w h(t) 6.02 Spring 2010 Lecture #14 DIGITAL C,OMMUNICATIOfi SblSTE:I'1S L, ---1~ Modulation Review Delay Issue JJl J r

Transcription

1 / w h(t) JJl J r L, ---1~ rf (free. domain) a M ~ "II, i\.il II j,!i ~.: ' II II N\ 1\!A (!! i.1 i ;,I \,) \~1 I 1'\ ii 1.1 I ~l '['Ii If I 1/" \!,~i \.!\:\4 "I.~ lfjl Y i! 1 I( \! \ J - 1'1 (!,fi:1 I',."!.ao~J- '.~ Ii...! lny -: :)(:1; I~Q 10 so 1m- 11R W,, -= :~, DIGITAL C,OMMUNICATIOfi SblSTE:I'1S "h~ 6.02 Spring 2010 Lecture #14 Modulation Review Delay Issue,~, r "~,,6:02 Spring 2010 Lecture 14, Slide #1,"1,~'

2 ""va -1\(. 't: \ c{\] ~\C(\J~() ~ t,(ct]c9)!'tcfl '1- ~ \, v L tf I-tP- -!::h.c~

3 W Reminder about Discrete-Time Fourier Series (Assume x[n] is periodic with period N) K L N-l n==o 6.02 Spring 2010 Lecture 14, Slide #2

4 Key DTFS Modulation Identity ~ / M Q.lJ.. ",.::h9/l P:re: 't '"e. i\ C'7 Et 1< Q [ "!<;J. '1.. < /(In x[n] COS Om [n] = L X[k]e J " k.s k~-k 1... n.~.,,,,, s : 1 f. ('l ').?"-i _. ej.. /. '...'.' ';'n4n+ -«:» 'tnn, J Y\Jt c,...r..-"-) p;' L-11 k - Z11. J! -::,-:~~ N tj +e.. ~ ilk =: N" Q(V1 - NM Ffe~telE 1 K 1 K 2 L X[k]ejcnk+n.n)n + 2 L X[klej(nk-r~'rr1,)Tl, k:::::::.-k k=...-l< 1<[(\J 5l nl4t [{\J -= r L "Z(bJ e.-l.(lt<- () i t:~q.., f\f'.j.e,-j s:t",il' R \ K::.-R z, / - ~ ~~"'LCK]~--\~K+.Q~)(\ -t ~ b~j~&.q K-JS k:::...,]s 6.02 Spring 2010 Lecture 14, Slide #3

5 Bw~l\Mr~'e-Jc ~1 c- ~. { t) ~\<h..r {l -::.. 1\ c. \0 DTFS of X (D ff Lr 11.11~ 5o.Clt f le \~+f'\-=- trc)o f K :::~b K~loLf SL l -< == Z If ~_k 8T\~-'~--"-~. " : \ ~ <- 11 x[n].' fs ~-...~ ,.---,.", '~--~'~, ~ ~ I 2 ~ O, -21-,j I ~:;f -----~' j' _ 0.." lq a!. 0.2! Q\ ~:~f -0...!2~O- X[k] ~----:t~-t--~-,---,~_..-j.._..-'-'-=1~5'"..._-_._-=1: :> c... ~ -l Q()\(~~../"IMHz ~ f c. ':: \ Q <) k H ~ r-~ ' "--' t-----l ::r n I......_m J <C ~:';L i MHZ" ',.e. '-I., Tc: ! lfoo 6.02 Spring 2010 Lecture 14, Slide #4

6 ( ) DTFS of modulated x /~9qKt-ttp[ nl=x] n]:cos(2*pi*( 4e5/4e6)[0 1 r~~l~~'~'-~~'~-""t""""-~'-'-"'---~~--~~ f =-~'~-'---~'"f""""==~=~".~w~~~-----~~~ s 1/11 ~~\ o P[k] 0.30, ,------,------,--+--~~--+_-, ,-----,----! 400-0,05: f _,,",...L "'-::_.I---, ~-----,-.---" '-0---' la i' r 0 '~2,O LO -O,~ v; \i. ' - ~{)b~r. MHz trt;>q K rt~ 0.2 1"""!""'" 0.1 ~ ool,, " ~O,2 r -2,0 f J-- U LO ,5 MHz J Spring 2010 Lecture 14, Slide #5

7 J '\, 1 e DTFS of x modulated by two carriers.-.\ [,1 X -r ~ -~XlJJ]. X n.j!-?~7..7tj~!~!: 'l-i'\~)r) / yy~l'--'-~'~f"". sampleslnl ~ C{'>; 2,:lT. \ 2 15 t ~ '. t- -;-~--~-, '-"'1 't"1 ~ t)~ _~\II~H~~M~ '-101,.' ".' _... o ,20 ~ ~:~~ [ a: I ~ \./\\ f ::r un OJ i......!! DTFS Coeffs[kJ ,00",,: --- -O,{)SI -0, " _._--"----_ ~ L) r MHz I.~--- -r-r+-v- -- -r- In... -'-, j_'.' ~,... i5 001 ~. U u u., ' "'----7" L _ ~.--O.O ,5... -'_ MHz - / 6.02 Spring 2010 Lf~Q -F \ t \. U \ 1 \'-. \~t)\(ti, ~ +c../! ' W ~ ~p kh7:- I II\~ ' ~, " "..J l... _ 1, "twk112- ~ : g'qokht- 2.0 ~OW H~-(ORki :f: Mt -T:' Lecture 14, Slide #6 C-.

8 @ x mod' d by two carriers and demod 15 i 101 I (OS dernod cos sarnpleslnl -.., , ,---, ;~~ T1 -,1 so 100,so demod coeffslk] I ' ~ 0.15, \ i>l A A A l' a: QJ 0.10' t rt \ r I \ t \, I =~:~~ ~_.c..._, ~.~.. f \---1 '---! '---' \...J J '---! \,. ~. --'- J )'L!,}Q K~=t -o.s!~i~z I'V1~~rtt ,....~-' , -r-r--r-rr-r-r-r-r-r-r- -' n 0.1" C'I I. 0.0 I""'H,-'". i -0.1 " -0.2' -2., MHz I"~ ~,-] VJe. MQJ\ 0y 4 v Od Klt~ \ Spring 2010 Lecture 14, Slide #7

9 @) Second Carrier Too Close tt-g Q k\-tt, 5 2. \)kti 6 ~~~ lol.-r-," " < sarnpleslnl,..--~ -rtvl1.,. ff"'l"t. -~~j -10 I~ '. I o DTfS Coeffs[kJ ~:~~I L~, I 1 '1 5' ~() t< fa:c 0,20, \., /?l ~ r' \ /"1, ~..,., I 0.00 t.mmm7:~~.mmm m -0,05 i-,,~ L.L. '" /' ---'--... L -4--'-!. ----'- -'-.._._... _,,_.. r \ -2,0 -l~----lo -0, ,01.5 2,0 ".._, MHz I ~Of<Ht'," r+".. 0,21 1-\ ni.~' 0.1 " I I 0,0.. I ~,.' I ' I I,," -0,1 ' i i l'~ lu'..., I. i, j ~0.2,\ I. of -2, , ,0 MHz.~ 6.02 Spring 2010 'bdweef\ C~c.f'f\J5 :rf"\ 1, -+tc::; =- ~QO R'l={=t > -f'm"2. - tc, -:: '+"l(j;( Ht; Lecture 14, Slide #8

10 I~~= ,0- Demod Second Carrier Too 12 et"'---'--~ ---,-_'_- 10 ~~j\\' I 411 ~rllt _~ I ~ji NiJ cos demod cos sarnpleslnl '~------~, ~ o ,, '.' ISO 200 _ 250 JOO 400 ~.:.;~f... demol ~~..f(fs[k],. ', I G.20 r 1\ ~ ~:~~ 1-1~.Ii'., 1'-\, _~:~~LU., u "5 -' ' ' =-65-\ (to i 0.5,, -,, -2, LO. MHz a: OO"t' '\ i t r.. I. \ 0,2 [ 0.11 Ill' ~ % 0.0 I ' \., C I i! - I i. LJ l'--""'~_~n_'.. ~. """~:.-.. _....._.."""" : ,.; _..,. r U "u i.. - -, ,..., ,0 I 6.02 Spring 2010 Lecture 14, Slide #9.. -:~.-

11 LPF transmitted signal before 'ijl-.---f-'" '" sernpleslnj ' - ", ". '1IIi-~ ~1~,~~~~v:~==J_=:=:J~J :'I~i o ~ ~ ~ _ m DBa DlFS Coeffs[k] :~I a.lor A A A A I \Ii ill \ tv' ~ ~~~~~ ~ -Q.05! -0 le 2 L. o ----.:-, -=-, ---.::-_:-----:_" ,~- ii\ I I! I,.~ -J..V -Y.:) v.v i.s 1.0 MH~ 0.2 If. r l n n n 0.1 I, " I' 'I I 001 1\ H!! \ I\! I -0').1 'Ill i ii!i -021 u J u l.l, L.._ _......_._._ _......_..~.._,, ~ _ J ],0 MHz L ~f:- ~ e.. ~-9 r-~ Lt-Q Q k. r\t 0..1\,), ; 1 Q \:'--1-/ "-e- MOd- :::.h,-++er_.b- ~E ~ t.. bo kin: tt ~\) K0t: ~ 60 K tf- ~ 6.02 Spring 2010 ~. ~ 1'M1 f'c 4,l!fl W!il I.,! 1\.\ o'lv F -41 (05 demod cos sarnplesln] P.1.~-w---.j.IJvvv.-~vvv..-,vvvv...A~.vN.."""~"~.~N.r'vvvV-- A~Y.HAViJli 1H ijr..n~ Jf i......_..._..~..,...:... L.--J ~ D ~ _ M _ m a demod (oeffs[k] O.,0,' , r ~ ,.... _.._.--::..- _.._--,-.._._....'1"._ '-"---""'1 (t1!ll f ~ ~r, ~ 0.20" r ' f \ i Iili M (Ir M ~~'~;2~-" --~.5.-.-:;:c--..-~: Lor---Q;,---U-.."..l~ _Jo.-_._ , , n t~.,: t:~' if Olr ng n'lfi. nn ~::~I. U '1 U J J.E 00, ili! 1!lV Hi""'! -----,,,2.,,, i.c -e.s i~~_~ ~!I:'v l~trt!~v~~ot'll <9t~c'\ if 0 ~ t< ~L:=6 cte rv1pel 'fbokji t.""""'-.. ".r., "'.w- -.. O".. '"..'-.. -, "'.. "'\ / ~-&~~t\ct5(~k\n '--~ f.-. '<.. T f tl\1.. 1+\ M<J.ft' no 0 "..l '. 'I ej \~f Lecture 14, Slide #10

12 (~ \.... DT~~n~f~~~d~,l,~~e~t~ S~4t.T\Q~A~0: r[nj= x[n ]sin(2 *pi*( 4e5/4e6)( nj r----r--w----r ~.-.-.-'--., ,..--w ,---~,-. :----'~---"-----"-. Sl~~-n~)~ :~,\~'\ ~~ -51 III ' fs ~, ' o R[k] =::~~_~ L~.~ ~ ~ ~2,O -1,5 -LO -0.5 CUi ,0 0..,3-r,----'-, -'" _ w.~'i:.d.{)kt..trt..mhz ~.' Kll(l 1,0 0,2 ~ '.. " -'.~.r:lj;;l 0,1,. I~' ~ \ 0,0[ " I \ \ - I -0,1..\.I""" ~ ,0 os.. ' MHz ' 1,0-2.(i Spring 2010 Lecture 14, Slide #11

13 6.02 Spring 2010 DTFS of cos demod' d sin mod' d x y.. [tlj 7C0--'~'-"~.,- ~-t; 'R --"-'-----,-- 1~~1\..~\~"'J~ ~h J'I c ss z.. ---"",, o /200 \ DT~S Coeffs[kJ &-::L 0.21, 0,1 _ l"""'!' ' -r: ' ' In' ( III \i:r. I...,I -0.2 I -20., _.. I...J' \ I I" ' I -1.0,I -(),:> i! 0.0 T( 'is n _. I\ MH'. Z 0.3 '. \ J 0.2 "\ ~. \. / - \ ~.~, I U 0.5 n -, \. \ \! ~ if - \\H. "H'~.,\ I., 1m.. I \ \ \ r \ I \v V i -O.3l_._~.-.~ ,-, ",,---.. _f-_...-l._...j..--l. _... -'-_.. ~J '. 0.0 I 0, \ MHz "J ~e\() <; l o.t' <)V f\~, ~.r-e ~.V ««cy 400 L~P -ffi--'::o Lecture 14, Slide #12

14 ( J -." I, }') I \ I, \ I, 1'l t, ~ I.../'. Modv of X [f\-'(). -e..- x C~-

15 I -- \ I -- -, (~ -,A ~ ~vrv\1f\c. X Cf\l is ~ ~ 1\ J LM \~c.rj 'Qr J L.. JL.- ~(~\ X Cf\\.~ II )~ 1.. jll. f) 7\ / '7 (...\J ~..,\(..., _,... Ii"- " t\. 7 i /~ h ~( ;~MjDr~ ~y, /'\ "- Ir( \ ) ("JS..HJI --~ r~ ----~-- ---L - Pt= --'..,---.-.i->: ~..,.---,...f'" "" /' <" J.1'. / i A \ / J~T? 0- ~ "'-~c.....-~ J -., rr-: ~ ;, - J. /\...;- \.r r- I,,' -!'Ul-Vj J \... - J i. ' '\.; ~' ~... I 7\ J.-", j J I 7 />::,. I. - ~\.- I I -~I~ ~ I C- 05.1[.j) - 2-.eli'. (1 A ~ 11\ fi V\ ~ 'I f 'X [~ - 0 "'".. USA.., \1 r.. v _ --..., ~ Ls-, I

16 In this example, consider using sine and cosine demodulation of cosine modulated signal x[n] where there is a delay. In, particular, for the case where Consider a delay of 5 samples (so that 25*Omega_1*D = pi/4) in the modulated signal. The following plots show the impact of a 5 sample delay on modulation and demodulation at four different modulation frequencies (but the same delay). The different modulation frequencies will change the phase shift introduced by the delay, and generate different outputs from sine and cosine demodulation. In particular, the modulation frequencies considered are 25, 50, 75 and 100 times Omega_ Fall 2009 Recitation Slide #1

17 Plot of x[n] and X[k] Note: 0.3 for coefficient 0 and ½*0.3 for coefficients -5, -4,-3 and 3, 4, Fall 2009 Recitation Slide #2

18 Cos25 Omega_1 n modulated x[n] Note: ½*0.3 for coefficient 25 and -25, 1/4*0.3 for coefficients -30, -29,-28, 22,-21, -20 and 30, 29, 28, 22, 21, Fall 2009 Recitation Slide #3

19 Plots of modulated x[n] and x[n-5] (note periodicity!) 6.02 Fall 2009 Recitation Slide #4

20 Cos25 Omega_1 n modulated x[n] after 5 sample delay Note impact of pi/4 phase shift (peaks are 0.3*½ *sqrt(2)/2 for real and imaginary part) Fall 2009 Recitation Slide #5

21 Cos25 Omega_1 n demod after delay 6.02 Fall 2009 Recitation Slide #6

22 Sin 25 Omega_1 n demod after delay 6.02 Fall 2009 Recitation Slide #7

23 Cos50 Omega_1 n modulated s[n] Note: ½*0.3 for coefficient 50 and -50, 1/4*0.3 for coefficients -55, -54, -53, -47,-46, -45 and 55, 54, 53,47,46, Fall 2009 Recitation Slide #8

24 Cos50 Omega_1 n modulation after 5 sample delay Note impact of pi/2 phase shift (peaks are 0.3*½ but are now in the imaginary part) Fall 2009 Recitation Slide #9

25 Cos50 Omega_1 n demod after delay 6.02 Fall 2009 Recitation Slide #10

26 Sin 50 Omega_1 n demod after delay 6.02 Fall 2009 Recitation Slide #11

27 Cos75 Omega_1 n modulated s[n] Note: ½*0.3 for coefficient 75 and -75 and 1/4*0.3 for coefficients -80, -79,-78, -72,-71,-70, 80, 79, 78, 72,71, Fall 2009 Recitation Slide #12

28 Cos 75 Omega_1 n modulated s[n] after 5 sample delay Note impact of approximately 3pi/4 phase shift (peaks are 0.3*½*sqrt(2)/2 for real and imaginary part, and opposite in sign from 25 Omega_1 case Fall 2009 Recitation Slide #13

29 Cos75 Omega_1 n demod after delay 6.02 Fall 2009 Recitation Slide #14

30 Sin 75 Omega_1 n demod after delay 6.02 Fall 2009 Recitation Slide #15

31 Cos100 Omega_1 n modulation Note: ½*0.3 for coefficient 100 and -100, and 1/4*0.3 for coefficients -105, -104, -103, -97, -96, -95, 105, 104, 103, 97, 96, Fall 2009 Recitation Slide #16

32 Cos 100 Omega_1 n modulation after 5 sample delay Note impact of pi phase shift (peaks are 0.3* ½ and real, but flipped in sign) Fall 2009 Recitation Slide #17

33 Cos100 Omega_1 n demod after delay 6.02 Fall 2009 Recitation Slide #18

34 Sin 100 Omega_1 n demod after delay 6.02 Fall 2009 Recitation Slide #19