EEE 304 Test 1 NAME: solutions

Transcription

1 EEE 4 NAME: olio Probl : For h oio i ih rafr fio. Fi h rgio of ovrg of orrpoig o:.. a abl... a aal.. Cop h i p rpo x, aig ha i i aal. / / / / } { } {R, }; {R, } {. } {R : }, R { :. L L L L Y L Y x L Caali Sabili RS RS Probl : For h ir i ih rafr fio. Fi h rgio of ovrg of orrpoig o:.. a abl... a aal.. Cop h i p rpo x of h aig ha i i abl. / / } { } { ;. } { }. {. Y Caali Sabili LS LS

2 EEE 4 Na: Probl : For h oio-i aal ih rafr fio op h 5 folloig:. h apli a h pha of h a-a rpo o a ioi ip x i o.. h ir-i qival of, a G, ig h Forar Elr Approxiaio a a aplig irval of... For G, op h apli a h pha of h a-a rpo o h ioi x apl a h i ia, i.., x i o. A a - a, i a a 8 a 5 / a 9. G [ ]/ [ ]/ G G G G A a a i G [o ] i ;.5 [o.5] i [o ] i.9 [o.5] i i /[o ] a i /[o.5] 8;.84 /[.54 ] a.84 /[.54.5] 8 a.4 a a 8 a No :agl i i ra o o.54, i.84 No :O gaiv ral par

3 EEE 4 NAME: Clo-boo, lo-o, allaor a rafor abl allo, 45 Probl : Sppo ha a oio i igal ha Forir rafor Z ih axi agi. h igal i o b ovr o ir i for proig, a ho i h figr blo, ih aplig i fix a. x Ai-Aliaig Filr SAMPLING D-Filr Rorio. Brifl i h rol of h ai-aliaig filr.. Sppo ha h., for largr ha h Nqi frq, h aliaig ff ar gligibl. Dig a orr ai-aliaig filr o hi pifiaio.. Aig ial rorio, fi h frq rpo of h abov fro x o, h h Dir-i Filr ha rafr fio..9. Ai-aliaig filr ar aalog lo-pa filr ha ar prior o aplig i orr o aa h high frqi of h aalog igal a r h aliaig ff.. A fir orr ai-aliaig filr ha a rafr fio of h for, hr W i h off frq. Si GZ a Z, h pifiaio. i aifi h G. bo h Nqi frq /. B, o.. hi iqali hol hol for all.. Alraivl, a fir orr filr roll off a -B/ afr i orr frq a a baih a -4B aaio a Nqi., h orr frq W hol b a blo Nqi.. Coir a xpoial ip. Afr aplig, g. hi xpoial ip pro a xpoial op afr filrig, i..,. Aig ha h frq i blo Nqi, h ror igal ill b h a frq, ih apli a oifi b h filr. ha i,. h frq rpo fro x o i.

4 Probl : Fi h larg aplig irval o allo prf rorio of h igal. i 7 o Forir pl[ 7,7] *[ δ δ ] ax 9 9. i i F ax ax, F. ax prf rorio i o poibl 4. i *i [ δ δ ][ δ δ ] ax a ill o F

5 EEE 4 ES 4 NAME: Probl: a. Dri h igal pro if h folloig q of opraio i prfor o a igal x ha i balii o i.., for.. Molaio ih a qar av arrir of frq a a o l, i..:, /, of prio h i a hr ohri. Bapa filrig ih a ial filr / for 4.. Molaio ih h a qar av arrir. 4. Lopa filrig ih a ial filr for. b. o o h l parar aff h op igal? a.. Fro Forir abl or Forir rafor via Forir Sri xpaio S S i * i i, δ δ h rplia of ar r a ah haroi frq, xig /- aro i.. h filrig ill allo ol h frqi aro h fir haroi o pa. I ill liia all ohr opo a h DC. h igal apli ill b h apli of h fir haroi lipli b / filr a h origial igal. [ ] ohri 4 i For vialiaio, ih h al riagl for, h filr igal ill b:. Molaig agai, g h a xprio a b ih i h pla of. [ ] S i i i * h lo frq igal i obai for a -.

6 Vialiaio: 4. Afr lo-pa filrig h lo frq igal i i LP b. h op igal i i LP h l parar a a val i, / hil i /. So h i arg rag fro o. A h o xr h raor of h offii of i ro i i o i i ola b h ro igal a i h ohr b a oa. Boh g filr o b h bapa filr hih aplifi h igal b /. h ovrall offii i ill ro i boh b i ri liarl far aro. Uig rial valaio, h pa appar aro /5, for hih h op apli i ~.9 of h ip.

7 EEE 4 5 Na: SOLUIONS_ Probl : For h fba ho blo, op h rafr fio /, /r r C P _, Probl : For h fba of Probl, ppo P /. a C K /. Dri K, o ha h roovr frq i a h Pha Margi i a la 6 o. Yo a h giv Bo plo o op h ar qaii graphiall. Magi B Bo Plo of P Pha g Frq ra/ A roovr, h pha oiio i K 9. 8 a 6 49 o.4 a a /. Uig hi val g h gai oiio K. K.4.4 K.67

8 EEE 4 W NAME Solio Probl : 4 Coir h filr ih ipl rpo h.. Fi h rafr fio. Fi h Lapla rafor of h op h x i5. Fi h op b aig h ivr Lapla rafor of or ar o par. 4. Ca o obai h a rl ig Forir rafor?., {R 4} 4 5. L{ x}, {R }; 5 5 Y, {R } A B C. L { Y } L L No : A 5 /[ 4 5] 5 / 4; B, C : A 5A B 4B C 4C 5 B 5 / 4, C / 4 5 / 4 5 / 4 / L L o5 i Y. B fiig h F{i5 } i vr ivolv a h ovolio F{i5} F{} a grop r approprial. h Forir approah ol fail if h -axi i o ii h Lapla or o i boar.

9 Probl : Coir h oio i aal filr ih rafr fio. Cop h rpo of h filr o x[] []. Cop h rpo of h filr o x[] [-]. Rpa par a for a abl ih h a rafr fio. ] [ ] [ / / } R { } {R } R { ;. : ] [ / / } R { } {R } R { ;.: } R {. fi. - o ll } { } {R } {R ;. / / } {R } {R } {R ; } {R. R R R R R R R R R Y Y Sabili Y Y Caali

10 Probl : Coir h ir i abl filr ih rafr fio.5. Cop h rpo of h filr o x[] [].. Rpa par for a aal filr ih h a rafr fio /.5 / } { } { ;.5 } { /.5 / } { } {.5 ;.5 } { Y Caali Y Sabili

11 EEE 4 oor Probl : Coir h folloig :.. rafr fio Coio i, aal.. rafr fio Dir i, aal.9 Cop h folloig:. Bo plo xprio, graph. Rpo o i for C a i/ for D. a. I, R. a a 6.8 i i., 8 a a. a Magi B Pha g Ω Ω Ω. Ω Ω.9 l8 a 5 i Ω a o Ω. i 5 5 o Ω. o Ω.9 i Ω i Ω i Ω Ω; o Ω.9.44 i 5.9 5, l # Ngaiv R par NOES: Bo Diagra - h D i a ir approxiaio of h oio o ih aplig i.. I i alo al b a gai of. h approxiaio i goo p o frqi ~.Nqi..4/.4 ra/. h, h rafr fio roll-off a high frqi o o appar i h D. Alo, or frq i bo hi arbirar lii b a faor of o. W xp o viaio of h D rl fro h C o -al agi.4 v.84, pha 6 o v o - I h opaio of agi a pha of h D h D frq Ω /5 ra/apl a. ovr, h ig MALAB bo oa, h frq Frq ra/ b ovr o ra/. ha i, f*[ -.],[ -.9 ],. [,p]bo,6.8, il h orr rl.44, p

12 Probl :. U forar a baar Elr approxiaio of rivaiv o riv h D orpar of h. ih rafr fio, for aplig i.,.... Forar Elr :..... Baar Elr :... U MALAB o opar h p rpo a frq rpo of h iriaio i P.. ih h C rafr fio, a i riaio ig h fio, ih i, oh a foh opio apl o i giv blo. Brifl, rib or obrvaio. - All approxiaio ar goo p o o orr of agi blo h Nqi frq. So oi o b raoabl p o / of Nqi frq. - Forar Elr fail h *pol i viola. f[-. ],[ ];. *[-.. ];ov[ -],[*,-];bf,, [-.*.**];ov[ -],[ -*];ff,, oh,,'oh',foh,,'foh',,,'i', bplo p,f,b,oh,foh, axi[,,- ] bplo bo,f,b,oh,foh, Sp Rpo -5 Bo Diagra Sp Rpo 4 Bo Diagra Apli.5.5 Magi B Apli.5.5 Magi B Pha g Pha g i - Frq ra/ - 5 i -6 - Frq ra/

13 EEE 4 W SOLUIONS Probl : Do Probl 7.8, 7. fro h xboo. 7.8 Coir h prioi igal,,. hi igal i filr b a ial Ai-Aliaig Filr ih off-frq 5., ol h r ih frq l ha h off ill pa ih h a apli a h r ill b r. For a xpoial o b i h pa ba hav 5 5/. So,. Wh hi igal i apl ih aplig i a iffr bol hr; h boo h a raig ofio h,. a. hi q i prioi i, ih h prio big N. b. Frhror, b oparig irl ih.94, x i i h for of a Forir Sri xpaio Dir i, o h offii ar ipl. 7. Coir a igal x xp, ih /. h, folloig h opraio i Fig.P.7., x x i x i a xpoial Lopa ba ; i, for a LI ih a xpoial ip x, Noi ha fro h la xprio, h rafr fio i /-xp- hih i o a fii iioal. Si h boo o o pif h apli of h lopa, i h rorio a ha h lo-pa ha h orr apli o rovr h igal i..,. If a a apli of, h rafr fio b ivi b.

14 Probl : Fi h larg aplig irval o allo prf rorio of h igal x* o ovolio i. i ax ig ovolio of Forir rafor. ax i. *i ax i, i i. ax. ax 5 i 4. *i 4 ax i,4,. I fa, if op h pro of h ax forir rafor g o a ill o.

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16 Probl 8.47 a Solio I hi probl a o oir h ff of a lo i hroiaio i pha a/or frq. h olaio a olaio ar ho i h figr blo. x[] [] [] [] LPF r[] o θ o θ For par a a b of hi probl, h iffr i frq i ro, a h iffr i h pha i o b θ θ - θ. a If h pr i ho i Figr P8.47b, h h pr of []. h pr i a ri riagl ha i balii a frq. b Sho ha a b ho o ha h op r[] i r[] x[] o θ. I parilar, ha i r[] if θ /? Solio Obrv h folloig [] x[] o θ [] [] o θ [] x[] o θ o θ [].5 x[] o θ.5 x[] o θ θ Obrv ha i h frq oai h fir r i h qaio for [] i a al vrio of h origial igal a ha h o r i a al a hif vrio of h origial igal. hi obrvaio a b o h h rl. hr ar aall o a hr, pig o h agi of h pha hif. O a ha ovrlap h ohr o o. Fiall, if h arrir frq i ho orrl, i i poibl o a ial lo-pa filr i h frq oai o filr o h o r i h qaio for []. I h pial a hr h pha hif i /, h h op r[] i ro.

17 Probl : 8.49 of xboo a prioi o i Forir rafor i op hrogh h Forir ri xpaio. ha i, / / i 4, i,, a a a a S a o o δ h a i ro for v, ohr ha. h ola hopp x ha Forir rafor a S * ilig h ba-pa filr-hopp igal, a v A A Aa Aa a V h axi alloabl frq o i x for hi xprio o b vali i M /. Nx, V i r-ola hopp a lo-pa filr. h olaio b pro o rplia of a, ah lipli b /, ilig a oal offii of / A. h r of h rplia ar a / ha ar filr o b h lo-pa filr. So, A V Y ih h a oiio o h axi frq i x, M /. b Fro h la xprio, h qival gai of h ovrall i / A.

18 EEE 4 W 5 SOLUIONS Probl : For h fba ho blo, op h rafr fio /r, /, /r, /. r - C P r PC, PC P, PC r C, CP CP, CP Probl : Lo Baih Corollr For h fba of Probl, ppo P /. a. a. Wh C K, ig K o ha h loop roovr frq i.., : PC i.5. Wha i h oribio of a oa i irba o h op? b. Wh C K /, ig K, o ha h roovr frq i.5 a h pha argi i.., h iffr b h loop agl a 8 a h roovr frq, P C 8 i a la 5 o. Wha i h oribio of a oa i irba o h op?.5 : P C K for abili C P -8 P /. /. PC K., li.47 b..5 : C P a 9 a.5 : P C K K K K for abili C P -8 P PC K.56 a.5, o.56 }.5.8 i.56 U, li Vrif i MALAB, ig pfbap,.,fbap,c. U K.56 C.56 /

19 Probl : igh Baih Corollr For h fba of Probl, ppo P /. a. Wh C K, ig K o ha h loop roovr frq i.., : PC i. Wha i h oribio of a oa i irba o h op? b. Wh C K /, ig K, o ha h roovr frq i a h pha argi i.., h iffr b h loop agl a 8 a h roovr frq, P C 8 i a la 5 o. Wha i h oribio of a oa i irba o h op? a. : P C K 4 K for abili C P -8 P / /.48 PC K, li.48 b. : C P a K : P C K U K for abili C P -8 P PC K K K a 7.85, o 4.5 } i4.5 U, li Vrif i MALAB, ig pfbap,,fbap,c\ K 7 C 7.54 / Probl 4: Opioal, % bo Sl a iabl aplig ra a or favori oio-o-ir ovrio ho o iri h orollr of P. a P. obai ir-i qival. Sila h rpo i SIMULINK ir-i orollr, oio i. For Pr..b, folloig h rl of 6 apl/ri-i, hav _r /BW ~ /.., h aplig ra i 6 apl/. or 6a/.7. Uig h Forar Elr ho o fa pol o aplig orai, g h ir qival C /-. Sili ill ap h oio of D a C a log a hr i ol o aplig i. For lira, ra ovrio blo b. h SIMULINK GUI i ho i h figr blo. Sp Sp Sbra Dir Filr Sbra rafr F Sop