Principles of Digital Data Transmission Chapter 7

Transcription

1 Priciples of Digital Data rasmissio Chapter 7 Lectured y Dr. Yu Q. Shi Dept. of Electrical & Computer Egr. New Jersey Istitute of echology shi@jit.edu ext used for the course: <Moder Digital ad Aalog Commuicatio Systems>, 4 th Editio, Lathi ad Dig, Oxford Itroductio A sigificat portio of commuicatio i 990s was i aalog. It has ee replaced rapidly y digital commuicatio. Now most of the commuicatios ecome digital, with aalog commuicatio playig a mior role. his chapter addresses the framework (several major aspects) of digital data trasmissio All we leart i this course so far is utilized here. Hece, this chapter is aturally a coclusio of this course. Dr. Shi Lathi & Dig-Digital Commu

2 Digital Commuicatio Systems: Source Digital Commuicatio Systems: Figure 7. Source ecoder, Basead modulatio (Lie codig), Digital carrier modulatio, Multiplexer, Chael, Regeerative repeater, Receiver (detectio). Source: he iput A sequece of digits We discuss maily: he iary case (two symols) Later i this chapter: he M-ary case (M symols) more geeral case Dr. Shi Lathi & Dig-Digital Commu 3 Figure 7. Fudametal uildig locks of digital commuicatio systems Figure 7. Fudametal uildig locks of digital commuicatio systems Dr. Shi Lathi & Dig-Digital Commu 4

3 Lie Coder (rasmissio Coder) he output of a multiplexer is coded ito electrical pulses or waveforms for the purpose of trasmissio over the chael. his process is called a lie codig or trasmissio codig May possile ways of assigig waveforms (pulses) to the digital data. O-off: a pulse p ( t ) 0 o pulse Dr. Shi Lathi & Dig-Digital Commu 5 Polar: 0 Lie Coder a a pulse pulse Bipolar (pseudoterary or alterate mark iversio (AMI)) is ecoded y -p (t) or p (t) 0 o pulse I short, pulses represetig cosecutive s alterate i sig. All the aove three could use half-width pulses. It is possile to select other width. Figure 7. (parts a, ad c). a pulse p (t) or - p (t) depedig o whether the previous Dr. Shi Lathi & Dig-Digital Commu 6 p ( t ) p ( t ) 3

4 Lie Codes Dr. Shi Lathi & Dig-Digital Commu 7 Lie Coder Full width pulses are ofte used i some applicatios. i.e., the pulse amplitude is held to a costat value throughout the pulse iterval (it does ot have a chace to go to zero efore the ext pulse egis). hese schemes are called o retur-to-zero (NRZ) i cotrast to retur-to-zero (RZ) Figure 7. shows A O-off NRZ sigal A Polar NRZ sigal i d ad e parts of the figure Dr. Shi Lathi & Dig-Digital Commu 8 4

5 Multiplexer Usually, the capacity of a practical chael >> the data rate of idividual sources. o utilize this capacity effectively, we comie several sources through a digital multiplexer usig the process of iterleavig. Oe way: A chael is time-shared y several messages simultaeously. Dr. Shi Lathi & Dig-Digital Commu 9 Regeerative Repeater Used at regularly spaced itervals alog a digital trasmissio lie to: ) detect the icomig digital sigal ad ) regeerate ew clea pulses for further trasmissio alog the lie. his process periodically elimiates, ad therey comats the accumulatio of oise ad sigal distortio alog the trasmissio path. he periodic timig iformatio (the clock sigal at R Hz) is required to sample the icomig sigal at a repeater. R (rate): pulses/sec Dr. Shi Lathi & Dig-Digital Commu 0 5

6 Regeerative Repeater he clock sigal ca e extracted from the received sigal. e.g., the polar sigal whe rectified a clock sigal at R Hz he o-off sigal A periodic sigal at R + a polar sigal (Figure 7.) Whe the periodic sigal is applied to a resoat circuit tued to R Hz, the output, a siusoid of R Hz, ca e used for timig. Dr. Shi Lathi & Dig-Digital Commu O-off Sigal Decompositio Dr. Shi Lathi & Dig-Digital Commu 6

7 Regeerative Repeater rectified he ipolar sigal a o-off sigal the clock sigal ca also e extracted. he timig sigal (the output of the resoat circuit) is sesitive to the icomig patter, sometimes. e.g. i o-off, or ipolar 0 o pulse If there are too may zeros i a sequece, will have prolem. o sigal at the iput of the resoat circuit for a while siusoids output of the resoat circuit starts decayig Polar scheme has o such a prolem. Dr. Shi Lathi & Dig-Digital Commu 3 rasparet Lie Code A lie code i which the it patter does ot affect the accuracy of the timig iformatio is said to e a trasparet lie code. he polar scheme is trasparet. he o-off ad ipolar are ot trasparet (otrasparet). Dr. Shi Lathi & Dig-Digital Commu 4 7

8 Desired Properties of Lie Codes. rasmissio adwidth: as small as possile. Power efficiecy: for a specific adwidth ad detectio error rate, trasmitted power should e as small as possile. 3. Error detectio ad correctio capaility: as strog as possile 4. Favorale power spectral desity: desirale to have zero PSD at (dc), called DC ull, ecause of ac couplig requires this. If at dc, 0, PSD 0, the dc waders i the pulse stream, which is ot desired. 5. Adequate timig cotet: should e possile to extract the clock sigal (timig iformatio). 6. rasparecy Dr. Shi Lathi & Dig-Digital Commu 5 PSD of Various Lie Codes Cosider a geeral PAM sigal, as show i Figure 7.4 (): R R t k the k th pulse i the pulse trai y( t) ak p( t) p (t): the asic pulse P () [P(f)]: Fourier spectrum of p (t) a k : aritrary ad radom he o-off, polar, ipolar lie codes are all special cases of the geeral pulse trai y(t), a k 0, +, - Dr. Shi Lathi & Dig-Digital Commu 6 8

9 A Radom PAM sigal All 3 lie codes ca e represeted this way New approach to determie PSD of y(t) Dr. Shi Lathi & Dig-Digital Commu 7 PSD of Various Lie Codes How to determie the PSD? Figure 7.4 A attractive geeral approach R x ( τ ), S x( ) eed to e studied oce he for differet p(t), differet PSD x(t): a impulse trai xˆ ( t) : a rectagular pulse trai ε h k whe ε 0, xˆ ( t) x( t) (Figure 7.5) ( a k Dr. Shi Lathi & Dig-Digital Commu 8 ) 9

10 Dr. Shi Lathi & Dig-Digital Commu 9 R R PSD of Various Lie Codes 0 ak k+ a k. a,ll, S ( ) y P( ) time average of R ak. ak+ R X ( τ ) Rδ ( τ ) S X ( ) a k j SX ( ) Re R + Differet lie codes differet P() differet () S y Dr. Shi Lathi & Dig-Digital Commu 0 [ R cos( ] 0 ) 0

11 p(t) 0 -p(t)} Polar Sigalig a ± equally likely a k,, k R R 0 a k a k. a k + R 0, 0 Qak. ak+ or equally likely Similar reasoig S ( ) y P( ) P( ). S. ( ) Dr. Shi Lathi & Dig-Digital Commu X Polar Sigalig o e specific, assume p(t) is a rectagular pulse of width (a half-width rectagular pulse) t p ( t ) rect P ( ) si c 4 S y ( ) 4 4 si c st zero cros si g: 4 4π π or f R Dr. Shi Lathi & Dig-Digital Commu

12 PSD of polar sigal (half-width rectagle) Dr. Shi Lathi & Dig-Digital Commu 3 Polar Sigalig Commet : he essetial adwidth of the sigal (mai loe) R For a full-width pulse, R Polar Sigalig is ot adwidth efficiet. Commet : No error-detectio or error-correctio capaility. Commet 3: No-zero PSD at dc( 0) his will rule out the use of ac couplig i trasmissio. Commet 4: Most efficiet scheme from the power requiremet viewpoit for a give power, the detectio-error proaility for a polar scheme is the smallest possile. Commet 5: rasparet Dr. Shi Lathi & Dig-Digital Commu 4

13 Achievig DC Null i PSD y Pulse Shapig Q P ( ) P (0) p ( t ) e p ( t ) dt jt dt If the area uder p(t) 0, P(0) 0. Dr. Shi Lathi & Dig-Digital Commu 5 Machester (split-phase, twied-iary) Sigal Dr. Shi Lathi & Dig-Digital Commu 6 3

14 4 Dr. Shi Lathi & Dig-Digital Commu 7 O-off Sigalig p(t) 0 o pulse + + j j X e e S all for R R ) ( Dr. Shi Lathi & Dig-Digital Commu 8 O-off Sigalig + X S π δ π 4 4 ) ( + y P S π δ π 4 ) ( ) ( For a half-width rectagular pulse, ( ) 6 4 y S sic π π δ + Note:. For some derivatio, refer to the ext four pages.. PSD is show i Figure PSD cosists of oth a discrete part ad a cotiuous part.

15 Some Derivatio (Lathi s ook, pp , Example.) A uit impulse trai: δ t) Com ( t) g( ) g ( t ) FS D ( 0 t 0 j t j t G ) ( f ) F { g ( t )} δ ( f f 0 0 π ( ) F { g ( t )} δ ( 0 e π G ) COMB Dr. Shi Lathi & Dig-Digital Commu ( 0 0 e ) cos( t ) Example. Dr. Shi Lathi & Dig-Digital Commu 30 5

16 Some Derivatio Fact (Lathi s ook, p.83, Eq. (3.0a, 3.0): δ e j πf t ( f ) δ ( f f ) jt [ e δ ( )] Fact : Also, δ δ ( t ) j ( t ) e Dr. Shi Lathi & Dig-Digital Commu 3 Some Derivatio Fact 3: 4 F ( t ) δ j t e FS j t π e F δ ( ) Dr. Shi Lathi & Dig-Digital Commu 3 6

17 PSD of A O-off Sigal Dr. Shi Lathi & Dig-Digital Commu 33 O-off Sigalig his is reasoale sice, as show i Fig. 7., a o-off sigal ca e expressed as the sum of a polar ad a periodic compoet Commet : For a give trasmitted power, it is less immue to oise iterferece. Q Noise immuity differece of amplitudes represetig iary 0 ad. If a pulse of amplitude or has eergy E, the a pulse of amplitude has eergy () E 4E. Dr. Shi Lathi & Dig-Digital Commu 34 7

18 O-off Sigalig For polar: Q digits are trasmitted per secod For o-off: Commet : polar sigal power E power 4 E Not trasparet E E Power is ow as large as twice of aove. Dr. Shi Lathi & Dig-Digital Commu 35 Bipolar Sigalig (Pseudoterary or Alterate Mark Iverted (AMI)) A. 0 o pulse p(t) or p(t) depedig o whether the previous was trasmitted y p(t) or p(t) B. [p(t), 0, -p(t)]: I reality, it is terary sigalig. C. Merit: a dc ull i PSD demerit: ot trasparet Dr. Shi Lathi & Dig-Digital Commu 36 8

19 Bipolar Sigalig D. R(0) R() R() R() R lim N N lim a k lim ( ± ) + (0) N N N k N N 3N lim ( ) + (0) N N a, 0, 0, 0, 00, 00, N lim N N k a k N k a N () + 8 k a k + N 5N ( ) , > (0) 0 Dr. Shi Lathi & Dig-Digital Commu 37 Bipolar Sigalig E. S y ( ) P( ) S X ( ) P ( ) j R e P ( ) R 0 R cos + P ( ) + ( ) cos 4 P ( ) P ( ) ( cos ) si Dr. Shi Lathi & Dig-Digital Commu 38 (7.0) (7.0c) 9

20 Bipolar Samplig S y ( ) 0 As 0 (dc) regardless of P() A dc ull (desirale for ac couplig) π si ( ) 0 at, f R adw idth R H z, regardless of P ( ) For a half-width rectagular pulse,, ( ) Dr. Shi Lathi & Dig-Digital Commu 39 P ( ) t P t r e c t s i c 4 S y ( ) sic si Zero: 4 4 ɷ 4π ɷ π 4π π f R f R R essetial adwidth Half that of polar or o-off sigalig. wice that of theoretical miimum adwidth (chael /w). ale 3.: t τ rect τ si c τ si c τ τ Dr. Shi Lathi & Dig-Digital Commu 40 0

21 Dr. Shi Lathi & Dig-Digital Commu 4 Merits of Bipolar Sigalig.Spectrum has a dc dull..badwidth is ot excessive. 3.It has sigle-error-detectio capaility. Sice, a sigle detectio error a violatio of the alteratio pulse rule. Dr. Shi Lathi & Dig-Digital Commu 4

22 Demerits of Bipolar Sigalig. Requires twice as much power as a polar sigalig. Distictio etwee A, -A, 0 vs. Distictio etwee A/, -A/. Not trasparet Dr. Shi Lathi & Dig-Digital Commu 43 S ( ) y P( ) Pulse Shapig S X ( ) P( ) ( R0 + R cos Differet sigalig (lie codig, S X () ) diff. S y () Differet pulse shapig (P()) diff. S y ( ) A additioal factor. Itersymol Iterferece (ISI) (refer to the ext slide) ime-limited pulses (i.e., trucatio i time domai) ot ad-limited i frequecy domai Not time-limited, causes prolem ad-limited sigal (i.e., trucatio i frequecy domai) Dr. Shi Lathi & Dig-Digital Commu 44 ) [Slide 9]

23 Itersymol Iterferece (ISI) i Detectio B. Sklar s Digital Commuicatios 988 Back to slide 44 Dr. Shi Lathi & Dig-Digital Commu 45 Phase Shapig Itersymol Iterferece (ISI) Whether we egi with time-limited pulses or ad-limited pulses, it appears that ISI caot e avoided. A iheret prolem i the fiite adwidth trasmissio. Fortuately, there is a way to get away: e ale to detect pulse amplitudes correctly. Dr. Shi Lathi & Dig-Digital Commu 46 3

24 Nyquist Criterio for Zero ISI he first method proposed y Nyquist. p( t) 0,, Example 6. (p.56) t 0 t ± R (7.) Miimum adwidth pulse that satisfies Nyquist Criterio Fig. 7.0 ( ad c) Dr. Shi Lathi & Dig-Digital Commu 47 : separatio etwee successive trasmitted pulses p( t) sic( π Bt), B R Sice πb π, is the st zero crossig. Miimum Badwidth Pulse (Nyquist Criterio) Dr. Shi Lathi & Dig-Digital Commu 48 4

25 Nyquist Criterio for Zero ISI t 0 p( t) sic( π Rt) 0 t ± P ( ) R rect π R R R adwidth ale 3.: F W Sic ( Wt ) π rect W Dr. Shi Lathi & Dig-Digital Commu 49 Prolems of his Method Impractical. < t <. It decays too slowly at a rate /t. Ay small timig prolem (deviatio) ISI Solutio: Fid a pulse satisfyig Nyquist criterio (7.), ut decays faster tha /t. Nyquist: Such a pulse requires a adwidth R k, k Dr. Shi Lathi & Dig-Digital Commu 50 5

26 6 Dr. Shi Lathi & Dig-Digital Commu 5 Let he adwidth of P() is i (R /, R ) p(t) satisfies Nyquist criterio Equatio (7.) Samplig p(t) every secods y multiplyig p(t) y a impulse trai ) ( ) ( P t p (t). δ s s P G Equatio P F t t t p t p ) ( ) ( ) 4. 6 ( ) ( ) ( ) ( ) ( ) ( δ δ Dr. Shi Lathi & Dig-Digital Commu 5

27 Derivatio P() is odd symmetric i y -o-y system. Dr. Shi Lathi & Dig-Digital Commu 53 he adwidth of P() is : the excess adwidth x γ roll offfactor he adwidth of P() is Derivatio + P(), thus derived, is called a Vestigial Spectrum. Dr. Shi Lathi & Dig-Digital Commu 54 x excess adwidth theoretica l mi imum adwidth x x 0 r R rr B + R ( + γ ) 7

28 8 Dr. Shi Lathi & Dig-Digital Commu 55 Realizaility A physically realizale system, h(t), must e causal, i.e., h(t) 0, for t < 0. (he.c. & s.c.) I the frequecy domai, the.c. & s.c. is kow as Paley-Wieer criterio Note that, for a physically realizale system, H() may e zero at some discrete frequecies. But, it caot e zero over ay fiite ad So, ideal filters are clearly urealizale. < + d H ) ( l Dr. Shi Lathi & Dig-Digital Commu 56 From this poit of view, the vestigial spectrum P() is urealizale. However, sice the vestigial roll-off characteristic is gradual, it ca e more closely approximated y a practical filter. Oe family of spectra that satisfies the Nyquist criterio is < + > < x x x x P π 0 ) ( si / ) (

29 Realizaility x 0 x 4 x ( r 0 ) ( r 0.5) ( r ) Show i the figure o ext slide x Commet : Icreasig (or r) improves p(t), i.e., more gradual cutoff reduces the oscillatory ature of p(t), p(t) decays more rapidly. Dr. Shi Lathi & Dig-Digital Commu 57 Pulses satisfyig Nyquist Criterio Dr. Shi Lathi & Dig-Digital Commu 58 9

30 Commet : As Realizaility, i.e., P( ) + cos rect R 4πR cos rect 4R 4πR his characteristic is kow as: γ x he raised-cosie characteristic he full-cosie roll-off characteristic Dr. Shi Lathi & Dig-Digital Commu 59 Realizaility cos( π R t ) p ( t ) R Sic ( π R t ) 4 R R t A. Badwidth is (γ ) B. p(0) R C. p(t) 0 at all the sigalig istats & at poits midway etwee all the sigalig istats D. p(t) decays rapidly as /t 3 relatively isesitive to derivatios of R, samplig rate, timig jitter ad so o. Dr. Shi Lathi & Dig-Digital Commu 60 30

31 E. Closely realizale Realizaility F. Ca also e used as a duoiary pulse. G. H c () : chael trasfer factor : trasmitted pulse P ) H ( ) P P i ( k ) i ( c ( ) Received pulse at the detector iput should e P() [p(t)]. Dr. Shi Lathi & Dig-Digital Commu 6 Sigalig with Cotrolled ISI: Partial Respose Sigals he secod method proposed y Nyquist to overcome ISI. Duoiary pulse: 0, p( ) 0 for all other Dr. Shi Lathi & Dig-Digital Commu 6 3

32 Sigalig with Cotrolled ISI Use polar sigalig p( t) 0 p( t) he received sigal is sampled at t p(t) 0 for all except 0,. Clearly, such a pulse causes zero ISI with all the pulses except the succeedig pulses. Dr. Shi Lathi & Dig-Digital Commu 63 Sigalig with Cotrolled ISI Cosider two such successive pulses located at 0 ad. If oth pulses are positive, the sample value at t. If oth pulses are egative, - If pulses are of opposite polarity, 0 Decisio Rule: If the sample value at t is Positive, Negative 0, 0 Zero 0, or, 0 Dr. Shi Lathi & Dig-Digital Commu 64 3

33 Sigalig with Cotrolled ISI ale 7. rasmitted Seq Sample of x() Detected seq Dr. Shi Lathi & Dig-Digital Commu 65 Sigalig with Cotrolled ISI: Duoiary Pulses If the pulse adwidth is restricted to e it ca e show that the p(t) must e: R p ( t ) P ( ) si( π R t ) π R t ( R t ) R cos R rect π R e j R Note: he raised-cosie pulse with x, ( r ), satisfies the coditio (7.3) to e sigalig with cotrolled ISI refer to Figure 7.3 (slides 57). Dr. Shi Lathi & Dig-Digital Commu 66 33

34 Sigalig with Cotrolled ISI Duoiary Pulses Dr. Shi Lathi & Dig-Digital Commu 67 Use of Differetial Codig For the cotrolled ISI method, 0-valued sample 0 to or to 0 trasitio. A error may e propagated. Differetial codig helps. I differetial codig, is trasmitted y a pulse idetical to that used for the previous it. 0 is trasmitted y a pulse egative to that used for the previous it. Useful i systems that have o sese of asolute polarity. Fig 7.7 Dr. Shi Lathi & Dig-Digital Commu 68 34

35 Differetial Code Dr. Shi Lathi & Dig-Digital Commu 69 Scramlig Purpose: A scramler teds to make the data more radom y Removig log strigs of s or 0 s Removig periodic data strigs. Used for prevetig uauthorized access to the data. Example. (Structure) 3 5 Scramler: Icomplete versio S D D D : : delayed y uits Modulo Sum Dr. Shi Lathi & Dig-Digital Commu 70 35

36 Dr. Shi Lathi & Dig-Digital Commu 7 Scramlig Descramler: Icomplete versio R ( D 3 D 5 ) [ ( ( D 3 F ) D 5 )] F D 3 D 5 Dr. Shi Lathi & Dig-Digital Commu 7 36

37 Example 7. he data stream is fed to the scramler i Fig. 7.9a. Fid the scramler output, assumig the iitial cotet of the registers to e zero. From Fig. 7.9a we oserver that iitially S, ad the sequece S eters the register ad is retured as 3 5 ( D D ) S FS through the feedack path. his ew sequece FS agai eters the register ad is retured as F S, ad so o. Hece, Complete versio S ( F FS F F S F 3 F 3 S...) S... (7.4) Dr. Shi Lathi & Dig-Digital Commu 73 Example 7. Recogizig that We have F D 3 D 5 F ( D D )( D D ) D D D D Because modulo- additio of ay sequece 8 8 with itself is zero, D D 0 ad Similarly, F 3 F D 6 D ( D D )( D D ) D D D D Dr. Shi Lathi & Dig-Digital Commu

38 ad so o Hece, Example ( D D D D D D D D D...) S Because D S is simply the sequece S delayed y its, various terms i the precedig equatio correspod to the followig sequeces: Dr. Shi Lathi & Dig-Digital Commu 75 S Example 7. D S D S D 6 D S D S D D D D S S S S S Dr. Shi Lathi & Dig-Digital Commu 76 38

39 Example 7. Note that the iput sequece cotais the periodic sequece 0000, as well as a log strig of 0 s. he scramler output effectively removes the periodic compoet as well as the log strigs of 0 s. he iput sequece has 5 digits. he scramler output up to the 5 th digit oly is show, ecause all the output digits eyod 5 deped o the iput digits eyod 5, which are ot give. We ca verify that the descramler output is ideed S whe this sequece is applied at its iput. Homework: Lear to e ale to determie where o terms eeds to e cosidered. Dr. Shi Lathi & Dig-Digital Commu 77 Regeerative Repeater hree fuctios:. Reshapig icomig pulses usig a equalizer.. Extractig timig iformatio Required to sample icomig pulses at optimum istats. 3. Makig decisio ased o the pulse samples Dr. Shi Lathi & Dig-Digital Commu 78 39

40 Regeerative Repeater Dr. Shi Lathi & Dig-Digital Commu 79 Preamplifier ad Equalizer A pulse trai is atteuated ad distorted y trasmissio medium, say, dispersio caused y a atteuatio of high-frequecy compoets. Restoratio of high frequecy compoets icrease of chael oise Fortuately, digital sigals are more roust. Cosiderale pulse dispersio ca e tolerated Mai cocer: Pulse dispersio ISI icrease error proaility i detectio Dr. Shi Lathi & Dig-Digital Commu 80 40

41 Zero-Forcig Equalizer Detectio decisio is ased solely o sample values. No eed to elimiate ISI for all t. All that is eeded is to elimiate or miimize ISI at their respective samplig istats oly. his could e doe y usig the trasversal-filter equalizer, which forces the equalizer output pulse to have zero values at the samplig (decisiomakig ) istat. (refer to the diagram i ext slide) Dr. Shi Lathi & Dig-Digital Commu 8 Dr. Shi Lathi & Dig-Digital Commu 8 4

42 Let c c Zero-Forcig Equalizer 0 k 0 P ( t) P ( t N 0 P ( t) P ( t) 0 r r tap settig k 0 ) If igore delay. Fig 7. () idicates a prolem : a, a -, a, a -, are ot egligile due to dispersio. Now, wat to force a a - a a - 0 Cosider c k s assume other values (other tap settig). Dr. Shi Lathi & Dig-Digital Commu 83 Zero-Forcig Equalizer Cosider c k s assume other values (other tap settig). p ( t) 0 p ( k 0 ) For simplicity of otatio: Nyquist Criterio: N p ( k) 0 0 N p ( k) 0 c p ( t ) N r N c p [( k ) ], r k 0 for k 0 k 0, ±, ±,... Dr. Shi Lathi & Dig-Digital Commu 84 p ( ) ( ) 0 k c pr k 4

43 Zero-Forcig Equalizer A set of ifiitely may simultaeous equatios: N+: c s Impossile to solve this set of equatios. If, however, we specify the values of p 0 (k) oly at N+ poits: k 0 p0( k) 0 k ±, ±,..., ± N the a uique solutio exists. Dr. Shi Lathi & Dig-Digital Commu 85 Zero-Forcig Equalizer Meaig: Zero ISI at the samplig istats of N precedig ad N succeedig pulses. Sice pulse amplitude decays rapidly, ISI eyod the N th pulse is ot sigificat for N> i geeral. Dr. Shi Lathi & Dig-Digital Commu 86 43

44 Page 7.37 Zero-Forcig Equalizer hik aout whe,, 0, -, -; a example i ext slide Dr. Shi Lathi & Dig-Digital Commu 87 Zero-Forcig Equalizer Page 7.37 Dr. Shi Lathi & Dig-Digital Commu 88 44

45 Eye Diagram A coveiet way to study ISI o a oscilloscope Figure Dr. Shi Lathi & Dig-Digital Commu 89 imig Extractio he received sigal eeds to e sampled at precise istats. imig is ecessary. hree ways for sychroizatio:. Derivatio from a primary or a secodary stadard (a master timig source exists, oth trasmitter ad receiver follow the master).. rasmittig a separate sychroizig sigal (pilot clock) 3. Self sychroizatio (timig iformatio is extracted from the received sigal itself). Dr. Shi Lathi & Dig-Digital Commu 90 45

46 imig Extractio Way : Suitale for large volumes of data high speed comm. Systems. High cost. Way : Part of chael capacity is used to trasmit timig iformatio. Suitale for a large availale capacity. Way 3: Very efficiet. Examples: O-off sigalig (decompositio: Fig. 7.) Bipolar sigalig rectificatio o-off sigalig Dr. Shi Lathi & Dig-Digital Commu 9 imig Jitter Small radom deviatio of the icomig pulses from their ideal locatios. Always preset, eve i the most sophisticated commuicatio systems. Jitter reductio is ecessary aout every 00 miles i a log digital lik to keep the maximum jitter withi reasoale limits. Figure 7.3 Dr. Shi Lathi & Dig-Digital Commu 9 46

47 Figure 7.3 Dr. Shi Lathi & Dig-Digital Commu 93 Detectio Error Proaility Received sigal desired + AWGN Example: Polar sigalig ad trasmissio (Fig. 7.4, the ext slide) Ap : Peak sigal value Polar: Istead of ± Ap, received sigal ± Ap + Because of symmetry, the detectio threshold 0. If sample value > 0, If sample value < 0, 0 Error i detectio may take place Dr. Shi Lathi & Dig-Digital Commu 94 47

48 Figure 7.4 Dr. Shi Lathi & Dig-Digital Commu 95 Error Proaility for Polar Sigal With respect to Figure 7.4 P ( / 0) proaility that > A p P( /) proaility that < -A p σ : AWGN p( ) e π σ σ P( 0) e d A σ π p ( x) A p A e dx Q p π σ σ x σ Dr. Shi Lathi & Dig-Digital Commu 96 48

49 Error Proaility for Polar Sigal where Similarly, Summary: x Q( y) e y π [ P( 0) + P( ) ] Q σ Q(x): Complemetary error fuctio: erfc (x) A p dx A P( ) Q p to σ Symmetric P( ) P(,0) + P(,) Here, assume P(0) P()/. P( 0) P(0) + P( ) P() A p 0.7 Q( x) e x π x Dr. Shi Lathi & Dig-Digital Commu 97 x, x > Error Proaility for Polar Sigal A p If k, P( 0) P( ) P( ) Q( k) σ k Q(k)P(ε) e.g oe of pulses will e possily detected wrogly Dr. Shi Lathi & Dig-Digital Commu 98 49

50 Error Proaility for O-off Sigals I this case, we have to distiguish etwee A p ad 0. hreshold: A p Error Proailities: P( ) P ( 0) pro. P ( ) pro. [ P( 0) + P( ) ] Q σ P ( ) > P( ε ) (i the case of polar sigalig) Dr. Shi Lathi & Dig-Digital Commu 99 A p of of Ap > A p Q σ Ap < A p Q σ Also assume 0 ad equally proale Dr. Shi Lathi & Dig-Digital Commu 00 50

51 5 Dr. Shi Lathi & Dig-Digital Commu 0 Error Proaility for Bipolar Sigals v A or A p p 0 0 If the detected sample value:.. 0, o w A A p p > < + > > p p p p p A Q A pro A pro A pro A pro P σ.... 0) ( Dr. Shi Lathi & Dig-Digital Commu 0 Error Pro. for Bipolar Sigals [ ] sigalig off o for P A Q P P P P P P P A Q used pulse egative whe A pro or used pulse positive whe A pro P p p p p > + + > < ) (.5 ) ( 0) ( () ) ( (0) 0) ( ) (.. ) ( ε σ σ

52 Detectio Error Proaility Summary, polar is the est, Aother factor: P( ) o-off is i middle, ipolar is the worst, Ap Q σ Ap Q σ Ap.5Q σ Assumptio: 0 ad Equally likely decreases expoetially with the sigal power. Dr. Shi Lathi & Dig-Digital Commu 03 Compariso amog three lie codes o otai: P( ) We eed: A p σ 5 A p σ 0 A σ p for polar case QQ(5) A for o-off case p Q P( ) Q σ for ipolar case Ap Q P( ).5Q σ Ap c 5.08 σ Dr. Shi Lathi & Dig-Digital Commu

53 Compariso amog three lie codes For the same error rate, required SNR, For the same SNR Polar < O-off < Bipolar A p σ Polar < O-off < Bipolar (from previous example), the caused error rate (from ext example) Polar is most efficiet i terms of SNR vs. error rate Betwee, o-off ad ipolar: Oly 6% improvemet of o-off over ipolar Performace of ipolar case case i terms of error rate. A p σ performace of o-off Dr. Shi Lathi & Dig-Digital Commu 05 Example a) Polar iary pulses are received with peak amplitude A p mv. he chael oise rms amplitude is 9.3 µv. hreshold detectio is used, ad ad 0 are equally likely. ) Fid the error proaility for (i) the polar case, ad the o-off case (ii) the ipolar case if pulses of the same shape as i part (a) are used, ut their amplitudes are adjusted so that the trasmitted power is the same as i part (a) Dr. Shi Lathi & Dig-Digital Commu 06 53

54 a) For the polar case Dr. Shi Example 3 A p (0 From ale 8. (4 th ed), we fid 7 P( ) Q (5.) ) Because half the its are trasmitted y opulse, there are, o the average, oly half as may pulses i the o-off case (compared to the polar). Now, doulig the pulse eergy is accomplished y multiplyig the pulse y σ Lathi & Dig-Digital Commu 5. ) (i order to keep the power dissipatio same) 07 hus, for o-off A p is Example herefore, from Equatio (7.53) times the A p i the polar case. Ap P( ) Q (3.68).66 0 Q σ As see earlier, for a give power, the A p for oth the o-off ad the ipolar cases are idetical. Hece, from equatio (7.54) A p P( ).5Q σ Dr. Shi Lathi & Dig-Digital Commu

55 Digital commuicatios use oly a fiite umer of symols Iformatio trasmitted y each symol icreases with M. I M log M its rasmitted power icreases as M, i.e., to icrease the rate of commuicatio y a factor of log M, the power required icreases as M. (see a example elow) M-ary Commuicatio I M : iformatio trasmitted y a M-ary symol Dr. Shi Lathi & Dig-Digital Commu 09 M-ary Commuicatio Most of the terrestrial digital telephoe etwork: Biary he suscrier loop portio of the itegrated services digital etwork (ISDN) uses the quarterary code BIQ show i Figure. 7.8 Dr. Shi Lathi & Dig-Digital Commu 0 55

56 Dr. Shi Lathi & Dig-Digital Commu Pulse Shapig i Multi-amplitude Case Nyquist Criterio ca e used for M-ary case Cotrolled ISI Figure 7.8: Oe possile M-ary Scheme Aother Scheme: Use M orthogoal pulses: ϕ( t), ϕ( t),..., ϕ M ( t) Defiitio: c ϕ i ( t) ϕ j ( t) 0 0 Dr. Shi Lathi & Dig-Digital Commu i i j j 56

57 Pulse Shapig i Multi-amplitude Case he figure i ext slide: Oe example i M orthogoal pulses: π k t si, 0 < t < k,,..., M ϕk ( t) 0, otherwise I the set, pulse frequecy: k :,,..., M M times that of the iary scheme (see a example elow) Dr. Shi Lathi & Dig-Digital Commu 3 Dr. Shi Lathi & Dig-Digital Commu 4 57

58 Pulse Shapig i Multi-amplitude Case I geeral, it ca e show that the adwidth of a orthogoal M-ary scheme is M times that of the iary scheme I a M-ary orthogoal scheme, the rate of commuicatio is icreased y a factor of log M at the cost of a icrease i trasmissio adwidth y a factor of M. Dr. Shi Lathi & Dig-Digital Commu 5 Digital Carrier Systems So far: Basead digital systems Sigals are trasmitted directly without ay shift i frequecy. Suitale for trasmissio over wires, cales, optical fiers. Basead sigals caot e trasmitted over a radio lik or satellites. Sice it eeds impractically large size for ateas Modulatio (Shiftig sigal spectrum to higher frequecies is eeded) Dr. Shi Lathi & Dig-Digital Commu 6 58

59 Digital Carrier Systems A spectrum shift to higher frequecies is also required whe trasmittig several messages simultaeously y sharig the large adwidth of the trasmissio medium, FDM: Frequecy-divisio Multiplexig. (FDMA: Frequecy-divisio Multiplexig Access) Dr. Shi Lathi & Dig-Digital Commu 7 Several ypes of Modulatio Amplitude-Shift Keyig (ASK), also kow as ooff keyig (OOK) m( t)cos( ct) : m(t) : o-off asead sigal (modulatig sigal) cos( c t) : carrier Dr. Shi Lathi & Dig-Digital Commu 8 59

60 Modulatio ypes Phase-Shift Keyig (PSK) m(t) : polar Sigal p( t)cos( t),0 p( t)cos( t) p( t)cos( π t) p( t)cos( t π) c c c c Dr. Shi Lathi & Dig-Digital Commu 9 Modulatio ypes Frequecy-Shift Keyig (FSK) Frequecy is modulatio y the ase ad sigal., 0 Iformatio it resides i the carrier frequecy. c c 0 Dr. Shi Lathi & Dig-Digital Commu 0 60

61 PSD of ASK, PSK & FSK Dr. Shi Lathi & Dig-Digital Commu FSK FSK sigal may e viewed as a sum of two iterleaved ASK sigals, oe with a modulatig frequecy, the other, c0 c Spectrum of FSK Sum of the two ASK Badwidth of FSK is higher tha that of ASK or PSK Dr. Shi Lathi & Dig-Digital Commu 6

62 Demodulatio A. Review of Aalog Demodulatio Basead sigal ad adpass sigal he ed of the d part of the slides of Ch. ad Ch. 3 Doule-sidead Suppressed Carrier (DSB-SC) Modulatio m( t) M( ) message sigal cos( ct ) [ δ( + c ) + δ( c )] carrier m( t)cos( c t) [ M( + c ) + M( c )] modulated sigal Dr. Shi Lathi & Dig-Digital Commu 3 Figure 4. (a), (), (c) DSB-SC Modulatio 4 6

63 DSB-SC Demodulatio Demodulatio: Sychroizatio Detectio (Coheret Detectio) Figure 4. (e) ad (d) Usig a carrier of exactly the same frequecy & pulse Dr. Shi 5 DSB-SC Demodulatio [ m( t)cos( t) ] cos( t) e( t) [ m( t) + m( t)cos( t) ] E( ) Aftera c M ( ) + 4 LPF M ( ) c [ M ( + ) + M ( )] m( t) c c c Dr. Shi 6 63

64 Figure 4. (d), (e) DSB-SC demodulatio 7 Motivatio: AM Sigal Requiremets of a carrier i frequecy ad phase sychroism with carrier at the trasmitter is too sophisticated ad quite costly Sedig the carrier a. eases the situatio. requires more power i trasmissio c. suitale for roadcastig Dr. Shi 8 64

65 Amplitude Modulatio (AM) ϕ ( t) Acos( t) + m( t)cos( t) ϕ AM t) c [ A + m( t) ] cos( t) [ M ( + ) + M ( )] AM ( c c + π A c [ δ + ) + δ ( )] ( c c c Dr. Shi 9 AM Sigal ad its Evelope 30 65

66 Demodulatio of AM sigals Rectifier Detectio: Figure 4. Output: m( t) π Evelope Detector: Figure 4. Output: A + m(t) Dr. Shi

67 33 B. Back to Digital Commuicatio System: Demodulatio Demodulatio of Digital-Modulated Sigals is similar to Demodulatio of Aalog-Modulated Sigals Demodulatio of ASK Sychroous (Coheret) Detectio No-coheret Detectio: Say, evelope detectio Dr. Shi Lathi & Dig-Digital Commu 34 67

68 Demodulatio of PSK Caot e demodulated o-coheretly (evelope detectio). Sice, evelope is the same for 0 ad Ca e demodulated coheretly. PSK may e demodulated o-coheretly if use: Differetial coheret PSK (DPSK) Dr. Shi Lathi & Dig-Digital Commu 35 Differetial Ecodig Differetial codig: ecoded y the same pulse used to ecode the previous data it o trasitio 0 ecoded y the egative of pulse used to ecode the previous data it trasitio Figure 7.30 (a) Modulated sigal cosists of pulses with a possile sig amiguity ± Acos( t) c Dr. Shi Lathi & Dig-Digital Commu 36 68

69 Differetial Ecodig Figure Dr. Shi Lathi & Dig-Digital Commu 37 Differetial Ecodig : oe-it iterval If the received pulse is idetical to the previous pulse, A Y ( t) A cos ( t) [ + cos( t) ] A z( t) If 0 is trasmitted c A Y ( t) A cos ( ct) c A z( t) Dr. Shi Lathi & Dig-Digital Commu 38 c [ + cos( t) ] 69

70 Demodulatio of FSK FSK ca e viewed as two iterleaved ASK sigals with carrier frequecy c0 ad c, respectively. Hece, FSK ca e detected y o-coheret detectio (evelope) or coheret detectio techique Dr. Shi Lathi & Dig-Digital Commu 39 Demodulatio of FSK Dr. Shi Lathi & Dig-Digital Commu 40 70

71 Demodulatio of FSK I (a), o-coheret detectio (evelope), H 0 () H () ured to c0 c respectively Compariso: aove or ottom Whose output is large 0 or Dr. Shi Lathi & Dig-Digital Commu 4 7