EE140 Introduction to Communication Systems Lecture 2

Transcription

1 EE40 Introduction to Communication Systms Lctur 2 Instructor: Prof. Xiliang Luo ShanghaiTch Univrsity, Spring 208 Architctur of a Digital Communication Systm Transmittr Sourc A/D convrtr Sourc ncodr Channl ncodr Modulator Absnt if sourc is digital ois Channl Usr D/A convrtr Sourc dcodr Channl dcodr Dtctor Rcivr 2

2 Sourc Information Mssag: gnratd by sourc Information: th unprdictabl part in a mssag Signal: a function that convys information about th bhavior or attributs of som phnomnon Analog signal vs. digital signal Transducr: convrt snsing signal to lctric signal 3 Contnts Dtrministic signals Classification of signals Rviw of Fourir Transform Frquncy-domain proprtis Tim-domain proprtis Vctor spac and orthogonality 4 2

3 What is Signal? In communication systms, a signal is any function that carris information. Also calld information baring signal. 5 Classification of Signals Analog, discrt-tim and digital signals Analog signal: both tim and valu ar continuous Discrt-tim signal: discrt tim and continuous valu Digital signal: both tim and valu ar discrt Discrt-tim signal Digital signal 6 3

4 Classification of Signals Priodic and non-priodic signals Random and dtrministic Dtrministic signal: no uncrtainty in valu. It can b modld or xprssd by an xplicit mathmatical function of tim. Random (stochastic) signal: its valu is uncrtain or unprdictabl. Probability distribution MUST b usd to modl it. 7 Classification of Signals Enrgy and powr signals Enrgy of a signal: E t 2 t f (t ) 2 dt jouls Signal f (t ) is an nrgy signal if Avrag powr of a signal: Signal f (t ) is a powr signal P t f (t ) 2 dt t2 2 f (t ) t 2 t 0 lim T T T / 2 T / 2 dt f (t ) 2 watts dt 8 4

5 Exrcis: Qustion 9 Exrcis: Solution 0 5

6 Contnts Dtrministic signals Classification of signals Rviw of Fourir Transform Fourir Transform Discrt-tim Fourir Transform Discrt Fourir Sris Discrt Fourir Transform Fast Fourir Transform Frquncy-domain proprtis Tim-domain proprtis Vctor spac and orthogonality Fourir Sris A priodic function = suprposition or linar combination of simpl sin and cosin functions Jan-Baptist Josph Fourir: Studnt of Laplac and Lagrang 807: introducd th Fourir sris xpansion 2 6

7 Fourir Sris In th first frams of th animation, a function f is rsolvd into Fourir sris: a linar combination of sins and cosins (in blu). Th componnt frquncis of ths sins and cosins sprad across th frquncy spctrum, ar rprsntd as pas in th frquncy domain, shown in th last frams of th animation. Th frquncy domain rprsntation of th function is th collction of ths pas at th frquncis. Sourc: 3 Fourir Sris Priodic Signal x(t), priodic with priod, cos sin Constant componnt of x(t) Cosin cofficints of x(t) 2 Sinusoid cofficints of x(t) 2 cos sin 4 7

8 An altrnativ form for th Fourir Sris arctan cos sin cos 5 Complx Exponntial Fourir Sris Eulr s Formula Rvrsd sin 2 cos 2 cos sin 6 8

9 Complx Exponntial Fourir Sris Lt dnot th complx cofficints of, rlatd to and by Exampl Signal sin( t) s( t) s( t ) 0 t t s(t) t Fourir Analysis C n T T / 2 j 2 πnf 0t T / 2 s( t ) dt sin( πt ) j 2 πnt 2 dt π( 4n ) 0 2 Fourir Synthsis s( t ) 2 π n 2 4n j 2 πnt 8 9

10 Fourir Transform Fourir Transform of a continuous-tim signal S( f ) Y s ( t) s( t) j 2ft dt if s(t) is absolutly intgrabl s (t ) dt Eulr s formula: j 2ft cos(2ft) j sin(2ft) Invrs Fourir Transform s( t) Y S ( f ) S( f ) - j 2ft dt 9 Rctangular wavform Exampl g a ( t ) 0 t τ / 2 t τ / 2 G (f ) a τ / 2 τ / 2 j 2πft dt ( j2πf jπfτ jπfτ sin( πfτ) ) τ τ sinc( πfτ) πfτ g a (t) G a (f) 0 (a) g a (t) t (b) G a (f) 20 0

11 Dirac Dlta Function Dirac dlta function δ( t ) 0 and t 0, t 0, - Fourir transform S(f ) Y δ( t )dt δ( 0) δ(t ) j 2πf 0 δ(t dt 0 ) j 2πft dt 2 Dirac Dlta Function (cont d) Sifting proprty: bcaus f (t 0 ) f (t ) δ(t t0 Th impuls function slcts a particular valu of th function f (t ) in th intgration procss Unit stp function Au(t-t 0 ) t t0 A u(t t0 ) 0 t t0 Rlationship btwn δ (t ) and u(t ) d δ( t t0 ) u(t t0 ) dt ) dt f (t ) δ (t t0 )dt f (t ) δ 0 (t t0 )dt f (t0 ) u(t t 0 ) t δ ( τ t 0 t 0 ) dτ t 22

12 Powr Signals: Cosin Wavform Cosin wavform: s(t)=cos2f 0 t τ / 2 j 2πft τ sin[ π(f f0 ) τ ] sin[ π(f f0 ) τ ] S(f ) lim cos 2 f0t dt lim π τ τ / 2 τ 2 π(f f0 ) τ π(f f0 ) τ τ lim sincπτ(f f0 ) sincπτ(f f0 ) τ 2 [ δ(f f0 ) δ(f f0 )] 2 t -f 0 0 f 0 (a) wavform (b) spctrum 23 Th comb function: Th Comb Function FT: Y δ T ( t ) δ(t nt ) n 2π T δ T ( t ) δω nω0 ω0 δω nω0 n n 24 2

13 Fourir Transforms cos 2 2 sin 2 2 xp2 xp2 xp, Proprtis of Fourir Transform Opration. Scaling 2. Tim shifting xp2 3. Frquncy shifting xp2 4. Tim diffrntiation 2 5. Frquncy diffrntiation 6. Tim intgration Tim convolution 8. Frquncy convolution 26 3

14 Contnts Dtrministic signals Classification of signals Rviw of Fourir Transform Fourir Transform Discrt-tim Fourir Transform Discrt Fourir Sris Discrt Fourir Transform Fast Fourir Transform Frquncy-domain proprtis Tim-domain proprtis Vctor spac and orthogonality 27 Discrt-Tim Fourir Transform Many squncs can b xprssd as a wightd sum of complx xponntials as j jn xn X d (invrs transform) 2 whr th wighting is dtrmind as j jn xn (forward transform) X n is th Fourir transform of th squnc It spcifis th magnitud and phas of th squnc Th phas wraps at 2 hnc is not uniquly spcifid DTFT transform pair j jn j j n X xn and xn X d 2 n 28 4

15 Som Proprtis of DTFT Ral part and imaginary part Amplitud and phas is continuous in Priodicity: rpat vry 2 29 Squnc: 0.2 Exampl j X n0 0.2 n 0.2 n jn j x n jn X( jω ) H(j) frquncy 30 5

16 Exampl Squnc: cos cos j X n x n jn n j(20.0 n) j(20.0 n) j(20.02 n) j(20.02 n) amplitud jn frquncy 3 Existnc of DTFT For a givn squnc, th DTFT xist if th infinit sum convrgnc j jn X xn or n j X, j jn jn X xn xn xn n n n Th DTFT xists if a givn squnc is absolut summabl All stabl systms ar absolut summabl and hav DTFTs 32 6

17 Absolut and Squar Summability Absolut summability is sufficint condition for DTFT Som squncs may not b absolut summabl but only squar summabl n 2 xn To rprsnt squar summabl squncs with DTFT W can rlax th uniform convrgnc condition Convrgnc is in man-squard sns M j jn j jn X xn XM xn n j j 2 M lim X X 0 M nm Error dos not convrg to zro for vry valu of Th man-squard valu of th rror ovr all dos convrg 33 Contnts Dtrministic signals Classification of signals Rviw of Fourir Transform Fourir Transform Discrt-tim Fourir Transform Discrt Fourir Sris Discrt Fourir Transform Fast Fourir Transform Frquncy-domain proprtis Tim-domain proprtis Vctor spac and orthogonality 34 7

18 Discrt Fourir Sris Givn a priodic squnc with priod x% [ n] x% [ n r] Th Fourir sris rprsntation can b writtn as xn %[ ] X % j2 / Th Fourir sris rprsntation of CT priodic signals rquir infinit complx xponntials For discrt-tim priodic signals w hav j 2 / m n j 2 / n j 2 mn j 2 / n Du to th priodicity of th complx xponntial w only nd xponntials for DT Fourir sris n xn %[ ] X % 0 j2 / n 35 Discrt Fourir Sris Pair A priodic squnc in trms of Fourir sris cofficints j2 / n xn %[ ] X % Th Fourir sris cofficints can b obtaind via For convninc Analysis quation Synthsis quation 0 j2 / x[ n] X% % W n0 j2 / X% x% [ n] W n0 0 n xn %[ ] X % W n n 36 8

19 Exampl DFS of a priodic impuls train n r xn %[ ] nr r 0 ls Sinc th priod of th signal is j 2 / n j 2 / n j 2 / 0 X% x% [ n] [ n] n0 n0 W can rprsnt th signal with th DFS cofficints as xn %[ ] nr r j2 / 0 n 37 Exampl DFS of an priodic rctangular puls train Th DFS cofficints 4 n0 j2 /05 j 2 /0 n j 4 /0 j2 /0 X% sin / 2 sin /0 38 9

20 Linarity Proprtis of DFS DFS x% n X% DFS x% 2n X% 2 DFS % % ax% n bx% n ax bx 2 2 Shift of a Squnc xn % DFS X% x% nm DFS j2 m/ X% j2 nm/ x% n DFS X% m Duality DFS X% DFS x% n X% n x% 39 Priodic Convolution Ta two priodic squncs DFS x% n X% DFS x% n X% 2 2 Lt s form th product X% X% X% 3 2 Th priodic squnc with givn DFS can b writtn as x% n x% mx% n m 3 2 m0 Priodic convolution is commutativ x% n x% m x% n m 3 2 m0 x% n x% m x% n m 3 2 m

21 Priodic Convolution (Proof) Substitut priodic convolution into th DFS Th innr sum is th DFS of shiftd squnc Substituting X% 3 x% [ m] x% 2[ nm] W n0m0 x% [ m] x% [ nmw ] 2 m0 n0 n0 n n m x% 2[ nmw ] W X% 2 X% 3 x m x2 n mw x mw X% 2 X% X% 2 m0 n0 m0 n m %[ ] %[ ] %[ ] n 4 Graphical Priodic Convolution 42 2

22 Proprtis of Discrt Fourir Sris 43 Contnts Dtrministic signals Classification of signals Rviw of Fourir Transform Fourir Transform Discrt-tim Fourir Transform Discrt Fourir Sris Discrt Fourir Transform Fast Fourir Transform Frquncy-domain proprtis Tim-domain proprtis Vctor spac and orthogonality 44 22

23 Th Fourir Transform of Priodic Signals Priodic squncs ar not absolut or squar summabl Hnc thy don t hav a Fourir Transform Combin DFS and Fourir transform Fourir transform of priodic squncs Priodic impuls train with valus proportional to DFS cofficints j 2 2 X % X % This is priodic with 2 sinc DFS is priodic Th invrs transform can b writtn as 2 2 j j n 2 2 j n X% d X% d j n jn X% d X% Exampl Considr th priodic impuls train pn %[ ] nr r Th DFS was calculatd prviously to b Thrfor th Fourir transform is P % for all j 2 2 P% 46 23

24 Rlation btwn Finit-lngth and Priodic Signals Finit lngth signal x[n] spanning from 0 to - Convolv with priodic impuls train x% [ n] x[ n] p% [ n] x[ n] nr xnr Th Fourir transform of th priodic squnc is j j j j 2 2 X% X P% X This implis that r 2 2 j j 2 X% X r DFS cofficints of a priodic signal can b thought as qually spacd sampls of th Fourir transform of on priod 47 2 X% X X j j 2 Considr th squnc 0n 4 xn [ ] 0 ls Th Fourir transform X j j 2 sin 5 / 2 sin / 2 Th DFS cofficints j4 /0 sin / 2 X% sin /0 Exampl 48 24

25 Sampling th Fourir Transform Apriodic squnc with a Fourir transform x[ n] Sampling th DTFT DTFT j 2 / Rsulting squnc is also priodic and could b th DFS of a corrsponding squnc, which is j X X% X X 0 xn %[ ] X % j2 / n j2 / 49 Sampling th Fourir Transform (Cont d) Th only assumption mad on th squnc is that th DTFT xists j jm j2 / X xm X X % j2 / n xn %[ ] X % m Combin quation to gt Trm in th parnthsis is 2 / % So w gt xn %[ ] x m 0 m 2 / 2 / j m j n 0 j2 / nm xm x m p n m m 0 m j nm 0 r % pnm nmr x% [ n] x n nr x nr r r 50 25

26 Sampling th Fourir Transform (Cont d) 5 Sampling th Fourir Transform (Cont d) Sampls of th DTFT of an apriodic squnc can b viwd as DFS cofficints of a priodic squnc obtaind by summing priodic rplicas of original squnc Th original squnc can b rcovrd by xn % 0 n xn 0 ls If th squnc is of finit lngth and w ta sufficint numbr of sampls of its DTFT, it is not ncssary to now th DTFT at all frquncis to rcovr th discrt-tim squnc in tim domain. Discrt Fourir Transform Rprsnting a finit lngth squnc by sampls of DTFT 52 26

27 Discrt Fourir Transform Considr a finit lngth squnc of lngth xn 0 outsid of 0 n Corrsponding priodic squnc x% n x n r r Th DFS cofficints ar sampls of th DTFT of Thr is no ovrlap btwn trms of and w can writ th priodic squnc as xn % xn mod xn To maintain duality btwn tim and frquncy choos on priod of as th Fourir transform of X% 0 X X% X mod X 0 ls 53 Discrt Fourir Transform (cont d) Th DFS pair % j2 / n j2 / x% [ n] xn %[ ] X % X n0 Th quations involv only on priod, w hav j2 / n xn [ ] 0 X % j2 / n X % 0 n0 xn [ ] 0 0 ls 0 ls 0 n Th Discrt Fourir Transform j2 / n j2 / x[ n] xn [ ] X X n0 0 Th DFT pair can also b writtn as X DFT x[ n] n 54 27

28 Exampl: DFT of a Rctangular Puls is of lngth 5 W can considr of any lngth gratr than 5 For 5, th DFS of th priodic form of X% 4 n0 j2 /5n j2 2 /5 j 5 0, 5, 0,... 0 ls 55 For 0, w gt a diffrnt st of DFT cofficints Still sampls of th DTFT but in diffrnt placs Exampl (cont d) 56 28

29 Proprtis of DFT Linarity DFT xn X DFT x2n X2 DFT ax nbx n ax bx 2 2 Duality DFT X DFT xn X n x Circular shift of a squnc DFT xn X DFT j2 / 0 n - x n m X m 57 Exampl: Duality 58 29

30 DFT Proprtis 59 Circular Convolution Circular convolution of two finit lngth squncs 3 2 m0 x n x m x n m 3 2 m0 x n x m x n m 60 30

31 Exampl Circular convolution of two rctangular pulss 6 0 n L xn x2n 0 ls DFT of ach squnc 2 j n 0 X X2 n0 0 ls Multiplication of DFTs 2 0 X3 XX2 0 ls Th invrs DFT 0n x3 n 0 ls 6 Exampl If 2 2 Th DFT of ach squnc 2 L j 2 2 j X X Multiplication of DFTs X 2 L j 3 2 j

32 Contnts Dtrministic signals Classification of signals Rviw of Fourir Transform Fourir Transform Discrt-tim Fourir Transform Discrt Fourir Sris Discrt Fourir Transform Fast Fourir Transform Frquncy-domain proprtis Tim-domain proprtis Vctor spac and orthogonality 63 Discrt Fourir Transform Th DFT pair was givn as j2 / n j2 / x[ n] xn [ ] X X n0 Baslin for computational complxity: Each DFT cofficint rquirs complx multiplications complx additions All DFT cofficints rquir 2 complx multiplications complx additions Complxity in trms of ral oprations 4 2 ral multiplications 2 ral additions 0 n 64 32

33 Fast Fourir Transform Most fast mthods ar basd on symmtry proprtis Conjugat symmtry 2 / 2 / 2 / 2 / j n j j n j n Priodicity in and 2 / 2 / 2 / j n j n j n 65 Th Gortzl Algorithm Mas us of th priodicity Multiply DFT quation with this factor Dfin j 2 / j2 With 0 for 0 and X y n n can b viwd as th output of a filtr to th input Impuls rspons of filtr: is th output of th filtr at tim j2 / j2 / rn j2 / rn [] [] X x r x r r r0 r0 2 / [] j n r y n x r u nr 66 33

34 Gortzl Filtr H z 2 j Th Gortzl Filtr z Computational complxity 4 ral multiplications 2 ral additions Slightly lss fficint than th dirct mthod Multiply both numrator and dnominator H z 2 2 j j z z z z 2cos z 2 2 j j 2 2 z 67 Scond Ordr Gortzl Filtr Scond ordr Gortzl Filtr H z 2 j z 2 2cos z Complxity for on DFT cofficint z 2 Pols: 2 ral multiplications and 4 ral additions Zros: d to b implmnt only onc, 4 ral multiplications and 4 ral additions Complxity for all DFT cofficints Each pol is usd for two DFT cofficints, approximatly ral multiplications and 2 ral additions Do not nd to valuat all DFT cofficints Gortzl Algorithm is mor fficint than FFT if lss than DFT cofficints ar ndd and log 68 34

35 Dcimation-In-Tim FFT Algorithms Mas us of both symmtry and priodicity Considr spcial cas of an intgr powr of 2 Sparat into two squnc of lngth /2 Evn indxd sampls in th first squnc Odd indxd sampls in th othr squnc j n j n j n 2 / 2 / 2 / [ ] [ ] [ ] X x n x n x n n0 n vn n odd Substitut variabls 2for vn and 2 /2 /2 for odd 2r 2r X x[2 r] W x[2r] W r0 r0 /2 /2 r r x[2 r] W/2 W x[2r ] W/2 0 r0 r GWH and ar th /2-point DFT s of ach subsqunc 69 8-point DFT xampl using dcimation-in-tim Two /2-point DFTs 2/22 complx multiplications 2/22 complx additions Combining th DFT outputs complx multiplications complx additions Total complxity 2 /2 complx multiplications 2 /2 complx additions Mor fficint than dirct DFT Rpat sam procss Divid /2-point DFTs into two /4-point DFTs Combin outputs Dcimation In Tim 70 35

36 Dcimation In Tim (cont d) Aftr two stps of dcimation in tim Rpat until w r lft with two-point DFT s 7 Dcimation-In-Tim FFT Algorithm Final flow graph for 8-point dcimation in tim Complxity: log complx multiplications and additions 72 36

37 Buttrfly Computation Flow graph constituts of buttrflis W can implmnt ach buttrfly with on multiplication Final complxity for dcimation-in-tim FFT /2log 2 complx multiplications and additions 73 In-Plac Computation Dcimation-in-tim flow graphs rquir two sts of rgistrs Input and output for ach stag ot th arrangmnt of th input indics Bit rvrsd indxing X00 x0 X0000 x000 X0 x4 X000 x00 X02 x2 X000 x00 X03 x6 X00 x0 X04 x X000 x00 X05 x5 X00 x0 X06 x3 X00 x0 X 7 x 7 X x

38 Dcimation-In-Frquncy FFT Algorithm Th DFT quation X x[ n] W n0 Split th DFT quation into vn and odd frquncy indxs /2 n2r n2r n2r X2 r x[ n] W x[ n] W x[ n] W Substitut variabls to gt Similarly for odd-numbrd frquncis n n0 n0 n/2 /2 /2 /2 n2r n/22r nr /2 n0 n0 n0 X 2 r x[ n] W x[ n /2] W x[ n] x[ n /2] W /2 2 [ ] [ /2] X r xn xn W n0 n2r /2 75 Dcimation-In-Frquncy FFT Algorithm Final flow graph for 8-point dcimation in frquncy 76 38

39 IFFT DFT and IDFT pair n X x[ n] W 0 r0 n xn [ ] XW 0n r0 Mthod, modify th FFT algorithm x X W n W n 77 IFFT Mthod 2 xn [ ] r0 * * n X W r0 FFT X X W * * ( ) n 78 39

40 Thans for your ind attntion! Qustions? 79 40