4. 4. Nyquist Criterion for Distortionless Baseband Binary Transmission

Transcription

1 4. 4. Nyquist Criterio or Distortioless Basebad Biary rasmissio Objective: o desig h ( t ad hd ( t uder the ollowig two coditios: (a. here is o ISI at the samlig istats (Nyquist criterio, this sectio. (b. A cotrolled amout o ISI is allowed (correlative codig, ext Gog

2 Desig o Badlimited Sigals or Zero ISI - Nyquist criterio Recall the outut o the receivig ilter, samled at t k, is give by y(k μbk + μ b ( k + o (k k hus, i time domai, a suiciet coditio or μ(t such that it is ISI ree is ( ( Questio. What is the coditio or P( i order or (t to satisy ( (Nyquist, Gog

3 heorem. (Nyquist A ecessary ad ad suiciet coditio or (t to satisy ( is that the Fourier trasorm P( satisies P( ( his is kow as the Nyquist ulse-shaig criterio or Nyquist coditio or zero ISI. Gog 3

4 Proo. Whe we samle (t at t k, k, ±, ±,L, we have the ollowig ulses ( t ( t δ ( t k δ k ( k δ ( t k k he Fourier trasorm o is give by P δ ( F( F δ k k (t δ ( t ( k δ ( t k P( k ( 3 O the other had P ( ( k F( δ ( t k δ ( kex( jπk k k (k ( is costat or t. ( rom ( ( 4 From (3 ad (4, ISI ree k P( k which gives the result i (.

5 Ivestigate ossible ulses which satisy the Nyquist criterio Suose that the chael has a badwidth o W, the H c ( or > W Sice P( H ( H ( H (, we have P ( or > W c d We write Z ( P( / ad distiguish the ollowig three Gog 5

6 Z( -/-W -/ -/+W -W W /-W / /+W Fig. 4. Z( or the case < /(W Z( -/ / Fig. 4. Z( or the case /(W Z( W -/ -W /-W W / -/+W Fig. 4.3 Z( or the case > /(W

7 . <, or > W (i.e., bit rate > W, imossible! No W choices or P( such that Z(.., i.e., (the Nyquist rate W I this case, i we choose W W P( i.e., P( rect otherwise W t ( t si c which results i his meas that the smallest value o or which the trasmissio with zero ISI is ossible is W R W (, bit rate his is called the ideal Nyquist chael.

8 I other words, Ideal Nyquist chael: R B W B o ( R R o b, Gog 8

9 Disadvatages: (a a ideal LPF is ot hysically realizable. (a Note that ( t si c t t hus, the rate o covergece to zero is slow sice the tails o (t decay as / t. Hece, a small mistimig error i samlig the outut o the matched ilter at the demodulator results i a iiite series o ISI Gog 9

10 @G. Gog 3. For, i.e,, i this case, there exists umerous choices or P( such that Z(. he imortat oe is so called the raised cosie sectrum. W > W < he raised cosie requecy characteristic is give by + + < + < ( ( ( ( ( cos 4 ( ( B B B B B B B B P α α α α α π α α [,] where is called the rollo actor ad ( i.e.,. R B B

11 ( B +α ( B α ( B α ( B +α B B B ( ( ( B B B P B P P Z( by the ollowig sum o three terms at ay iterval o legth B o : 3 ( ( ( B B B P B P P + + +

12 he time resose (t, the iverse Fourier trasorm o P(, is give by cosπαbt ( t sicb t 6α B t (5 his uctio has much better covergece roerty tha the ideal Nyquist chael. he irst actor i (5 is associated with the ideal ilter, ad the secod actor that decreases as / t or large t. hus ( t 3 Gog

13 @G. Gog 3

14 Summary: Nyquist Criteria + P( R, R -B o B o Ideal Nyquist Chael Raised Cosie Gog 4

15 Examle he Fourier trasorm P( o the basis ulse (t emloyed i a certai biary commuicatio system is give by P( 6 6 i 6 otherwise (Hz 6. From the shae o P(, exlai whether this ulse satisies the Nyquist criterio or ISI ree trasmissio.. Determie (t ad veriy your result i art. 3. I the ulse does satisy the Nyquist criterio. What is the trasmissio rate ( i bits/sec. ad what is the roll-o Gog 5

16 Solutio:. he Nyquist criterio is Z ( P( R, R / I we choose R 6 Hz, the (t satisies Nyquist criterio or ISI ree trasmissio, show as Figure. P( 6 6 Z ( 6 6 (Hz 6 6 R b R b Figure Gog 6

17 . We have Cosequetly, ( t F [ P( ] sic ( 6 t (, ad ( where ±, ±,... hereore, at will i the t, ±, ±,..., the ulses ( t ( ot iterere received sigal is with each other, samled i.e.,, ±, ± ISI - ree,... trasmissio. ( t sic ( 6 t t ( μ Gog 7

18 3. he relatioshi betwee the badwidth ad the roll-o actor is W ( + α B α ( B B 6 where the trasmissio rate R (bits/s. I this case, we have, R 6 ad B.5R.5 Hz Gog 8

19 5. 5. Correlative Codig ad Equalizatio A. Correlative Codig For zero ISI, the symbol rate R / < W, the Nyquist rate. We may relax the coditio o zero ISI i order to achieve R W. he schemes which allow a cotrolled amout o ISI to achieve the symbol rate W are called correlative codig or artial resose sigalig Gog 9

20 he coditio or zero ISI is, (, ( Suose that we desig the bad - limited sigal ( t have cotrolled ISI at oe time istat, additioal ozero value i the samles or examle, ( i.e., { ( }, ad (, otherwise to allow oe Gog

21 ( t has a larger time duratio tha P ( F[ o requecy domai tha ( t] has a smaller badwidth P ( F[ ( t; Sectral eiciecy is icreased by usig ( t]; ( t. Note. he ISI we itrodece by usig (t is determiistic or cotrolled ad, hece, its eect o sigal detectio at the receiver ca be removed, as discussed Gog

22 (A Duobiary Sigalig (Class I artial resose he reix duo imlies doublig o the trasmissio caacity o a straight biary system. Figure shows a duobiary ecoder. { } a re-coder { } b + { } y Ideal chael ( H Nyquist { } ˆ / y { b } duobiary decoder ˆ Postcoder { aˆ } Delay overall chael Duobiary ecoder Figure. Block diagram o duobiary ecoder

23 Leged i Figure :. with { a }, a {, } hesamle(bit or symbol { b }, (or level the NRZolar,theoutut o coverter b a ad + k duratio + d d a is beig i i. the a a ideedet or k memoryless recoder.. 3 he requecy resose o the duobiary ecode is give by H ( H( H ( I Nyquist which is a cascaded two ilters, sice we have Delay + δ (t h ( t δ( t + δ( t H ( Figure. raser uctio o delay oerator

24 4 the he Notice that reset iut ulse y ilter outut b y + b b + b d d he eect o (3: { b }, two level {d, -d}, ucorrelated, y } three level {-d,, d}, correlated. { y ca be b ad reresete d i i Gog 4 its a a a revious a a a (3 by the value sum o Note that here we cosider oiseless chael ad μ which is rom the ollowig ormula: yk Y ( k μ bk + μ b ( t + o ( k (3-a k b (4,

25 4 O the requecy domai, the traser uctio o the duobiary ecoder is: H I ( H( H ( H ( F[ δ ( t + δ ( t ] H Nyquist H Nyquist Nyquist Nyquist ( [ + ex( jπ] ( ex( jπ[ex( jπ + ex( jπ] ( ex( jπcos( π H Nyquist Sice a ideal Nyquist chael o badwidth B / traser uctio H Nyquist he, (, H I i / i > / cos( πex( jπ (, i / i > /. Gog 5

26 H I (. R bit rate R / R / arg R / π [ π H I ( ] R / Figure 3. he requecy resose o the duobiary Gog 6

27 ] ( ( [ sic t t t + δ δ ( H I 5 he imulse resose corresodig to cosists o two sic (Nyquist ulses that are time-dislaced by secods with resect to each other, which ca be derived as ollows. ] ( [ ] ( [ ] ( [ ( t H F t H F t H F t h Nyquist I I ( / si( ( / si( / si( t t t t t t t π π π π π π + t t sic sic + t t t t / ( / ( si( / / si( π π π π

28 Note : he tails o h I ( t decay as / t which is aster rate o decay tha the is / t ecoutere d i the ideal Nyquist cahel. hereore, the ISI due to bit sychroizatio error is reduced by the duobiary sigalig., h I (t t Figure Gog 8

29 (B. Decodig o the Duobiary Sigalig y he decoder the origial rom the estimate the receiver estimate bˆ based two- level duobiary- coded o o Eq. the at time rom Delay y origial, sequece (3. we Decisio eedback sequece get bˆ ulse t.he, { b } { y } Seciicaly, subtractig bˆ Gog 9 bˆ y b as Figure 5 be usig let bˆ detected a received by the eedback rereset revious Drawback: error roagatio

30 Duobiary Scheme with Precoder o uiquely determie the source bit i the kth sigalig iterval, eve i a error is made o the (k-th bit, the kth source bit, we itroduce the recodig: b * * a b (5 where is modulo oeratio. { } a { } b * Level coverter { } b Duobiary ecoder / { } ŷ Delay Precoder Figure 6

31 he recoded sequece b * he biary sequece * { b } alied a corresodig two- level sequece duratio as beore. he y b b b * *,, + b is { * b } i i is a a give by. (6 where the bar reresets the comlemet o the symbol. to { b } a level coverter, roducig, where b (7 ± d with samle y From (4, we have d i i d i b b b * * * b b b * * * Combiig with (6, y i ± d i a a Gog 3

32 From (8 we decisio sequece i i i y y y rule { a } rom { y } deduce the ollowig or detectig the origial < d, the symbol > d, the symbol d, radomly guessig : a a is is a biary Gog 3

33 A useul eature o this detector isthat o kowledge o ay iut samle other tha the reset oe is required. Hece, error roagatio caot occur i the detector. { } ŷ { } Rectiier ŷ Decisio { â } device threshold d Figure Gog 33

34 Summary: Correlative codig ca achieve a trasmissio rate o W symbols er secod by usig the duobiary scheme together with the recodig. { } a * { } b + LC { } b + { } y Ideal chael ( H Nyquist { } ˆ y { b } duobiary decoder ˆ Postcoder { aˆ } D D overall chael Duobiary Scheme with Gog 34

35 Examle. Precodig with memory ad duobiary codig. Cosider the biary data sequece. o roceed with the recodig o this sequece, which ivolves eedig the recoder outut back to the iut, we add ad extra (iitializatio bit to the recodig outut. his extra bit is chose arbitrarily to be. Hece, usig (4, we id that the sequece {b * } at the recoder outut is as show i row o the ollowig table. he olar ormart {b } o the sequece {b * } is show i row 3 o the table. Fially, usig (7, we id the duobiary ecoder outut has the amlitude levels give i row 4 o the table. o detect the origial biary sequece, we aly the decisio rule, give by (A, so, obtai the biary sequece give i row 5 i the table. he last row shows that, i absece o oise, the origial biary sequece is detected Gog 35

36 Examle. Duobiary codig with recodig. Biary Sequece { } a Precoded sequece { } b * wo-level sequece +d +d +d -d -d +d -d -d { } b Duobiary Ecoder outut +d +d -d -d { } y Detected biary sequece { } Gog 36

37 Examle. Fid the error robability o the duobiary sigalig i AWGN where the symbols are equally likely. Solutio. From (3-a ad (8, we have y k o ( k ± d + o ( k i i a a k k Sice or {a k }, ad are equally likely, the outut levels ±d each occur with ¼ ad the outut level occurs with rob. ½ assumig o oise. I the thresholds are set at ±d, errors occurs as ollows: I a k, the (i (ii d + o ( k < d or d + ( k > d or o o ( k < d ( k > d o -d -d d d We write N o (k. Cosequetly, hus, error occurs whe d y k N < d or N > d i ak < y k < d < d or y k > d i i a a k Gog 37

38 P( e P{ d < yk < d } + [ P{ yk < d } + P{ yk > d 4 }] [ P { N > d} + P{ N < d}] + [ P{ N < d} + P{ N > 4 3 [ P { N < d} + P{ N > d}] 4 3 Q d / N d}] Remark. I the actor o 3/ is igored, the ractio o F (4/π amouts to a degradatio i sigal-to-oise ratio o. db o duobiary over direct biary. hat is, to achieve the same error robability, the trasmissio ower or duobiary must be. db greater tha that or direct biary, assumig ideal chael ilterig ad AWGN. his is the sacriice that aid or the smaller badwidth required by duobiary Gog 38

39 5. Correlative Codig ad Equalizatio (Cot. B. Eye Patter Eye atter is a exerimetal tool to evaluate the combied eect o receiver oise ad ISI o overall system erormace i a oeratioal eviromet. It is deied as the sychroized suerositio o all ossible realizatios o the sigal o iterest (e.g. received sigal, receiver outut viewed withi a articular sigalig iterval. he eye atter deserves its ame rom the act that it resembles the huma eye or biary waves. he iterior regio o the eye atter is called the eye Gog 39

40 Biary data Fig. 8 (a Distorted biary wave with oisy, but o ISI t Biary data Fig. 8 (b Eye atter t t Fig. 9 (b Eye atter t Fig 9. (a Distorted biary wave with oisy ad ISI

41 Gog 4

42 Remark. A eye atter rovides a great deal o useul iormatio about the erormace o a data trasmissio system, as described i Figure. Seciically, we may make the ollowig statemets:. he width o the eye oeig deies the time iterval over which the received sigal ca be samled without error rom ISI. It is aaret that the reerred time or samlig is the istat o time at which the eye is oe the widest.. he sesitivity o the system to timig errors is determied by the rate o closure o the eye as the samlig time is varied. 3. he height o the eye oeig, at a seciied samlig time, deies the oise margi o the system. 4. Whe the eect o ISI is severe, traces rom the uer ortio o the eye atter cross traces rom the lower ortio, with the result that the eye is comletely closed. I such a situatio, it is imossible to avoid errors due to the combied resece o ISI ad oise i the system.

43 Remark. I the case o a M-ary system, the eye atter cotais (M - eye oeigs stacked u vertically oe o the other, where M is the umber o discrete amlitude levels used to costruct the trasmitted sigal. I a strictly liear system with truly radom data, all these eye oeigs would be idetical. Figures ad show the eye diagrams or a basebad PAM trasmissio system usig M ad M 4 resectively, uder the idealized coditios: o chael ose ad o badwidth limitatio (i.e., oiseless ad zero ISI, ad Figures 3 show the eye diagrams with a badwidth limitatio. Note. For how to geerate eye diagrams, see Hadout Gog 43

44 .5 Eye Diagram.5 Eye Diagram.5.5 Amlitude Amlitude ime ime Samle istace Figure. M Samle istace Figure. M 4 Noiseless ad zero Gog 44

45 Eye Diagram.5 Eye Diagram Amlitude Amlitude ime ime.5 Eye Diagram Eye Diagram Amlitude Amlitude ime ime Figure 3. Bad-width Gog 45

46 Equalizatio I the recedig sectios, we discussed that i a bad-limited chael H c ( is kow, the it is ossible to achieve ISI-ree trasmissio by usig a suitable air o x ad Rx. I ractice we ote ecouter chaels whose requecy resose characteristics are either ukow or chage with time. he methodology to overcome this roblem is to emloy chael equalizers. Chael equalizers: o comesate or the chael distortio, a liear ilter with adjustable arameters may be emloyed. he ilter arameters are adjusted o the basis o measuremets o the chael characteristics. hese adjustable ilters are called chael equalizers or, simly, equalizers. (Figure Gog 46

47 (k+n (k+ (k (k- (k-n Delay Delay Delay Delay w N w N + w w w w N wn + y(k Figure 4 Eective Chael (t Equalizer w(t Figure 5 N w ( t w δ ( t k kn k

48 Recall that the outut o the overall ilter may be samled eriodically to roduce the sequece y + μ b k + k μb k ok k ( where y k y(k ( ad (k ok he middle term o the equatio ( reresets the ISI. I the ractical system, it is reasoable to assume that the ISI aects a iite umber o symbols. Hece the ISI observed at the outut o the receivig ilter may be viewed as beig geerated by assig the data sequece though a liear ilter. Gog 48

49 Zero-orcig equalizer Suose that the equalizer is coected i cascade with the eective chael (which cosists o the x ilter, hysical chaels ad Rx ilter, as show i Figure 5. Let g(t deote the imulse resose o the equalized systems, the g ( t ( t w( t w ( t N N N At the time istace t k, g k N w k where ad ( (8 g k g(k { g Note that ay term i the sequece } is the weighted sum o cosecutive N + terms o { }. o elimiate the ISI, accordig to the Nyquist criterio or the distortioless trasmissio, we should satisy g k i i k k From (8, we may orce the coditios g k or or k k ±, ±,..., ± N (9 From (8 ad (9, we obtai a set o liear equatios: N N w k or or k k ±, ±,..., ± N (

50 @G. Gog 5 ( : : : : : : : : : : : : : : N N N N N N N N N N N N N N N N w w w w w Equivaletly, we have the ollowig matrix orm A taed-delay-lie equalizer described by Eq. ( or ( is reerred to as a zero-orcig equalizer. Such a equalizer is otimum i the sese that it miimizes the eak distortio (ISI.

51 I summary, (i i resece o additive white Gaussia oise, a matched ilter is the otimum detector; ad (ii i the resece o ISI, a equalizer is the desired structure to mitigate ISI. Ituitively, the otimum receiver should cosist o a matched ilter ad a equalizer i tadem, as show i Figure 6. x(t + r(t Matched ilter equalizer Frot-ed (t receiver Figure Gog 5

52 Questio: What should the roted o the receiver match to? { } a { } b recoder ( H ( H ( H ( H ( C It should match to the x ad the hysical chael: H ( H ( H C ( (Fig. 7. matched ilter j πt H C H ( e H ( e (t jπt + equalizer oise ath x ilter hysical chael Figure 7 Sigal ath: the eective chael has the imulse resose: a urther stretchig o the ulse the matched ilter accetuates ISI the equalizer eeds to work harder. Sigal ath Noise ath: the ilter oise must ass through the equalizer which is ot equalizig the matched ilter equalizig will ehace residual additive oise.

53 wo tyes o equalizers Preset equalizers: O chaels whose requecy-resose characteristics are ukow, but time-ivariat, we may measure the chael characteristics, adjust the arameters o the equalizer, ad oce adjusted, the arameters remai ixed durig the trasmissio o data. Adative equalizers: udate their arameters o a eriodic basis durig the trasmissio o data. Miimum mea-square error (MSE equalizer: :he ta weights are chose to miimize MSE o all the ISI terms lus the oise ower at the outut o the equalizer. Remark. Most high-seed telehoe lie modems use a MSE weight criterio, because it is suerior to a zero-orcig criterio, ad it is more robust i the reset o oise ad large Gog 53

54 Remark o equalizatio o digital data trasmissio he zero-orcig equatios ( or ( i Sec do ot accout or the eect o oise. I additio, a iite-legth ilter equalizer ca miimize worst-case ISI oly i the eak distortio (i.e., the magitude o the dierece betwee the chael outut ad desired sigal is suicietly small. he sequece {w k }, i Figure i Sec. 4. 5, ca be chose i a way such that oe ca miimize the mea-square error (MSE o all the ISI terms lus the oise ower at the outut o the equalizer. his is called miimum MSE equalizer. (Here, MSE is deied as the exected value o the squared dierece betwee the desired data symbol ad the estimated data symbol.

55 A Few Remarks o the Deiitio o badwidth ad Relatio betwee Chael Badwidth ad rasmissio Rate he badwidth dilemma For all badlimited sectra, the waveorms are ot realizable, ad or all realizable waveorms, the absolute badwidth is iiite. he mathematical descritio o a real sigal does ot ermit the sigal to be strictly duratio limited ad strictly badlimited. All badwidth criteria have i commo that attemt to seciy a measure o the width, W, o a oegative real-valued sd deied or all requecies Gog 55

56 Geeral shae o sd H X ( si c π ( c c c (a c + (b (c (d (e 35 db (e 5 Gog 56

57 (a Hal-ower badwidth. his is the iterval betwee requecies at which H X ( has droed to halower, or 3 db below the eak value. (b Equivalet rectagular or oise equivalet badwidth. It is deied by W N PX / H X ( c, where P X is the total sigal ower over all requecies. (c Null-to-ull badwidth. It is deied as the width o the mai sectral lobe, where the most o the sigal ower is cotaied (the most oular measure o badwidth. ( Absolute badwidth. his is the iterval betwee requecies, outside o which the sectrum is zero. (Useul abstractio. For all realizable waveorms, this is iiite. (d Fractioal ower cotaimet badwidth. Federal Commuicatio Commissio (FCC Rules ad Regulatios Sectio.. It states that the occuied badwidth is the bad that exactly.5% o the sigal ower above the uer bad limit ad exactly.5% o the sigal ower below the lower bad limit. hus 99% o the sigal ower is iside the occuied bad. (e Bouded ower sectral desity. Everywhere outside the seciied bad, H X ( must have alle at least to a certai stated level below that oud at the bad ceter. yical atteuatio levels might be 35 or 5 db.

58 Examle. Digital elehoe Circuits. Comare the system badwidth requiremets or a terrestrial 3-kHz aalog telehoe voice chael with that o a digital oe. For the digital chael, the voice is ormatted as a PCM bit stream, where the samlig rate is 8 samles/s ad each voice samle is quatized to oe o 56 levels. he bit stream is the trasmitted usig a PAM waveorm ad received with zero ISI. Solutio. he resultig o the samlig ad quatizatio rocess yields PCM words such that each word has oe o L 56 levels. I each samle were set as a 56-ary PAM ulse (symbol. hus the required system badwidth without ISI or sedig R s symbols/s would be W R s /. Sice each PCM word is coverted to 8 bits. hus, the system badwidth required usig PCM is W PCM (8 bits/symbol(8symbols/s 3 khz. hereore, the PCM ormat, usig 8-bit quatizatio ad biary sigalig with biary PAM, requests at least eight times the badwidth required or the aalog chael.

59 A Note o Relatio betwee Chael badwidth ad trasmissio rate Questio: I the ideal Nyquist chael, W R/. How ca it be ossible or the chael badwidth W to be smaller tha the trasmissio rate R? Aswer: he chael badwidth W (Hz ad the trasmissio rate R (bit er secod, or bs are two dieret hysical quatities. I geeral, they are roortioal to each other, but it is NO ecessary or them to be equal.

60 rasmitter Filter Physical chael Receiver ilter H ( H C ( ( H R rasmitted sigal Received sigal Eective chael P ( H ( H C ( H R ( (assumig μ he sd o the trasmit ted sigal H ( the badwidth o the trasmi tted sigal is the same as the badwidth o H (

61 Let he B i order to 3 B B B : badwidth system should B C R C : badwidth : badwidth B R achieve high sectrum high trasmitted o ( badwidth o o be H H H ( o ( ( desiged high trasmissio accuracy C R i such a utilizatio eiciecy the trasmitted sigal ower utilizatio eiciecy way that 3. For the discrete PAM sigal ormats, the sigal badwidth (e.g., deied as the requecy iterval which cotais 99% o the total ower, Deiitio (d may ot be equal to the trasmissio rate / Gog 6

62 4 Cosider a secial case where H ( H C ( H R ( P(,, < W > W he ideal Nyquist chael We have the Fourier air : Wsic(Wt H ( Usig Wsic(Wt or symbol "" ad Wsic(Wt or symbol "", we ca trasmit a biary iormatio sequece at a rate R W without ISI, as the symbol iterval ca be as small as / W uder the costrait o ISI - ree Gog 6

63 Summary o Chater 4 (Chater 6 i the textbook.pam sigals ad ower sectrum desity o S X ( H ( RB ( ex( jπ biary PAM sigals. ISI due to badlimited chael 3. Nyquist criteria or ISI - ree, (, trasmissio P( R R / Ideal Nyquist Chael R W W Raised Cosie Sectrum R Gog 63

64 4. Correlative codig ad equalizatio Duobiary sigalig: achievig the maximum trasmissio rate W with zero ISI i Pre-coder with memory: a b, y ecodig ad decodig (error roagatio * ii Pre-coder without memory: a b, b, y ecodig ad decodig (o error roagatio iii Psd o the duobiary PAM sigals ad error robability Eye Patters: Equalizatio: to mitigate the eects o ISI, zero-orcig equalizer.

65 { } a * { } b + CL { } b + { } y - Ideal chael ( H Nyquist { } ˆ y { b } duobiary decoder ˆ Postcoder { aˆ } D D overall chael Modiied Duobiary Scheme with Precodig