6.02 Fall 2010Lecture #14• Frequency Domain Sharing• Spectrum and the Fourier Series• Fourier Series Examples• Rise Time and SpectrumFrequency Domain Sharing – Big Picture• Questions – What is Modulation (Mod in figure)?– What do typical X’s look like?– What should the Demodulator be?– What is the relation between and Channel+XModModModx1[n]x2[n]xP[n]...YDeModDeModDeMod...y1[n]y2[n]yP[n]xi[n]yi[n]xP[n]cos -1nCosine Modulationx[n] =PPi=1xi[n] cos -in...cos -Pnx1[n]Ideal Channel Case ?Jl•'\\?IIV\/'J,/t.-----.....r:L'------~(~~~iH/0::- .[OJ~Xrz:I~~i....d"i}<;-l '.. \';.0\\C• 'V)('-<:I. \..J.V'.<I~ \"-~'..... "'"•...--.....J/'"-'?,f',djn!Ot' ~[YJ/~~~ "r'OU'0~tJ0\0'(JM6H--.. r> '.db'055eJVVI1'1-1 .1.J- 1(,\ ~ tl#'>.yy-c;.'\AIfl41+'.A"JI/t:>r~J~ ~-+,~ ~ \~l.".~ <f.\.~rlJ"1171rd. LV]'X/~lVJX~'-..,~-...".•,..if"S'~s-~fV'5tL.f.~~~~::.\')v}..c ()W~)16-l~j'dS1j{)f11s~.~!A/ot)7J() h~..J~,, ~ iJ\J'Cvr7~,r-I .tI,.",r-r?,(-,t;'"rd1--LV]f\-V ••.•.\.'I'•\!C)111r:I\rOlVU'l?,~ .IJ..JV'(),,-+-+1'>Q\V )"---;'1+S-1.Il--'~ \'d~[lJ],~-d=l--~ v..J~00c)l\vI/}o~-+~'1\"\@)'.~Q =v./\'d :::::rd...--./(\'"?U+\1>1-/-\\.--xN----~~_6-·~.--------~~fl---~. ·i-~~·IIr. "8~ ,~-. -19~\-l: ~. J10l-\,\D=VII-II-e•....•......•..-II~-I~-I-I-II,I-I1111-~-II_ \"?::>\J'lb?-f-5'lo~#..J0J~I-~.. V-- ' >---.\--\~/'. \/:,V)->I .-r.-(\AT \-J-\ <,.».r/j~'/'I;\,-.--.,;;-~~. JI~- . .>"r: .,.-r- ~-~<.~.J~ .- '" 'I_ y#,,'W'" •• '"( 1\"I. I.-'" 'VA_~': ()~\)' p\' W ~'--7!-\ ~- I. -+~FSO~~--O=.V.--II\-~C'd~/IL...~\II-,lJ..'-.I-r---,-~If--Jt ~ .....,J ~V.1~v \~ .~ ~ Q\(J. (\ ~-----If)}GV():;. \J---V \o-~.\'>1 -\ ~~~-(QfJS;\t-.'1(\1\-I' .fl.2\-NIICosine in Time x[n] = cos¼3n =12ej¼3n+12e¡j¼3nSine in Timex[n] = sin¼3n =¡j2ej¼3n+j2e¡j¼3nSine Fourier Series, 2 Imaginary valuesx[n] =N2¡1Xk=¡N2X[k]ej2¼Nkn=¡j2ej¼3n+j2e¡j¼3nX[k]2¼NkNote horizontal axis ranges over–pi to pi!Fast Rise Pulsex[n]nSpectrum of Fast Rise Pulsex[n] =N2¡1Xk=¡N2X[k]ej2¼NknX[k]2¼NkSlow Rise Pulsex[n]nSpectrum Slow Rise Pulsex[n] =N2¡1Xk=¡N2X[k]ej2¼NknX[k]2¼NkKey Fourier Series Advantagex[n] =N2¡1Xk=¡N2X[k]ej2¼NknHXYy[n] =N2¡1Xk=¡N2H(ej2¼Nk)X[k]ej2¼Nkny[n] =Pnm=0h[m]x[n ¡ m]ORH(ej-) =PLm=0h[m]e¡j-m--+------II~·-----1,-I,,<,/.....•.•.\u~\JlS---. ~ YJl+-vi: )S"6 -...)~tOf\J--I~-•••••~ov.I'].~~..•~..•...---"--...•.•••--..•....--tiI!/-~•---.-v., \J/','vv'Ci'--, .I.'1/""",.....•.=.\~")~)\;'\J~\.~",/-~o ;::::([>1J'XJ'CJ./rr-,~f::~r_c::P\c-~r~-YJL-~t3\'--..•,...,I,0,Q2~"'"<.,~-~'f"-~\I~C:ll!-vtJ,.~1""~-.1..1-.~..,..,.I.-.•._rE..,.i.,..•••• 'W .••. \I"~ .•..-'Iff,~.•.JIL/r...., I --'"l~l~'-1) ~~-\<#.0=(>1~/wl0--p" .~<1'-,.~. .\...,I@bI~III-h--"""'"-~t -JR\0("----'iI-"-..-\~"", I.2J,.."-1_-•••--'.-,..d..•.-to.~-V.•..''"--.....w-.~f6"-W'..~.~ '\I •.••••. ~ •-...,.-,II' ••....-~-~j ...••.\-~IV'''''1J,J. (!--H..;rl..J'.PO-""-\~rI(L.o>\jX) v:I.~"tfIG()l__.~AJl..1tI-.--..•.... ...•.~""{7rT"--"'I"••••V~-lW•-,-'".••.v.•..~t~...J.,I "A~I~~.tJ' ~~-~J'" ~/-/([.,,'J'><~~'-/f,...i,~( 1-h-t,=:"c6rdc:~---~-~~~,I\. ,)!-,\) Tf ~-"11'd/L~J--~t.. -""t1\"? -\(z:rI.r- ~;-",...I-~./"\®"-\-1I"i'-'"'I}A'$OJ""l~I~ \J~~N-(\..3'OJ., \~ ~\)~ 0:41 -- lL7'-/'\ II\~.J-((>l·~I ~~ )SOJL4-r-rvc z:.-1 -\1\J.!,.f.../v\....-1'\-";"d :-)1lLL.,+,1f./~1\"jIf\d+d'--vi.. ~'"0"(.tr?"",-\--....J••.Ib~'j/..2.)1n'-/-.lIt-.. _I~f ~ ~(U7,>J~\--ll::71JJ..G- ~r;'7!I--..-,-..I,)j~ZJ'~d.-~~I().'lj ~r,01 ., ~)--~-...--.vJ,J.<..\O-:=LJiN~d[D]X('I'\';;1].-:&.o<r..•....••-"-->I ••••.,u./1{t~ ,-\:NT...•.•. -.-...v-,.>:",,-~J\ }F:>\ft.t--!~~J,~~ !f001-ct.J....~~.~~..---'---_....----~_-"2.",\?~:7.)1151~-"2~b'\~l-~-..•..'-7\•••,..:I...."...0lbci\4~Ip.P ..,...,r:~~T.~~.),.~IL~-//'-~-t'""-.--;,-;,-v/.~l-Vyf1(62-=-10)-V'5~o~l(ld~YJnJ05\~.('L-z )J ~f\;"j ~ .~"\(n7JIr-' ,\""~~~1Ie '" ,-tr-- '\r~"'~,l1"';...,.(1\ \ ., 11+/1""-. ~-r\ '\ )l2.)'"' ~c,\J \rt-d•.JI'I/'",}7 ~-I ",,/)~ h'i;-\()'(..~\'" 1L1..'-"" lL:f)\-,,","(Y'jl .r. :J~71-<) \7"~t.--4-, J\'"r-,~.->lIt. v--J..\ •...1LL \ ~I....•.-;\I~//.-,""\~6~f'.L.-;>+).\.~'?-,.r~tJ+ \I-.)I.....J~\,l.LL-z.. \' ,it.(/1/;....'fIC:""l)~.~..\...\'\~G..\I407 _\1>\~'\~ -~ -lL7\"-\)I\t---\./\e--I15'eJ ~ )f\f7I- ~ -.jU11~SJ•-d,-"siI+vt"r~0+--;)-1Q~ \II)0:;.J-t0Nr ::,,>.?'ft\')! ~~k)O)r"G+~(t;I.so?~4-~I-~-'-'--\"'jI\~I;'i\;(.~.t'~dZ+- "\I~~f'-d~+-'. 'bl".ll2""\ .i1-.u.-Z'\"J\1(i2.f'dolr+"0"2 ~r - -;) ·l,+-l;•~)~:~Js:'1~ ~" ]['"2..J/t..,tI-'--.ii/t.~ 12Ol~JitS""h~-L ~~,I-1.-~-.-'>.f-..-~....---........••\L ~-~.."~il~.,."~\I-t!\rt,, ?(,.1; (!fj""I"tvr't1\I,\I~\..•..-~\f1Yf./---_ ..J"L~.~~~".• i><,OZ:=N;oi~-Jd$\(Jeft,hJ'0 \r'?V0\-.\-L~"""---1.00.8fX[~J,0.6!0.4--1\,0.2,".,-t-.•.--0.0o51015205GC..l~'e.- ~~t$e,~TF.s CQ efTrcteo+.,)0.15Real Part0.00•,•4••I•-••4T T• ••4 •~,ill-2-1a1230.10R~~o.os0.15IIImag Part,I0.00 ~ • • • • • • • • • • • • • • • • •• •tl0.10~~~~ 0.05·-2-1o123~ ll__~_l_~_.O_~_J_\_~r~_J__ls_~ C__~_s_~_· _3.0r--------,--------,-------.-----------2.52.0-)([ftj-~--I-------------------------------------I-~I------1.0 ...0.5InReal Part0.1It-_.-•~,,-;.ii,T2.;-1o1230.4r \~~~~JjO.30.20.0Imag
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