Unformatted text preview:

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


View Full Document

MIT 6 02 - Lecture Notes

Download Lecture Notes
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Lecture Notes and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Lecture Notes 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?