DC Digital Communication MODULE III PART2

BASE BAND DATA

TRANSMISSION

BASE BAND DATA BASE BAND DATA

TRANSMISSIONTRANSMISSIONMODULE 3 MODULE 3

PART IIPART II

DIGITAL COMMUNICATION

�� Module III Module III –– Part IIPart II

Base band data transmission:Base band data transmission: - Discrete PAM

signals – Power spectra of discrete PAM signals -

Inter symbol interference- Nyquist’s criterion for

distortion less baseband binary transmission - Eye

diagram - Adaptive equalization.

Line Codes or Digital Formats

� To send the encoded digital data over a communication channel we

require the use of a format or waveform for representing the binary

data.

� Such digital formats are also called line codes.

� Line codes specify how 0’s and 1’s are represented in a

communication channel.


0 1 1 0 1 0 0 0 1

UNIPOLAR NRZ

SIGNALING

POLAR NRZ SIGNALING

UNIPOLAR RZ SIGNALING

BIPOLAR AMI SIGNALING


0 1 1 0 1 0 0 0 1 1

UNIPOLAR SIGNALING

SPLIT PHASE OR

MANCHESTER ENCODING

POLAR RZ SIGNALING

POLAR QUARTERNARY

SIGNALING


�� UnipolarUnipolar NRZ format:NRZ format: Symbol 1 is represented by transmitting a

pulse of constant amplitude for the duration of the symbol and the

symbol 0 is represented by switching off the pulse. This type of

format is referred to as on off signaling.

�� Polar NRZ format:Polar NRZ format: Symbols 0 and 1 are represented by pulses of

equal positive and negative amplitudes.

�� UnipolarUnipolar RZ format:RZ format: A rectangular pulse of half the symbol duration

is used for a 1 and no pulse for 0.

�� Bipolar NRZ format:Bipolar NRZ format: Positive and negative pulses of equal

amplitudes are used alternatively to represent symbol 1 and no

pulse for symbol 0.

�� Split phase or Manchester encoding:Split phase or Manchester encoding: Symbol 1 is represented by a

positive pulse followed by a negative pulse, with both pulses being

of equal amplitude and half symbol wide. For sym,bol 0 the polarities

are reversed.

Line Codes or Digital Formats�� Polar RZ format:Polar RZ format: A rectangular pulse of half the symbol duration and

positive amplitude is used to represent symbol 1. A similar

rectangular pulse of equal negative amplitude is used to represent

symbol 0.

�� Polar quaternary format:Polar quaternary format: Here the bits are grouped in to pairs and

the pairs of bits are represented by 4 voltage levels.

�� MM--ary coding:ary coding: In this format we group together k bits and hence we

get M=2k distinct levels. According to the occurrence of a particular

group of bits we transmit the corresponding symbol.

1/210

3/211

-1/201

-3/200

VoltageBit Pair

Properties of Line Codes

�� Power efficiency:Power efficiency: For a given bandwidth and a specified detection error

probability the transmitted power should be as small as possible.

�� Error detection and correction capacity:Error detection and correction capacity: It should be possible to detect

and correct errors. In the bipolar case a single error will cause bipolar

violation and can be easily detected.

�� Average DC value should be as small as possible:Average DC value should be as small as possible: When DC and AC are

transmitted simultaneously through cable pairs AC coupling need to be

used. If the DC value is small ac coupling can easily be used.

�� Adequate timing content:Adequate timing content: It should be possible to extract timing or clock

information from the signal.

�� Transparency:Transparency: It should be possible to transmit a digital signal correctly regardless of the pattern of 1’s and 0’s. If the data is coded in such a way that for every possible sequence of data the coded signal is received faithfully the code is transparent.

�� Transmission BW:Transmission BW: Line codes should make the BW as small as possible

Power spectra of discrete PAM signals

�

Base band transmission of PAM modulated

signals.

PULSE

GENERATOR

TRANSMITTING

FILTERCHANNEL

RECEIVING

FILTER

DECISION

DEVICE

TRANSMITTER

RECEIVER

Clock Pulses

kb

)(tx

( )fHT ( )fHC

( )fHR )(tybiTttimeatsample =

)( ityThreshold

Output

binary data

Input binary

Data


signals.

� Consider a discrete pulse amplitude modulation system in which the

amplitude of the transmitted pulses is varied in a discrete manner in

accordance with the given digital data.

� The signal applied to the input of the system consists of a binary

data sequence bk with a bit duration Tb , bk is in the form of 1 or 0.

This signal is applied to a pulse generator, producing the pulse

waveform.

� g(t) denotes a shaping pulse with its value at time t=0 defined by

g(0) =1. The amplitude Ak depends on the identity of the input bit bk.

∑∞

−∞=

−=k

bk kTtgAtx )()(

−

+=

0

1

symbolisbifa

symbolisbifaA

k

k

k


signals.� The PAM signal is passed through a transmitting filter of transfer

function of HT(f).

� The resulting filter output defines the transmitted signal which is modified in a deterministic fashion as a result of transmission through the channel of transfer function HC(f).

� At the receiver the signal is passed through a receiver filter of transfer function HR(f).

� This filter output is sampled synchronously with the transmitter with the sampling instants being determined by a clock or timing signal that is usually extracted from the receiving filter output.

� The sequence of samples thus obtained is used to reconstruct theoriginal data sequence by means of a decision device.

� The amplitude of each sample is compared to a threshold. If the threshold is exceeded a decision is made in favour of symbol 1. If the threshold is not exceeded a decision is made in favour of symbol 0.

Inter Symbol interference

1 00

1 1 10

Pulse dispersion causes incorrect determination of bits at receiver


•

•

•

••

•

•

•

••

••

•

BIT STREAM

OUTPUT FROM

PERFECT FILTER

OUTPUT FROM

IMPERFECT

FILTER

•


� In the base band PAM system assume that the channel is noiseless. Even then some errors occur in the bit determinations due to thedispersive nature of the communication channel.

� The receiving filter output of the system may be written as

� Where µ is a scaling factor and p(t) is a pulse whose shape is different from g(t).

� When the pulse Ak g(t) is applied to the input of the system we get µAkp(t) at the output of the system after passing through the cascade connection of transmitting filter, channel and receiving filter.

� Where P(f) and G(f) are the Fourier transform of p(t) and g(t) respectively.

( ) ( )k b

k

y t A p t kTµ∞

=−∞

= −∑

( ) ( ) ( ) ( ) ( )k k T C RA P f A G f H f H f H fµ =

)()()()()( fHfHfHfGfP RCT=µ


� The receiving filter output y(t) is sampled at time ti=iTb (with i taking

integer values) yielding

� The first term represents the contribution of the ith transmitted bit.

� The second term represents the residual effect of all other

transmitted bits on the decoding of the ith received bit.

� This residual effect is called inter symbol interference.

( ) ( )i k b b

k

y t A p iT kTµ∞

=−∞

= −∑ [ ]( )k b

k

A p i k Tµ∞

=−∞

= −∑

[ ]( )

(0) ( )i k b

kk i

A p A p i k Tµ µ∞

=−∞≠

= + −∑

[ ]( )

assuming ( ) (0) 1i k b

kk i

A A p i k pTµ µ∞

=−∞≠

= + =−∑


� In the absence of ISI, y(ti)=µAi i.e., ith transmitted bit can be

decoded correctly.

� The unavoidable presence of ISI introduces errors in the decision

device at the receiver output.

� The channel transfer function Hc(f) and the pulse spectrum G(f) are

specified and so we have to adjust HT(f) and Hc(f) so as to enable

the receiver to correctly decode the received sequence of sample

values y(ti).

Nyquist’s criterion for distortion less baseband binary data transmission� When the frequency response of the channel HC(f) and transmitted

pulse response G(f) are specified the problem is to determine the

frequency responses of the transmit and receive filters HT(f) and

HC(f) so as to reconstruct the original binary data sequence bk.

� The correct decoding requires that y(ti)=µAi and the contribution

produced by other pulses be zero.

� For this the pulse p(t) should be controlled such that

� If p(t) satisfies the condition in eqn (1), y(ti)=µAi for all i which

implies zero intersymbol interference.

[ ] ........( 11

( )0

)b

for i kp i k T

for i k

=− =

≠

Nyquist’s first method

� Time limited pulses cannot be band limited and vice versa. If we are using perfect time limited pulses part of its spectra are suppressed by a band limited channel. This causes pulse distortion. i.e., spreading of pulses.

� One solution is to use pulses which are bandlimited so that they can be transmitted through a bandlimited channel. But bandlimited pulses cannot be time limited and there is spreading of pulses.

� So we cannot avoid ISI even if we are using bandlimited or timelimited pulses. It is inherent in a finite BW channel.

� Pulse amplitudes can be detected correctly if there is no ISI at the decision making instants. This can be accomplished by a properlyshaped bandlimited pulse.

Nyquist’s criterion for zero ISI

� We choose a pulse shape that has a non zero amplitude at its

centre t = 0 and zero amplitudes at t = ±nTb, (n=1,2,3,….) where Tb

is the separation between successive transmitted pulse.

� A pulse satisfying this criterion causes zero ISI at all the remaining

pulses centers.

� Consider the several pulses shown below centered at t=0,Tb, 2Tb,

3Tb,….. If we sample at 0, Tb,2Tb…. the sample value consists of

the amplitude of one pulse only with no interference from the

remaining pulses.

� A pulse satisfying the above criteria is the sinc pulse.

1 0( )

0 b

for tp t

for t nT

==

= ±

1 0sinc( )

0

b

b

for tf t

for t nT

==

= ±


1 1 1 1 1

0 Tb 2Tb 3Tb 4Tb 5Tb


� The Fourier transform of this pulse is

� It has a bandwidth of 1/2Tb. Using this pulse we can transmit at a

rate of 1/Tb pulses per second without ISI over a bandwidth of 1/2Tb.

� The parameter B0 is called Nyquist BW and it defines the minimum

transmission BW for zero ISI.

( ) ( )b bP f T rect fT=0 0

1

2

B 2

frectO

Br

021 B

0B− 0B

0 0

0

1 2 0( )

0

B f BP f

f B

≤ ≤=

>

( )0 0( ) 1 2 2 P f B rect f B=

0

1

2 b

BT

=

Nyquist’s criterion for distortion less

baseband binary data transmission

021 B

0B− 0B

-4Tb -3Tb -2Tb -1Tb 0 1Tb 2Tb 3Tb 4Tb

1

Nyquist’s criterion for zero ISI1 0 1 1 0 0 1 1

Nyquist’s criterion for zero ISI� Nyquist method solves the problem of ISI with minimum BW

possible but there are two practical difficulties in it.

� The magnitude characteristics of P(f) should be flat from –B0 to

+B0 and zero else where. This is physically unrealizable because of

the abrupt transitions at the band edges ±B0.

� The function p(t) decreases slowly because of the abrupt

discontinuity of P(f) at B0. So there is no margin of error in sampling

times at the receiver.

� Since the sine function extends to infinity the tails of several pulses

get superimposed and we get large amplitudes at points away from

pulse centers.

� So a little error in sampling time causes incorrect decision and bit

errors.

Sampling Errors Produces wrong results1 1 0 1 1

Superposition

Of several sidelobes produces

large amplitude near to the

sampling point

Incorrect sampling time

produces wrong decisions

about the identity of the bit

Raised cosine spectrum� The ideal Nyquist channel solution of zero ISI is practically

unacceptable because of the following reasons.

� (i) It is impossible to achieve a perfectly rectangular response.

� (ii) Due to abrupt discontinuities in the spectrum the sinc

functions extends to infinity and it does not decay abruptly. It

leaves a tail extending to infinity. When such tails are

superimposed large amplitudes are available slightly near to

sampling points. So the sampling should be performed

without error.

� One solution to these problems is to obtain a sinc pulse that decays fast. For that the ideal spectrum with fast transitions should be converted to a spectrum with edges rolling off to zero.

� A spectrum meeting the above conditions is raised cosine spectrum which is defined as follows:

Raised cosine spectrum

0

1

0 0 1

1 0

1

1

0 1

1

2

11 cos

4 2 2

( )

0

0

2

2

for

P f for

fo

f f

f f

B

f f

B B fB

fr

f

f B

π − +

≤ ≤

≤ ≤ −

=

≥ −

−

� The frequency parameter f1 and BW are related by

11 0

0

1 (1 ) andf

f BB

α α= − = −


0=α

5.0=α

1=α0=α

1=α

5.0=α

0 1 2/3 21−2/3−2−

)(2 0 fPB

)(tp 0/ Bf

0 11 22 t

0B0B− 02B02B−


� Parameter α is called roll off factor. It indicates the excess BW over

the ideal solution, Bo.

� BW is extended from B0 to an adjustable value between B0 and 2B0

� 2B0-f1 is defined as transmission BW.

� The frequency response P(f) normalized by multiplying by 2B0 is

plotted for various values of α ie,0, 0.5 and 1.

� The time response p(t) is the inverse FT of the frequency response

P(f).

0 1 2 TTransmission BW B B f= −0 02 (1 )B Bα= − −

0 0B Bα= + 0(1 )B α= +

00 2

0

cos2( ) sinc(2 )

1 16

B tp t B t

B t

παα

= −


� The time response p(t) consists of the products of two factors, the factor sinc 2Bt characterizing the ideal Nyquist channel and a second factor which decreases as 1/t2 for large t.

� The first factor ensures that the zero crossings of p(t) are at desired sampling instants of time iTb.

� The second factor reduces the tails of the pulses considerably below that obtained from the ideal Nyquist channel.

� So transmission of binary waves using such pulses is relatively insensitive to sampling time errors.

Eye Pattern

� Eye pattern is used to study inter symbol interference.

� For this we apply the received wave to the vertical deflection plates

of an oscilloscope and a saw tooth waveform at the transmitted

symbol rate 1/T to the horizontal deflection plates.

� The waveforms in successive symbol intervals will be translated in

to one interval in the oscilloscope display.

� The resulting display is called eye pattern.

� The interior region of the pattern is called eye opening.

� The eye pattern provides a number of information as given below:

� 1. The width of the eye opening defines the time interval over

which the wave can be sampled without error from intersymbol

interference. The preferred time for sampling is the instant of

time at which the eye is open widest.

Eye Pattern

� 2. The sensitivity of the system to timing error is determined by the

rate of closure of the eye as the sampling time is varied.

� 3. The height of the eye opening, at a specified sampling time,

defines the margin over channel noise.

� When the effect of inter symbol interference is severe, traces from

the upper portion of the eye pattern cross traces from the lower

portion, with the result that the eye is completely closed.

� In such a situation it is impossible to avoid errors due to the

combined effects of ISI and channel noise.

� In the case of an M-ary system the eye pattern consists of M-1 eye

openings stacked vertically one on the other where M is the number

of discrete amplitude levels used to construct the input signal.

Eye Pattern

1 0 1 1 0 0 1

Eye

Pattern

Tb

Tb

Eye Pattern

� Eye pattern:Display on an oscilloscope which sweeps the system

response to a baseband signal at the rate 1/Tb (Tb symbol duration)

time scale

amplitude scale

Noise margin

Sensitivity to

timing error

Distortion

due to ISI

Timing jitter

Eye Pattern

Time interval during

which waveform can be

sampled

Margin over

noise

Distortion of

zero crossings

Distortion at

sampling time

Best sampling

time

Slope indicates

sensitivity to

timing error

time scale

amplitude scale

Eye Pattern

� Perfect channel (no noise and no ISI)

Eye Pattern

�AWGN (Eb/N0=20 dB) and no ISI

Eye Pattern� AWGN (Eb/N0=10 dB) and no ISI

Equalization� There are various amplifiers, filters and reactive circuit elements

throughout a communication system.

� All these components are band limited to a specific bandwidth W beyond which it cannot faithfully communicate a signal.

� Most of the practical communication channels can be modeled as Linear Time Invariant Systems. An LTI system has constant amplitude and linear-phase frequency response

� If we transmit the digital symbols at a rate that require slightly greater BW than available BW, attenuation may occur but no interference.

� But for practical systems that are non ideal the amplitude response is not flat or the phase response linear with frequency.

� Transmission of digital symbols through such non ideal channel at a transmission rate exceeding BW results in interference among a number of adjacent symbols. Such distortion is called Inter Symbol Interference.

Equalization� Intersymbol interference arises because of the spreading of a

transmitted pulse due to the dispersive nature of the channel, which

results in overlap of adjacent pulses.

� If the channel is known precisely it is always possible to make the

ISI small by using a suitable pair of transmit and receive filters so as

to control the over all pulse shape.

� Regardless of which particular pulse shape has been chosen some

amount of residual ISI remains in the output signal as a result of

imperfect filter design, incomplete knowledge of channel

characteristics etc.

� The goal of equalizers is to eliminate intersymbol interference (ISI)

and the additive noise as much as possible and to overcome the

negative effects of the channel

Equalization� In general, equalization is partitioned into two broad categories

� (i) Maximum Likelihood Sequence Estimation (MLSE) which entails making measurement of channel impulse response and then providing a means for adjusting the receiver to the transmission environment. (Example: Viterbi equalization)

� (ii) Equalization with filters, uses filters to compensate the distorted pulses.

� These type of equalizers can be grouped as preset or adaptive equalizers.

� Preset equalizers assume that channel is time invariant and try to find H(f) and design equalizer depending on H(f). The examples of these equalizers are zero forcing equalizer, minimum mean square error equalizer, and decision feedback equalizer.

� Adaptive equalizers assume that channel is a time varying channel and try to design equalizer filter whose filter coefficients are varying in time according to the change of channel, and try to eliminate ISI and additive noise at each time. The implicit assumption of adaptiveequalizers is that the channel is varying slowly.

Adaptive Equalization

� In the design of equalizers we assume that the channel

characteristics, either impulse response or frequency response

were known at the receiver.

� In most communication systems that uses equalizers the channel

characteristics are not known before hand and the channel response

is time variant.

� In such a case the equalizers are designed to be adjustable to the

channel response and to be adaptive to the time variations in the

channel response.

� The process of equalization is said to be adaptive when the

equalizer is capable of adjusting its coefficients continuously during

the transmission of data.

� The equalization is performed according to a well defined algorithm

to modify the received data.

� This type of equalizers are called adaptive equalizers.


DELAY

T

DELAY

T

DELAY

T

DELAY

T

DELAY

T

ww

SAMPLED

INPUT

SIGNAL

DESIRED

RESPONSEERROR

VARIABLE

WEIGHTS

)(nTx

)( TnTx − )2( TnTx − )3( TnTx − )( TMTnTx +−

0w 1w 2w 2−Mw 1−Mw

)(nTd)(nTy

)(nTe

Adaptive Equalization� Figure shows a popular structure used to design adaptive

equalizers.

� The structure is a tapped delay line filter that consists of a set of delay elements, a set of multipliers connected to delay line taps, a corresponding set of adjustable tap weights and a summer for adding the multiplier outputs.

� Let the sequence x(nT) appearing at the output of the receiving filter be applied to the input of the tapped delay line filter producing the output

� The M tap weights constitute the adaptive filter coefficients.

� The tap spacing is chosen equal to the symbol duration T of the transmitted signal.

� The following steps are carried out in the adaptation process.

1

0

( ) ( )M

i

i

y nT w x nT iT−

=

= −∑ tapitheofWeightw th

i ⇒ tapsofnumberTotalM ⇒

Adaptive Equalization� (i) A known sequence d(nT) is transmitted, and in the receiver the

resulting response sequence y(nT) is obtained by measuring The

filter output at the sampling instants.

� (ii) Viewing the known transmitted sequence d(nT) as the desired

response, the differences between it and the response sequence

y(nT) is computed. The difference is called error sequence,

denoted by e(nT)

� (iii) The error sequence e(nT) is used to estimate the direction in

which the weights wi of the filter are changed so as to make

them approach their optimum settings.

1,......2,1,0 ),()()( −=−= NnnTynTdnTe

sequencethelength ofTotalN ⇒

Adaptive Equalization� A criterion used for optimization is the total energy of the error sequence

defined by

� The optimum values of the tap weights result when

the total energy E is minimized.

� So an algorithm is required that adjusts the tap weights of the filter in a

recursive manner.

� The present estimate of each tap weight is updated by incrementing it by

a correction term proportional to the error signal at that time.

� A commonly used algorithm is Least Mean Square Algorithm.

� According to LMS algorithm the tap weights are adapted as follows.

∑−

=

=1

0

2 )(N

n

nTeE

1210 .,,.........,, −oMooo wwww

UPDATED

VALUE OF

THE Kth TAP

WEIGHT

OLD VALUE

OF THE Kth

TAP WEIGHT

STEP SIZE

PARAMETER

INPUT SIGNAL

APPLIED TO THE

Kth TAP WEIGHT

ERROR

SIGNAL= + x x


� The adaptation constant µ controls the amount of correction applied

to the old estimate to produce the updated estimate

� The correction depends on the filter input and the error

signal both measured in time nT

� By a proper choice of the adaptation constant, the use of the

recursive equation helps the adjustment of the tap weights move

toward their optimum settings in a step by step fashion.

ˆ ˆ( ) ( ) ( ) ( )iw nT T w nT e nT x nT iTµ+ = + −

ˆ ( )i oi w nT Present estimate of the optimum weight w for tap i at time nT.⇒

ˆ ( )w nT T Updated est e imat+ ⇒ Adaptation constant µ ⇒i 0,1,2,.............,M -1=

)(ˆ nTw )(ˆ TnTw +)( iTnTx −

)(nTe


OLD VALUE

CORRECTION

UPDATED

VALUE

UNIT DELAY

w

++

)()( nTeiTnTx −µ

)(ˆ TnTw +)(ˆ nTw i

Signal Flow Graph Of

Least Mean Square

Algorithm

Adaptive Equalization� LMS algorithm requires a knowledge of the desired response d(nT) and

the filter response y(nT) to form the error signal e(nT).

� For this, prior to data transmission, the equalizer is adjusted under the

guidance of a training sequence transmitted through the channel.

� Thus there are two modes of operation for the adaptive equalizer

(i) Training mode (ii) Decision directed mode.

� During the training mode a known sequence is transmitted and a

synchronized version of it is generated at the receiver where it is

applied to the equalizer as the desired response.

� The tap weights of the equalizers are then adjusted in accordance with

LMS algorithm.

� When the training process is completed, the adaptive equalizer is

switched to its second mode of operation.

� In this mode of operation, the error signal is defined by

)()(ˆ)( nTynTdnTe −=d(nT) symboldtransmitte the of estimate correct final(nT)d ⇒ˆ


DECISION

DEVICE

TRAINING

SEQUENCE

GENERATOR

W

ADAPTIVE

EQUILIZER

)(nTx )(nTy )(nTd

)(ˆ nTd

)(nTe

+

−

12

DECISION

DIRECTED

MODE

TRAINING

MODE

Duo Binary Encoding� Consider a binary input sequence b(k) consisting of uncorrelated

binary symbols 1 and 0 each with duration Tb.

� This sequence when applied to a pulse amplitude modulator

produces a two level sequence whose amplitude ak is denoted by

� When this sequence is applied to a duo binary encoder, it is

converted in to a three level output consisting of -2, 0, +2

� One of the effects of the transformation described above is to

change the input sequence bk of uncorrelated binary digits in to

sequence ck of correlated digits.

� This correlation between adjacent transmitted levels may be viewed

as introducing ISI in to the transmitted signal in a deliberate manner.

� This type of coding is called correlative coding also.

−

+=

0 1

1 1

is bif symbol

is bif symbola

k

k

k

Duo Binary Encoding

DELAY

Tb

IDEAL

CHANNEL

Hc(f)Sample at

time t=kTb

Output

Sequence ck+

+

Filter

TRANSFER

FUNCTION

OF SYSTEM

H(f)

kb kc

BIPOLAR

CONVERTER

kakb

Duo Binary Encoding- An Example

1−+= kkk aac

+1+1+1-1-1

a5a4a3a2a1

011011 =+−=+= aac

211122 −=−−=+= aac

011233 =−=+= aac

211344 =+=+= aac

211455 =+=+= aac

1ˆˆ −−= kkk aca

10 +=a 1ˆ0 +=a

220-20

c5c4c3c2c1

110ˆˆ011 −=−=−= aca

112ˆˆ122 −=−−−=−= aca

110ˆˆ233 +=−−=−= aca

112ˆˆ344 +=−+=−= aca

112ˆˆ455 +=−+=−= aca

ENCODING DECODING

Duo Binary Encoding

� The polar sequence ak is first passed through a simple filter

consisting of the parallel combination of a direct path and an ideal

element producing a delay of Tb seconds.

� For every unit impulse applied to the input of this filter we get two

unit impulses spaced Tb seconds apart.

� The output of this filter in response to the incoming polar sequence

ak is then passed through the channel of transfer function Hc(f).

� A continuous waveform is thus produced at the channel output. The

resulting waveform is sampled uniformly every Tb seconds, thereby

producing the duo binary encoded sequence.

� The effect of the channel is included in this encoding operation.

Duo Binary Encoding� The cascade connection of the delay line filter and the channel is

called a duo binary conversion filter.

� An ideal delay element producing a delay of Tb seconds, has the

transfer function so that the transfer function of the delay

line filter is

� Hence the over all transfer function of this filter connected in

cascade with the ideal channel Hc(f) is

bfTje

π2−

bfTje

π21

−+

[ ]bfTj

C efHfHπ2

1)()(−+=

[ ]bbbb fTjfTjfTjfTj

C eeeefHππππ −−− += ..)(

( )[ ]bbb fTjfTjfTj

C eeefHπππ −−+= )(

( ) bfTj

bC efTfHππ −= cos)(2

Duo Binary Encoding� For an ideal channel with band width B0=1/2Tb

� Thus the over all frequency response has the form of a half cycle

cosine function as shown by

≤

=otherwise

T/f fH

b

C 0

21 1)(

( ) ≤

=−

Otherwise

TfefTfH

b

fTj

bb

0

2/1 cos2)(

ππ

bT21bT21−

2)( fH

bT21

2/π−

)( fH∠

0bT21

2/π

Duo Binary Encoding� The corresponding value of the impulse response consists of two

sinc pulses time displaced by Tb seconds as given by

bb

bb

b

b

TTt

TTt

Tt

Ttth

/)(

)/)(sin(

/

)/sin()(

−−

+=ππ

ππ

bb

b

b

b

TTt

Tt

Tt

Tt

/)(

)/sin(

/

)/sin(

−+=

ππ

ππ

)(

)/sin(2

tTt

TtT

b

bb

−=

ππ

)(

)/sin()(

2

tTt

TtTth

b

bb

−=

ππ

Duo Binary Encoding

bT bT2 bT3bT−bT2−bT3− 0

b

b

Tt

Tt

/

)/sin(

ππ

bb

bb

TTt

TTt

/)(

)/)(sin(

−−

ππ

Duo Binary Encoding


b

b

Tt

Tt

/

)/sin(

ππ

bb

bb

TTt

TTt

/)(

)/)(sin(

−−

ππ


0.1

)(th

Duo Binary Encoding

bT21bT21−

2)( fH

TRANSFER

FUNCTION

OF SYSTEM

H(f)

kb kc

Duo Binary Encoding� The over all impulse response has only two distinguishable values

at the sampling instants.

� The original data ak may be detected from the duo binary coded

sequence ck by subtracting the previous decoded binary digit from

the currently received digit ck .

� Let represent the estimate of the original digit ak as detected by

the receiver at time t = kTb. Then

� If ck is received without error and if the previous estimate at

time t = (k-1)Tb also corresponds to a correct decision, then the

current estimate will be correct too.

� This method of using a stored estimate of the previous symbol in the

estimation of the current symbol is called decision feedback.

ka

1ˆˆ −−= kkk aca

ka

1ˆ −ka

Duo Binary Encoding With Precoder� In the case of duo binary decoding once errors are made they tend to

propagate as a decision on the current binary digit bk depends on the

correctness of the decision made on the previous binary digit bk-1

� A practical means to avoid this error propagation is to employ

precoding before duo binary coding.

� The precoding operation converts the input binary sequence bk to

another binary sequence dk as below

� The pre-coder output is converted to ak with polar representation as

� The resulting output ak is then applied to duo binary coder thereby

producing the sequence ck as follows

1−⊕= kkk dbd

1−+= kkk aac

=+

=−=

11

01

k

k

k if d

, if da

Duo Binary Encoding With Precoder

� What ever the value of dk-1, we get

=

=±=

10

02

k

k

k b if

, if bc

DELAY

Tb

POLAR

CONVERTER

DUOBINARY

ENCODER

kb kd

1−kd

ka

bkTt

atSample

=

kc

PRE-CODER


0 1 =−kdLet

0 =kbLet

000 =⊕=kd

1−+= kkk aac

211 −=−+−=kc

1 =kbLet

101 =⊕=kd

1−+= kkk aac

011 =−=kc

1−⊕= kkk dbd

=+

=−=

11

01

k

k

k if d

, if da

1−⊕= kkk dbd

ENCODING WITH dk-1=0

=+

=−=

−

−−

11

01

1

1

1

k

k

k if d

, if da


1 1 =−kdLet

0 =kbLet

110 =⊕=kd

1−+= kkk aac

211 =+=kc

1 =kbLet

011 =⊕=kd

1−+= kkk aac

011 =+−=kc

=+

=−=

11

01

k

k

k if d

, if da

1−⊕= kkk dbd1−⊕= kkk dbd

ENCODING WITH dk-1=1

=+

=−=

−

−−

11

01

1

1

1

k

k

k if d

, if da

Duo Binary Encoding With Precoder� So the following decision rule may be adapted for constructing the

decoded binary sequence bk at the receiver output.

� The detector consists of a rectifier, the output of which is compared

to a threshold of 1V, and the original binary sequence is thus

detected.

� No knowledge of any input sample other than the present sample is

required. Hence error propagation cannot occur in the detector.

≤

>=

Vc, if symbol

Vc, if symbolb

k

k

k11

10 ˆ

RECTIFIERTHRESHOLD

DETECTOR

kc kckb

Threshold = 1Vc,if b kk 10 ˆ >=

Vc,if b kk 11 ˆ ≤=


An Example

kbsequencebinaryInput

kd uencebinary seq Precoded

ka tionrepresentaPolar

kc outputcoder binary Duo

0 1 1 0 1 0 0

0 0 1 0 0 1 1

1 11 1 1 11 1 - - - - ++++

2- 0 0 2- 0 2 2

kb uencebinary seq Decoded ˆ 0 1 1 0 1 0 0

1

Let us Start with dk-1= 1

1−⊕= kkk dbd

=+

=−=

11

01

k

k

k if d

, if da

1−+= kkk aac

≤

>=

Vc, if symbol

Vc, if symbolb

k

k

k11

10 ˆ


An Example

kbsequencebinaryInput

kd uencebinary seq Precoded

ka tionrepresentaPolar

kc outputcoder binary Duo

0 1 1 0 1 0 0

1 1 0 1 1 0 0

111 1 1 11 1 ++−++−−−

2 0 0 2 0 2 2 −−

kb uencebinary seq Decoded ˆ 0 1 1 0 1 0 0

0

WE GET THE SAME DECODED

OUTPUT WHEN WE START

WITH dk-1=0 OR dk-1=1

Let us Start with dk-1= 0

Documents

DC Digital Communication MODULE III PART2