CHAPTER CONTENTS. Notes. 9.0 Line Coding. 9.1 Binary Line Codes


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1 Noes CHAPTER CONTENTS 9. Line Coding 9. inary Line Codes 9. ipolar and iphase Line Codes 9.. AMI 9... inary N Zero Subsiuion 9..3 lock Line Codes 9.3 Mary Correlaion Codes 9.3. Q 9.3. Correlaion Coding 9.4 Nyquis Channels 9.4. rick Wall Filer (Ideal LPF) 9.4. Cosine Channel (Duobinary Channel) ipolar Encoding Differenial Encoding Raised Cosine Channel Sine or Modified Duobinary Channel 9.5 Gaussian Pulse Shaping Review Quesions For Furher Research Digial Communicaions Principles 9 
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3 Noes 9. Line Coding Objecives This secion will: Review he basic ypes of binary line codes Examine bipolar line codes Examine Mary correlaion codes Inroduce he idea of baud reducion by using conrolled ISI Hewle Packard has produced an excellen applicaion noe summarizing he various forms of digial modulaion. Digial Modulaion in Communicaions Sysems  An Inroducion by hp 9. inary Line Codes The erm line code refers o he physical shape of he signal ha is placed on he loop. Some of he more common wo level or binary line codes include: Signal NRZ L NRZ M NRZ S RZ iphase L iphase M iphase S Differenial Mancheser ipolar Commens Non reurn o zero level. This is he sandard posiive logic signal forma used in digial circuis. forces a high level forces a low level Non reurn o zero mark forces a ransiion does nohing Non reurn o zero space does nohing forces a ransiion Reurn o zero goes high for half he bi period does nohing Mancheser. Two consecuive bis of he same ype force a ransiion a he beginning of a bi period. forces a negaive ransiion in he middle of he bi forces a posiive ransiion in he middle of he bi There is always a ransiion a he beginning of a bi period. forces a ransiion in he middle of he bi does nohing There is always a ransiion a he beginning of a bi period. does nohing forces a ransiion in he middle of he bi There is always a ransiion in he middle of a bi period. does nohing forces a ransiion a he beginning of he bi The posiive and negaive pulses alernae. forces a posiive or negaive pulse for half he bi period does nohing A bipolar signal is no acually a binary signal since i has 3 disinc levels. Digial Communicaions Principles 9 
4 Noes Line Coding inary Line Code Waveforms NRZ L NRZ M NRZ S RZ iphase L iphase M iphase S Differenial Mancheser ipolar Each line code has advanages and disadvanages. The paricular line code used is chosen o mee one or more of he following crieria: Minimize ransmission hardware Faciliae synchronizaion Ease error deecion and correcion Minimize specral conen Eliminae a dc componen 9. ipolar and iphase Line Codes ipolar line codes have wo polariies, are generally implemened as RZ and have a radix of hree since here are hree disinc oupu levels One of he principle advanages of his ype of code, is ha i can compleely eliminae any DC componen. This is imporan if he signal mus pass hrough a ransformer or a long ransmission line. iphase line codes require a leas one ransiion per bi ime. This makes i easier o synchronize he ransceivers and deec errors however; he baud rae is greaer han ha of NRZ codes. 9  Digial Communicaions Principles
5 Line Coding Noes NRZ and iphase Specral Densiy NRZ codes are more bandwidh efficien han bipolar RZ ones since heir baud rae is half ha of RZ codes and heir specral componens go all he way down o Hz. CMI In CMI, marks are encoded as alernae polariy, full period pulses. Spaces are encoded by half a period pulse a he negaive volage and half period pulse a he posiive volage. This coding scheme has he advanage ha i uses only wo volage levels insead of hree, as does AMI. inary NRZ inary RZ CMI 9.. AMI AMI is a bipolar line code. Each successive mark is invered and he average or DC level of he line is herefor zero. This sysem is used on Tcarrier sysems, bu canno be used on fiber opic links. inary NRZ inary RZ AMI AMI is usually implemened as RZ pulses. Digial Telephony (nd ed.), John ellamy, Figure 4.3 Coded Mark Inversion Alernae Mark Inversion Digial Communicaions Principles 93
6 Noes Line Coding One of he weaknesses of ransmiing only marks is ha long srings of zeros cause he receivers o loose lock. I is essenial o mainain lock because he T carrier muliplexing scheme organizes daa as a series of concaenaed channels. If synchronism is los, he specific channels canno be idenified. I is herefore necessary o impose addiional rules on he signal o eliminae long srings of zeros inary N Zero Subsiuion Long srings of zeros can be prevened by subsiuing a sequence of 3, 6, or 8 zeros wih a special code of he same lengh. The subsiuion code conains bipolar violaions, which alers he receiver o noe he changes. In an AMI sysem, wo consecuive pulses of he same polariy consiue a violaion. Therefor, if conrolled violaions are subsiued for srings of zeros, he receiver can disinguish beween subsiuions and errors. 6ZS 6ZS is used on 6.3 Mbps, T AMI ransmission links. Since he las mark preceding a sring of zeros may have been eiher posiive or negaive, wo ypes of subsiuions are used: Polariy of previous mark Subsiuion These subsiuions force wo consecuive violaions. A single bi error does no creae his condiion. 6ZS Example: Original daa: AMI daa:   6ZS daa: inary RZ AMI RZ 6ZS inary 6 Zero Subsiuion 94 Digial Communicaions Principles
7 Line Coding Noes 8ZS This scheme uses he same subsiuion as 6ZS. Since he example above has a sring of 7 zeros, no subsiuion would be made. Polariy of previous mark Subsiuion ZS 3ZS is more involved han 6ZS, and is used on DS 3 carrier sysems. The subsiuion is no only dependen on he polariy of he las mark, bu also on he number of marks (even and odd) since he las subsiuion. I should be remembered ha he number zero is by definiion even. Previous Mark Polariy Number of marks since he las subsiuion Odd Even ZS Example: Assuming an odd number of marks since he las subsiuion, we obain: inary RZ AMI RZ 3ZS From he above example, i seems ha here are more negaive pulses han posiive ones, hus craing a DC componen. I should be noed however, ha his is saisically eliminaed over longer daa sequences. HD3 HD3 is used in Europe and inroduces bipolar violaions when four consecuive zeros occur. The second and hirds zeros are lef unchanged, bu he fourh zero is given he same polariy as he las mark. The firs zero may be modified o a one o make sure ha successive violaions are of alernae polariy. inary 3 Zero Subsiuion High Densiy inary 3 Freeman, Telecommunicaion Handbook, & Sallings W., ISDN an Inroducion Digial Communicaions Principles 95
8 Noes Line Coding Previous Mark Polariy Number of marks since he las subsiuion Odd Even HD3 Example: inary RZ AMI RZ HD3 From he above example, i seems ha here are more posiive pulses han negaive ones, hus craing a DC componen. I should be noed however, ha his is saisically eliminaed over longer daa sequences. The las pulse in he subsiuion is he V or violaion pulse. If here have been an even number of subsiuions, a or balancing pulse is added o preven a dc buildup lock Line Codes lock codes wih a radix of or 3 can be used on digial loops. These schemes operae on byes raher han bis. Some ransmi he signal as binary levels, bu mos use mulilevel pulses. A binary block code has he designaion nm, where n inpu bis are encoded ino m oupu bis. The mos common of hese is he 34 code. 34 Coding Inpu Oupu or or  In Europe 43T, which encodes 4 binary bis ino 3 ernary levels, has been seleced as he RA for ISDN. Some block codes do no generae mulilevel pulses. For example, 4P or 45 simply adds a P or pariy bi o a 4bi block. 3 Freeman, Telecommunicaion Handbook, & Sallings W., ISDN an Inroducion 96 Digial Communicaions Principles
9 Line Coding Noes 9.3 Mary Codes Line codes wih a radix of 4 or more are also block codes, and have he poenial o significanly reduce he baud rae Q In Norh America, Q which encodes binary bis ino quaernary level has been seleced for RA. Q Coding Inpu Oupu Correlaion Coding One of he chief aims of digial communicaions echnology is o pack as many bis of informaion hough a sysem as possible. This is paricularly rue in long haul, fixed bandwidh sysems such as digial microwave radio and saellies. Alhough i is no possible o exceed he ShannonHarly limi, he Nyques bi rae [ bis per Hz] can be exceeded. This is accomplished by inroducing a conrolled amoun of inersymbol inerference in he signal. Parial response signaling alers he shape of a daa pulse o conrol he signal specrum and make efficien use of he ransmission channel. In order o do his, i is necessary o deermine he ime and frequency domain characerisics of he communicaions channel or sysem. This requires he use of he Fourier ransform. In mos cases, signal shapes are specified in he ime domain, bu communicaions channels are specified in he frequency domain. Since he physical channel represens he real world, i is necessary o firs examine he channel frequency characerisics and hen deermine he mos suiable shape of he daa signal in he ime domain. A communicaions ransmission link or channel can be regarded as a sor of filer. The filer characerisic may be defined by he physical aribues of he channel or by governmen regulaions and indusry sandards. Radio based sysems for example, have severe resricions placed on user frequency bands and each broadcas channel may be viewed as a bandpass filer. I is ofen difficul o find a mahemaical expression defining he frequency response of acual filers and ransmission sysems. Consequenly, he analysis is generally performed on simpler funcions, which can hen be used o approximae complex sysems. The mos common family of channel ypes are Nyquis channels. Digial Communicaions Principles 97
10 Noes Line Coding 9.4 Nyquis Channels The principle channel ypes of ineres are he cosine, raised cosine, and sine. The brick wall response is also of ineres since i provides he basic ool needed o evaluae he oher channel ypes. Nyquis Communicaions Channel Caegories 4 Class Channel Radix Name Noaion f[δ] Ideal LPF Cosine [Duobinary] Raised Cosine 3 4 Sine [Modified Duobinary] 5 Channel Response H ( ω) Impulse ω cos ω 4 cos ω ω cos cos ω ω jsin sin ω sin ω 4 sin The oupu of each channel ype is naurally slighly differen., 3,, 5,, 5,, 3,,,, 5 Fourier Analysis Simplificaions Fourier analysis can ofen be simplified by observing he funcion symmery: For even funcions, use he inverse cosine ransform For odd funcions use he inverse sine ransform Funcions ha do no exhibi symmery mus use boh pars of he ransform h ( ) = H ( ω ) cos( ω) j sin( ω) Real par for even funcions dω Imaginary par for odd funcions This is ofen called he impulse response because i is obained by injecing a Dirac Dela pulse [δ] ino he inpu. 4 Digial, Analog, and Daa Communicaion, William Sinnema, Table Digial Communicaions Principles
11 Line Coding Noes The complex ransform is also used in a more mahemaically rigorous regime when he negaive frequency domain is considered rick Wall Filer (Ideal LPF) The ideal low pass channel has an infinie rolloff a he cuoff frequency. This is of course no echnically achievable. Ideal LPF (rick Wall Filer) f c ω H ( ω) = ω (Some exbooks ake a more rigorous approach and include negaive frequencies. From an engineering perspecive, he idea of LPFs having a negaive frequency response is no paricularly meaningful. For a more horough discussion of he Fourier Transform, please see Appendix 3) The ime domain response (also called he impulse response) is found by aking he inverse Fourier ransform of he channel: h =  ( ) F { H ( ω) } = H ( ω ) cos( ω) = sinω = sin A plo of his funcion resembles: dω..8.6 h( ) = sin ( f c ) f c 3 f c f c f c 3 4 Digial Communicaions Principles 99
12 Noes Line Coding This funcion is he familiar sync or sampling funcion and forms he basic ime domain elemen used o analyze Mary pulses. I should be noed, ha if a ime domain pulse of his exac shape were creaed, i would have an ideal cuoff in he frequency domain. Time domain pulses of his exac ype however, are no pracical, since he leading and railing ails never compleely vanish. The daa inpu sream produces he following oupu response: rick Wall Response (Ideal LPF) Since he impulse response of an ideal LPF consiss of one sinc pulse, i is someimes wrien as f[δ] =. δ Filer Inpu Ideal LPF Filer Oupu Acual ime domain response 9  Digial Communicaions Principles
13 Line Coding Noes If wo δ pulses separaed by = f c occur a he filer inpu, he peak of he second sync response will occur a a zero crossing of he firs response. This suggess ha a ha precise momen, i is possible o disinguish beween boh pulses even hough a grea deal of overlap or ISI has occurred. y normalizing he bandwidh o uniy, i can be observed ha he maximum bi rae wih no ISI is: Maximum i Rae = = = bis/hz.5. Massive ISI a his insan No ISI a his insan. Sampling Insans [andwidh Normalized o Uniy] Conrolling ISI forms he bases of Mary signaling heory and allows he Nyques rae o be exceeded. If a ransmied pulse waveform consiss of sinc componens, i is possible o separae he sinc componens a he receiver, hus exceeding he Nyques rae Cosine Channel (Duobinary Channel) The cosine shape is well defined mahemaically, being he firs quarer of a cosine waveform. Is impulse response consiss of sinc componens. For he impulse response o be comprised of uni sinc pulses, he cosine filer is ofen modified by giving i an ampliude of : Digial Communicaions Principles 9 
14 Noes Line Coding H( ω).5 Cosine Filer H( ω) = cos ω ω H ω ( ω ) = cos ω The daa inpu produces he following oupu responses: Cosine Channel Response Cosine filers are also known as duobinary filers and are used RD3 digial microwave radio sysems and ISDN ransmission sysems using he 43T line forma. 9  Digial Communicaions Principles
15 Line Coding Noes RD3 Radio Transmier 5 Noice he use of a cosine filer a he ransmier oupu. The cosine impulse response is found by aking he real (or even) par of he inverse Fourier ransform: h ( ) cos cos( ω) = = sin = sin( ) ( ) ( ) ( ) ω ω sin sin( ) ( ) ( ) ( ) dω ω sin = ( ) ( ) sin ( ) ( ) h u = ω and ω = f normalizing o ( ) c f c [( ) ] 4 ( ) c =, we obain : c [( ) ] 4 ( ) 4 sin sin = 4 a sync pulse a sync pulse 5 Digial, Analog, and Daa Communicaion, William Sinnema, Figure 84a Digial Communicaions Principles 93
16 Noes Line Coding A duobinary pulse can be decomposed ino wo sinc pulses:.73 COS Impulse Response.5 Ampliude h( ) h( ) h( ) Toal Response s Sinc Pulse nd Sinc Pulse Since he overall response is composed of wo sinc pulses he impulse response is wrien as f[δ] =,. Since he maximum bi rae is / and he firs zero crossing occurs a = 3/4f c, and i would appear ha he maximum bi rae is: Time 4 f c Apparen Maximum i Rae = = = 3.33 f This means ha.33 bis/hz can be ransmied hrough his filer or channel. When he bandwidh is normalized o uniy, he overall pulse shape is composed of wo sinc pulses shifed by = ±.5 This is idenical o ha obained from a brick wall filer excied by wo impulses separaed by =.5, applied o he inpu! c δ Inpu Cosine Filer Oupu δ.5t δ Ideal LPF Oupu Inpu If wo impulses are applied o a cosine channel, he response is: 94 Digial Communicaions Principles
17 Line Coding Noes δ δ Cosine Filer Inpu Oupu If hey are spaced.5t apar. The leading sinc envelope of one pulse compleely overlaps he railing sinc envelope of he previous pulse. Combined Response δ.5t δ Inpu Cosine Filer Oupu If he leading sinc envelope of he second pulse compleely overlaps he railing sinc envelope of he firs pulse, a correlaion over bi period occurs and he overall oupu resembles:.. f c 3 f c f c f c A firs glance, i would appear ha his represens only one bi of informaion, bu if a receiver can decompose his pulse ino is sinc componens, wo bis of daa can be exraced. To do his, he RD3 radio receiver has a bi delay feedback loop. This performs a bi correlaion since i relaes he curren signal sae o he previous sae. 3 4 Digial Communicaions Principles 95
18 Noes Line Coding RD3 Radio Receiver 6 Injecing a conrolled amoun of ISI has given rise o a mulilevel waveform: Inpu Peak Oupu Zero or space Single mark. or more consecuive marks. Using dela pulses as inpu marks while leaving spaces as zero, causes synchronizaion problems in he receiver when long srings of zeros occur. This can be avoided by using a very simple bipolar encoding scheme ipolar Encoding In order o eliminae synchronizaion problems associaed wih long srings of zero, a mark [] is used o creae a posiive impulse [δ] response, and a space [] is used o creae a negaive impulse [δ] response. This generaes a minimum of hree oupu levels. Example: An inpu daa sream of would resemble: Mark Space The corresponding cosine channel oupu response is: 6 Digial, Analog, and Daa Communicaion, William Sinnema, Figure 84b 96 Digial Communicaions Principles
19 Line Coding Noes I is a lile difficul o picure wha he overall oupu shape is, bu if each oupu pulse is decomposed ino is sinc pulse pairs, i becomes clear. Recall ha for every inpu δ pulse, here are sinc pulses a he oupu The acual oupu is: Approximae oupu Noice ha here is a reducion in he number of zero crossings in he signal, and herefore a reducion in he channel bandwidh requiremens. Noe also ha here are hree possible levels a he sampling insan: Inpu Peak Signal or more consecuive marks. Alernaing marks and spaces or more consecuive zeros or spaces . There is a leas inerval a when swiching beween ± levels. If he signal reurns o is original or  level, i mus remain a for an even number or inervals. In order o correcly sample he signal, he sampling insan mus be shifed slighly o correspond o he peak of he Sinc pulse. Digial Communicaions Principles 97
20 Noes Line Coding Convering his back o a binary signal is somewha of a challenge since: The binary value of any paricular pulse depends on he pas sample The zero level can correspond o a mark or a space Errors can propagae. To overcome hese problems, he binary inpu signal can be differenially encoded before being shaped Differenial Encoding Differenially encoding he signal allows he oupu signal magniude o be direcly relaed o he inpu daa. inpu A τ Cosine Filer oupu The exclusive OR gae goes high only when he curren inpu and he pas signal inpu are differen, a form of differenial encoding. A mark will now correspond o a oupu level, and a space o he ± level. ecause of he delay, i is necessary o esablish an iniial condiion when differenial encoding. In he following example, we will assume ha he iniial logic sae a poin was. The example in he above MahCAD file, can be inuiively approximaed: 98 Digial Communicaions Principles
21 Line Coding Noes daa inpu A iniial condiion The approximaed XOR oupu resembles: Sampling Insans Noe ha by leing a mark equal zero and a space equal a posiive or negaive pulse, he original daa inpu can be read direcly a he cosine channel oupu. The exac oupu is: In ISDN applicaions, 4 binary bis can be mapped ino 3 ernary levels, resuling in 43T encoding. This resuls in some surplus saes since 4 binary symbols represen 6 possible condiions, bu 3 ernary symbols represen 7 possible condiions. These addiional saes can be used by he service provider o mainain housekeeping funcions wihou reducing he cusomer s bi sream Raised Cosine Channel This shape is also known as a Hanning window, and is a beer approximaion of an analog filer since he response curve gradually apers off. H ω ( ω ) = cos ω This funcion is ofen wrien slighly differenly so ha he impulse response will consis of uni sinc pulses: 4 inary 3 Ternary Digial Communicaions Principles 99
22 Line Coding Noes 9  Digial Communicaions Principles ω Raised Cosine Filer H ω ( ) H ω ( ) = 4 cos ω 4 The filer impulse funcion is found by: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ω ω ω ω ω ω ω ω ω ω ω ω ω sin sin sin sin sin sin cos cos cos cos cos cos 4cos = = = = = d d d h u = f c and by normalizing frequency o f c =, we obain: ( ) ( ) sin sin sin = h
23 Line Coding Noes The overall response is composed of hree overlapping sinc pulses: RACOS Impulse Response h( ) Ampliude h( ) h( ) h3( ) Toal Response s Sinc Pulse nd Sinc (doubled) 3rd Sinc Pulse Using his funcion would appear o be a sep backwards since he firs zero crossing occurs a T =.. This however is no he case. Noice ha he ail porion of his response is quie small. The raised cosine impulse response is composed of hree sinc pulses separaed by T =.5, he middle one of which is wice as large as he ouer ones. Therefor his funcion is noed as f[δ] =,,. In he cosine filer, he firs sinc envelope of he second pulse was allowed o compleely overlap he second sinc envelope of he firs pulse. Thus a correlaion over bi period occurred. Time Digial Communicaions Principles 9 
24 Noes Line Coding Firs pulse . Posiion of nex pulse for zero ISI Posiion of nex pulse wih conrolled ISI f c 3 f c f c f c 3 4 A correlaion over bi periods can be obained by carefully conrolling he ISI and allowing wo sinc envelopes on consecuive pulses o overlap. This resuls in a 4 level ransmission scheme: Inpu Peak Oupu Zero or space Single mark consecuive marks 3 3 or more consecuive marks 4 The daa inpu produces he following oupu responses: 9  Digial Communicaions Principles
25 Line Coding Noes Raised Cosine Channel Response One disadvanage of his sysem is ha he frequency componens go all he way down o DC. Some ransmission mehods canno handle low frequency or DC. One mehod which reains 4 level or quaernary signals found in a raised cosine channel bu has no DC componen, is he modified duo binary echnique and is characerized by sine filers. I should be noed ha he Raised Cosine channel is acually consiss of a whole family of curves. Each of hese differs by he rolloff facor Sine or Modified Duobinary Channel Sine Filer H ( ω ) = sin ω ω H ω ( ω ) = sin ω In order o obain uniy sinc pulses, his funcion is normally wrien as: H ω ( ω ) = sin ω This is also known as a modified duobinary sysem. I is used on single sideband radio sysems where he DC levels associaed wih cosine responses canno be oleraed. Digial Communicaions Principles 93
26 Noes Line Coding SS Radio Transmier 7 Noice ha wo bi precoding is used in he ransmier. This allows he receiver o make each binary decision based on he presen received value. I also eliminaes he possibiliy of error propagaion. SS Radio Receiver 8 Noice ha wobi correlaion is performed in he receiver. The impulse response of he sine filer is found by aking he imaginary [or odd] par of he inverse Fourier ransform. 7 Digial, Analog, and Daa Communicaion, William Sinnema, Figure 83a 8 Digial, Analog, and Daa Communicaion, William Sinnema, Figure 83b 94 Digial Communicaions Principles
27 Line Coding Noes h = ( ) sin sin( ω) sin = sin = Normalizing his funcion we obain: h ( ) sin = ω ([ ] ω ) ( ) ( ) ( ) ([ ] ) ( ) sin sin dω ([ ] ω ) ( ) ( ) ( ) ([ ] ) sin ( ) This response is composed of wo sinc pulses separaed by =...4 Sine Impulse Response Ampliude h( ) h( ) h( ) Toal Response s Sinc Pulse nd Sinc This funcion is usually is presened wih he posiive cycle appearing firs [which is acually he inverse of he above expression]. In he previous ypes of channels, he sinc componens were only separaed by =.5. This funcion is wrien as f[δ] =,, . The above waveform represens a mark. A space is represened by he inverse response. Each inpu daa pulse creaes sinc pulse cenered hus: Time Mark Space T = The modified duobinary sysem can be implemened by cascading a delay elemen equal o T, and inverer, wih a sandard duobinary filer as follows: Digial Communicaions Principles 95
28 Noes Line Coding τ Σ  Duo inary Filer A Modified Duo inary Sysem This sysem is used on digial single sideband radio and can carry 4 bis per Hz of bandwidh. Example A daa sream of, in erms of δ pulses resembles: Mark.5 Space The sinc envelope impulse equivalen resembles: Approximae Oupu Or more accuraely, he ransmied oupu is: 9 The daa inpu produces he following oupu responses: Noe ha his resuls in a quaernary or 4 level sysem. 96 Digial Communicaions Principles
29 Line Coding Noes In ISDN applicaions, binary bis can be mapped ino quaernary level, resuling in Q encoding. There are no surplus saes since binary symbols represen 4 possible condiions, as does quaernary symbol. The process of correlaion or sinc envelope overlap can heoreically be coninued indefiniely. However, in pracice his approach is limied o 5 ampliude levels. This is because he S/N raio requiremens become more sringen as he number of levels increases. 9.5 Gaussian Pulse Shaping Gaussian channels are no he same as Nyquis channels, since here is no iming crieria ha can guaranee zero inersymbol inerference. However, his kind of shape does offer he advanage of bandwidh efficiency and clock recovery. inary Quaernary Digial Communicaions Principles 97
30 Noes Line Coding Review Quesions Quick Quiz. Wha is he Nyques bi rae in a KHz ideal low pass filer?. The [sine, cosine] channel has a bipolar impulse response. 3. The [sine, cosine] channel has a correlaion over bis. 4. A modified duobinary SS microwave ransmier has a correlaion span of [,, 3] bis. Analyical Problems. Given he frequency response H(ω) = for ω, derive he impulse response of an ideal brick wall filer: Ideal LPF (rick Wall Filer) ω. Given a modified duobinary channel or sine filer: a) Skech he frequency response b) Why is his funcion someimes wrien as: f[δ] =,, ? c) Skech he approximae ime domain response d) How does i differ from a duobinary sysem? e) How many bis per herz can his sysem ransmi? f) Where is his sysem used? Composiion Quesions. Describe he mehods employed o remove long srings of zeros in binary ransmissions. 98 Digial Communicaions Principles
31 Line Coding Noes. Wha is he purpose of pulse shaping in daa ransmission sysems? 3. Where is he duobinary ransmission scheme used? 4. Discuss he operaing principles and applicaions in correlaion based communicaions links. Digial Communicaions Principles 99
32 Noes Line Coding For Furher Research Hermann J. Helger, Inegraed Services Digial Neworks, AddisonWesley, New York, (99) hp:// hp:// hp://pilo.msu.edu/user/hsuhsuni/adsl.hm hp://ironbark.bendigo.larobe.edu.au/courses/bcomp/c/lecures/ hp://sap.colorado.edu/~hinonm/prs/moreinfo.hm hp:// hp://pioneer.hannam.ac.kr/doc/saffs/noe/comm.hml hp:// hp:// hp:// Gaussian Processes hp:// 93 Digial Communicaions Principles
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