Introdution to Analog And Digital Communiations Seond Edition Simon Haykin, Mihael Moher
Chapter 6 Baseband Delta Transmission 6. Baseband Transmission o Digital Data 6.2 The Intersymbol Intererene Problem 6.3 The Nyquist Channel 6.4 Raised-Cosine Pulse Spetrum 6.5 Baseband Transmission o M-ary Data 6.6 The Eye Pattern 6.7 Computer Experiment : Eye Diagrams or Binary and Quaternary Systems 6.8 Theme Examples : Equalization 6.9 Summary and Disussion
The transmission o digital data over a physial ommuniation hannel is limited by two unavoidable ators. Intersymbol intererene 2. Channel noise Lesson : Understanding o the intersymbol intererene problem and how to ure it is o undamental importane to the design o digital ommuniation systems Lesson 2 : The raised osine spetrum provides a powerul mathematial tool or baseband pulse-shaping designed to mitigate the intersymbol intererene problem Lesson 3 : The eye pattern is a visual indiator o perormane, displaying the physial limitations o a digital data transmission system in an insightul manner 3
6. Baseband Transmission o Digital Data In this hapter, we emphasize the use o disrete pulse-amplitude modulation. Disrete pulse-amplitude modulation is simple to analyze 2. It is the most eiient orm o disrete pulse modulation in terms o both power and bandwidth use 3. The analyti tehniques developed or handling disrete pulseamplitude modulation may be extended to other disrete-pulse modulation tehniques using phase or requeny In disrete PAM (Pulse-Amplitude Modulation The amplitude o transmitted pulses is varied in a disrete manner in aordane with an input stream o digital data Fig. 6. 4
Fig.6. Bak Next 5
The level-enoded signal and the disrete PAM signal are a k + The hannel output is i the input b i the input b k k k is symbol (6. is symbol s ( t a g( t (6.2 k kt b x( t s( t h( t (6.3 y( t x( t q( t (6.4 6
6.2 The Intersymbol Intererene Problem We may express the reeive-ilter output as the modiied PAM signal y ( t a p( t (6.5 k k kt b p( t g( t h( t q( t (6.6 P ( G( H ( Q( (6.7 k y ( it a p[( i kt ], i, ±, ± 2,... b k b y i p i y p ( itb ( itb y i k a k pi k, i, ±, ± 2,... (6.8 7
p p( E (6.9 y i Ea i + a,,, ± ± k pi k i k k i 2,... (6. Residual phenomenon, intersymbol intererene (ISI y i Ea i, or all i Without ISI Pulse-shaping problem Given the hannel transer untion, determine the transmit-pulse spetrum and reeive-ilter transer untion so as to satisy two basi requirements. Intersymbol intererene is redued to zero 2. Transminssion bandwidth is onserved 8
6.3 The Nyquist Channel Nyquist Channel The optimum solution or zero intersymbol intererene at the minimum transmission bandwidth possible in a noise-ree environment the ondition or zero ISI, it is neessary or the overall pulse shape p(t, the inverse Fourier transorm o the pulse spetrum P(, to satisy the ondition p i p( it b E,, or i or all i (6. ( i p t p sin (2B (6.2 t i i 2B B (6.3 2T b P ( t E sin (2B t opt E sin(2πb t 2πB t (6.4 9
The overall pulse spetrum is deined by the optimum brik-wall untion p opt ( E, 2B, or B < otherwise < B (6.5. The brik-wall spetrum deines B as the minimum transmission bandwidth or zero intersymbol intererene 2. The optimum pulse shape is the impulse response o an ideal low-pass hannel with an amplitude response in the passband and a bandwidth B Fig. 6.2
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Two diiulties that make its use or a PAM system impratial. The system requires that the spetrum P( be lat rom B to B, and zero else-where 2. The time untion p(t dereases as / t or large t, resulting in a slow rate o deay To pursue the timing error problem under point 2, onsider Eq. (6.5 and sample the y(t at t t y( t E a p( t kt, or it y t a sin [2B ( t ] y( t E Ea k k a k k sin (2B sin (2B b t + t k E k k a k b sin(2πb t πk 2πB t πk ( k kt b k sin( 2πB t πk sin(2πb tos( πk os(2πb tsin( πk y( t sin(2πb t sin (2 Ea B t + E π k k ( a 2B t k k k (6.6 2
3 6.4 Raised-Cosine Pulse Spetrum To ensure physial realizability o the overall pulse spetrum P(, the modiied P( dereases toward zero gradually rather than abruptly. Flat portion, whih oupies the requeny band or some parameter to be deined 2. Roll-o portion, whih oupies the requeny band 2B - One ull yle o the osine untion deined in the requeny domain, whih is raised up by an amount equal to its amplitude The raised-osine pulse spetrum (6.7 2, 2, 2( ( os 4, 2 ( < + B B B B E B E p π
The roll-o ator α (6.8 B os(2παb t p( t E sin (2B t 2 2 2 6α B t (6.9 The amount o intersymbol intererene resulting rom a timing error t dereases as the roll-o ator is inreased orm zero to unity. For speial ase o α sin (4B t p( t E 2 2 6B t (6.2 Fig. 6.3 4
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Transmission-Bandwidth Requirement The transmission bandwidth required by using the raised-osine pulse spetrum is 2B B T B T B ( + α (6.2 Exess hannel The transmission bandwidth requirement o the raised-osine spetrum exeeds that o the optimum Nyquist hannel v αb (6.22. When the roll-o ator is zero, the exess bandwidth is redued to zero 2. When the roll-o ator is unity, the exess bandwidth is inreased to B. 6
7
8 Two additional Properties o the Raised-Cosine Pulse Spetrum Property The roll-o protion o the spetrum P( exhibits odd symmetry about the midpoints ±B A unique haraterization o the roll-o portion o the raised-osine spetrum (6.23 ( ( ( opt P P P v (6.24 2B or 2B, 2 or B, 2( ( os 4 or, 2( ( os 4 or, ( + B B B E B B B E P v π π (6.25 ( ( ' ' P P v v (6.26 ' B
Fig. 3.24(a and 6.4(a, they are basially o an idential mathematial orm, exept or two minor dierenes :. The baseband raised-osine pulse spetrum P( o Fig. 6.4(a is entered on the origin at, whereas the vestigial sideband spetrum o Fig. 3.24(a is entered on the sinusoidal arrier requeny 2. The parameter v in Fig. 6.4(a reers to the exess bandwidth measured with respet to the ideal brik-wall solution or zero intersymbol intererene, whereas the parameter v in Fig. 3.24(a reers to the exess bandwidth measured with respet to the optimum bandwidth attainable with single sideband modulation.. Fig. 3.24 Fig. 6.4 9
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Fig.6.4 Bak Next 2
Property 2 The inite summation o replias o the raised-osine pulse spetrum, spaed by 2B hertz, equals a onstant m P( 2mB 2 E B (6.27 n n P 2B δ t n 2B 2B m P( 2mB (6.28 Sampling the modiied pulse response p(t at the rate /2B, P sin( nπ.sin ( n nπ, orn, orn ±, ± 2,... n os( πnα E sin ( n 2 2 2B 4n α 2.os( πnα or n, 22
Finally, nothing that the Fourier transorm o the delta untion is unity, Eq. (6.29 is merely another way o desribing the desired orm shown in Eq. (6.27 n E, or n P 2B, or n ±, ± 2,... ( t 2B P( 2mB m E δ E ( δ t P( 2mB (6.29 Given the modiied 2B m pulse shape p(t or transmitting data over an imperet hannel using disrete pulse-amplitude modulation at the data rate /T, the pulse shape p(t eliminates intersymbol intererene i, and only i, its spetrum P( satisies the ondition m P m T onstant, 2T (6.3 23
Root Raised-Cosine Pulse Spetrum A more sophistiated orm o pulse shaping or baseband digital data transmission is to use the root raised-osine pulse spetrum G ( H ( P / 2 ( (6.3 Q ( P / 2 ( (6.32 G ( H ( Q( P( The pulse shaping is partitioned equally between two entities The ombination o transmit-ilter and hannel onstitutes one entity. With H( known and P( deined by Eq. (6.7 or a presribed roll-o ator, we may use Eq. (6.3 to determine the requeny response o the transmit ilter. The reeive ilter onstitutes the other entity. Hene, or the same roll-o ator we may use Eqs. (6.7 and (6.32 to determine the requeny response o the reeive-ilter. 24
6.5 Baseband Transmission o M-ary Data The output o the line enoder takes on one o M possible amplitude levels with M>2. Signaling rate (symbol rate /T Symbols per seond, bauds The symbol duration T o the M-ary PAM system is related to the bit duration T b o a binary PAM system T T b log 2 M (6.33 25
6.6 The Eye Pattern Eye Pattern Be produed by the synhronized superposition o suessive symbol intervals o the distorted waveorm appearing at the output o the reeive-ilter prior to thresholding From an experimental perspetive, the eye pattern oers two ompelling virtues The simpliity o generation The provision o a great deal o insightul inormation about the harateristis o the data transmission system, hene its wide use as a visual indiator o how well or poorly a data transmission system perorms the task o transporting a data sequene aross a physial hannel. 26
Timing Features Three timing eatures pertaining to binary data transmission system, Optimum sampling time : The width o the eye opening deines the time interval over the distorted binary waveorm appearing at the output o the reeive-ilter Zero-rossing jitter : in the reeive-ilter output, there will always be irregularities in the zero-rossings, whih, give rise to jitter and thereore non-optimum sampling times Timing sensitivity : This sensitivity is determined by the rate at whih the eye pattern is losed as the sampling time is varied. Fig. 6.5 27
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The Peak Distortion or Intersymbol Intererene In the absene o hannel noise, the eye opening assumes two extreme values An eye opening o unity, whih orresponds to zero intersymbol intererene An eye opening o zero, whih orresponds to a ompletely losed eye pattern; this seond extreme ase ours when the eet o intersymbol intererene is severe enough or some upper traes in the eye pattern to ross with its lower traes. Fig. 6.6 29
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Noise margin In a noisy environment, The extent o eye opening at the optimum sampling time provides a measure o the operating margin over additive hannel noise Eye opening ( peak Eye opening D (6.34 Fig. 6.7 Plays an important role in assessing system perormane Speiies the smallest possible noise margin Zero peak distortion, whih ours when the eye opening is unity Unity peak distortion, whih ours when the eye pattern is ompletely losed. The idealized signal omponent o the reeive-ilter output is deined by the irst term in Eq. (6. The intersymbol intererene is deined by the seond term y i Ea i + k k i a p, i, ±, ± 2,... k i k (6. 3
Fig.6.7 Bak Next 32
(Maximum ISI D peak k k k i p ik p( i k T p i k m k i b (6.35 Eye pattern or M-ary Transmission M-ary data transmission system uses M enoded symbols The eye pattern or an M-ary data transmission system ontains (M- eye openings staked vertially one on top o the other. It is oten possible to ind asymmetries in the eye pattern o an M-ary data-transmission system, whih are aused by nonlinearities in the ommuniation hannel or other parts o the system. 33
6.7 Computer Experiment : Eye Diagrams or Binary and Quanternary Systems Fig. 6.8(a and 6.8(b show the eye diagrams or a baseband PAM transmission system using M2 and M4. Fig. 6.9(a and 6.9(b show the eye diagrams or these two basebandpulse transmission systems using the same system parameters as beore, but this time under a bandwidth-limited ondition. + ( / H ( 2 N Fig. 6.8 Fig. 6.9. N 3, and.6hz or binary PAM 2. N 3, and.3hz or 4 PAM B T.5( +.5. 75Hz 34
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Fig.6.9 Bak Next 36
6.8 Theme Example : Equalization An eiient approah to high-speed transmission o digital data Disrete pulse-amplitude modulation (PAM, Linear modulation sheme Transversal ilter Delay line, whose taps are uniormly spaed T seond apart; T is the symbol duration Adjustable weights, whih are onneted to the taps o the delay line Summer, whih adds suessively delayed versions o the input signal, ater they have been individually weighted. Adjustable transversal equalizer (transversal equalizer With hannel equalization as the untion o interest and the transversal ilter with adjustable oeiients as the struture to perorm. Fig. 6. 37
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Zero-Foring Equalization To proeed with the solution to the equalization problem, onsider then the omposite system depited in Fig. 6. The irst subsystem haraterized by the impulse response (t represents the ombined ation o the transmit-ilter and ommuniation hannel The seond subsystem haraterized by the impulse response h eq (t aounts or pulse shaping ombined with residual-distortion equalization in the reeiver. h eq N ( t wkδ ( t k N kt (6.36 p( t ( t h eq ( t ( t N k N w δ ( t k kt (6.37 p( t N k N w ( t δ ( t k kt N k N w ( t k kt (6.38 Fig. 6. 39
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disrete onvolution sum, N p( it wk(( i k T k N (6.39 p i N k N w k ik (6.4 p i E,or i,or all i p i E,, i i ±, ± 2,..., ± N (6.4 N k N w k ik E,, i i ±, ± 2,..., ± N (6.42 4
42 Equivalently, in matrix orm we may write Sine the zero-oring equalizer ignores the eet o additive hannel noise, the equalized system does not always oer the best solution to the intersymbol intererene problem (6.43 2 2 2 2 + + + + E w w w w w N N N N N N N N N N N N N N N N
How Could the Reeiver Determine the { k }? A pilot-assisted training session For the binary data sequene applied to the transmitter input, use a deterministi sequene o s and s that is noise-like in harater, hene the reerene to this sequene as a pseudo-noise (PN sequene. The PN sequene is known a priori to the reeiver. Aordingly, with the reeiver synhronized to the transmitter, the reeiver is enabled to know when to initiate the training session Finally, knowing the transmitted PN sequene and measuring the orresponding hannel output, it is straight-orward matter or the reeiver to estimate the sequene { k } representing the sampled impulse response o the transmit-ilter and hannel ombined. 43
6.9 Summary and Disussion Baseband data transmission, or whih the hannel is o a low-pass type Band-pass data transmission, or whih the hannel is o a band-pass type The intersymbol intererene problem, whih arises due to imperetions in the requeny response o the hannel ISI reers to the eet on that pulse due to ross-talk or spillover rom all other signal pulses in the data stream applied to the hannel input A orretive measure widely used in pratie is to shape the overall pulse spetrum o the baseband system, starting rom the soure o the message signal all the way to the reeiver. The eye pattern portrays the degrading eets o timing jitter, ISI, hannel noise ISI is a signal-dependent phenomenon, it thereore disappears when the inormation-bearing signal is swithed o. Noise is always there, regardless o whether there is data transmission or not. Another orretive measure or dealing with the ISI; hannel equalization 44
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