Bandlimited channel. Intersymbol interference (ISI) This non-ideal communication channel is also called dispersive channel

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1 Inersymol nererence ISI ISI s a sgnal-dependen orm o nererence ha arses ecause o devaons n he requency response o a channel rom he deal channel. Example: Bandlmed channel Tme Doman Bandlmed channel Frequency doman BT.33 Inersymol nererence ISI Ths non-deal communcaon channel s also called dspersve channel The resul o hese devaon s ha he receved pulse correspondng o a parcular daa symol s aeced y he prevous symols and susequen symols. BT.34

2 Example aveorm o 101 BT.35 Inersymol nererence ISI Two scenaros I. The eec o ISI s neglgle n comparson o ha o channel nose. use a mached ler, whch s he opmum lnear menvaran ler or maxmzng he pea pulse sgnalo-nose rao. II. The receved S/N rao s hgh enough o gnore he eec o channel nose For example, a elephone sysem conrol he shape o he receved pulse. BT.36

3 ISI Consder a nary sysem, he ncomng nary sequence { } consss o symols 1 and 0, each o duraon T. The pulse amplude modulaor modes hs nary sequence no a new sequence o shor pulses approxmang a un mpulse, whose amplude a s represened n he polar orm a 1 0 { } Pulseamplude modulaor { } a Transm s ler g Channel h x o x w he nose BT.37 ISI Example:{ } 1101 a δ T { }: a T BT.38

4 ISI The shor pulses are appled o a ransm ler o mpulse response g, producng he ransmed sgnal s a g T The sgnal s s moded as a resul o ransmsson hrough he channel o mpulse response h. In addon, he channel adds random nose o he sgnal. x a g T h + n { } Pulseamplude modulaor { } a Transm s ler g Channel h x o w x he nose BT.39 ISI The nosy sgnal x s hen passed hrough a receve ler o mpulse response c.the resulng oupu y s sampled and reconsruced y means o a decson devce. x The recever oupu s y Receve ler c Sample a T Decson devce y µ a p T + n λ 1 y > λ 0 y < λ where µ p g h c and µ s a consan. BT.40

5 Example: { } 1101 ISI { }: a a1δ a δ T y T assume n 0 µ a1 p µa p T y µ a p T + n BT.41 The sampled oupu s y µ µ a a + µ ISI p[ T a ] + n p[ T ] + n µa : conruon o he h ransmed. µ a p[ ] : T The resdual eec o all oher ransmed s. Ths eec s called nersymol nererence BT.4

6 Example:{ } 1101 y T ISI y µ a + µ a p[ T ] + n assume n 0 µa µa1 p T, 1 µa3 p T 0, 3 y µ a p T + n.e. BT.43 Dsoronless Transmsson In a dgal ransmsson sysem, he requency response o he channel h s speced. e need o deermne he requency responses o he ransm g and receve ler c so as o reconsruc he orgnal nary daa sequence }. { } Pulseamplude modulaor { } { a Transm s ler g Channel h x o w x he nose x Receve ler c y Decson devce 1 0 y > λ y < λ Sample a T λ BT.44

7 BT.45 Dsoronless Transmsson The decodng requres ha T T p Ignore he nose ] [ ] [ n T p a a n T p a y µ µ µ y 0 assume n y µa BT.46 Dsoronless Transmsson I can e shown ha he condon s equvalen o T T p 0 1 n T T n P /

8 Example p p0 p T Sample pons BT.47 Example p T snc p 1/T T T p p 1/ T p / T n p n / T T BT.48

9 The smples way o sasyng Ideal Nyqus Channel n P n / T T s a recangular uncon: 1 < < p 0 > 1/ T 1/ 1/ T BT.49 Ideal Nyqus Channel p snπ π The specal value o he rae R 1 / T s called he Nyqus rae, and s called he Nyqus andwdh. Ths deal aseand pulse sysem s called he deal Nyqus channel BT.50

10 Example Samplng nsans BT.51 Ideal Nyqus Channel In praccal suaon, s no easy o acheve due o The sysem characerscs o P e la rom -1/T up o 1/T and zero elsewhere. Ths s physcally unrealzale ecause o he ransons a he edges. The uncon decreases as 1/ or large, resulng n a slow rae o decay. Thereore, here s praccally no margn o error n samplng mes n he recever. BT.5

11 BT.53 Rased Cosne Specrum e may overcome he praccal dcules encounered y ncreasng he andwdh o he ler. Insead o usng we use > < < p 0 1 > < < P p p 0 1 T 1/ BT.54 Rased Cosne Specrum A parcular orm s a rased cosne ler

12 Rased Cosne Specrum The requency characersc consss o a la amplude poron and a roll-o poron ha has a snusodal orm. The pulse specrum p s speced n erms o a roll o acor α as ollows: 1 0 < 1 1 π p 1 sn 1 < > 1 The requency parameer and andwdh are relaed y α 1 1 / 1 BT.55 Rased Cosne Specrum where α s he rollo acor. I ndcaes he excess andwdh over he deal soluon Nyqus channel where 1/T. The ransmsson andwdh s 1+ α BT.56

13 Rased Cosne Specrum The requency response o α a 0, 0.5 and 1 are shown n graph elow. e oserved ha α a 1 and 0.5, he uncon P cuo gradually as compared wh he deal Nyqus channel and s hereore easer o mplemen n pracce. BT.57 Rased Cosne Specrum The me response p s oaned as cosπα p sn c 1 16α The uncon p consss o wo pars. The rs par s a snc uncon ha s exacly as Nyqus condon u he second par s depended on α. The als s reduced α s approachng 1. Thus, s nsensve o samplng me errors. BT.58

14 BT.59 BT.60 Example For α 1, 1 0 he sysem s nown as he ull-cosne rollo characersc. > < < + p 0 0 cos π

15 Example snc p 1 16 BT.61 Example Ths me response exhs wo neresng properes: A ± T / ± 1/4 we have p 0.5; ha s, he pulse wdh measured a hal amplude s exacly equal o he duraon T. T / BT.6

16 Example There are zero crossngs a ± 3T /, ± 5T /,... n addon o he usual crossngs a he samplng mes ± T /, ± T /,... 3T / 5T / BT.63 Example These wo properes are exremely useul n exracng a mng sgnal rom he receved sgnal or he purpose o synchronzaon. However, he prce pad or hs desrale propery s he use o a channel andwdh doule ha requred or he deal Nyqus channel correspondng o α 0. BT.64

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