Dgtal Transmsson Most modern communcaton systems are dgtal, meanng that the transmtted normaton sgnal carres bts and symbols rather than an analog sgnal. The eect o C/N rato ncrease or decrease on dgtal transmssons appears n the orm o errors n the transmtted dgtal normaton. Derence between a ymbol and a Bt symbol s a pulse wth duraton T seconds (ths s a perod wth unts o seconds not temperature as was used beore) that may contan a bt or more o data. For: a symbol that takes one o 2 values, each symbol contans 1 bt T = T B & = B a symbol that takes one o 4 values, each symbol contans 2 bts T = 2T B & = B /2.. symbol that takes one o M = 2 N values, each symbol contans N bts. T = N T B & = B /N There s an nherent advantage to transmttng data usng symbols wth large values o M. The advantage s that more data s transmtted by usng the same BW. That s, the element that determnes the bandwdth n dgtal transmsson s number o transtons between symbols not the number o levels over whch we change. o, two sgnals wth 10 changes per second but one o them changes between 2 levels whle the other changes among 10 levels wll requre the same bandwdth. Types o gnalng (Lne codes) The realty s that normaton sources that provde dgtal data actually provde numbers that are not sutable or transmsson on any channel as they are. n analog to dgtal converter (DC) that converts an analog audo sgnal to a dgtal ormat provdes sample values every 50 to 100 mcroseconds. The sample values must be ormatted n a proper way to make them sutable or transmsson through the communcaton channel. When each dgtal value s represented usng a pulse or each bt (n bnary communcaton) or a pulse or multple bts smultaneously (or M-ary communcaton), the sgnal generated by stackng derent pulses s called a lne code. To put t n a smple way, the process o lne codng s usng the dgtal data obtaned rom an normaton source to
generate a voltage sgnal that represents the normaton. There are derent orms o lne codes that can be used to represent the normaton. The terms Return to Zero (RZ) and Non Return to Zero (NRZ) wll be used n descrbng these sgnal. ome lne codes are shown next along wth ther power spectral denstes, advantages and dsadvantages. We wll consder only bnary lne codes although many lnes codes can work wth bnary and M-ary transmssons: 1. Unpolar (On O) Non-Return to Zero (NRZ): In ths orm o lne codes, a bt o 1 s represented by some postve voltage (+5 volts or example) and a bt o 0 by 0 volts (justyng callng ths sgnal On O). The pulses correspondng to bnary 1 reman at the postve voltage or the whole duraton o the bt perod (t does not return to zero at any tme durng the bt perod justyng callng ths lne code NRZ). T 0 R 1 2 2T = R ω dvantages: 1. Very smple to generate (has only two levels whch can be easly generated usng smple dgtal electroncs) 2. pectrally ecent (requres the mnmum amount o bandwdth or a specc bt rate) Dsadvantages: 1. Does not provde any orm o bt or rame synchronzaton (or long sequences o ones or zeros, the transmtter and recever may get unsynchronzed) 2. Does not provde any orm o error detecton. 3. Generally has non-zero DC (even voltages o -5 V and +5 V are used, unless the number o zeros and number o ones are equal). Ths may be a problem or some communcaton systems that cannot transmt DC values. 2. Bpolar Non-Return to Zero (NRZ): In ths lne codes, a bt o 1 s represented n an alternatng orm by some postve voltage (+5 volts or example) once and the next tme a bt
o 1 appears t wll be represented by the same voltage but wth a negatve value ( 5 volts). bt o 0 s represented by zero volts. The pulses correspondng to bnary 1 reman at the postve and negatve voltages or the whole duraton o the bt perod (they do not return to zero). The advantage o ths lne code over the Unpolar (On O) NRZ s that t has zero DC value because bts o 1 alternate n usng the postve and negatve voltages. lne code wth zero DC s desred n some applcatons that requre that the transmtted sgnal to have no DC. T 0 R 1 2 2T = R ω dvantages: 1. pectrally ecent (requres the mnmum amount o bandwdth or a specc bt rate) 2. Provde synchronzaton between the transmtter and recever or long sequences o Logc 1. 3. llows some orm o error detecton (snce two consecutve logc 1 s are represented by a postve and negatve pulses, the recever detects two consecutve pulses that have the same polarty (both are postve or both are negatve), t can easly detect that there must have been an error n the transmsson). 4. Has zero DC value regardless o the number o Logc 1s and Logc 0s n the normaton to be transmtted. Dsadvantages: 1. Does not provde synchronzaton normaton or long sequences o zeros. 2. Requres more sophstcated electroncs to be generated because t uses sgnals wth 3 levels (or example, +5, 0, and -5 V) 3. Manchester (B-Phase): In ths lne code, a bt o 0 s represented by some postve voltage or the rst hal o the bt perod and some negatve voltage or the second hal o the bt perod. bt o 1 s smply the negatve o the zero bt so t s represented by the negatve voltage or the rst hal o the bt perod and the postve voltage or the second hal o the bt. Unlke prevously dscussed lne codes, whch carry the normaton n the levels o
the pulses, snce each o the two bnary values (0 and 1) n ths lne code are transmtted usng pulses that have hal o ther duraton beng a hgh voltage (or postve voltage) and the other hal beng a low voltage (or negatve voltage), the normaton s not carred n the levels but n the transton rom hgh to low voltage or vce versa n the mddle o the pulse representng each dgtal bt. transton rom hgh to low may represent a zero whle a transton rom low to hgh would then represent a one. Ths lne code s very good or nsurng synchronzaton between the transmtter and recever. For consecutve bts that are equal, a transton may occur at the border o bts. These transtons are smply gnored and do not carry normaton. R 1 2 2T = R 1.5R dvantages: 1. Provdes ull synchronzaton normaton or long sequences o zeros and long sequences o ones. lso, ths lne code can easly be used to also provde rame synchronzaton normaton by smply transmttng bts wth both o ther two parts beng hgh voltage or both are low voltage, where the transmtter can use ths method to sgnal to the recever that a byte, or example, has ended and a new byte t startng. 2. llows some orm o error detecton (snce a bt has two parts, so the two receved parts o a bt have the same value, an error may have happened (ths assumes that rame synchronzaton as dscussed n pont 1 above s not used) 3. Has zero DC value regardless o the number o Logc 1s and Logc 0s n the normaton to be transmtted. Dsadvantages: 1. Requres more bandwdth or transmsson (approxmately 1.6 tmes the bandwdth o the prevous lne codes) snce there are more transtons n the sgnal o ths lne code compared to the prevous lne codes or the same bt rate. 2. lthough ths lne code has only two levels, t s slghtly more complcated to generate compared to Unpolar NRZ lne code.
Dgtal Modulaton Methods everal types o dgtal modulatons exst. The three basc types o dgtal modulatons are named: 1. mpltude ht Keyng (K) 2. Phase ht Keyng (PK) 3. Frequency ht Keyng (FK) (Wll not be dscussed here) Next we descrbe several o these modulatons and shown ther tme doman pulses and ther constellaton. mpltude ht Keyng (K) Bnary mpltude ht Keyng (BK) In K, the derent pulses all have the same phase but derent ampltudes. In the bnary ampltude sht keyng (BK) modulaton technque we transmt one o two pulses or each bt: 1) or logc 0, we transmt nothng ( s 0 ( t) = 0 or 0 t T ) 2) or logc 1, we transmt a modulated pulse wth magntude s () t = cos2π t or 0 t T ) ( ( ) 1 c nce all the pulses have the same phase (you can thnk o the rst sgnal as s0 ( t) = 0 cos( 2π ct) or 0 t T ), the constellaton o ths modulaton technque (the representaton o the pulses o the modulaton algorthm) becomes M-ary mpltude ht Keyng (M-ary K) In ths dgtal modulaton technque, we transmt one o M pulses, where M s a power o two number such that
M = 2 n. The quantty n here s equal to the number o bts that are carred by each transmtted pulse. To determne whch pulse to transmt n each case, we wll have to dvde the sequence o normaton bts nto groups o n consecutve bts. The combnaton o bts wll be one o M = 2 n possble combnatons that wll determne the pulse to be transmtted or these n bts. o, n ths modulaton we wll transmt one o the ollowng pulses: 1) or bt sequence 00..00, transmt s () 0 or 0 00 00 t = t T 2) or bt sequence 00..01, transmt 00 01 ( π 3) or bt sequence 00..10, transmt 00 10 ( π 4) or bt sequence 00..11, transmt ( π ) s () t = cos2 t or 0 t T s () t = 2 cos2 t or 0 t T s () t = 3 cos2 t or 0 t T 00 11 : : M) or bt sequence 11..11, transmt ( ) ( π ) 11 11 c s ( t) = M 1 cos 2 t or 0 t T nce all pulses o ths modulaton technque have the same phase, the representaton o all pulses n the constellaton all on a straght lne. The constellaton o ths modulaton algorthm s shown below. c Clearly n ths case, the power requred or transmttng derent pulses s derent and the average power o transmsson can be obtaned easly by averagng all powers assumng that derent bt sequences have equal probabltes. Phase ht Keyng (PK) Bnary Phase ht Keyng (BPK) In ths dgtal modulaton technque, we transmt one o two pulses or each bt that are smlar n magntude but have a 180 phase sht between them (one o them s the negatve o the other): 1) or logc 1, we transmt the pulse 1 ( π 2) or logc 0, we transmt the pulse ( π ) The constellaton o the modulaton algorthm n ths case becomes. s () t = cos 2 t or 0 t T s () t = cos 2 t or 0 t T 0 c
It s worth mentonng that although ths algorthm s very smlar to the BK, ths s better because wth the same probablty o error as a BK, ths modulaton technque would requre less power (very ths). Quadrature Phase ht Keyng (QPK) In ths dgtal modulaton technque, we transmt one o our pulses that correspond to a group o 2 bts. ll pulses have equal magntudes but are separated n phase by 90 (each two are negatve o each other). Because o the phase sht o 90 between the derent pulses, two o the pulses wll be represented by dots on the x-axs and two wll be represented by dots on the y-axs. Ths means that the 4 pulses are 1) or logc 00, we transmt the pulse 00 ( π 2) or logc 01, we transmt the pulse 01 ( π 3) or logc 10, we transmt the pulse 10 ( π 4) or logc 11, we transmt the pulse ( π ) s () t = cos 2 t or 0 t T s () t = sn 2 t or 0 t T s () t = cos2 t or 0 t T s () t = sn 2 t or 0 t T 11 c The constellaton now s a 2-D constellaton that looks lke the ollowng
8-ary Phase ht Keyng (8-ary PK) The QPK modulaton dscussed above can be extended urther where more symbols are added so nstead o havng 4 symbols n the QPK, we now have 8 symbols. We transmt one o 8 pulses such that each pulse now carres 3 bts. ll pulses wll have equal magntudes but are separated n phase by 45. Thereore, the transmtted pulses are 0π 1) s000( t) = cos 2π ct or 0 t T 2 π 2) s001( t) = cos 2π ct or 0 t T 2π 3) s010( t) = cos 2π ct or 0 t T 3π 4) s011( t) = cos 2π ct or 0 t T 4π 5) s100( t) = cos 2π ct or 0 t T 5π 6) s101( t) = cos 2π ct or 0 t T 6π 7) s110( t) = cos 2π ct or 0 t T 7π 8) s111( t) = cos 2π ct or 0 t T o the constellaton o ths modulaton can easly be shown to be the ollowng (remember that ( α β) ( α) ( β) ( α) ( β) cos ± = cos cos sn sn ):
PK Modulatons wth hgher power o two values o M are possble but as the value o M ncreases, the modulaton algorthm becomes more necent as derent ponts n ts constellaton become very close to each other resultng n hgh probablty o error compared to the requred power o transmsson. We can easly mody the pulses that are requred to transmt the normaton such that the average power o transmsson s reduced and at the same tme, the probablty o error s ether lowered or at least mantaned at ts level. In ths case, other modulaton technques n whch the constellaton pnts are dstrbuted more evenly are used. These modulaton technques are called Quadrature mpltude Modulaton (QM), and rom ther name, they are a mx o both K and PK. Quadrature mpltude Modulaton (QM) Oten, we need to transmt more bts per symbol than what smple dgtal modulaton technques descrbed n above and n the prevous lecture can practcally do. We have seen that M-ary PK can be used or transmttng 3 bts per symbol or hgher bts per symbol but once the number o bts per symbol ncreases, the ecency (probablty o error) o the modulaton ncreases sgncantly because the ponts o the constellaton are conned to a crcle o a specc radus. Redstrbutng the constellaton ponts n a derent type o modulaton gves better perormance n terms o probablty o error at the expense o a more complcated system and a no longer constant ampltude or the derent pulses as t s the case wth all PK modulatons. We can generate M sgnals that have derent ampltudes and/or phases such that each carres n bts ( n= log2 M ) by havng the derent pulses n the orm
where {, } ( π ) ( π ) s ( t) = a cos 2 t + b sn 2 t or 0 t T c c s and = 1, 2,, M a b s a par o coecents that are gven by a matrx o pars dependng on the desred M-ary QM modulaton scheme (or sometmes called M-QM or short). For example, 16-QM wll have a matrx or {, } { a, b} a b gven by ( 3,3) ( 1,3) (1,3) (3,3) ( 3,1) ( 1,1) (1,1) (3,1) = ( 3, 1) ( 1, 1) (1, 1) (3, 1) ( 3, 3) ( 1, 3) (1, 3) (3, 3) o n 16-QM, each 4 bts are combned and based on the pattern o these 4 bts (gvng 16 possbltes), the correspondng par {, } pulse or these 4 bts s generated and transmtted. The constellaton o the above 16-QM s shown below: a b s selected rom the above matrx and the correspondng More complcated constellatons are obtaned or hgher levels o QM. Once the pulses have been transmtted, the receved pulses can be plotted on the same constellaton. The pulse assumed to have been transmtted s the one wth the pont n the constellaton that s closest to that o the receved pulse. QM s wdely used n dgtal communcaton systems ncludng the V.92 56k bps modem.