Lecture 2. Basic Digital Communication Principles

Transcription

1 Lcr Basic Principls Signals Basd on class nos by Pro: Amir Asi Basic Digial Commnicaion Principls In his lcr w prsn a rviw abo basic principls in digial commnicaion, som o i yo migh hav sn bor Digial vs. analog. Spcral dnsiy Aocorrlaion Random signals Transmission hrogh linar sysms Bandwidh o digial daa

2 Nomnclar Classiicaion o Signals Drminisic Signals W know h val a any poin o im no ncrainy. Modld as an plici prssion Random Signals W don know h val o h signal bor w rciv i. May hibi som rglariis ha can b dscribd in rms o probabilisic modls.

3 Classiicaion o Signals Priodic signal is a signal ha rpas isl ar som priod T h smalls T is h priod. Ohrwis apriodic + T < < Analog signal is a coninos ncion o im spciid or all. Discr signal is a signal ha is val is known only a discr poin o im kt, k is an ingr, and T is a id im inrval. Odd/vn Classiicaion o Signals

4 Enrgy and Powr Signals p v E P T T T T / / R i T / T / T / d d R Usally normaliz R Enrgy dissipad in h inrval -T/,T/ Avrag powr dissipad dring h sam inrval Enrgy and Powr Signals Enrgy Signals I i has a nonzro b ini nrgy or all im Signal nrgy drmin h prormanc o a comm. Sysm. ighr nrgy mans asir dcion and lss rrors In ral world, w always ransmi signals wih ini nrgy T / E lim d d, < E T T / <

5 Enrgy and Powr Signals Powr signals In ordr o dal wih priodic signals, and random signals ha hav inini nrgy, as opposd o drminisic and non priodic signalsw inrodc h concp o powr signal. A powr signal: i i has a ini b nonzro powr or all im P lim T T T / T / d Uni Impls Fncion AKA Dirac dla ncion δ d δ or δ nbondd a δ d

6 Forir ransorm/sris An Enrgy apriodic signal can b rprsnd by is Forir Transorm A powr priodic signal can b rprsnd by is Forir Sris c k k jπ jπ jπ k d and, whr cn T priod d jπ k d Spcral Dnsiy Th spcral dnsiy o a signal indicas h disribion o h signal nrgy powr in h rqncy domain. For nrgy signals, h nrgy spcral dnsiy ESD, and or powr signals, h powr spcral dnsiy PSD.

7 Enrgy Spcral Dnsiy Toal nrgy in h signal is E d Th wavorm nrgy spcral dnsiy ESD o h signal is d ψ and E ψ d Jols/z ψ d For a ral signal is Forir ransorm is an vn ncion o Powr Spcral Dnsiy is a priodic signal wih a priod o T T / P d T T / Th powr spcral dnsiy is G P G n c n δ G n I is a non priodic powr signal, i may no hav a Forir ransorm. In his cas, w s a rncad vrsion o T hn h PSD can b calclad by aking h limi as T gos o nℵ c n

8 Aocorrlaion Aocorrlaion is a masr o maching bwn h signal and a dlayd vrsion o isl. For a ral vald Enrgy signal whr R τ + τ d R τ R τ R τ R R τ ψ R d FT τ < τ < Aocorrlaion and Enrgy Spcral Dnsiy ESD orm a Forir ransorm pair Val a origin nrgy in signal Aocorrlaion For a priodic signal powr signal Aocorrlaion is dind as R τ lim + τ d < τ < T T T / i priodic wih T T / R τ + τ d < τ < T whr R τ R R τ G R T / τ R τ R T / d τ

9 Random Variabls. Sampl Spac: is a s o all possibl ocoms Eampl I: S {, T, T, TT} in ossing o a coin wic. Eampl II: S {NNN, NND, NDN, NDD, DNN, DND, DDN, DDD} in sing hr lcronic componns wih N dnoing nondciv and D dnoing dciv.. Random variabl is a ncion ha associas a ral nmbr wih ach ocom o an primn. Eampl I: In ossing o a coin, w con h nmbr o hads and call i h RV Possibl vals o,,. Eampl II: In sing o lcronic componns, w associa RV Y o h nmbr o dciv componns. Possibl vals o Y,,, Discr RV: aks discr s o vals. RV and Y in abov ampls ar discr 4. Coninos RV: aks vals on an analog scal. Eampl III: Disanc ravld by a car in 5 hors Eampl IV: Masrd volag across a rsisor sing an analog mr. Random Variabls Probabiliy dnsiy ncion o a discr RV: is h disribion o probabiliis or dirn vals o h RV. Eampl I: S {, T, T, TT} in ossing o a coin wic wih no. o hads Val P /4 / /4 Eampl II: S {NNN, NND, NDN, NDD, DNN, DND, DDN, DDD} in sing lcronic componns wih Y nmbr o dciv componns. Propris: Val P a. b. c. p /8 p 3/ 8 P p 3/ 8 3 / 8 always posiiv adds o probabiliy

10 Random Variabls Probabiliy dnsiy ncion o a coninos RV: is rprsnd as a coninos ncion o. Eampl III: Disanc ravld by a car in 5 hors has an niorm disribion bwn 5 and 5 km Proporis 5 5 p. probabiliy. o ingras. always posiiv. < < b a d p b a P c d p b p a Random Variabls RV or coninos or discr RV d p F p F RV or coninos or discr RV } { d p p E n n n i d p F F P d df p F F F F F Man pcd val is dind as m E{}. Varianc E{-m } Varianc var{} E{ } m.

11 Random Procsss A random procss is a ncion o variabls A,, an vn A, and im For a spciic vn A, j w hav a im ncion j Th oaliy o all h sampl ncions, ar calld nsmbl. For ampl: h nmbr o arriving packs a a swich in im -> For a im k k is a random variabl. Random Procsss A random procss can b hogh o as a collcion o random variabls.

12 Random Procsss In commnicaion sysms, i is sicin o hav a parial dscripion o h procss. I is sicin o hav h man and aocorrlaion E{ } k P d Th man is dind as k Whr, k is h random variabl obaind by obsrving h random procss a im k and h pd o k, h dnsiy ovr h nsmbl o vns a im k is m k P k Random Procsss Man is dins as E E { } k p or discr - im random procss { } p d or coninos - im random procss k k k k k { } R, E Aocorrlaion, ;, R d d Eampl Find h man and aocorrlaion o Acosπ + φ whr A and ar consans, whil θ is a niormly disribd RV ovr, π. Calcla h man and aocorrlaion or h aormniond procss.

13 Classiicaion o Random Procsss Random Procsss WSS Saionary Ergodic. Wid Sns Saionary WSS Procss: A random procss is said o b WSS i is man and aocorrlaion is no acd by a shi in h im origin E{ } m consan and R, R. Sric Sns Saionary SSS Procss: A random procss is said o b SSS i non o is saisics chang wih a shi in h im origin p p,, K, k,, K, k ;,, K, k,, K, k,, K, k ; + T, + T, K, k + T 3. Ergodic Procss: Tim avrags qal h saisical nsambl avrags. m lim T T / d T / R τ lim T T / + τ d T / R, R R τ WSS Procsss Js as varianc provids a masr o randomnss or random variabls, aocorrlaion provids h sam or random procsss P R τ R τ R τ R R τ G R E G FT G G G Evn ncion w. r.. Forir ransorm pairs { } Corrlaion R τ G FT d Maimm occrs a τ Powr spcral dnsiy Evn ncion Forir ransorm pairs Varianc τ G is hpowr spcraldnsiy G lim T T T Forir ransorm o

14 Ergodic Procsss For rgodic procsss, w can calcla h nsmbl avrag by calclaing h im avrag ovr a singl sampl ncion. A random procss is rgodic in h man, i m lim / T T T / d T / A random procss is rgodic in h corrlaion sns i T / R τ lim / T + τ d T T / Ergodic Procsss Tsing or rgodiciy is vry diicl. Usally i is sicin o s or rgodiciy in h man and aocorrlaion ha is sally sicin in h absnc o ransin cs.. m qal h dc lvl in h signal. m is h normalizd powr in h dc componn 3. Th scond momn E{ }is h oal avrag normalizd powr 4. SqrE{ } is h rms val o volag or crrn

15 Ergodic Procsss 5. Varianc σ is qal o h avrag normalizd powr in h ac componn o h signal. 6. I h procss has man,h scond momn and h varianc ar h sam, and varianc rprsns h rms val o h normalizd powr 7. Sandard dviaion σ is h rms val o h ac componn. 8. I man is, hn sandard dviaion is h rms val o h signal.

16 Aocorrlaion Th ara ndr h PSD is h signal powr. I w wan o ransmi rciv a rasonabl porion o h powr, hn h PSD givs s h signal bandwidh. I w considr h BW o b h main spcral lob, hn h widh o h main spcral lob is h rqird BW. B also h aocorrlaion ncion alhogh i is a ncion o im, i givs s an ida abo h BW. In h irs igr, h slop is mor gradal han h scond igr, ha mans lss BW. Nois Nois is h nwand lcrical signals in h sysm i is sprimposd on h signal and limi h rcivr abiliy o making h corrc symbol dcision. Nois cold b man-mad swiching, igniion,.. or naral lighning, sn, galacic vns,. Good nginring dsign can limina mch o h nois. Thrmal Johnson nois is h hards o limina rsls rom h moion o lcrons in h marial.

17 Nois Thrmal nois can b dscribd as a zroman Gassian random procss. A random ncion n whos val n a any arbirary im cold b saisically characrizd as p N n [ ] n p σ π σ Nois Varianc is σ Th normalizd Gassian dnsiy ncion has a varianc o. A random signal is rprsnd as h sm o a Gassian nois and a dc componn. Cnral limi horm: Undr a vry gnral condiions, h sm o a larg nmbr o saisically indpndn random variabls approachs h Gassian disribion [ ] z a + n random variabl p za p z A + n random procss Z z σ π σ

18 Whi Nois Th powr spcral dnsiy or hrmal nois is h sam or all rqncis all rqncis o inrs. Whi nois is h sam or all rqncis, i.. manas an qal amon o nois powr pr ni bandwidh Whi Nois N Gn was/z Rn N τ I { G } δ n Dla ncion mans ha nois is oally ncorrlad rom is im shid vrsion. For Addiiv Whi Gassian Nois AWGN, h nois acs ach symbol indpndnly. Throgho h cors, w assm ha h sysm is corrpd by zro-man AWGN

19 Linar Tim-Invarian Sysms Impls rspons, yh i δ Th sysm is assmd o b casal no op bor. In his cas LTI Sysm h Inp Signal Op Signal y α α α d h h y α α α α α α d h d h h y Linar Tim-Invarian Sysm To masr h, s as inp Acosπ, h op is LTI Sysm Inp Signal Op Signal Y { } { } R an, IM Y Y j θ θ [ ] cos A y θ π +

20 Random Procsss and LTI Sysms Man : Aocorrlaion : PSD : µ R G y y y µ τ R G h d τ h τ h τ I h inp is a random procss. Th op is also a random procss. Disorionlss Transmission. For a disorion-lss ransmission, h op may b a dlayd and scald down vrsion o h inp js chang in scal. jπ jπ y K, Y K K For idal ransmission, ransr ncion ms hav a consan magnid, and linar phas shi linar wih rqncy. All signal componns ms arriv a h sam im, sinc θ radians sconds π radians/scond τ dθ π d Ms b proporional o Envlop dlay or grop dlay a masr o disorion

21 Idal Filr Th problm wih h prvios nwork, is ha i is no ralizabl by diniion, i rqir an inini BW. I w can limi or anion o a band o rqncis bwn l and, h pass band o h ilr. Low pass ilr l, ini, high pass ilr l ini,, band pass ilr l, Idal Filr < π j Lowpass Filr : < π l l j ighpass Filr : < < π j Bandpass Filr : l l Calcla h impls rspons or hs 3 ilrs, casal? Ralizabl?

22 Idal Filr sinc sin ilr idal d d h j j j j j j I < π π π π π π θ θ Non-casal

23 Ec o idal LPF on Whi Nois τ τ Y n Y N R N G G sinc ohrwis / h op Dnsiy o Powr Spcral < Th aocorrlaion ncion is no longr a whi nois no a dla ncion. I has zro corrlaion only a mlipl o Ralizabl ilrs RC RC RC J j π θ π π θ an, + +

24 n Ralizabl Filrs P V log log log db P V Brworh Filr, is h ppr - 3db co rqncy n + / Ec o RC ilr on Whi Nois G R Y Y G n N N τ 4RC + πrc p Again, no longr whi nois. τ RC For a narrowband ilr a larg RC, h op nois has highr corrlaion bwn nois sampls o a id im shi han h op rom a widband ilr

25 Ergodic Procsss hrogh LTI Filr Y µ E Y Y µ µ Y h τ τ dτ [ ] h τ E µ [ τ ] h τ dτ Impls rspons h E h τ τ dτ dτ h τ µ τ dτ T is saionary, h man is consan Y is h zro - rqncy DC rspons o h sysm Aciviy: ind h aocorrlaion o Y Idal Pls hrogh a LPF

26 Bandwidh Bandwidh al-powr BW: Inrval bwn rqncis a which h powr in h signal has droppd o hal is pak val. Eqivaln rcanglar or nois qivaln BW: W N P /G c. Or i is h BW o a iciios idal rcanglar ilr ha wold pass h sam amon o whi nois powr as h acal sysm. Nll-o-nll BW: Th widh o h main spcral lob Fracional powr conainmn BW :is h rqncy band cnrd arond c conaining 99% o h signal powr Bondd powr spcral dnsiy BW: h widh o h band osid which h PSD has droppd o a crain spciid lvl 35dB, 5dB o h pak val. Absol BW: