Handout 10: Pulse-Code Modulation

Transcription

1 ENGG 31-B: Priciples of Commuicaio Sysems Hadou 1: Pulse-Code Modulaio Firs Term Isrucor: Wig-Ki Ma November 1, 17 Suggesed Readig: Chaper 7 of Simo Hayki ad Michael Moher, Commuicaio Sysems (5h Ediio), Wily & Sos Ld; or Chaper 7 of B. P. Lahi ad Z. Dig, Moder Digial ad Aalog Commuicaio Sysems (h Ediio), Oxford Uiversiy Press. Pulse-code modulaio (PCM) is a echique ha aims a represeig a aalog message sigal by a sequece of biary pulses. By doig so, we ca perform digial rasmissio of aalog sigals. Noe ha PCM follows esseially he same priciple i aalog-o-digial ad digial-o-aalog coversios i digial sigal processig. Also, PCM is a file forma for sorig voice ad music. 1 Sysem Descripio Figure 1: A PCM sysem diagram. (a) The rasmier. (b) The receiver. Figure 1 shows a sysem block diagram of PCM. Some key compoes of PCM are described as follows. Samplig: The process is he same as he samplig process i PAM rasformig he icomig coiuous-ime sigal io a discree-ime sigal. The sampler obais sample ampliudes m(t s ) from a message sigal a imes = T s, wih akig ieger values. I order o saisfy he perfec recosrucio requireme by he samplig heorem, he samplig rae f s = 1/T s is assumed o be a leas wice of he message badwidh W. For a similar reaso, a lowpass filer may be applied before he sampler o exclude frequecy compoes greaer ha f s /. Quaizaio: Quaizaio aims a discreizig he sigal ampliudes. The quaizaio process may be described by he followig formula v(t s ) = Q[m(T s )], (1) 1

2 where v(t s ) deoes he quaizer oupu, ad Q[ ] is he quaizer fucio. The quaizer oupu v(t s ) has is values ake from a fiie se of sigal ampliudes or levels. Specifically, he values of v(t s ) are ake from a sigal level se {v 1, v,..., v L }, where each v i is a prespecified sigal level ad L is he umber of levels. Figure shows a example, where he umber of levels is four ad he levels are v 1 = 3, v = 1, v 3 = 1, v = 3. The quaizer fucio i his example is give by 3, m(t s ) 1, < m(t Q[m(T s )] = s ) 1, < m(t s ) 3, < m(t s ) More geerally, he quaizer produces a oupu Q[m(T s )] = v k if v k is closes o m(t s ); ha is, he disace m(t s ) v k is he smalles amog all oher disaces m(t s ) v i for all i = 1,..., L, i k. m(ts) v(ts) (a) (b) 3 1 v m (c) Figure : A example of he quaizaio process. (a) The sampled sigal m(t s ). (b) The correspodig quaized sigal. (c) The quaizaio fucio. The dashed lies i (a) represe he decisio hresholds of he quaizer. Ecodig: The purpose of ecodig is o represe each quaized sigal level v i o a biary codeword. A example of he ecodig process correspodig o he example i Figure is

3 show i Table 1. For coveiece, suppose ha he umber of levels L is a power of. The we have L = R, () or equivalely, where R is he umber of bis per sample. R = log (L), (3) Level v i Biary codeword Table 1: A example of he ecodig able for he four-level case. Lie codes: The ecodig process resuls i a biary daa sream (like, i form of ). A lie code describes a scheme of coverig he biary daa sream o elecrical (or physical) sigals for rasmissio over he chael. There are may lie codes, ad some examples are show i Figure 3. The mos aural lie codes are uipolar oreur-o-zero (NRZ) sigalig ad polar NRZ sigalig. Uipolar NRZ sigalig is simply a o-off sigalig scheme, usig zero ad a posiive ampliude o represe ad 1 respecively. Polar NRZ sigalig uses a egaive ampliude, say, A, ad a posiive ampliude A o represe ad 1 respecively. Moreover, uipolar reur-o-zero (RZ) sigalig is a varia of uipolar NRZ sigalig where a half-widh pulse is used i place of he full-widh pulse i uipolar NRZ sigalig. The reaso for doig so is o allow he receiver o exrac imig from he received sigals. See he exbooks for more deails o various lie codes, icludig he bipolar RZ sigalig scheme ad he Macheser code show i Figure 3. Regeeraio: A chael is rarely ideal i pracice. The sigal a he receiver is ofe subjeced o oise corrupios ad sigal disorios iroduced by he chael. I should be oed ha i he previous hadous, we have assumed ideal chaels so as o faciliae our udersadig of modulaio ad demodulaio; i real world, his is o rue. The ask of he regeeraio circui is o recover he daa sream from he received sigal. I is impora o oe a his poi ha digial rasmissio allows us o perform sigal recovery quie accuraely. Simply speakig, he idea is o apply a decisio-makig device, where for each bi ierval he device would observe he received sigal ad decide wheher he received bi is a or a 1. As a addiioal oe, he receiver also has o have imig sychroizaio wih he rasmier i order o recover he daa sream. Timig sychroizaio ca be accomplished by a variey of ways; he problem will o be cosidered here, bu we should oice ha he imig sychroizaio ask is also doe i he regeeraio circui. We should also meio ha i) decodig a he receiver is he iverse of ecodig, coverig received codewords back io sigal levels; ad ha ii) he recosrucio filer a he receiver is o impleme he ierpolaio or perfec recosrucio formula i he samplig heorem. 3

4 (a) Uipolar oreur-o-zero sigalig. (b) Polar oreur-o-zero sigalig. (c) Uipolar reur-o-zero sigalig. (d) Bipolar reur-o-zero sigalig. (e) Macheser code. Figure 3: Illusraio of some lie codes.

5 Quaizaio Noise While quaizaio eables biary represeaio of aalog sigals, i also iroduces errors. To describe he problem, le q(t s ) = m(t s ) v(t s ) () defie he quaizaio oise, he error caused by quaizaio. Figure provides a illusraio of he quaizaio oise i he eigh-level case. We are ieresed i quaifyig he quaizaio oise effecs. To do so, we assume uiform quaizaio where v i v i+1 =, for every i = 1,..., L 1, (5) where is called he sep-size. I his uiform case, he quaizer oupu Q[m(T s )] produces a oupu v k if v k is such ha v k / < m(t s ) v k + /. Also, he sep-size is chose as = m max, (6) L where m max = max m() is he maximum ampliude of m(). I follows ha q(t s ) = m max L. (7) Also, by recallig ha L = R, he above equaio ca be furher expressed as q(t s ) = m max R. (8) Eq. (8) provides us wih he followig implicaio: fixig he sigal dyamic rage specified by m max, he quaizaio oise ampliude reduces expoeially wih he umber of bis per sample R. I oher words, every ime we add oe bi, he quaizaio oise ampliude is reduced by a facor of. v(ts) D q(ts) D (a) (b) Figure : Illusraio of quaizaio oise. The red dashed lie i (a) is he origial message sigal m(). 3 Furher Discussio Digial rasmissio allows us o recover or deec rasmied daa from a received sigal ha has bee subjeced o impairmes by he chael, ad his is accomplished via exploiaio of 5

6 he discree aure of he rasmied sigal (of course, he recovery accuracy is up o a cerai limi which depeds o how worse he chael impairmes are). I fac, his is oe of he key reasos why digial commuicaio is popular. For he same reaso, we ca also recover he rasmied daa from he icomig (ad impaired) digially modulaed sigal ad he regeerae a clea versio of he modulaed sigal for furher rasmissio a device ha performs he above meioed ask is geerally called a regeeraive repeaer. A illusraio for he applicaio of regeeraive repeaers is give i Figure 5. The sceario is ha of a log rasmissio pah, where regeeraive repeaers are placed a iermediae pois o deec ad re-rasmi he sigal. By doig so, coverage ca be exeded or performace may be ehaced. Figure 5: Illusraio of usig regeeraive repeaers o exed rasmissio disaces. 6