NAME: SOLUTIONS EEE 203 HW 1

Transcription

1 NAME: SOLUIONS EEE W Problm. Cosir sigal os grap is so blo. Sc folloig sigals: -, -, R, r R os rflcio opraio a os sif la opraio b. - - R - Problm. Dscrib folloig sigals i rms of lmar fcios,,r, a comp a. r r r

2 NAME: EEE ES Problm. 4ps Cosir sigal os grap is so blo. Sc folloig sigals: -, -, R, r R os rflcio opraio a os sif la opraio b. - - R R Problm. 6ps Dscrib folloig sigal i rms of lmar fcios,,r, a comp a. r r /

3 NAME: EEE ES Problm. 6ps Cosir sigal -rr-. Sc folloig sigals:, -, -, -, - R, r R os rflcio opraio a os sif la opraio b. Problm. 4ps Dscrib folloig sigal i rms of lmar fcios,,r, a comp a.

4 NAME: SOLUIONS EEE W Problm. Comp covolio * r-r---, -. is i sp Uforal, so is a la vrsio of prssio: r--r---. * -* - r--r--- Problm. Cosir filrs: A. B.. i a grap ir impls rsposs. A. B.. i a grap ir sp rsposs. A. sp B. sp L. Wic filrs ar casal? Jsif A. No casal, - is o zro. B. No casal, is ozro i -,. 4. Wic filrs ar sabl? Jsif A. Sabl sic i is IR. is absoll smmabl. Σ 4 < if. B. Usabl sic ivrgig implis a ivrgs. sp

5 NAME: SOLUIONS EEE ES W. 9/7/6, clos boos&os Problm. Comp covolio * - -,. r r Problm. Cosir filrs: A. B.. i a grap ir impls rsposs. A. B.. Wic filrs ar casal? Jsif A. Casal bcas, for all <. B. Casal bcas, for all <.. Wic filrs ar sabl? Jsif A. Sabl bcas < B. Usabl bcas lim, ic ivrgs.

6 EEE W NAME: Problm : L b prioic sigal so i figr blo sqar av i offs. Comp cofficis a of orir sris pasio of. -.5 L b saar sqar av i abls, i 4,.,.5.5 S{ }.5.5 S{ 4 si }.5S{} for for No: S{} for, a oris. Also, prssio for ca b frr simplifi o si.5 Problm : L X b orir rasform of. i X a X. Usig fiiio of orir rasform: X X X No: I is cas, i is also possibl o comp orir rasform of : { } { } { } R{ } R{ 4 } R 4

7 NAME: SOLUIONS EEE ES //6,, clos boos a os, rasform abls allo Problm : L b prioic saoo av sigal so i figr blo. Comp cofficis a of orir sris pasio of. {} { } {} > < Dirc compaio :, S S S S S S S a No: Parsval s orm sas > < a. Appli o or cas: > < 4 4. Compl compaio o riv ll-o formla for sris /.

8 NAME SOLUIONS EEE W 4 /8/6 Problm : Cosir filr i impls rspos.. i rasfr fcio. Sc Bo Plo. i orir rasform of op 4. i orir rasform of op si 5. i op a sig covolio, a b aig ivrs orir rasform of or asr o par. 6. i op si a sig covolio, a b aig ivrs orir rasform of or asr o par 4. } { * 5. * } { }* {si } { 4., } {. s aac plos a,.. X X

9 si cos par R } { si cos Imagiar par si * al m m R R R I I

10 NAME SOLUIONS EEE s 4 CLOSED BOOK & NOES. RANSORM ABLES ALLOWED. Problm : Cosir filr i impls rspos..p i rasfr fcio.p i orir rasform of op si.p i op X X. { } R R R 45 si a si si AL.: 45 si 8 a cos o o o Problm : Cosir filr i impls rspos si..p i rasfr fcio.p i orir rasform of op si.p i op... X X. { } { }

11 EEE W 5 NAME: SOLUIONS Problm : i largs samplig irval s o allo prfc rcosrcio of sigals: NOE: * os covolio of a si si. * si si si si * { } { }{ } ma. frq. ra/s, Nqis frq. ra/s, mi.samplig irval si. si si si * { r r r } s ma { }*{ } ma. frq. ra/s, Nqis frq. 4 ra/s, mi. samplig irval si. *si si si * si { si } { } ma. frq. ra/s, Nqis frq. ra/s, mi.samplig irval s ma Nq. i.., zro or a cosa fcio ca b sampl i arbiraril larg samplig irvals Nq s ma. Problm : or a samplig procss i ra ms, a is coff frqc of ial lo-pass filr for rcosrcio? filr coff frqc sol b sam as maimm allo sigal frqc: s ms ma ma ra/s 5 z. s Nq.

12 EEE ES 5 NAME: SOLUIONS CLOSED BOOK & NOES. RANSORM ABLES ALLOWED. Problm : i largs samplig irval s o allo prfc rcosrcio of sigals: NOE: * os covolio of a si. *cos. si si *cos { } { cos} { }{ } ma. frq. ra/s, Nqis frq. ra/s, mi.samplig irval s ma Nq si. cos. si si cos * cos { } { } { }*{ } ma. frq. ra/s, Nqis frq. 6 ra/s, mi.samplig irval s ma Nq Problm : frqc spcrm of a sigal is so i figr blo. Drmi samplig ra a coff frqc of ial lo-pass filr for rcosrcio. X 4 z maimm frqc i sigal is approimal 4 z, or, 4 ra/s. rfor maimm samplig im, o allo rcosrcio, is /8 /8 s. /f ma ial lo-pass filr coff is sam as maimm frqc i sigal, c 8 ra/s 4 z.

13 EEE W 6 NAME: Problm : Cosir casal filr scrib b iffrc qaio 4. Drmi rasfr fcio. Comp rspos of filr o. Comp sa sa rspos o 4. Comp sa sa rspos o si / 6 - Problm : Cosir coios im casal filr i rasfr fcio s s s Comp rspos of filr o Problm : Cosir iscr im sabl filr i rasfr fcio z z z.9 z. Comp rspos of filr o

14 EEE W 6 NAME: SOLUIONS_ Problm : Cosir casal filr scrib b iffrc qaio 4. Drmi rasfr fcio. Comp rspos of filr o. Comp sa sa rspos o 4. Comp sa sa rspos o si / 6 - No: or a sabl ssm if covrgs o a prioic sigal _s, covrgs o a prioic sigal a is ssm rspos o _s. is ca b comp sig orir or. A sfl simplificaio is: o o o cos cos or Coios im: o o o o Ωo Ωo Ωo Ωo Ωo or Discr im: cos Ωo cos Ωo r, s, z ar coios a iscr ssm rasfr fcios, rspcivl... zz zz zz zz zz 4 zz zz 9 zz 4 4 zz zz 4 zz 4 zz zz 4 9 zz zz ZZ 4 9 zz 4 zz 9 zz 4 zz filr is sabl sic pol /4 as magi lss a o. c, sa-sa rspos is ll-fi. or cosa sa-sa, zz cos ssss Ω si Ω Ω, Ω 6, ssss si aa si 6.4 si 6 6 9oo cos 6 4 si 6 cos 6 4 Problm : Cosir coios im casal filr i rasfr fcio s s s Comp rspos of filr o ss ss ss ssss ss LL ss / ss / ss / /

15 Problm : Cosir iscr im casal filr i rasfr fcio z z z.9 z. Comp rspos of filr o zz zz zz zz ZZ zz. zz.9. zz. zz zz.9zz zz ZZ zz.9 5zz zz

16 EEE ES 6 NAME: SOLUIONS CLOSED BOOK & NOES. RANSORM ABLES ALLOWED. 6 Problm : or coios im ssm i rasfr fcio s s s s. i ROC, assmig a ssm is sabl.. i sa-sa rspos o.. i ampli of sa-sa op for a sisoi ip of frqc ra/s si.. pols ar a,. Sic ssm is sabl ROC icls -ais a s o ars pol lf a rig. So, ROC R s > - s. ss ss s s s s. ss si Ampl.. 45 s s 4 s Problm : or iscr im ssm i rasfr fcio z z.5 z.5. i ROC, assmig a ssm is sabl.. i ROC, assmig a ssm is casal. or sabl z, comp rspos o a sp.. pols ar a.5,.5. Sic ssm is sabl ROC icls i circl a s o ars pol. So, ROC {z : z >.5}.. ROC of a casal rasfr fcio s from largs i magi pol, o ifii. So, ROC {z : z >.5}. z Z{ } z z. z z X z z.5 z.5 z Z { z} z z z