GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING ECE6 Fall Quiz # Writt Eam Novmr, NAME: Solutio Kys GT Usram: LAST FIRST.g., gtiit Rcitatio Sctio: Circl t dat & tim w your Rcitatio Sctio mts ot La: Failig to circl t corrct sctio will cost you poits Mo Tu Wd Tu 9:-: L6 Batti :-: L7 Causy :-: L9 Causy L Batti :-6: L Moor L Davport L Juag L Walorst 6:-7: L Agasi L Davport L Juag L Walorst Importat Nots: Writ your am o t frot pag ONLY. DO NOT UNSTAPLE t tst. Closd oo, ut a calculator is prmittd. O pag 8. of HAND-WRITTEN ots prmittd. OK to writ o ot sids. Partial crdit for icorrct aswrs may gratd ONLY w you JUSTIFY your corrct rasoig CLEARLY. You must writ your aswr i t spac providd o t am papr itslf. Oly ts aswrs will gradd. Circl your aswrs, or writ tm i t os providd. If mor spac is dd for scratc wor, us t acs of prvious pags. Prolm Valu Scor Eard No/Wrog Rc - Total
PROBLEM Fall--Q..: T followig figur dpicts a cascad coctio of two liar tim-ivariat systms; i.., t output of t first systm is t iput to t scod systm, ad t ovrall output is t output of t scod systm. Suppos tat Systm # is a -pt avrag, dfid y t diffrc quatio: ad Systm # as t followig frqucy rspos:. a Dtrmi t impuls rspos of t d systm, ; prss your rsult i trms of tim-siftd uit impuls fuctios i.., Rfrrig to t aov figur, otai a sigl diffrc quatio tat rlats y to. c T aov cascad systm is driv y a iput squc, = u u, wr u is t uit stp fuctio. Otai ad prss t output y i a proprly lald stm plot i t o. LTI Systm # LTI Systm # y w w si H si H w 6 6 y 7 6 8 6 y
PROBLEM Fall--Q..: /.8-6 - / 6 f Hz t Idal C-to-D T spctrum of a ral sigal t is sow as aov. T sigal is ig sampld at a rat of 7 sampls/s wit a idal C-to-D, procssd y a LTI systm, ad fially rcostructd wit a idal D-to-C opratig at t sam samplig rat as t C-to-D, to produc t cotiuous-tim output yt. a Writ t sampld squc i matmatical prssio. t.6cos t cos t.6cos.6cos 7 cos cos 7.6cos.6cos sampld at f s = 7 y cos Idal D-to-C yt.6cos.6cos T LTI systm is spcifid y t diffrc quatio: y = +. Otai yt. H cos H cos / H cos.6 cos y.6cos To otai yt, rplac wit f st, i.., 7t.6cos 7 y t t.6cos t y t.6cos t
td t tal if cssay PROBLEM Fall--Q..: A sampld squc is giv: cos7 / 6 / 6cos7 /8 for =,,,, 7. Now, w ta t first quartr of t squc, startig from =, ad comput its -poit DFT. List all o-zro DFT cofficits, X, ot i trms of tir idics ad t corrspodig cofficit valus. 7 9 X 8 6 6 First d to ma sur of t priodicity of t squc. 7 7 Tr is a commo factor of amog 6, 8,. Trfor, t squc as a priod of. Taig t first quartr, i.., poits, would covr actly o full priod. So, w ca ust pursu t DFS ad ma quivalc to fid t corrspodig DFT cofficits. 6 6 cos7 / 6 / 6cos7 /8 7 cos 7 6cos cos 7 6cos 6 8 7 7 7 9 8 T fial stp aov accouts for t scalar factor of i t frot of t summatio of all compl potials for a poit IDFT. T DFS cofficits must scald up y. 7 9 8
PROBLEM Fall--Q..: Fid t Discrt-tim Fourir trasform DTFT X of t squc: si. si.. You oly d to provid valus of t trasform at tr frqucis:,, ad. X. X. X. X... si. si. si.. si. si.. si. si...