Eye Pattern and Equalization

F.1

Baseband Data Transmission III

References– Ideal Nyquist Channel and Raised Cosine Spectrum

• Chapter 4.5, 4.11, S. Haykin, Communication Systems, Wiley.

– Equalization• Chapter 9.11, F. G. Stremler, Communication Systems,

Addision Wesley.

F.2

Ideal Nyquist ChannelThe simplest way of satisfying

∑∞

−∞=

=−n

bb TTnfP )/(

is a rectangular function:

>

<<−=Wf

WfWWfp

||021

)(

bTW 2/1=

Equation for zero ISI

F.3

Ideal Nyquist Channel

WtWttp

ππ

2)2sin()( =

The special value of the bit rate WRb 2= is called theNyquist rate, and W is called the Nyquist bandwidth.

This ideal baseband pulse system is called the idealNyquist channel

F.4


F.5


In practical situation, it is not easy to achieve it due to

1. The system characteristics of P(f) be flat from -1/2T up to 1/2T and zero elsewhere. This isphysically unrealizable because of the transitions atthe edges.

2. The function )(tp decreases as 1/|t| for large t,resulting in a slow rate of decay. Therefore, there ispractically no margin of error in sampling times inthe receiver.

F.6

Raised Cosine Spectrum

We may overcome the practical difficultiesencountered by increasing the bandwidth of the filter.

Instead of using

>

<<−=Wf

WfWWfp

||021

)(

we use

>

<−=++−+Wf

fWWWfPWfpfp

||021

)2()2()(

F.7


A particular form is a raised cosine filter.

F.8

Raised Cosine SpectrumThe frequency characteristic consists of a flatamplitude portion and a roll-off portion that has asinusoidal form. The pulse spectrum p(f) is specifiedin terms of a roll off factor α as follows:

−>

−<≤

−−

−

<≤

=

1

111

1

20

222

(sin1

41

021

)(

fWf

fWfffW

WfW

ffW

fpπ

The frequency parameter 1f and bandwidth W are relatedby

Wf /1 1−=α

F.9


where α is the rolloff factor. It indicates the excessbandwidth over the ideal solution (Nyquist channel)where W=1/2Tb.

The transmission bandwidth is W)1( α+

F.10

Raised Cosine SpectrumThe frequency response of α at 0, 0.5 and 1 are shown ingraph below. We observed that α at 1 and 0.5, thefunction P(f) cutoff gradually as compared with the idealNyquist channel and is therefore easier to implement inpractice.

F.11


The time response p(t) is obtained as

)161

)2cos())(2((sin)( 222 tWWtWtctp

απα

−=

The function p(t) consists of two parts. The first part is asinc function that is exactly as Nyquist condition but thesecond part is depended on α. The tails is reduced if α isapproaching 1. Thus, it is insensitive to sampling timeerrors.

F.12


F.13


Example:For α = 1, (f1 = 0) the system is known as the full-cosinerolloff characteristic.

>

<<

+

=Wf

WfWf

Wfp20

202

cos141

)(π

and 22161)2sinc()(

tWWttp

−=

F.14


This time response exhibits two interesting properties:• At t = ± Tb/2 = ± 1/4W we have p(t) = 0.5; that is,

the pulse width measured at half amplitude isexactly equal to the bit duration Tb.

• There are zero crossings at t = ± 3Tb/2, ± 5Tb/2, ... inaddition to the usual crossings at the sampling timest= ,2, bb TT ±±

These two properties are extremely useful inextracting a timing signal from the received signal forthe purpose of synchronization. However, the pricepaid for this desirable property is the use of a channelbandwidth double that required for the ideal Nyquistchannel corresponding to α = 0.

F.15


Example:A sequence of data transmitting at a rate of 33.6 Kbit/sWhat is the minimum BW at Nyquist rate.BW=W=33.6/2=16.8KHzIf a 100% rolloff characteristic BW=W(1+α)=33.6KHz

F.16


ExampleBandwidth requirement of the T1 systemT1 system: multiplex 24 voice inputs, based on an 8-bit PCM word.

Bandwidth of each voice input (B) = 3.1 kHz

Nyquist sampling rate kHz 2.62 == Bf Nyquist

Sampling rate used in telephone system kHz 8=sf

F.17


With a sampling rate of 8 kHz, each frame of themultiplexed signal occupies a period of 125µs. Inparticular, it consists of twenty-four 8-bit words, plusa single bit that is added at the end of the frame for thepurpose of synchronization.

Hence, each frame consists of a total of 193 bits.Correspondingly, the bit duration is 0.647 µs.

F.18


The minimum transmission bandwidth iskHzTb 7722/1 = (ideal Nyquist channel)

The transmission bandwidth using full-cosine rolloffcharacteristics is

MHzkHzW 544.17722)1( =×=+α

F.19

Eye Patterns

This is a simple way to give a measure of how severethe ISI (as well as noise) is. This pattern is generatedby overlapping each signal-element.

Example: Binary system

1 0 1 1 0 0 1 1 bT2

F.20

Eye Patterns

Eye pattern is often used to monitoring the performance ofbaseband signal. If the S/N ratio is high, then the followingobservations can be made from the eye pattern.

• The best time to sample the received waveform iswhen the eye opening is largest.

• The maximum distortion and ISI are indicated bythe vertical width of the two branches at samplingtime.

• The noise margin or immunity to noise isproportional to the width of the eye opening.

• The sensitivity of the system to timing errors isdetermined by the rate of closure of the eye as thesampling time is varied.

F.21

Eye Patterns

F.22

Equalization

In preceding sections, raised-cosine filters were used toeliminate ISI. In many systems, however, either the channelcharacteristics are not known or they vary.

ExampleThe characteristics of a telephone channel may vary as afunction of a particular connection and line used.

It is advantageous in such systems to include a filter thatcan be adjusted to compensate for imperfect channeltransmission characteristics, these filters are calledequalizers.

F.23

EqualizationExample

F.24

EqualizationTransversal filter (zero-forcing equalizer)

kx

F.25

Equalization

The problem of minimizing ISI is simplified by consideringonly those signals at correct sample times.

The sampled input to the transversal equalizer iskxkTx =)(

The output iskyyTx =)(

For zero ISI, we require that

≠=

=0001

kk

yk …(*)

F.26

Equalization

The output can be expressed as

∑−=

−=N

Nnnknk xay

There are 2N+1 independent equations in terms of na . Thislimits us to 2N+1 constraints, and therefore (*) must bemodified to

±±±==

=Nk

kyk ,...,2,10

01

F.27

Equalization

The 2N+1 equations becomes

=

−

+−

−

−−

−−−−−

−−−−

+−−+−

−−−−−

00

00

1

0

1

01122

1212212

101

122101

2121

M

M

LL

LL

MM

LL

MM

LL

LL

N

N

N

N

NNN

NNN

NNNN

NNN

NNNo

aa

a

aa

xxxxxxxxxx

xxxxx

xxxxxxxxxx

F.28

Equalization

ExampleDetermine the tap weights of a three-tap, zero-forcingequalizer for the input where

1.0,3.0,0.1,2.0,0.0 21012 =−==== −− xxxxx ,0=kx for 2>k

The three equations are

03.01.012.03.002.0

101

101

01

=+−=++−=+

−

−

−

aaaaaa

aa

Solving, we obtain8897.0,2847.0,1779.0 011 ==−=− aaa

F.29

Equalization

The values of the equalized pulse are

0285.0,0036.0,0.0,0.1,0.0

,0356.0,0.0

32

101

23

=====

−==

−

−−

yyyyy

yy

This pulse has the desired zeros to either side of the peak,but ISI has been introduced at sample points farther fromthe peak.

F.30

Equalization

F.31

Duobinary Signaling

Intersymbol interference is an undesirablephenomenon that produces a degradation in systemperformance.

However, by adding intersymbol interference to thetransmitted signal in a controlled manner, it is possibleto achieve a signaling rate equal to the Nyquist rate of2W symbols per second in a channel of bandwidth WHz.

F.32

Coding and decodingConsider a binary input sequence }{ kb consisting ofuncorrelated binary symbols 1 and 0, each havingduration bT . This sequence is applied to a pulse-amplitude modulator producing a two-level sequenceof short pulses (approximating a unit impulse), whoseamplitude is

−

=0 is symbol if11 is symobl if1

k

kk b

ba

This sequence is applied to a duobinary encoder asshown below:

{ }ka

bTDelay

Nyquistchannel

{ }kc

1−+= kkk aac

F.33

Coding and decoding

One of the effects of the duobinary encoding is tochange the input sequence }{ ka of uncorrelated two-level pulses into a sequence }{ kc of correlated three-level pulses. This correlation between the adjacentpulses may be viewed as introducing intersymbolinterference into the transmitted signal in an artificialmanner.

F.34

Coding and decoding

Example: Consider { } 0010110=kb where the first bitis a startup bit.

Encoding:{ }kb : 0 0 1 0 1 1 0{ }ka : -1 -1 +1 -1 +1 +1 -1{ }kc : -2 0 0 0 +2 0

Decoding:Using the equation 1−−= kkk aca , i.e.,

If 2+=kc , decide that 1+=ka .If 2−=kc , decide that 1−=ka .If 0=kc , decide opposite of the previousdecision.

F.35

Duobinary Signaling: Impulse response and frequency spectrum

Let us now examine an equivalent model of theduobinary encoder. The Fourier transfer of a delay canbe described as bfTe π2− , therefore, the transfer functionof the encoder is )( fH I is

bfTjI efH π21)( −+=

The transfer function of the Nyquist channel is

<

=otherwise0

2/11)( b

NTf

fH

F.36


The overall equivalent transfer function )( fH of theis then given by

bfTj

fTjfTjfTj

fTjbNI

fTe

eee

e

TffHfHfH

b

bbb

b

ππ

πππ

π

cos2

)(

)1(

2/1||for )()()(2

−

−−

−

=

+=

+=

<=

H(f) has a gradual roll-off to the band edgewhich can be easily implemented

F.37


The corresponding impulse response h(t) is found by takingthe inverse Fourier transform of H(f)

)()/sin(

/)()/sin(

/)/sin(

/)()/)(sin(

/)/sin()(

2

b

bb

bb

b

b

b

bb

bb

b

b

TttTtT

TTtTt

TtTt

TTtTTt

TtTtth

−=

−+=

−−

+=

ππ

ππ

ππ

ππ

ππ

F.38


Notice that there are only two nonzero samples, at T-secondintervals, give rise to controlled ISI from the adjacent bit.The introduced ISI is eliminated by use of the decodingprocedure.

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Eye Pattern and Equalization