ANALOG COMMUNICATION (2)

Transcription

1 DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING ANALOG COMMUNICATION () Fall 03 Oriinal slids by Yrd. Doç. Dr. Burak Klli Modiid by Yrd. Doç. Dr. Didm Kivan Turli OUTLINE Th Invrs Rlaionship bwn Tim and Frquny Dira Dla Funion Appliaions o Dira Dla Funion Fourir Transorm o Priodi Sinals

2 THE INVERSE RELATIONSHIP BETWEEN TIME AND FREQUENCY I h im-domain dsripion o a sinal is hand, hn h rquny-domain dsripion o h sinal is hand in invrs mannr, and vi vrsa. This invrs rlaionship prvns arbirary spiiaions o a sinal in boh domains. I a sinal is srily limid in rquny, hn h im-domain dsripion o h sinal will rail on indinily. Exampl: sin puls is srily limid in rquny, bu i is asympoially limid in im ranular puls is srily limid in im, bu i is asympoially limid in rquny. BANDWIDTH Th bandwidh o a sinal provids a masur o h xn o siniian spral onn o h sinal or posiiv rqunis. Whn h sinal is srily band-limid, h bandwidh is wll dind. Howvr, whn h sinal is no srily band-limid, whih is nrally h as, i is diiul o din h bandwidh o h sinal. Baus h manin o siniian spral onn is mahmaially impris. A sinal is a low-pass sinal i is siniian spral onn is nrd around h oriin (zro rquny) A sinal is a band-pass sinal i is siniian spral onn is nrd around ± whr is nonzro rquny.

3 NULL TO NULL BANDWIDTH NULL TO NULL BANDWIDTH 3

4 3-DB BANDWIDTH 3-DB BANDWIDTH 4

5 TIME-BANDWIDTH PRODUCT For any amily o puls sinals ha dir in im sal, h produ o h sinal s duraion and is bandwidh is always onsan. (duraion) (bandwidh) = onsan This produ is alld h im-bandwidh produ. This onsany is anohr manisaion o h invrs rlaionship bwn im-domain and rquny-domain. DIRAC DELTA FUNCTION Th Dira dla union is dind as havin zro ampliud vrywhr xp =0, whr i is ininily lar in suh a way ha i onains uni ara undr is urv. 0 d 0 0 lim 0 d d 5

6 DIRAC DELTA FUNCTION Considr h produ o () and im-shid dla union (- 0 ). Th inral o his produ 0 d 0 F sin d is an vn union F F usin h dualiy d DELTA FUNCTION AS A LIMITING FORM OF GAUSSIAN PULSE Considr a Gaussian puls o uni ara G 6

7 APPLICATIONS OF DIRAC DELTA FUNCTION DC Sinal: A d sinal is ransormd in h rquny domain ino a dla union. sin os d F Usin h Fourir Transorm is ral valud, and usin Eulr rlaionship d x os x sin x APPLICATIONS OF DIRAC DELTA FUNCTION Complx Exponnial Funion: L s apply h rquny shiin propry o DC sinal F Complx xponnial union o rquny is ransormd in h rquny domain ino a dla union a = Sinusoidal Funion: L s us Eulr propry os sin os sin F F 7

8 SPECTRUM OF COSINE SPECTRUM OF SINE 8

9 9 APPLICATIONS OF DIRAC DELTA FUNCTION Sinum Funion: Sinum union dos no saisy Diril ondiions, sin is nry is no ini. W may din is Fourir ransorm by viwin i as h limiin orm i h anisymmri doubl xponnial puls lim lim sn sn 0 0 a a a a a F a G F a sn 4 lim sn 4 0 SPECTRUM OF SIGNUM FUNCTION

10 APPLICATIONS OF DIRAC DELTA FUNCTION Uni Sp Funion: Th Fourir ransorm o uni sp an b drivd usin h Fourir ransorm o sinum union and linariy propry. u u u sn F LINEAR TIME INVARIANT (LTI) SYSTEMS x() h() y x h X( ) H() Y X H In nral w din an LTI sysm by is impuls rspons. I h inpu o his sysm is δ() [and h prior sa o h sysm is 0] hn h oupu o h sysm is h(). I h inpu o his sysm is x() hn h oupu is y x h h x I h Fourir ransorm o h inpu is X(), and h Fourir ransorm o h impuls rspons is H(), hn h Fourir ransorm o h oupu is Y X H 0

11 TRANSMISSION OF SIGNALS THROUGH LINEAR SYSTEMS Th Fourir Transorm o h impuls rspons is alld h ranr union. Th ransr union H() is, in nral, a omplx quaniy H H H H() is alld h ampliud rspons and H() is h phas rspons. Th ain may also xprssd in dibls (db) 0lo0 H LINEAR TIME INVARIANT (LTI) SYSTEMS Linar Tim Invarian Sysms ar omplly hararizd by hir impuls rspons. I you know h impuls rspons or h ransr union o h sysm hn you an prdi h oupu or any inpu o h sysm. Bu wha dos linar man?

12 LINEAR SYSTEM x () x () h() h() y y ax () + bx () h() ay () + by () TIME INVARIANT SYSTEM Today: x () h() y hours lar x () Tomorrow: x () h() h() y y

13 CONVOLUTION Whn h iniial ondiions ar qual o zro, h oupu o h LTI sysm is h onvoluion o h inpu and h impuls rspons: x h y * x h d h x d MORE IMPORTANT SYSTEM PROPERTIES Causaliy and Sabiliy: A sysm is ausal i i dos no rspond bor h inpu is applid. For a linar im-invarian sysm, h nssary and suiin ondiion or ausaliy is h 0 0 A sysm is sabl i h oupu sinal is boundd or all boundd inpu sinals. (boundd inpu-boundd oupu (BIBO)) Th nssary and suiin ondiion or BIBO sabiliy is h d 3

14 TRANSMISSION OF SIGNALS THROUGH LINEAR SYSTEMS Frquny Rspons: Considr a linar im-invarian sysm o impuls rspons h() drivn by a omplx xponnial inpu o uni ampliud and rquny y H h d h h d Th Fourir Transorm o h oupu sinal y() is Y H X d TRANSMISSION OF SIGNALS THROUGH LINEAR SYSTEMS Th ransr union H() is, in nral, a omplx quaniy H H H H() is alld h ampliud rspons and H() is h phas rspons. Th ain may also xprssd in dibls (db) 0lo0 H 4

15 FILTERS A ilr is a rquny-sliv dvi ha limis h sprum o a sinal o som spiid band o rqunis. IDEAL FILTERS Low-Pass Filr H(w) Band-Pass Filr H(w) w 0 w Sopband Passband Sopband wu wl 0 wl wu Sopband Passband Sopband Passband Sopband Hih-Pass Filr H(w) Band-Sop Filr H(w) Passband w 0 w Sopband Passband wu wl 0 wl wu Passband Sopband Passband Sopband Passband 5

16 WHAT DO FILTERS DO? Exampl: Idal Low Pass Filr X( ) H( ) rquny rquny Y( ) : uo rquny rquny WHAT DO FILTERS DO? Exampl: Idal Hih Pass Filr X( ) H( ) rquny rquny Y( ) rquny : uo rquny 6

17 WHAT DO FILTERS DO? Exampl: Idal Hih Pass Filr X( ) H( ) rquny rquny Y( ) rquny : uo rqunis MORE REALISTIC-LOOKING FILTERS hp:// 7

18 FILTERS Dsin o Filrs: A ilr is hararizd by spiyin is impuls rspons h() or rquny rspons H(). Mos o h im h ilrs ar usd o spara sinals on h basis o hir rquny onn. Thror, h ilrs ar usually dsind in rquny domain. Usin h Lapla, h ilr ransr union is wrin as pols and zros H s K s z s z s zm s p s p s p For sabiliy h pols mus b insid h l hal o h s-plan. n COMMON ANALOG FILTERS Burworh: Th pols o h ransr union H(s) li on a irl wih oriin nrd and B as h radius, whr B is h 3-dB bandwidh. Chbyshv: Th pols li on an llips. Th hbyshv ilrs has asr roll-o han Burworh ilr at h xpns o hihr phas disorion. Ellipi: I has asr roll-o han hbyshv a h xpns o rippl in h passband and sopband. 8

19 HILBERT TRANSFORM Shiin phas anls o all omponn o a ivn sinal ±90 drs is known as h Hilbr Transorm o h sinal. Th Hilbr Transorm o h sinal () d Th Hilbr Transorm is a linar opraion. Th invrs Hilbr Transorm d HILBERT TRANSFORM No ha h Hilbr ransorm may b inrprd as h onvoluion o () wih h im union /() Th Fourir ransorm /() F sn G sn G Hilbr ransorm o a sinal is basially passin i hrouh a ilr whos ransr union is qual o sn() This ilr produs a phas shi o -90 drs or all posiiv rqunis and +90 drs or all naiv rqunis. 9

20 HILBERT TRANSFORM Exampl: Considr h osin union G G os sn sin G sn PROPERTIES OF HILBERT TRANSFORM A sinal () and is Hilbr ransorm ĝ() hav h sam ampliud sprum. h maniud o sn() is qual o on I ĝ() is h Hilbr ransorm o (), hn h Hilbr ransorm o ĝ() is () sn or all A sinal () and is Hilbr ransorm ĝ() ar orhoonal. d 0 0

21 LOW-PASS AND BAND-PASS SIGNALS Low-Pass Sinals: Frquny onn is nrd a h oriin and limid by <W Communiaion usin low-pass sinals is rrrd o as basband ommuniaion Limid o wird or abld ommuniaion sysms Band-Pass Sinals: Frquny onn is nrd a a rquny (arrir rquny) muh hihr han h bandwidh. Sprum o a Band-Pass sinal COMPLEX BASEBAND REPRESENTATION Th sprum o omplx sinals is no omplx onua lik h sprum o ral sinals. Thror, mor inormaion an b snd usin omplx sinalin. Considr a nral sinal a os a() is h nvlop and () is h phas o h sinal. W may rwri his quaion in rms o osin and sin os sin I Q a os a sin I ar alld in-phas and quadraur omponns o () Q

22 COMPLEX BASEBAND REPRESENTATION Th nvlop and phas o h omplx sinal is drivd usin h ollowin rlaionship a I Q Q an I PHASOR REPRESENTATION OF A BAND- PASS SIGNAL G(T) Phasor rprsnaion o a band-pass sinal () Phasor rprsnaion o h orrspondin omplx nvlop ()

23 PHASORS x os x sin x You may rall ha Aos ~ w A w R A w R A Phasor rprsnaion o h sinal. I Q Aos Asin in phasomponn quadraur omponn THE PHASOR ROTATES AS THE PHASE OF THE SIGNAL CHANGES Hr is ixd, θ is hanin I Aos Aosw Q Asin Sour: hp://n.wikipdia.or/wiki/phasor 3

24 4 COMPLEX BASEBAND REPRESENTATION () may b wrin in omplx orm a Q I Q I Q I Q I ~ ~ R R R sin os os Th omplx nvlop orrsponds o a phasor ha has h onsan phas roaion is supprssd. x x x sin os sinal hband -pass h omplx nvlop o is ~ COMPLEX BASEBAND REPRESENTATION Fourir Transorm o () an b obaind shiin h rquny o h Fourir ransorm o omplx nvlop G G G * * ~ ~ ~ ~ no ha R[x]=x+x*

25 COMPLEX BASEBAND REPRESENTATION In-Phas and Quadraur omponns Fourir ransorms o i () and q () ar symmri abou h oriin Corrspondin band-pass sprum Fourir ransorm o band-pass sinal is symmri abou h oriin bu i is no uarand o b symmri abou Corrspondin omplx nvlop GENERATING BAND-PASS SIGNAL FROM COMPLEX ENVELOPE 5

26 DEMODULATING COMPLEX ENVELOPE FROM BAND-PASS SIGNAL PHASE AND GROUP DELAY Whn a sinal is ransmid hrouh rquny sliv sysm, som dlay is inrodud ino h oupu sinal in rlaion o h inpu sinal. Phas dlay: I is h dlay o h pur sinusoidal sinal. Th dlay is qual o H( )/ sonds. Group dlay: Th dlay bwn h nvlop o h inpu sinal and oupu sinal. Phas and Group dlays an b drivd by xpandin h phas rspons o Taylor sris. H H H p H H p H 6