ECE Introduction to Comm/SP, Fall 2011 Homework #5 Due: 11/04/11

Transcription

1 2 O ECE Intoduction to Comm/SP, Fll 2011 omewok #5 Due: 11/04/11 1. Conside the following system to pocess continuous-time signls with discete-time pocessing. Smple: Feq = Convet to Sequence Rte Lowpss () Suppose tht!#"%$'&)(+*-,./" $'&, 789(:7<;=(?>@6666 z, nd the discete-time filte fequency esponse, which you ecll is peiodic with peiod 032, is given in ABC2ED'2GF s: "1IJ'KG&L(N O D P PRQ2 Find the output ST"%$'&. (b) Suppose tht!#"%$'& hs Fouie Tnsfom UV"174&=(XWE"174YR>@6666&, 7Z8[(\7<;](^>@6666 z, nd the discete-time filte fequency esponse, which you ecll is peiodic with peiod 032, is given in ABC2ED'2GF s: "1I J_K &`( >3D bo QcP PQ2 6D othewise Find the Fouie tnsfom de"174& of the output Sf"%$_&, nd then use it to find Sf"%$_&. (c) Suppose tht!#"%$'&g(:*-,./" $_&, 7 8 (h7 ; (\>@6666 z, nd the discete-time filte is defined by the diffeence eqution: SfA itfg(j6kl5<sfa imb>nfop!#a itf Find the output ST"%$'& [Note: You my get some weid tigonometic thing (e.g. *-,.qfs"-tu& ) tht you cnnot evlute without you clculto. Just leve it in its tigonometic fom (e.g. *-,.Zqfv" tuz& ) in you nswe.]

2 t 2. Conside the following system to pocess continuous-time signls with discete-time pocessing. Smple: Feq = Convet to Sequence Discete Pocesso Rte Lowpss You discete-time pocessing toolkit on you compute hs the following fou blocks fo possible use (when fequency esponse is given, it is vlid fo ABC2ED'2GF, nd it epets outside of tht nge, of couse): Block 1: "1IJ'KT&`( I q J'K <D whee you cn choose ny i tht you wish Block 2: O "1I J'K &`( Block 3: "1I J'K &`( >3D P 6D else PXbO Block 4: The fouth block povides output SfA itft(x" B >s&!ea itf fo input!ea itf. Using ny combintions of these blocks (including multiple vesions of ny, if needed) nd specifying the smpling fequencies 78 nd 7<;, povide septe system to implement ech of the following desied continuous-time opetions with the block digm bove. () The lowpss filte Kz to yield the signl ST"%$'&. "174& (VW#"174Y303666&, to be pplied on input signls!#"%$'& of bndwidth up to >3kl5 (b) A system tht delys the input!#"%$'&, of bndwidth less thn >@6 Kz, by 2 C>@6Gq t seconds to yield the signl Sf"%$_&. (c) A system tht yields ST"%$'&`(!#"%$'& fo inputs!#"%$'& of bndwidth less thn >@6 Kz. (d) A system tht implements the bndpss filte: "174&`( >3D 53666]QcP 7LPRQ >@6666 6D else on inputs signls!#"%$'& of bndwidth less thn >@6 Kz to yield the signl ST"%$'&.

3 7 3. A voice signl "%$'& of bndwidth 5 Kz is smpled t 10 Kz to yield the discete-time signl A itf. We desie to tnsmit this discete-time voice signl using double-sidebnd suppessed cie with some pssbnd filteing by filte "%$'&. One possible method of doing such is: A itf "%$'& 78 ( >@6 Kz *-,./" $'& whee "%$'& hs Fouie Tnsfom: 2 "174& B 7EB 5 kz B 7 B 74o 5 kz 7#B 5 kz o 5 kz -1 oweve, we hve tough time building bndpss filtes nd thus would like to do the filteing t (complex) bsebnd. Futhemoe, since A itf is ledy discete-time, we might s well do the filteing in discete-time, which llows us to use the following cicuit: A itf Discete Pocessing 78 *-,./" $'& bo 7 8 O Find 78, 78 O, nd the discete-time pocessing such tht this cicuit yields the sme output s the cicuit in the fist digm bove.

4 4. The compny you wok fo builds d to collect dt on signl!e"%$_& with Fouie tnsfom: U "174& 1-10 Kz 10 Kz 7 The d smples (using n idel smple) the signl!#"%$'& t te of 25 Kz to poduce the discetetime sequence!#a itf, which is ecoded. Note tht the wvefom!e"%$_& is no longe vilble, since the d only ecoded!#a itf. () Find Ü "1I J_K &, the Fouie Tnsfom of the discete-time sequence!#a itf. You boss comes to you nd tells you tht she wishes tht you hd lowpss filteed!#"%$'& with the filte 8 "%$'& befoe you smpled it. The filte 8 "%$'& hs fequency esponse: 1 8 "174& -10 Kz 10 Kz 7 You sech the WWW nd e ble to find coefficients fo vey good discete-time filte fequency esponse, which you ecll is peiodic with peiod 032, given in ABC2ED'2GF s: A itf with "1I J_K & 1 B t b (b) Using discete-time pocessing with only upsmples, downsmples, pefect discete-time lowpss filtes nd the discete-time filte A itf, give block digm fo pocessing!ea itf to obtin the sequence SfA itf tht you boss desies: tht which would hve been the esult of smpling 8 "%$_&!#"%$'& t 25 Kz. t b

5 5. In this poblem, you e going to conside how to do multipliction of two continuous-time signls using discete-time pocessing. () Find the Fouie tnsfom UV"174& of:!#"%$'&`( sinc" >@66663$'& nd the Fouie tnsfom de"174& of Sf"%$_&`( sinc" 66663$_& (b) Find the bndwidth (you do not hve to find the full Fouie tnsfom unless you would like) of the signl "%$'&L(!E"%$_&Sf"%$'&. (c) You buddy smples the signl!e"%$_& t 15 Kz to obtin the sequence!ea itf nd the signl Sf"%$_& t 45 Kz to obtin the signl SfA itf. Give the discete-time pocessing of!#a itf nd STA itf, nd the smpling fequency 73;, fo the convesion such tht the cicuit below outputs "%$_&[(?!#"%$'&Sf"%$_&. [Notes: (1) Be sue tht you cicuit would wok fo ny!e"%$_& nd ST"%$'& of the sme bndwidth s the signls given. (2) You hve ll discete-time blocks fo you use, including discete-time filtes of ny type, upsmples, downsmples, mthemticl opetions of ny type, etc.] Discete Pocessing Rte Lowpss