LPF time contnt: = RC =. ω H H ( ) ( z) = + = + z + 38
Finite ifference pproximtion 39 Finite ifference pproximtion 4
Finite ifference pproximtion higher ccurcy! 4 y t 4. Biliner rnform t = x() t y() t = x( ) H ( ) = Input ignl Are A n (ABCD)~ integrl I n : numericl metho (trpezoil rule) n n ( n ) ( ) I = x 42
n ( ) (( ) ) ( ) I = y n y n = x ( n ) ( AB CD) AD ( ) + (( ) ) + x n x n = 2 2 yn [ ] yn [ ] = ( xn [ ] + xn [ ] ) 2 + z H z H ( ) = = ( ) 2 z 2 z = z + n Biliner trnform: 2 z = ; H z = H + z ( ) ( ) 2 z = z + 43 Reltion of the n z plne for the biliner trnformtion metho + 2 z = z = 2 + z 2 z 2 2 + σ + ω 2 2 = σ + ω 2 2 2 2 σ< z <. σ= z = σ> z >. 44
σ, left hlf plne ( plne) unit ic (z plne) So: < z < σ= z =, imginry xi ( plne) unit circle (z plne) σ> z >, right hlf plne ( plne) outie the unit ic 45 Reltion between frequency repone for the biliner trnform Anlog n igitl frequencie connection: 2 Ω ω= tg or 2 ω Ω= 2rctg 2 46
ω + j ω j2rctg 2 2 jω = jω z = = e = re ω j 2 ω r = z = n Ω= 2 rctg 2 Imginry xi ( plne) unit circle (z plne). Anlog n igitl frequencie: 2 Ω ω ω= tg or: Ω= 2rctg. 2 2 47 Ditorte igitl frequency repone ue to non-liner reltion between frequencie 48