EE 421: Communicaions I Dr. Mohammed Hawa Inroducion o Digial Baseband Communicaion Sysems For more informaion: read Chapers 1, 6 and 7 in your exbook or visi hp://wikipedia.org/. Remember ha communicaion sysems in general can be classified ino four caegories: Analog Baseband Sysems, Analog Carrier Sysems (using analog modulaion), Digial Baseband Sysems and Digial Carrier Sysems (using digial modulaion). Baseband Carrier Analog Analog Baseband Communicaion Sysems Analog Carrier Communicaion Sysems Digial Digial Baseband Communicaion Sysems Digial Carrier Communicaion Sysems Digial baseband and digial carrier ransmission sysems have many advanages over heir analog counerpars. Some of hese advanages are: 1. Digial ransmission sysems are more immune o noise (due o hreshold deecion a he receiver; and he availabiliy of regeneraive repeaers, which can be used insead of analog amplifiers a inermediae poins hroughou he ransmission channel). 2. Digial ransmission sysems allow muliplexing a boh he baseband level (e.g., TDM) and carrier level (e.g., FDM, CDMA and OFDMA), which means we can easily carry muliple conversaions (signals) on a single physical medium (channel). 3. The abiliy o use spread specrum echniques in digial sysems help overcome jamming and inerference and allows us o hide he ransmied signal wihin noise if necessary. In addiion, he use of orhogonaliy is easier and allows increasing he ransmission rae by overcoming impairmens such as fading. 4. The possibiliy of using channel coding echniques (i.e., error correcing codes) in digial communicaions reduces bi errors a he receiver (i.e., i effecively improves he signal-o-noise raio (SNR)). 5. The possibiliy of using source coding echniques (i.e., compression) in digial communicaions reduces he amoun of bis being ransmied, and hence, allows for more bandwidh efficiency. Encryping he bis can also lead o privacy. 1
6. Digial communicaion is inherenly more efficien han analog sysems in exchanging SNR for bandwidh, and allows such exchange a boh he baseband and carrier levels. 7. Digial hardware implemenaion is flexible and permis he use of microprocessors. Using microprocessors o perform digial signal processing (DSP) eliminaes he need o build expensive and bulky discree-componen devices. In addiion, he price of microprocessors coninues o drop every day. 8. Digial signal sorage is relaively easy and inexpensive. Also he reproducion of digial messages can be exremely reliable wihou deerioraion, unlike analog signals. Acually, due o hose imporan advanages mos of communicaion sysems nowadays are digial, wih analog communicaion playing a minor role (we sill, for example, lisen o analog AM and FM radio). This aricle provides a very quick overview of some of he main conceps ha are relevan o digial baseband ransmission. You will sudy more abou his opic in he EE422 Communicaions II course. The main conceps o be emphasized here will be he analog-o-digial conversion process and he differen line coding echniques. Digiizaion (and Analog-o-Digial (A/D) Conversion): Signals ha resul from physical phenomena (such as voice or video) are almos always analog baseband signals. Such analog baseband signals can be convered ino digial baseband signals using he following seps: 1. Sampling (in which he signal becomes a sampled analog signal, also called a discree analog signal). 2. Quanizaion (he signal becomes a quanized discree signal, bu no a digial baseband signal ye). 3. Mapping (he signal becomes a sream of 1 s and s). Mapping is someimes confusingly called encoding. 4. Encoding and Pulse Shaping (afer which he signal becomes a digial baseband signal). These four seps are shown in he Figure below and are explained in he following subsecions. Analog Baseband Signal A/D 1's & 's Encoding & Pulse Shaping Digial Baseband Signal Digial Modulaion Digial Carrier Signal Source Encoding Channel Encoding Line Encoding Pulse Shaping Sampling Quanizaion Mapping 2
\ I. Sampling: Sampling is he process in which only a relaively-small se of values, called discree samples {mn}, are aken o represen he signal m() insead of he (ime-coninuous) infinie se of values included in he analog signal (see he following Figure). Samples Analog Signal, m() m p T s Sampling Time -m p T s In uniform sampling, he ime inerval beween successive samples is se o a consan value equal o Ts, called he sampling ime. In his case, he sampling frequency is fs = 1/Ts. Nyquis Shannon sampling heorem saes ha for he samples {mn} o ruly represen he original signal m(), we need he sampling frequency fs o be a leas wice as high as he bandwidh B of he band-limied analog signal m() (i.e., fs 2B). Such a condiion will preven aliasing. Aliasing should be avoided a all coss since i means ha he signal m() canno be recovered from he discree samples {mn} by simple low-pass filering (LPF) a he receiver. As a specific example, elephone conversaions are sampled a 8 khz (wice he 4 khz bandwidh of he human voice signal 1 ), while compac disc (CD) audio is sampled a 44.1 khz (more han wice he 2 khz bandwidh of music signals 2 ). I is ineresing o noice ha pracical A/D inegraed circuis (such as ADC81, ADC82, ADC832, TDA991, ec) canno generae impulses as shown in he Figure above since impulses require infinie energy (impulses are heoreical signals by definiion). Hence, such A/D ICs generae insead pracically-sampled signals in which recangular pulses (of widh Ts) are used o represen he samples insead of impulses, as shown in he Figure below 3. 1 Subjecive ess show ha signal ariculaion (inelligibiliy) is no effeced if all componens above 34 Hz are suppressed. Hence, sricly speaking, voice bandwidh is 3.4 khz no 4 khz. 2 Some music insrumens generae signals wih bandwidhs exceeding 2 khz bu a human ear canno hear sounds above he 2 khz mark. 3 Typical IC-based ADC chips perform sampling, quanizaion and mapping all on he same chip. 3
Samples Values Sampled Signal, m s () m p T s Sampling Time -m p T s II. Quanizaion: Quanizaion is he process in which each sample value is approximaed or limied o a relaively-small se of discree quanizaion levels. For example, in uniform quanizaion if he ampliude of he signal m() lies in he range (-mp, mp), we can pariion his coninuous range ino L discree inervals, each of lengh v = 2 mp /L, and he value of each sample is hen approximaed o only one of hese L levels. Noice ha quanizaion can be done in several differen ways. In one mehod he value of each sample can be runcaed o he quanizaion level jus below i. This is shown in he Figure below for L = 5 levels. Sample Values now Quanized m p m() error v L = 5 Levels, Rule = Truncae -m p Quanizaion Error = m s() m sq() = [, v] m sq() Noice ha he quanizaion error (i.e., he difference beween he original sample value and he quanized sample value) is limied o he range [, v]. This quanizaion error is a deliberae error inroduced by he ransmier o conrol he ransmied bi rae. Noice, however, ha his error can be conrolled by reducing he value of v, which can be achieved by increasing he number of quanizaion levels L as shown below where L = 1 levels now. 4
Sample Values now Quanized m p m() error v L = 1 Levels, Rule = Truncae Quanizaion Error = m s() m sq() = [, v] -m p m sq() Anoher valid mehod of quanizaion is where samples are rounded off o he neares quanizaion level eiher below i or above i. This is shown in he Figure below. Noice ha in his mehod he quanizaion error is now limied o [- v/2, v/2]. m p error m() Sample Values now Quanized v L = 1 Levels, Rule = Approx. Quanizaion Error = m s() m sq() = [- v/2, v/2] -m p m sq() The number of quanizaion levels L is an imporan parameer in digial sysems because i decides (see nex secion) how many bis will be used o represen he value of each sample. For example, if L = 256, he value of each sample can be in one of 256 possibiliies, which means ha each sample mus be mapped (encoded) ino 8 bis. This is because 8 binary bis can be in 2 8 = 256 possible saes (, 1, 1, 11,..., 11111111). For L = 65,536, we need 16 bis o encode each sample value. In voice elephony, for example, he number of quanizaion levels is chosen o be L = 2 8 = 256 since inelligibiliy (raher han high fideliy) is required, while for compac disc (CD) audio, he number of quanizaion levels is L = 2 16 = 65,536 possible values per sample. Of course, a bigger value of L means a smaller quanizaion error range v, and hus beer qualiy. III. Mapping: Digial sysems use binary sreams, in which each of he quanized sample values (e.g., 6V, 1V, ec) is mapped o a corresponding binary code (e.g., 11, 11, ec). The resul is a sream of 1 s and s. 5
Noice ha each sample is represened by n = log2 (L) bis, and hence he bi ime T is given by T = Ts / log2 (L), where Ts is he sample period. For example, if L = 256, we have T = T s 8 (see Figure below). The oal number of bis generaed in one second is called he daa bi rae f = 1 / T measured in bis/s (or bps) and is given by: f [bps] = fs [samples/s] log2 (L) [bis/sample] m sq () T s T s 1 1 1 1 1 1 1 m digial () T T T s = log 2 (L) T T s = log 2 (L) T IV. Encoding: The encoding sage akes a bi sream of 1 s and s and convers i ino volages appropriae for ransmission on a physical channel. Encoding schemes are usually divided ino: source encoding, channel encoding and line encoding (see Figure in page 2). A. Source coding: Source coding (someimes called enropy encoding) refers o he process of compressing daa. This is ypically done by replacing long binary codes (named codewords) ha occur frequenly by shorer ones, and hose ha occur less frequenly by longer codes. For example, a 4-bi sequence 11 occurring frequenly can be mapped ino he shorer 2-bi 1 sequence, while anoher 4-bi sequence 111 occurring less frequenly can be mapped o he longer 7-bi sequence 111. This makes sure ha sequences ha occur more ofen in he bi sream are he shorer ones. In informaion heory, Shannon's noiseless coding heorem places an upper and lower bounds on he expeced compression raio. Examples of source codes currenly in use are: Shannon codes, Huffman codes, run-lengh encoding (RLE), arihmeic coding, Lempel-Ziv coding, MPEG-2 and MPEG-4 video coding 4, Linear Predicion Coding (LPC) and Code-Excied Linear Predicion (CELP) coding, ec. 4 MPEG is shor for Moion Picure Expers Group. 6
B. Channel coding: Channel coding refers o error correcing codes. Such codes are used o proec daa sen over he channel from corrupion even in he presence of noise. In oher words, channel codes can improve he signal-o-noise raio (SNR) of he received signal. The mos obvious example of such codes is he simple pariy bi sysem. The heory behind designing and analyzing channel codes is called Shannon s noisy channel coding heorem. I pus an upper limi on he amoun of informaion you can send in a noisy channel using a perfec channel code. This is given by he following equaion: C = B ch log 2 1 + SNR where C is he upper bound on he capaciy of he channel (bi/s), Bch is he bandwidh of he channel (Hz) and SNR is he Signal-o-Noise raio on he channel (uniless). Examples of channel codes currenly in-use include: Hamming codes, Reed-Solomon codes, convoluional codes (usually decoded by an ieraive Vierbi decoder), Turbo codes, ec. C. Line coding: Line coding refers o he process of represening he bi sream (1 s and s) in he form of volage or curren variaions opimally uned for he specific properies of he physical channel being used. The selecion of a proper line code can help in so many ways: (a) One possibiliy is o aid in clock recovery a he receiver. A clock signal is recovered by observing ransiions in he received bi sequence, and if enough ransiions exis, a good recovery of he clock is guaraneed, and he signal is said o be self-clocking. Selfclocking is imporan in digial sysems as all digial receivers require he exisence of he clock o funcion properly (his is similar o he synchronous deecion in DSB-SC demodulaors). (b) Anoher advanage is o ge rid of DC shifs. The DC componen in a line code is called he bias or he DC coefficien. Unforunaely, mos long-disance communicaion channels canno ranspor a DC componen 5. This is why mos line codes ry o eliminae he DC componen before being ransmied on he channel. Such codes are called DC balanced, zero-dc, zero-bias, or DC equalized. Oher advanages of proper line coding include he (c) possibiliy of ransmiing a a higher daa bi rae while requiring smaller bandwidh for he resuling baseband signal, and (d) reducing he amoun of power a low-frequency componens of he specrum, which is imporan in elephone line applicaions, where he ransfer characerisic has heavy aenuaion below 3 Hz. Some ypes of line encoding in common-use nowadays are unipolar, polar, bipolar, Mancheser, MLT-3 and Duobinary encoding. These codes are explained here: 5 DC-values creae excessive hea generaion in he channel, hey cause baseline drif and also do no fi sysems ha carry an addiional small direc curren o power inermediae line amplifiers (an example is elephone neworks). 7
1. Unipolar (Unipolar NRZ and Unipolar RZ): Unipolar is he simples line coding scheme possible. I has he advanage of being compaible wih TTL logic. Unipolar coding uses a posiive recangular pulse p() o represen binary 1, and he absence of a pulse (i.e., zero volage) o represen a binary. Two possibiliies for he pulse p() exis 6 : Non-Reurn-o-Zero (NRZ) recangular pulse and Reurn-o-Zero (RZ) recangular pulse. The difference beween Unipolar NRZ and Unipolar RZ codes is ha he recangular pulse in NRZ says a a posiive value (e.g., +5V) for he full duraion of he logic 1 bi, while he pule in RZ drops from +5V o V in he middle of he bi ime. The figure below shows he difference beween Unipolar NRZ and Unipolar RZ for he example bi sream 1111111. Clock Daa 1 1 1 1 1 1 1 Code T Bi ime T s Sampling ime (assuming 8 bis per sample) Unipolar NRZ Code Clock Daa 1 1 1 1 1 1 1 Code T Bi ime Unipolar RZ Code A drawback of unipolar (RZ and NRZ) is ha is average value is no zero, which means i creaes a significan DC-componen a he receiver (see he impulse a zero frequency in he corresponding power specral densiy (PSD) of his line code shown in he diagram below). As we explained earlier, a DC-value is no desired in longdisance communicaion sysems. Anoher disadvanage of such unipolar (RZ and NRZ) signaling is ha i does no include clock informaion especially when he bi sream consiss of a long sequence of s. 6 Acually here are so many possibiliies for he pulse shape p(); no jus a recangular NRZ or recangular RZ pulses. Changing p() waveform is called Pulse Shaping and affecs he characerisics of he line code as will be explained laer. 8
The disadvanage of unipolar RZ compared o unipolar NRZ is ha each recangular pulse in RZ is only half he lengh of NRZ pulse. This means ha unipolar RZ requires wice he bandwidh of he NRZ code. This can be seen from he PSD of boh signals shown below 7. S m (w) Unipolar NRZ S m (w) Unipolar RZ -8pf -6pf -4pf -2pf 2pf 4pf 6pf 8pf f -4f -3f -2f -f f 2f 3f 4f w -8pf -6pf -4pf -2pf 2pf 4pf 6pf 8pf f -4f -3f -2f -f f 2f 3f 4f w 2. Polar (Polar NRZ and Polar RZ): In Polar NRZ line coding binary 1 s are represened by a pulse p() (e.g., +5V) and binary s are represened by he negaive of his pulse -p() (e.g., -5V). Polar (NRZ and RZ) signals are shown in he diagram below. Using he assumpion ha in a regular bi sream a logic is jus as likely as a logic 1, polar signals (wheher RZ or NRZ) have he advanage ha he resuling DCcomponen is very close o zero. Polar NRZ Code Polar RZ Code 7 The above specra were calculaed based on he assumpion ha logic 1 s and logic s are equally likely in he ransmied bi sequence. This is a simplifying assumpion ha we use hroughou his aricle. 9
In addiion, he rms value of polar signals is bigger han unipolar signals, which means ha polar signals have more power han unipolar signals 8, and hence have beer SNR a he receiver. Acually, polar NRZ signals have more power compared o polar RZ signals. The drawback of polar NRZ, however, is ha i lacks clock informaion especially when a long sequence of s or 1 s is ransmied. This problem does no exis in polar RZ signals, since he signal drops o zero in he middle of each pulse period. The power specral densiies (PSD) of boh polar NRZ and polar RZ are shown below. S m (w) Polar NRZ Polar RZ -8pf -6pf -4pf -2pf 2pf 4pf 6pf 8pf w f -4f -3f -2f -f f 2f 3f 4f Signals ransmied on a compuer moherboard ofen use Polar NRZ code. Anoher useful applicaion of his encoding is in Fiber-based Gigabi Eherne (1BASE-SX and 1BASE-LX). Noe: Polar NRZ is ofen jus called NRZ. Polar RZ is ofen jus called RZ. A varian of Polar NRZ is Non-Reurn-o-Zero-Level (NRZ-L) in which he 1 s and s are represened by -p() and p(), respecively. This is Polar NRZ using negaive logic. An example where NRZ-L is used is he legacy RS-232 serial por communicaion. NRZ-L Code 8 2 Remember ha he average power in a signal is he square of is rms value (P av = x rms ). 1
3. Non-Reurn-o-Zero, Invered (NRZI): NRZI is a varian of Polar NRZ. In NRZI here are wo possible pulses, p() and -p(). A ransiion from one pulse o he oher happens if he bi being ransmied is a logic 1, and no ransiion happens if he bi being ransmied is a logic. NRZI Code 9 This is he code used on compac discs (CD), USB pors, and on fiber-based Fas Eherne a 1-Mbi/s (1Base-FX). NRZI can achieve synchronizaion beween he ransmier and receiver, if we make sure ha here are enough umber of 1 s in he ransmied bi sream. 4. Bipolar encoding (also called Alernae Mark Inversion (AMI)): Bipolar (or AMI) is a hree-level sysem ha uses p(), -p(), and he absence of pulses (e.g. +5V, -5V, V) o represen logical values. A logic is represened wih an absen pulse, and a logic 1 by eiher a posiive or negaive pulse. The direcion of he pulse is opposie of he pulse sen for he previous logic 1 (mark) (see he Figure below). Bipolar (AMI) Code The alernaing code in bipolar encoding prevens he build-up of a DC volage in he cable. You can also observe he absence of low frequencies (including he DC componen) from he PSD for AMI shown below. 9 NRZI is always polar no unipolar. 11
S m (w) Duobinary Bipolar NRZ (AMI) Polar NRZ Polar RZ Line Code Bandwidh Unipolar NRZ f Unipolar RZ 2 f Polar NRZ f Polar RZ 2 f Bipolar NRZ f Duobinary f / 2 2pf 4pf f 2f AMI coding was used exensively in firs-generaion digial elephony PCM neworks. AMI suffers he drawback ha a long run of 's produces no ransiions in he daa sream, and a loss of synchronizaion is possible. This was solved in elephony by adoping improved varians of AMI encoding o ensure regular ransiions in he baseband signal even for long runs of s. The Binary-wih-8-Zero- Subsiuion (B8ZS) is he line coding scheme ha was adoped for Norh America T1 sysem, while High-Densiy Bipolar 3-Levels (HDB3) was he line coding scheme used in he European E1 sysem. This is no par of he exam maerial Noe: A very similar encoding scheme o AMI, wih he logical posiions reversed, is also used and is ofen referred o as pseudoernary encoding. This encoding is essenially idenical o AMI, wih marks (1 s) being zero volage and spaces ( s) alernaing beween posiive and negaive pulses. Noe: Coded Mark Inversion (CMI) is anoher variaion of AMI, where bis are represened by a ransiion in he middle of he bi ime insead of zero volage. 5. Duobinary: In a duobinary line code a bi is represened by a zero-level elecric volage; a 1 bi is represened by a p() if he quaniy of bis since he las 1 bi is even, and by -p() if he quaniy of bis since he las 1 bi is odd. An illusraion of he duobinary line code is shown below. Clock Daa 1 1 1 1 1 1 1 Code Duobinary Code 12
For a bi rae of f, duobinary line code requires f/2 bandwidh, which is he minimum possible (heoreical) bandwidh for any digial baseband signal (called Nyquis bandwidh). In addiion, he duobinary line code permis he deecion of some ransmission errors wihou he addiion of error-checking bis. However, duobinary line codes have significan low frequency componens as seen by he PSD shown earlier. The differenial version of he duobinary line code is common in he 2 Gbi/s and 4 Gbi/s uncompensaed opical fiber ransmission sysems. I is imporan, however, ha you do no confuse a duobinary line code (explained above) wih somehing compleely differen called a duobinary pulse (shown below). This pulse (which you are going o sudy in he Communicaions II course) is commonly used in conrolled iner-symbol inerference (ISI) scenarios. Confusingly, duobinary signaling refers o using he duobinary pulse wih a polar line coding rule (no a duobinary line coding rule). p() = Duobinary Pulse 1 V -3T -2T -T T 2T 3T 6. Muli-Level Transmission 3-Levels (MLT-3): MLT-3 encoding is used mainly in 1BASE-TX Fas Eherne, which is he mos common ype of Eherne nowadays. MLT-3 cycles hrough he saes -p(),, p(),, -p(),, p(),,... ec. I moves o he nex sae o ransmi a 1 bi, and says in he same sae o ransmi a bi. MLT-3 has many advanages including emiing less elecromagneic inerference, requiring less bandwidh han unipolar, polar, and bipolar (AMI) signals operaing a he same daa bi rae. The PSD of MLT-3 code is shown on he nex page. MLT-3 Code 7. Mancheser: Currenly here are wo opposing convenions for he represenaion of Mancheser codes: 13
The firs convenion of hese was firs published by G. E. Thomas in 1949 and is followed by numerous auhors (e.g., Andrew S. Tanenbaum). I specifies ha for a bi he signal levels will be Low-High wih a low level in he firs half of he bi period, and a high level in he second half (see figure below). For a 1 bi he signal levels will be High-Low. The second convenion is also followed by numerous auhors (e.g., Sallings) as well as by IEEE 82.4 and IEEE 82.3 (Eherne 1 Mbps 1Base-T) sandards. I saes ha a logic is represened by a High-Low signal sequence and a logic 1 is represened by a Low-High signal sequence. If a Mancheser encoded signal ges invered somewhere along he communicaion pah, i ransforms from one varian o anoher. In his aricle, we will adop he firs convenion (see figure below). Mancheser Code 1 In Mancheser code each bi of daa is signified by a leas one ransiion. Mancheser encoding is herefore considered o be self-clocking, which means ha accurae clock recovery from a daa sream is possible. In addiion, he DC componen of he encoded signal is zero. Alhough ransiions allow he signal o be self-clocking, i carries significan overhead as here is a need for essenially wice he bandwidh of a simple Polar NRZ or NRZI encoding (see he PSD below). This is he main disadvanage of he Mancheser code. S m (w) MLT-3 Polar NRZ Mancheser Polar RZ Line Code Bandwidh Polar NRZ f Polar RZ 2 f MLT-3.9 f Mancheser 2 f 2pf 4pf f 2f w f 1 Mancheser is always polar no unipolar. 14
This is no par of he exam maerial Differenial Mancheser encoding In Differenial Mancheser encoding a 1 bi is indicaed by making he firs half of he signal equal o he las half of he previous bi s signal i.e. no ransiion a he sar of he bi-ime. A bi is indicaed by making he firs half of he signal opposie o he las half of he previous bi's signal i.e. a zero bi is indicaed by a ransiion a he beginning of he bi-ime. Differenial Mancheser Code Because only he presence of a ransiion is imporan, differenial schemes will work exacly he same if he signal is invered (wires swapped). In he middle of he bi-ime here is always a ransiion, wheher from high o low, or low o high, which provides a clock signal o he receiver. Differenial Mancheser is specified in he IEEE 82.5 sandard for IBM Token Ring LANs, and is used for many oher applicaions, including magneic and opical sorage. 8. M-ary Coding In binary line coding, he number of bis per second is idenical o he number of symbols per second (called baud rae). We say ha for binary signaling: f, daa bi rae [in unis of bi/s] = fsymb, symbol rae [in unis of baud] Noice ha a symbol is defined as a waveform paern ha he line code has for a cerain period of ime before swiching o anoher waveform paern (i.e., anoher symbol). In M-ary signaling, a cluser of bis is grouped o represen one symbol. For example, in he 4-ary (also called Quaernary) case, wo bis are grouped ino one symbol. The wo bis can be in one of 4 possible saes, which means ha he symbol can ake M=4 differen symbols. The following able shows a possible mapping of bi values o symbols of a Quaernary signal. An example of how such line code works for he bi sream 1111111 is shown nex. 15
Bis Symbol -5 V 1-1 V 1 5 V 11 1 V M=4 levels Clock Daa 1 V 1 1 1 1 1 1 1 5 V Code -5 V -1 V T Bi Time T symb Symbol Time T s Sampling Time (assuming 8 bis per sample) Quaernary Code Noice ha a symbol ime Tsymb is now wice he bi ime T. This means ha here are half as much symbol ransiions as here are bi ransiions. We can say ha: symbol rae [in unis of baud] = (½) daa bi rae [in unis of bi/s] For a general M-ary coding scheme, we have: symbol rae [in unis of baud] = (1/log2 (M)) daa bi rae [in unis of bi/s] where M is he number of levels (possibiliies) for a symbol. Such a drop in ransiion rae in he resuling signal will reduce he bandwidh of he signal by a facor of log2 (M), because he bandwidh of digial signal is acually dependen on is baud rae fsymb no is bi rae f. Remember: An M-ary is a baseband signal ha has a bandwidh equal o is baud rae (fsymb) as shown below. S m (w) M-ary -4pf symb -2pf symb 2pf symb 4pf symb f -2f symb -f symb f symb 2f symb w 16
V. Pulse Shaping: In explaining he above line codes, we limied ourselves o pulses p() shaped as eiher recangular NRZ or recangular RZ pulses. I is essenial ha you undersand ha hese wo are no he only choices you have; insead a variey of pulse shapes p() can be used wihou compromising he informaion ransfer process. In his secion, we will discuss wo possibiliies: he riangular pulse and he raised cosine pulse. The Figure below shows he bi sream 1111111 represened using a polar line code combined wih riangular pulse shape. Clock Daa 1 1 1 1 1 1 1 Code You migh be wondering a his poin abou he advanages of choosing a pulse shape differen han he simple and familiar recangular pulse? Well, here is a number of advanages for doing so, he main of which is being able o conrol he shape of he power specral densiy PSD (and hence he bandwidh) of he resuling digial baseband signal. To undersand his, recall ha he line code you chose earlier (unipolar, polar, bipolar, ec) has affeced he bandwidh of he corresponding code. However, he PSD shape always looked like a sinc 2 () funcion. The reason his was he case is ha he Fourier ransform of a recangular pulse has he sinc() shape and because he PSD is he square of he Fourier ransform, he PSD looked like a sinc 2 () funcion (see Figure in page 18). You also recall ha using a RZ recangular pulse insead of a NRZ recangular pulse resuled in expanding he bandwidh by a facor of 2. This is an immediae resul of he Time compression, Frequency expansion propery of he Fourier ransform, and is also illusraed in he Figure in page 18. The quesion ha arises now is his: If we pick a daa sream encoded using a paricular line code (say polar encoding), can we conrol is bandwidh by changing he pulse shape p() while sill keeping he polar line code rules? The answer o his quesion is YES; picking he righ pulse shape can resul in smaller bandwidh compared o he bandwidh of he recangular pulse which is, sricly-speaking, infiniy (because he sinc() funcion exends from minus infiniy o posiive infiniy). If we choose a smooher pulse (compared o he recangular pulse), he high frequencies in he resuling PSD of he daa sream are eliminaed. For example, if we pick a riangular pulse insead of a recangular pulse (see Figure in page 15) he high frequency componens are heavily reduced. (This is more apparen if you look a he specrum using a log scale). Remember ha he Fourier ransform of a riangular pulse is sinc 2 () and, hence, he PSD has he shape of sinc 4 () funcion. 17
P(w) S p(w) p() Fourier Transform Power Specral Densiy T Bi ime -8pf -6pf -4pf -2pf w -8pf -6pf -4pf -2pf 2pf 4pf 2pf 4pf 6pf 8pf 6pf w 8pf S p(w) [Log Scale] -8pf -6pf -4pf -2pf 2pf 4pf 6pf w 8pf P(w) S p(w) p() Fourier Transform Power Specral Densiy T Bi ime -8pf -6pf -4pf -2pf w -8pf -6pf -4pf -2pf 2pf 4pf 2pf 4pf 6pf 8pf 6pf w 8pf S p(w) [Log Scale] -8pf -6pf -4pf -2pf 2pf 4pf 6pf w 8pf P(w) S p(w) p() Fourier Transform Power Specral Densiy T Bi ime -8pf -6pf -4pf -2pf w -8pf -6pf -4pf -2pf 2pf 4pf 2pf 4pf 6pf 8pf 6pf w 8pf S p(w) [Log Scale] -8pf -6pf -4pf -2pf 2pf 4pf 6pf w 8pf P(w) S p(w) p() Fourier Transform Power Specral Densiy T Bi ime -8pf -6pf -4pf -2pf w -8pf -6pf -4pf -2pf 2pf 4pf 2pf 4pf 6pf 8pf 6pf w 8pf T S p(w) [Log Scale] -8pf -6pf -4pf -2pf 2pf 4pf 6pf w 8pf 18
Reducing high frequency componens in he PSD is imporan so ha he signal can pass hrough band-limied channels wihou oo much linear disorion. Such disorion usually creaes wha is called iner-symbol inerference (ISI) in digial sysems, where one bi value overlaps wih (and corrups) he adjacen bis. Using smooher pulses (and hence reducing heir high frequency conen) can be aken o he exreme if we pick a pulse ha is no limied in ime. In oher words, he pulse spells ouside he bi ime (T). This will limi is Fourier ransform (because an expansion in he ime-domain resuls in compression in he frequency-domain). One popular example used in many pracical sysems is he raised-cosine pulse. This is a pulse ha looks similar o (bu decays much quicker han) a sinc() funcion and has a Fourier ransform similar o a raised cosine shape in he frequency domain. This pulse and is corresponding Fourier ransform are shown a he boom of he above Figure. The raised-cosine pulse shape changes wih a parameer called he roll-off facor. The Figure below shows a =.5 raised-cosine pulse wih he corresponding Fourier ransform. 1 V p() = Raised Cosine Pulse P(w) T f /2-3T -2T -T T 2T 3T w - 2pf - pf pf 2pf f - f - f /2 f /2 f This ype of pulse saisfies wha is called he firs Nyquis ISI crierion, which saes ha i is OK for he pulse shape no o be limied o he bi ime (T), so long as i does no inroduce iner-symbol inerference (ISI) in he ransmied pulse sream. An example of using he raised-cosine pulse combined wih polar line code is shown in he nex page, where he binary sequence 111111 is ransmied. Noice ha we use p() for logic 1 and -p() for logic (i.e., he polar line code). The doed lines represen he pulses p() and -p() corresponding o individual bis, while he solid line represens he resul of adding hese pulses (i.e., he ransmied signal on he channel). A he receiver side, sampling he received signal a exacly he middle of he bi ime will rerieve he original bi sequence as shown in he Figure. Our example used raised-cosine pulses wih parameer =.5. In such case, he bandwidh of he resuling sream is given by he formula (1+ ) f / 2 = (1+.5) f / 2 =.75 f, where f is he daa rae. Hence, if f = 5 kbi/s, he resuling bandwidh = 37.5 khz. Wha would he bandwidh have been if you used a recangular NRZ pulse? How abou a recangular RZ pulse? 19
Clock Daa 1 1 1 1 1 1 1 V -1 V TX signal 1 V -1 V 1 V RX sampling -1 V 2