2.5. intarziere[s/km] 1.5

Transcription

1 Baseband ransmissions 1. Parameers of he physical lines - The physical lines are channels whose useful frequency band is limied only by heir physical characerisics; - Examples: copper wised wires, coaxial cables, UTP cables - They can be modeled as ransmission media wih disribued consans, characerized by he propagaion exponen γ, aenuaion a, phase shif b and L/ characerisic impedance Z c,: R/ R+jL = ( )+ j ( ); a = l; b= l; Zc( )= ; C G G+jC (1) R/ L/ - in fig. 1, R, L, C and G (leakage conducance) are Figure 1. a. Equivalen scheme of he wised wires per lengh uni. - - The group delay ime vs frequency g (ω) characerisic is defined as: g (ω) = dφ(ω)/dω (1 ) - Condiion required o ensure consan a(f) and linear β(f) (or consan τ g (f)) characerisics, is ().a. - Fulfillmen of ().a leads o an undisored ransmission of he daa pulses on hese channels. - The values of aenuaion and phase shif per lengh uniy are obained imposing ().a in (1); we ge ().b and ().c RC = 1; (a) GL R C G L = + ; (b) = LC; (c) () L C x 1 - Pracically, condiion ().a canno be -8.5 fulfilled over a whole frequency band; - he real characerisics look like figs. 1. b, c Figure 1 b. c. The a(f)-blue and g (f)-red characerisics UTP cable for wised copper wires he slope of a(f) is 1 seeper frecvena[hz] x For medium frequency ransmissions, he group-delay ime disorion is negligible; only he aenuaion disorion should be considered. The characerisic impedance Z c, varies in frequency; is ypical values range from 6 Ω a f < 4 khz o 13 Ω a f in he domain of MHz; - A special case is he DSL (Digial Subscriber Line DSL) ransmission; - Due o he large diversiy of employed cables, he ETSI sandards include some cable parameers and a channel model ha should be used when evaluaing he aenuaion vs. frequency; - For a greaer flexibiliy, he Inserion Loss is employed o specify he a(f) characerisic; - The Inserion Loss (IL) is he raio beween he power dissipaed by a generaor ino he load hrough he pair of wised cables and he power dissipaed by he same generaor direcly ino he same load. I is dependen on he lengh and physical characerisics of he cable. - ETSI characerizes he cables by heir elecrical lengh, i.e., he value of he inserion loss a a given frequency and by heir Z c ; - some brief consideraions regarding he ETSI classificaion of he cables are presened in Annex 1. aenuare[db/km]. Baseband Codes.1. General aspecs - The baseband ransmission signifies he ransmission of a signal in is original bandwidh. - The baseband (or line) coding means he correspondence, acc. o a cerain rule, beween he elecrical signal (wih a finie se of values) and he symbols of he source s alphabe, e.g., binary and 1 ). - In erms of he number of levels of he associaed elecric signal, he BB codes can be spli ino: Unipolar codes wih wo levels, ou of which one differs from zero (e.g., TTL V and 5V) inarziere[s/km] 1

2 Bipolar codes, levels, symmerical o V(e.g., CMOS -1V and +1V) Bipolar codes, 3 levels: V and wo values symmerical o V (e.g., V, -AV and +AV) Mulilevel codes he PAMs see nex chaper - he informaion-carrier signal is recangular simple srucure and implemenaion, bu some shorcomings: very large frequency BW a weak resilience o perurbaions - main aspecs ha should be considered when analyzing hese ransmissions: 1. he disribuion of he power specral densiy of he ransmied signal. he recovery and synchronizaion of he symbol (bi) clock a he receiving end 3. bi-error rae (BER) performance 4. implemenaion complexiy - he disribuion of he power specral densiy should be analyzed for he following reasons: 1) Due o he increase of aenuaion wih frequency, he Physical.Channels have a LP characerisic; heir a(f) characerisic may be approximaed 18 by, see fig.1.b: a(f) = K f; (3) aenuaion[db/km] absolue aenuaion a frequency f absolue aenuaion a he reference frequency f r ar(f) [db] ar frequency[hz] x 1 3 fr f - a signal wih large BW would be disored because is specral componens would be aenuaed differenly. The larger he signal BW, he greaer would be he difference beween he aenuaion of is exreme componens, leading o an increased signal disorion. ) he line-connecion circuis, implemened wih impulse ransformers, do no allow he passage of he d.c. componen and disor he very low frequency componens, e.g. due o he sauraion of he magneic core. - figure shows he a(f) characerisic in absolue values; - in pracice, a relaive aenuaion characerisic, described by (4), is used: a r (f) = a abs (f) - a abs (f ref ); (4) - he power specral disribuion of he daa signal should be modified prior o ransmission; i should be concenraed ino a BW as narrow as possible, placed as low as possible, bu far enough from he d.c. componen for he following reasons: he narrow BW leads o a reduced effec of he non-uniformiy of he a(f) characerisic; he posiioning on low frequencies would decrease he absolue aenuaion of he signal; he disance kep from he d.c. componen would cancel he effecs of he line-connecion circuis. - he processing ha ransforms he daa signal o fulfill, more or less, he above requiremens, is he baseband encoding; he codes employed are also called ransmission codes or line codes. - he corresponding baseband decoding, performed a he receiving end, requires a locally generaed clock signal ha has o be synchronized wih he received coded signal. - he synchronizaion capabiliy is anoher imporan parameer of he line codes. Long sequences conaining idenical bis lead o a synchronizaion loss; herefore, he baseband coded signal should have addiional ransiions insered, so ha no long sequence wihou ransiions would be ransmied.. Baseband codes - he BB codes modify he specral disribuion of he signal and help he synchronizaion of he bi clock. - he specrum of he coded signal would be posiioned in he same BW and he carrier signal would be recangular. - he mos employed BB codes are: he Non-Reurn o Zero (NRZ), biphasic, Miller, CMI, and AMI wih is varians: BnZs, HDB3, 4B3T; he combinaion of MLT-3 and b4b5 is also used. - he main parameers of hese codes are: he BW which concenraes he mos of he coded signal s power and is posiioning on he absoluefrequency axis; he presence/absence of he d.c. componen in specrum of he coded signal; 6

3 he synchronizaion capabiliy; he encoding-decoding complexiy. The Non-Reurn-o-Zero (NRZ) Codes - here is a class of NRZ codes, namely NRZ-L, NRZ-M, NRZ-S and NRZ-I codes. - he NRZ-L (Level) encodes a 1 (Mark) wih level High, and a (Space) wih level Low - he NRZ-M (Mark) and NRZ-S encodes a Mark (Space) alernaively wih levels High and Low, while he Space (Mark) are encoding by mainaining he level of he previously coded bi - he NRZ-I encodes a 1 by a ransiion a he midpoin of he bi inerval (HL or LH), while a is encoded alernaively as a H or L during he whole bi period, or equivalenly as HH or LL. - if he wo levels corresponding o H and L are posiive and negaive he codes are named bipolar codes, while if he L level corresponds o vols, he codes are denoed as unipolar codes. - he specral densiies of he unipolar NRZ codes are similar o he one presened in Figure 3, while he specral densiies of bipolar ones lack he d.c. componen. 1 - he specral disribuion of he average power of. 8 he digial unipolar NRZ-M signal is shown in. 6 figure Figure 3. Power specral densiy of a random unipolar NRZ. digial signal f/ fn - mos of he signal power is concenraed in he [; ] f N BW, f N = f bi /; his BW should be received as correcly as possible. - he daa signal has is energy concenraed in he d.c. and low frequency componens, which would be disored by he line-connecion circuis; W-densiaea specrala de puere - he specral disribuions of he biphasic, CMI, Miller and AMI are presened in fig S P (f/f N )/P o CMI [.; ]fn AMI [.3; 1.3]fn Bifazic [.6;.5] Biph Miller CMII AMI.4 MILLER [.3;1.3] f/f N Figure 4. Power specral densiies of he biphasic-s, Miller, CMI and AMI codes approximae represenaion The Biphasic (Mancheser) Codes - he paren biphasic code, he biphasic-l, has he following encoding rule, see figure 5: Figure 5. biphasic-l (Mancheser) code Encoding rule - 1 (mark-m) -coded wih a posiive ransiion in he middle of he bi-period (LH); - (space-s) - coded wih a negaive ransiion in he middle of he bi-period (HL) - his code exhibis an uncerainy of 18º, which requires a re-synchronizaion circui- explanaions: due o he line connecion - o avoid his, wo differenial varians were developed: biphasic-s and biphasic-m, also named differenial Mancheser codes. - he biphasic-s has he following encoding rule: - a ransiion is insered a every margin of he bi-period and, if he bi is, an addiional ransiion is insered a he middle of he bi period see figure 6 for a bipolar recangular carrier. - he biphasic-m has a similar encoding rule, bu he addiional ransiion is insered in 1 -bi period. - for he hree biphasic codes, he minimum level duraion would equal half bi-period, equivalen o a double oupu bi rae, leading o an increased frequency BW of he coded signal, see figure 4. 3

4 - considering he specral componens ha conain abou 8% of he coded signal s power, he useful FB of all hree biphasic codes is [.6.5] f N [.5,.5]f N - he specrum of his useful FB is approximaely symmerical around is cenral frequency f cb 1.5 f N - his BW is approximaely equal o he one of he non-coded sequence, bu he disribuion of he signal power wihin he BW is significanly modified; - he d.c. componen is removed and he power is concenraed in he higher-frequency componens; - he low frequency componens have small ampliudes, decreasing he disoring effec of he lineconnecion circui and simplifying is implemenaion - he relaively large BW of his code, makes i unadvisable for high bi raes. On applicaions employing coaxial or UTP cables, he biphasic is employed for bi raes up o ens of Mbps due o he large BW and shor lenghs of hese cables. - he synchronizaion capabiliy of he code is very good, due o he small maximum ime inerval beween wo subsequen ransiions, which equals one bi period. - since he iniial sae of he encoder could be or 1, here are wo biphasic-encoded sequences for he same informaion sequence. - considering he encoding rule, he minimum level duraion is half bi-period, or a period of a f bi clock. This would require he synchronizaion a he receiving end of a local clock f local = f bi. - If we consider he wo half-periods, k+1 and k+, of he (k+1)-h bi period, when he b k+1 bi is encoded (see figure 4), he biphasic-s can be expressed: s b(k 1) s b(k); s b(k ) s b(k 1) bk 1 (5) - he firs expression shows he occurrence of a ransiion in he coded signal a he beginning of every biperiod; he second one, he occurrence of a ransiion a he middle of he bi-period if he encoded bi is "". The Miller Code - he mahemaical expression of he Miller-encoding rule is complex [alex]; - a varian of he Miller code can be defined in a simpler manner using he biphasic-s encoding: Tx D b k b k+1 b k C od bifazic - S C od M iller T xc k - f b i T b i k+ 1 k+ k+ 3 k+ 4 Cod CM I Figure 6. Encoding wih he biphasic-s, Miller and CMI codes examples every oher ransiion is removed from he biphasic-encoded sequence see figure 6 - he minimum level duraion of he encoded signal is one bi-period; - he maximum ime inerval beween wo ransiions is wo bi-periods, leading o he decrease of he BW occupied by he Miller-encoded signal. - he Miller code concenraes he power wihin he [,3 1,3] f N BW, see figures 4 and 7, considering he componens ha conain approximaely 8% of he coded signal s power, while he cenral frequency is f cm.8 f N - his decreases he disoring effecs of he a(f) absolue value and non-uniformiy, allowing he code s use for higher bi raes. - bu his code exhibis a d.c. componen and low-frequency componens, which make i unadvisable for low bi raes. - figures 7 and 8 presen in a comparaive manner he effecs of absolue aenuaion and of he nonuniform a(f) characerisic upon he biphasic and Miller encoded signals. 4

5 - due o is narrower BW, he Miller-encoded sequence is less disored by he a(f) characerisic and is less affeced by disurbances. - due o is posiioning a lower frequencies, he Miller code is less aenuaed by a(f) han he biphasic code - he synchronizaion capabiliy is good, i.e. he maximum inerval beween wo subsequen ransiions being of four encoded levels or wo bi-periods. - because here are wo biphasic encoded sequences for every informaion sequence and he iniial sae of he Miller encoder could be or 1 and considering ha from he biphasic-encoded sequence we may eliminae eiher he posiive or he negaive ransiions, we would have 8 Miller-encoded sequences, ou of which pairs of wo are idenical, i.e. for he same informaion sequence here are four Miller-encoded sequences. I leads o he necessiy of a resynchronizaion circui in he receiver.,5 a a(f) characerisic of he channel w - power specral densiy of he biphasic code,6,4, Absolue aenuaion a he cenral frequency of he coded signal s bandwidh Aenuaion s variaion in he band of he biph. code a Absolue aenuaion a he lower margin of he coded signal s bandwidh w Absolue aenuaion a he upper margin of he coded signal s bandwidh w - Power specral densiy of he Miller code,5 w w,6,4, Aenuaion s variaion in he bandwidh of he Miller code Absolue aenuaion a he lower limi of he coded signal useful a a(f) characerisic of he channel a Absolue aenuaion a he cenral frequency of he coded signal s bandwidh Absolue aenuaion a he upper limi of he coded signal useful bandwih,5f N,6fN f N 1,5f N f N,5f N 3f N f Figure 7. Effecs of non-uniformiy of he a(f) characerisic upon he biphasic-encoded signal Figure 8. Effecs of non-uniformiy of he a(f) characerisic upon he Miller-encoded signal The Coded Mark Inversion Code - he encoding rule of he CMI code is, see figure 6: he bi is encoded wih a rising ransiion a he middle of he bi-period (LH); he 1 bi is encoded alernaively wih 1 (+V) or ( V), (LL or HH) - he ransiion a he beginning of he bi-period will no occur for all daa bis sequences; - he "" bis are encoded wih a ransiion a he middle of he bi-period, insering he informaion required by he bi-clock synchronizaion sysem a he receiving end - he power specral disribuion of he CMI code is asymmerical, see figure 4; he BW of he CMI code is approximaely [, f N, f N ] by considering he componens ha conain 8% of he coded signal s power and he coded signal has no d.c. componen The AMI (Alernae Mark Inversion) Code - he AMI code has he following encoding rule, see figure 9: he bi is encode wih he level of vols; - undesirable he 1 bi is encoded alernaively wih levels of +/- A for he whole bi-period. - i is a 3-level code and he ransmied bi rae equals he informaion bi rae 1 +A V In p u d a a,3f N f N 1,3f N 1,5f N f N,5f N 3f N f -A + A C ode A M I V -A Figure 9. Encoding rules for he AMI and B6ZS codes - he power specral densiy is shown in figure 4; he code concenraes is power in he BW [,3 1,3] f N, considering only he specral componens conaining 8% of he coded signal s power - he AMI s BW is narrower han he one of biphasic code, here is no d.c. componen and he energy is concenraed a higher frequencies han he energy of he Miller code. C ode B 6Zs C om pleion sequence V B V B 5

6 - he synchronizaion capabiliy of he code is very poor, since he long sequences are encode wihou any ransiions - o remove his inconvenience he BnZS (Bipolar wih n Zeros Subsiuion used in US and in Europe) code was developed. - he encoding rule is idenical o he one of he AMI code, excep for -bi sequences longer or equal o n, a naural number. Should such a sequence occur, i is replaced by a compleion sequence. - for n = 6, he compleion sequence is VBVB, see figure 9: - -bi coded wih vols V 1 -bi coded wih a +/- A level ha violaes he AMI encoding rule B - 1 -bi coded wih a +/- A level ha observes he AMI encoding rule - he violaions of he AMI encoding rule are employed for he idenificaion of hese sequences a he receiving end, where hey should be replaced by n-bi sequences. - The power specral densiy of he BnZs code is disribued similarly o he one of he AMI code - oher codes developed o improve he synchronizaion capabiliy of he AMI code are he HDB3 (high-densiy bipolar-3 zeros- used in Europe) and 4B3T (4-binary, 3-ernary). The MLT (Muli Level Transmi))-3 and 4b5b Codes - his concaenaion of wo codes is used in he Eherne 1Base-TX ransmissions a 1 Mbps - he MLT3 used he bipolar NRZ daa and encodes i according o he following rule, see Figure 1: a every bi of 1, he oupu level jumps o he nex level from he paern +V,, -V a every bi of, he oupu level keeps he level of he previous Clock symbol period, i.e., he level ransmied during he previous bi of 1 Figure 1 Encoding rule of MLT-3 - This code has a poor synchronizaion capabiliy because for long inpu series he oupu signal has no ransiions - o compensae for his shorcoming, he inpu daa flow is precoded wih he b4b5 code, see Table 1. - Every group of 4 bis is replaced by a group of 5 bis by selecing 16 combinaions wih a leas wo bis of 1 ou of he 5 = 3 combinaions; his leads o: a beer synchronizaion capabiliy, since only series of wo consecuive are provided a beer resilience o ransmission impairmens, since here are only 16 valid 5-bi combinaions, which is equivalen o insering one check bi a every 4 informaion bis (i.e., a coding rae of 4/5) he acually ransmied bi rae increases from 1 Mbps o 15 Mbps; his is a disadvanage Table 1. Encoding rule of he b4b5 code - he symbol frequency of his ransmission is f s = 15/5= 5 MHz and he specral componens above his frequency are raher small. Specrum of b4b5-mlt-3 for Eherne 1Base-TX f (MHz) Implemenaion of he BB encoding his secion will be used in he laboraory classes and is presened in Annex 6

7 Decoding of he baseband codes - he biphasic-s code - he decoding of he biphasic S code is based on he encoding equaions (5). s b(k 1) s b(k); s b(k ) s b(k 1) bk 1 (6) - considering ha bi b k+1 was encoded during he half-periods k+1 and k+ of he bi-clock; - using (3) and performing he addiions for hree half-periods separaed by a bi-period, we ge (7). - he bi b k+1, encoded during half-periods k+1 and k+, is decoded during half-periods k+ and k+3, insering a half bi-period delay. - he decoding of he nex daa bi b k+ is sared during he k+4 inerval. s b(k ) s b(k) s b(k 1) bk1s b(k) s b(k) bk1s b(k) 1bk1b k1; s b(k 3) s b(k 1) s b(k ) s b(k 1) s b(k 1) bk1s b(k 1) b k1; (7) s(k b 4) s(k b ) s(k b 3) bks(k b ) s(k b ) bks(k b ) 1bk b k ; - he decoding of he Miller code is accomplished by similar addiions, ha employ signals delayed wih 1,, 3 and 4 half-bi periods. - he decoding of he CMI code is performed by XOR-ing he coded signals delayed wih a half-bi period. - for all hree codes, he signals resuled from he summaion circuis should be sampled wih a bi clock, wih frequency f bi, whose phase differs from code o code. - he biphasic and CMI codes he sampling bi-clock should be shifed wih 7º, compared o he bi clock obained by dividing by he f sincro clock delivered by he dynamic synchronizaion circui. - for he Miller code he sampling clock has o be shifed wih 9º compared o he same reference signal Furher consideraions regarding he implemenaion of he BB decoding is presened in Annex 3 and will be used in he laboraory classes. Block diagram of a baseband modem - i is presened in figure 11 RTS Trans. Conrol CTS TxD Coder CE Code Selecţ Coded signal Line Uni w 4w- Tx TxCk CD RxD RxCk Decoder f b i f sincro - local f b i O sc. D iv. C.D. f aac f aac Synchronizaion In pu sage + EQ. - he TxD are delivered o he encoder, using he bi clock Tx Ck delivered by he Oscillaor-Divider block. The desired BB code is seleced by he inernal conrol Code Type. - he encoded signal is sen o he line uni which ransfers he signal ino he physical signal. - he Transmission Conrol block is commanded by he RTS inerface signal; i sends back o he compuer he confirmaion CTS signal, afer he RTS/CTS ime inerval, while delivering o he encoder he synchronizaion sequence. - depending on he w/4w connecion, he ransmission employs one or wo pairs of wised wires. - for he w half-duplex operaion, he ransmission has prioriy, so he Transmission Conrol circui disables he receiver s inpu sage when he RTS signal is acive. On he 4w full-duplex operaion his condiioning is no applied. - he inpu sage ensures he ransfer from symmerical o asymmerical circui, he amplificaion and filering of he received signal and he equalizaion of he aenuaion insered by he wires. - he signal is hen limied and used, as a reference signal, by he synchronizaion circui, which delivers he f sincro clock, required by he demodulaor and he receive clock RxCk. L I M R eceived signal - lim ied Figure 11. Simplified block diagram of a BB modem w 4w 4w - Rx 7

8 - he limied signal is also sen o he decoder, ogeher wih he f sincro clock, for he exracion of he received daa RxD. - he signal delivered by he inpu sage is employed by he Carrier Deecor block, which moniors he level of he inpu signal and enables/disables he receiver if is level is higher/smaller han some prese levels; his bloc delivers o he compuer he CD inerface signal. - he modem may conain circuis ha implemen he es loops, scramble-descrambler and circuis for he generaion and analysis of he es daa; hese circuis were no insered in fig. 1. Error probabiliy of he BB ransmissions - he frequency band available on he physical channels is usually larger han he useful BW of he BBcoded signals, herefore an inpu filering is required for he improvemen of SNR. - he evaluaion made below assumes ha he receiver has an adapable BP filer ha changes he passing band according o he code employed - he ransmission codes presened in his chaper are eiher -level codes (+/-A; M = ), i.e. biphasic, Miller and CMI, or 3-level codes (+/-A, V; M = 3), i.e. AMI, BnZs. - he average signal power P s, for he wo ypes of codes is given by (8). a and (8). b, respecively, and he power of he Gaussian noise, wih a N power specral densiy, a he oupu of he inpu filer wih a BW = B f is given by (8).c: P s = A ; a. P s3 = A /; b. P z = BW f N = σ ; c. (8) - we assume ha he received signal is affeced by he a Gaussian noise of null mean and variance, which is added o he useful signal, i.e.: r sr n (9) - herefore, he probabiliy ha he received signal would equal r a he sampling momen, if he ransmied level was m is: 1 r m pr m exp ; (1) - since he decided levels are obained based on he minimum Euclidean disance beween he received level and he permied levels, he symbol (bi)-error probabiliy equals he occurrence probabiliy of a noise ampliude which makes he level of he received signal o be closes o a permied level ha differs from he level ha was ransmied in ha symbol period. - if wihin a symbol period, he level m k is ransmied, wih probabiliy P mk, hen he probabiliy o have ha symbol misaken afer he decision is given by (11), where N k,a is he number of permied levels ha are placed a a disance A from he level m k and A is half of he minimum disance beween wo ransmied levels of ha code. - he probabiliy p r mk A e mk k k,a k p P p rm A N ; (11) acually represens he occurrence probabiliy of a noise-level greaer han A in he probing momen; his probabiliy is: 1 r mk A prmk A exp d rmkq ; A (1) where Q() 1 u exp du; (13) - since he power of a Gaussian noise equals σ, he signal/noise raio in linear expression is: Ps,3 Ps,3,3 P (14) z,3,3 - knowing he for he -level codes A = A and for he 3-lvel codes A = A/ and using (8) we express he average powers of he wo ypes of codes in erms of heir respecive A as: s s3 P A ; P A ; (15) - if we express A of each code in erms of is P s from (15) and inser i (1) and hen consider (14) we ge for he -level and 3-level codes: 8

9 p r mk A Q for level a.; p r mk A Q for 3 level b.; (16) - Q is a sricly decreasing funcion, i.e. he error probabiliy increases wih he decrease of he signal/noise raio ρ (in linear expression). - Recall ha: SNR = 1lg(P s /P z ); (17) - he Q funcion is a sricly descending funcion, i.e. he BER increases wih he decrease of he SNR. Consideraions regarding he BER of -level codes - he subsequen consideraions assume ha he average power of he received coded signal is A, for - level codes (+/- A), he power specral densiy of he noise equals N, and he noise power in he useful band of code x equals N BW x = x. - for he -level codes he half of he minimum disance beween wo levels equals A =A, i.e. he decision hreshold is placed a. - replacing (16).a in (11) and aking ino accoun ha each level has only one neighbour a disance A and ha all levels are equiprobable, we ge he average error-probabiliy of a level (symbol), which equals he bi-error probabiliy of a bi, as: pe.5q A 1.5 Q A 1 Q (18) A A - considering ha he useful BW of he biphasic, Miller and CMI codes are: B biph f N ; B Mil 1f N ; B CMI 1,8f N (19) -he SNR values, raed o he one of he biphasic code, are: Ar A r biph ; Mil biph; SNR Mil SNR biph[db] 3dB; fn N 1fN N () A r CMI 1,11 biph; SNR CMI SNR biph[db],46db; 1,8 f N N - and he BER is compued using (18), for M = : he approximae expression is obained by using he firs erm of he Taylor series decomposiion of Q funcion - e pe Q ; - consideraions regarding he Q() funcion and is approximaion will be discussed in he PSK chaper. - equaions (19) and () show ha, for he same values of N and of he received signal s power level, he SNR values of he Miller and CMI codes are greaer wih 3 db and.46 db respecively, han he SNR ensured by he biphasic code. Since he Q funcion is sricly descending, he BERs ensured by he Miller and CMI codes are smaller han he one provided by he biphasic code. - o compare he levels of he received signal and of N which ensure he same value of BER for he Biphasic and Miller codes, we should firs noe ha, due o he bijeciviy of he Q and square-roo funcions, we may wrie: p p Q( ) Q( ) () ebif emill bif Mill bif Mill - he wo raios ρ may be expressed in erms of energy-per-bi/power-specral-densiy raio, E b /N ; relaion (3) shows he compuaion for he Miller code and presens he final resul for he CMI code as well: Prbif PrMill Prbif PrMill Ebbif EbMill Nbif LBbif NMill LBMill Nbif fbi NMill,5fbi Nbif NMill Ebbif EbMill Eb Pr [db] [db] 3dB; where Nbif NMill N N fbi Ebbif EbCMI [db] [db],46db N N ; (3) bif CMI - relaion (3) shows ha, in order o ensure he same BER a he same bi rae D, he Miller and CMI codes (1) 9

10 allow he decrease of he received signal s power and/or he increase of he N,, so ha heir E b /N raios would be smaller wih 3 db, and.46 db respecively, han he E b /N raio required by he biphasic code. - Figure 1 shows he BER vs. SNR curves of he Biphasic-S, CMI and Miller codes Consideraions regarding he BER of 3-level codes - for he AMI code, he received levels +A and -A have occurrence probabiliies equalling.5, while he level has he occurrence probabiliy.5 (since he 1 and bis are equiprobable). Therefore he decision hresholds are se a +/- A/, so for hese codes A = A/. - he level has wo neighbours a a disance equalling A, while he oher wo levels have only one neighbour a he same minimum disance. - based on he above consideraions and replacing (16).b in (1), he error-probabiliy of he 3-level AMI codes can be expressed as (4), where ρ denoes he SNR of he used 3-level code in linear expression: pe3 Q 1 Q Q 1 Q (4) considering ha BW AMI f N and a bi rae wih f bi = f N, he SNR would be: A AMI ; (5) N fn - he BER of AMI is compued using (4) and (5)for ρ AMI - he facor 1/ under he square-roo in (4) is generaed by he fac he ha he code has 3 levels in he same range of ampliude [-A, +A] which makes A o be equal o A/. - o express AMI s signal/noise raio in a manner similar o (18), we define an equivalen signal/noise raio for AMI: 1 ' AMI AMI (6) - we express he ρ AM I in erms of he signal/noise raio of he biphasic code: 1 1 A 1 ' AMI AMI bif ; SNR ' AMI[dB] SNR bif [db] 3dB; Nf N (7) - equaions (3), (5) and (7) show ha, a he same values of N and received ampliude levels +/-A, and for he same bi rae, he value of he signal/noise raio of he ernary codes (AMI ype) is smaller wih 3 db, han he one of he biphasic code. This value should be furhermore increased wih abou.5 db o compensae he effec of he 1.5 facor from (4). Since he Q funcion is sricly descending, he BERs ensured by he AMI-ype codes are greaer han he BER provided by he biphasic code. - he consideraions above are also valid for he BnZs and HDB3 codes - o compare he levels of he received signal and of N which would ensure he same value of BER for he wo codes, we should firs noe ha, due o he bijeciviy of he Q and square-roo funcions, we may wrie (8), where he facor 3/ was approximaed by 1: p p Q( ) Q( ' ) ' (8) ebif eami bif AMI bif AMI - he wo raios ρ may be expressed in erms of energy-per-bi/power-specral-densiy raio, E b /N : Prbif 1 PrAMI Arbif 1 ArAMI Ebbif 1 EbAMI Nbif LBbif NAMI LBAMI Nbif fbi NAMI f bi / Nbif NAMI EbAMI Ebbif Eb Pr [db] [db] 3dB; where N N N N f AMI bif bi (9) 1

11 - relaion (9) shows ha, in order o ensure he same BER for he same bi rae D, he AMI-ype codes require he increase of he received signal s power and/or he decrease of he N, so ha he E b /N raio would be greaer wih 3 db han he E b /N raio required by he biphasic code. This value should be furhermore increased wih abou.5 db o compensae he effec of he 1.5 facor from (4). - finally, he AMI-ype ernary codes require a E b /N higher wih 3.5 db han he biphasic code o ensure he same BER, a he same bi rae, if he BW of he inpu filer is modified according he code used. - Figure 13 shows he BER vs. E b /N of AMI. compared o hose of Biphasic-S, CMI and Miller - able presens he useful BW and he relaive E b /N values, raed o he one of he biphasic code, required by he BB codes o ensure he same BER, if he inpu filer bandwidh B f is modified according o he employed code. Code Biphasic M = Miller, M = CMI; M = AMI, BnZs, HDB3; M = 3 BW f f N 1 f N 1,8 f N 1 f N E b /N [db] E b /N bif E b /N bif 3 db E b /N bif,46 db E b /N bif + 3,5 db Table Useful bandwidhs (raed o f N ) and E b /N values (relaive o he Biphasic code) of Miller CMI and AMI - he increase of he number of levels of he coded signal (M = 3), when he maxim level is kep consan (+/-A) o keep he peak power consan, leads o a smaller disance beween wo neighboring levels, from A for codes wih M =, o A for codes wih M = 3, and o a smaller average power of he coded signal. These facs explain he increase of he E b /N required by he M= 3 codes o ensure he same BER as he codes wih M =. - he BER compuaion for he MLT-3 code is more complex and is beyond he scope of his course; - he filering of he BB-coded signal, in order o mach he useful BW of he received code, migh exhibi adverse effecs which should be removed by he filering iself and by subsequen processing. - he compuaions above consider only he power of he received coded signal. Bu, due o heir differen posiioning of he absolue-frequency axis and o he differen aenuaion insered by he cable for differen frequencies, he signals coded wih hese codes suffer differen aenuaions; herefore, in he assumpion of he same ransmied power, he received power is depending on he code employed. - he compuaion of he received power in erms of ransmied power and he effecs of he differen aenuaions insered by he cable will be deal wih in he seminar classes. References Nicolae Dumiru Alexandru, Guner Morgensern; Digial Line Codes and Specral Shaping, Marix Rom Bucureși 1998 Ha H. Nguyen, Ed Shwedyk; A Firs Course in Digial Communicaions Cambridge Universiy Press 9 11

12 Annex 1 Brief descripion of he ETSI classificaion of copper cables - no required for he exam - ETSI characerizes he cables by heir elecrical lengh, i.e., he value of he inserion loss a a given frequency and by heir Z c ; IL [db] - I has defined five ypes (groups) of es -1 cables (loops), Q1,,Q5, by defining heir Q5 elecric lengh IL(f) see figure A.1, for - approximae represenaion. -3 Loops 1, 3 and - Each group conains 4 ypes of loops having -4 he same elecrical lengh, bu differen Z c. - he physical lenghs of he Q1 es loops -5 Q1 vary beween 199 and 1 m; he ones of Q5 have beween 5 and 3 m in -6 physical lengh. 5MHz 1MHz 15MHz - he perurbaions ha occur on hese physical channels are he Gaussian noise, impulse noise and crossalk. - he coaxial cables exhibi a similar behavior, bu he increase of he aenuaion wih frequency is significanly slower, leading o a larger usable frequency bandwidh. The levels of he above menioned disorions are significanly smaller. Annex Implemenaion of he BB encoding no required for he exam his secion will be used in he laboraory classes - he encoding of he BB codes can be implemened using he mahemaical relaions ha describe he encoded signal, e.g. relaion (3) for he biphase code. - he implemenaion can be significanly simplified if he some pariculariies of he encoding rules are considered. - due o he differen numbers of levels of he encoded signals and o he differen duraions of a level in he coded signals, half of he bi-period or a bi-period, he encoding of he biphase, Miller and CMI codes should be approached in a differen manner, as opposed o he encoding of he AMI-ype codes. - he biphase-s, and CMI have a common rule of encoding he 1 -bi, namely by alernaing he level H and L. This can be implemened wih J-K flip-flop, clocked by he bi clock, which has on is J and K inpus he daa signal. - he wo codes also ransmi he bi clock signal, during he -bi - he CMI code encodes he bi by he bi-clock signal. - he biphase-s encodes he -bi eiher by he bi-clock or by he invered bi-clock, depending on he level (H or L) which encoded he previous 1 -bi. - based on hese consideraions a common encoder for he biphase-s and CMI codes migh be buil using a J-K flip-flop and an elecronic swich; an addiional swich should be used o selec he desired code. - o ensure he bipolar carrier, C-MOS circuis ha operae using he +/- V levels should be employed. - he Miller can be generaed by dividing by (in frequency) he biphase-s coded signal. HOMEWORK: Design he elecric diagram of an encoder for he hree codes wih C-MOS digial circuis using he consideraions menioned above. - he encoding of AMI should consider ha he coded signal has hree ampliude levels. - I could be generaed using a level converer ha operaes similarly o a -bi D/A converer and whose sign bi is oggled alernaively, for he 1 inpu bis. - anoher approach employs a J-K flip-flop for he encoding of 1 (see consideraions above) and a swich ha has an inpu o he elecrical ground, for he -bi, conrolled by he inpu daa. - for he encoding of he BnZS codes, prior o he AMI encoder here should be insered a circui ha idenifies he n-bi series and replaces hem by he compleion (synchro) sequence of he code. 1

13 Annex 3 Implemenaion of he BB decoding no required for he exam; his secion will be used in he laboraory classes - he biphasic-s code - he decoding of he biphasic S code is based on he encoding equaions (5). - considering ha bi b k+1 was encoded during he half-periods k+1 and k+ of he bi-clock; - using (3) and performing he addiions for hree half-periods separaed by a bi-period, we ge (7). - he bi b k+1, encoded during half-periods k+1 and k+, is decoded during half-periods k+ and k+3, insering a half bi-period delay. - he decoding of he nex daa bi b k+ is sared during he k+4 inerval. s b(k ) s b(k) s b(k 1) bk1s b(k) s b(k) bk1s b(k) 1bk1b k1; s b(k 3) s b(k 1) s b(k ) s b(k 1) s b(k 1) bk1s b(k 1) b k1; (3) s(k b 4) s(k b ) s(k b 3) bks(k b ) s(k b ) bks(k b ) 1bk b k ; - he decoding of he Miller code is accomplished by similar addiions, ha employ signals delayed wih 1,, 3 and 4 bi half-bi periods. - he decoding of he CMI code is performed by XOR-ing he coded signals delayed wih a half-bi period. - for all hree codes, he signals resuled from he summaion circuis should be sampled wih a bi clock, wih frequency f bi, whose phase differs from code o code. - he biphasic and CMI codes he sampling bi-clock should be shifed wih 7º, compared o he bi clock obained by dividing by he f sincro clock delivered by he dynamic synchronizaion circui. - for he Miller code he sampling clock has o be shifed wih 9º compared o he same reference signal - he block diagram of a decoder for he biphase-s, Miller and CMI codes is shown in figure A.a. - he K 1 swich selecs he decoded signal; he K swich selecs he sampling clock, boh depending of he code o be decoded - he decoding of he AMI and BnZs codes employs he fac ha he 1 bi is encoded wih levels of modulus A, and he bi is encoded wih a level of vols. - he decoding of he AMI code requires he -wave recificaion of he received coded signal which is sampled wih a locally-recovered bi clock. - hen a comparison wih +A/ hreshold performs he decision of he received bi, see figure A.b. Before he comparison, an AGC ensures a consan level of he received signal. - he decoding of he BnZs and HDB3 codes require he inserion, besides he AMI decoding, of a dedicaed circui ha recognizes he synchronizaion sequence, insered a he ransmission end, and replace wih n or 4 bis of "". - he recovery of he symbol clock in AMI decoders migh be performed using he energeic mehod see he QAM lecures laer. S dec - b if DP Semnal rec. coda D Q Ck S k+1 S k+ S k+3 S k+4 D Q Ck D Q Ck D Q Ck S dec - C M I K 1 D Q Ck b k f sincro = f b i S dec - M iller f bi f bi -7º Bif sau CMI Σ K f bi-9º M iller a. S ign al A M I S. BnZs A G C R ecovery + Sincro f bi R ec al + V / f bi C o m p arao r D Ck Q b - A M I Id en ificaio n Synchro Seq. R eplacem en of Synchro Seq. b -BnZs b. Fig. A a. Block diagram of he biphase-s, CMI and Miller decoders; b. Block diagram of he AMI and BnZs decoders 13