Caper : Time-Domai Represeaios of Liear Time-Ivaria Sysems Ci-Wei Liu
Oulie Iroucio Te Covoluio Sum Covoluio Sum Evaluaio Proceure Te Covoluio Iegral Covoluio Iegral Evaluaio Proceure Iercoecios of LTI Sysems Relaios bewee LTI Sysem Properies a e Impulse Respose Sep Respose Differeial a Differece Equaio Represeaios Solvig Differeial a Differece Equaios Lec - cwliu@wis.ee.cu.eu.w
Oulie Caracerisics of Sysems Describe by Differeial a Differece Equaios Block Diagram Represeaios Sae-Variable Descripios of LTI Sysems Eplorig Coceps wi MATLAB Summary 3 Lec - cwliu@wis.ee.cu.eu.w
Iroucio Meos of ime-omai caracerizig a LTI sysem A IO-relaiosip a bo oupu sigal a ipu sigal are represee as fucios of ime Impulse Respose Te oupu of a LTI sysem ue o a ui impulse sigal ipu applie a ime =0 or =0 Liear cosa-coefficie iffereial or ifferece equaio Block Diagram Grapical represeaio of a LTI sysem by scalar muliplicaio, aiio, a a ime sif for iscree-ime sysems or iegraio for coiuous-ime sysems Sae-Variable Descripio A series of couple equaios represeig e beavior of e sysem s saes a relaig saes o e oupu of e sysem 4 Lec - cwliu@wis.ee.cu.eu.w
A arbirary sigal is epresse as a weige superposiio of sife impulses k k k = eire sigal; k = specific value of e sigal a ime k. 0 A series of ime-sife versios of e weige impulse sigal 5 0 k k Lec - cwliu@wis.ee.cu.eu.w k
Te Covoluio Sum a Te Impulse Respose A arbirary sigal ca be epresse as a weige superposiio of sife impulses Te weigs are jus e ipu sample values a e correspoig ime sifs Impulse respose of LTI sysem H{ }: H{} 6 LTI sysem Ipu Oupu y k k k k k k H k k k H H y } { } { Covoluio sum liear ime-ivaria y Lec - cwliu@wis.ee.cu.eu.w
Covoluio Sum Eample Fiie Impulse Respose FIR k k k k k k??? 7 Lec - cwliu@wis.ee.cu.eu.w
Covoluio Sum? k k k k k k 8 Lec - cwliu@wis.ee.cu.eu.w
Illusraio of Covoluio Sum/Iegral Coiuous-ime sigals f * g -- y Oupu of e covoluio - 0 3 9 Lec - cwliu@wis.ee.cu.eu.w
Illusraio of Covoluio Sum g- f g- f - - - + - - + - - - + - - + - y - -3 3 f g - 0 3 0 Lec - cwliu@wis.ee.cu.eu.w
Covoluio Sum Evaluaio Proceure Reflec a sif covoluio sum evaluaio. Grap bo k a k as a fucio of e iepee variable k. To eermie k, firs reflec k abou k=0 o obai k. Te sif by.. Begi wi large a egaive. Ta is, sif k o e far lef o e ime ais. 3. Wrie e maemaical represeaio for e iermeiae sigal ω k=k k. 4. Icrease e sif i.e., move k owar e rig uil e maemaical represeaio for ω k cages. Te value of a wic e cage occurs efies e e of e curre ierval a e begiig of a ew ierval. 5. Le be i e ew ierval. Repea sep 3 a 4 uil all iervals of imes sifs a e correspoig maemaical represeaios for ω k are ieifie. Tis usually implies icreasig o a very large posiive umber. 6. For eac ierval of ime sifs, sum all e values of e correspoig ω k o obai y o a ierval y w k.6 k Lec - cwliu@wis.ee.cu.eu.w
Eample. 3 Cosier a sysem wi impulse respose u. Use Eq..6 o 4 eermie e oupu of e sysem a ime = 5, 5, a 0 we e ipu is =u. Sol: =u Figure.3 a Te ipu sigal k above e reflece a ime-sife impulse respose k, epice as a fucio of k. b Te prouc sigal w 5 k use o evaluae y 5. c Te prouc sigal w 5 k o evaluae y5. Te prouc sigal w 0 k o evaluae y0. Lec - cwliu@wis.ee.cu.eu.w
Eample. coi.. k: k 3, k k 4 0, oerwise. Iermeiae sigal w k: For = 5: w k 5 0 Eq..6 y5 = 0 For = 5: 5k 3, 0 k 5 w5 k 4 0, oerwise 5 Eq..6 3 y5 k 0 4 5k y For = 0: 0k 3, 0 k 0 w0 k 4 0, oerwise Eq..6 y 4 5 5 k 5 3 4 3 3 5 3.88 4 k0 3 4 4 3 0 4 0 0k 0 0 k 0 3 3 4 3 3 k04 4 k03 4 4 3 3.83 6 3 Lec - cwliu@wis.ee.cu.eu.w
Eample.3 Movig-Average Sysem Te oupu y of e four-poi movig-average sysem is relae o e ipu by 3 y k 4 k 0 Deermie e oupu of e sysem we e ipu is u u 0 <Sol.>. Firs fi e impulse respose of is sysem by leig =, wic yiels u u 4 4. Reflec a sif covoluio sum evaluaio k y k k w k.6 k 4 Lec - cwliu@wis.ee.cu.eu.w
Eample.3 coi. For < 0 a > : y = 0. For 0 3: s ierval: < 0 ierval: 0 3 3 r ierval: 3 < 9 4 ierval: 9 < 5 ierval: > 5 Lec - cwliu@wis.ee.cu.eu.w y k 0 e For 3 < 9: k3 /4 4 y /4 3 4 f For 9 < : 9 3 /4 9 3 4 4 y k3
Eample.4 Ifiie Impulse Respose IIR Sysem Te ipu-oupu relaiosip for e firs-orer recursive sysem is give by Deermie e oupu of e sysem we e ipu is bu4, assumig a b a a e sysem is causal. <Sol.>. Firs fi e impulse respose of is sysem by leig =, wic yiels. Reflec a sif covoluio sum evaluaio k y k k w k y y.6 k Sice e sysem is causal, we ave = 0 for < 0. For = 0,,,, we fi a 0 =, =, =,, or u Ifiie impulse respose IIR 6 Lec - cwliu@wis.ee.cu.eu.w
Eample.4 coi. k k k b, 4k 0, oerwise k, k 0, oerwise w k k k b, 4k 0, oerwise For < 4: y = 0. For 4: y b 4 k k b b 4 k4 k k b b 5 7 Lec - cwliu@wis.ee.cu.eu.w
Eample.4 coi. 5 5 b y b 4 0, 4 b, 4 Figure.5c p. 0 c Te oupu y assumig a p = 0.9 a b = 0.8. 8 Lec - cwliu@wis.ee.cu.eu.w
Eample.5 Ivesme Compuaio y: e ivesme a e sar of perio Suppose a ieres a a fie rae per perio r%, e e ivesme compou grow rae is =+r% If ere is o eposis or wirawals, e y=y- If a ere is eposis or wirawals occurre a e sar of perio, says, e y=y-+ Please fi e value of a ivesme earig 8% per year if $000 is eposie a e sar of eac year for 0 years a e $500 is wiraw a e sar of eac year for 7 years. 9 Lec - cwliu@wis.ee.cu.eu.w
w k k 000.08, 0 k 0, oerwise 0 9. 0 6 7 k k 000.08, 0 k 9 000.08, 0 k 9 k k w k 500.08, 0 k w k500.08, 0 k 6 0, oerwise 0, oerwise 0 Lec - cwliu@wis.ee.cu.eu.w
Eample.5 coi. Figure.7 e Te oupu y represeig e value of e ivesme immeiaely afer e eposi or wirawal a e sar of year. Lec - cwliu@wis.ee.cu.eu.w
Coiuous Iegral Reflec-a-sif coiuous iegral evaluaio aalogous o e coiuous sum. Grap bo a as a fucio of e iepee variable. Begi wi e sif large a egaive, i.e. sif o e far lef o e ime ais o obai 3. Wrie e maemaical represeaio for e iermeiae sigal w =. 4. Icrease e sif i.e. move owar e rig uil e maemaical represeaio of w cages. Te value a wic e cage occurs efies e e of e curre se a e begiig of a ew se. 5. Le be i e ew se. Repea sep 3 a 4 uil all ses of sifs a e correspoig w are ieifie. Tis usually implies icreasig o a very large posiive umber. 6. For eac ses of sifs, iegrae w from = o =, o obai. w Lec - cwliu@wis.ee.cu.eu.w
Eample.6 Give a 3 uu u u Evaluae e covoluio iegral y =. <Sol.> w, 0, oerwise w, 3 0, oerwise 3 Lec - cwliu@wis.ee.cu.eu.w y 0,, 3 5, 3 5 0, 5
Impulse Respose of Coiuous-Time LTI Sysem H{} Lec - cwliu@wis.ee.cu.eu.w 4 Recall a LTI sysem Ipu Oupu y } { } { H H H y Liear Time-ivaria y Te oupu is a weige superposiio of impulse resposes ime sife by
Eample.&.7 RC Circui Sysem y Accorig o KVL, we ave Ri y, i.e. RC y If = u, i.e. e sep respose, e soluio is / RC y e u Please fi e impulse respose of e RC circui <Sol.> RC circui is LTI sysem. / RC y e, u u u / RC y e u, // RC // RC y e u / e u / // RC // RC u / u / e u / e u / 5 Lec - cwliu@wis.ee.cu.eu.w
lim 0 0 y lim y / RC e u / RC / RC e u u e e e u RC / RC / RC, Cacel eac oer! / RC y e u,.93 RC Eample.7 RC Circui Oupu i.e. e impulse respose of e RC circui sysem We ow assume e ime cosa i e RC circui sysem is RC = s. Use covoluio o eermie e volage across e capacior, y, resulig from a ipu volage u u. 6 Lec - cwliu@wis.ee.cu.eu.w
Eample.7 coi. <Sol.> RC circui is LTI sysem, so y =.. Grap of a :. Iervals of ime sifs:. For < 0, w = 0 7, 0 0, oerwise. For 0 <, w 3. For, a e, 0 w 0, oerwise 3. Covoluio iegral: For seco ierval 0 < : 0 3 For ir ierval : e, 0 0, oerwise y e e e e y e e e e e 0 Lec - cwliu@wis.ee.cu.eu.w, e e u 0, oerwise 0 0 --
Eample.8 Suppose a e ipu a impulse respose of a LTI sysem are u u 3 u u a Fi e oupu of e sysem. <Sol.> Tere are five iervals s ierval: < 0 ierval: 0 < 3 r ierval: < 3 4 ierval: 3 < 5 5 ierval: 5 Figure.4 a Te reflece a ime-sife impulse respose, epice as a fucio of. b Te prouc sigal w for 0 <. c Te prouc sigal w for < 3. Te prouc sigal w for 3 < 5. e Te prouc sigal w for 5. f Te sysem oupu y. 8 Lec - cwliu@wis.ee.cu.eu.w
Oulie Iroucio Te Covoluio Sum Covoluio Sum Evaluaio Proceure Te Covoluio Iegral Covoluio Iegral Evaluaio Proceure Iercoecios of LTI Sysems Relaios bewee LTI Sysem Properies a e Impulse Respose Sep Respose Differeial a Differece Equaio Represeaios Solvig Differeial a Differece Equaios 9 Lec - cwliu@wis.ee.cu.eu.w
Iercoecio of LTI sysems Te resuls for coiuous- a iscree-ime sysems are early ieical. Parallel Coecio of LTI Sysems Disribuive propery Coiuous-ime case Discree-ime case Lec - cwliu@wis.ee.cu.eu.w 30 Equivale sysem. y
Iercoecio of LTI sysems. Cascae Coecio of LTI Sysems Associaive propery Equivale sysem Lec - cwliu@wis.ee.cu.eu.w 3 z v v z z y Cage variable by =
Iercoecio of LTI sysems Lec - cwliu@wis.ee.cu.eu.w 3. Cascae Coecio of LTI Sysems Commuaive propery Cage variable by =
Eample. Cosier e iercoecio of four LTI sysems. Te impulse resposes of e sysems are u, u u,,. 3 a 4 u Fi e impulse respose of e overall sysem. <Sol.> u u u u a. Parallel combiaio of a : 3 u u b. is i series wi 3 : c. 3 is i parallel wi 4 : 4 u u 33 Lec - cwliu@wis.ee.cu.eu.w
Relaio Bewee LTI Sysem Properies a e Impulse Respose Te impulse respose compleely caracerizes e IO beavior of a LTI sysem. Usig impulse respose o ceck weer e LTI sysem is memory, causal, or sable.. Memoryless LTI Sysems Te oupu epes oly o e curre ipu Coiio for memoryless LTI sysems c simply perform k c k scalar muliplicaio o e ipu. Causal LTI Sysem Te oupu epes oly o pas or prese ipus y 0. Coiio for causal LTI sysems k 0 for k 0 0 for 0 cao geerae a oupu before e ipu is applie 34 Lec - cwliu@wis.ee.cu.eu.w
Relaio Bewee LTI Sysem Properies a e Impulse Respose 3. BIBO sable LTI Sysems Te oupu is guaraee o be boue for every boue ipu. y M k k k k k k k M k M k k Coiio for memoryless LTI sysems k k Te impulse respose is absoluely summable/iegrable 35 Lec - cwliu@wis.ee.cu.eu.w
Eample. Firs-Orer Recursive Sysem Te firs-orer sysem is escribe by e ifferece equaio y y a as e impulse respose u Is is sysem causal, memoryless, a BIBO sable? <Sol.>. Te sysem is causal, sice = 0 for < 0.. Te sysem is o memoryless, sice 0 for > 0. 3. Sabiliy: Ceckig weer e impulse respose is absoluely summable? k k k iff < k k0 k0 A sysem ca be usable eve oug e impulse respose as a fiie value for all. Eg: Ieal iegraor: 36 y wi e impulse respose: = u. Ieal accumulaor: y k wi e impulse respose: = u k Lec - cwliu@wis.ee.cu.eu.w
Relaio Bewee LTI Sysem Properies a e Impulse Respose 4. Iverible Sysems A sysem is iverible if e ipu o e sysem ca be recovere from e oupu Associaive law y iv iv iv Coiio for memoryless LTI sysems iv i.e. Decovoluio a Equalizer iv Easy coiio, bu ifficul o fi or impleme 37 Lec - cwliu@wis.ee.cu.eu.w
Summary 38 Lec - cwliu@wis.ee.cu.eu.w