Correlative level coding

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Hyeong-Seok YuHyeong-Seok YuVada Lab.Vada Lab.

[email protected]@vada1.skku.ac.kr

Baseband Pulse TransmissionBaseband Pulse TransmissionCorrelative-Level CodingCorrelative-Level Coding

Baseband M-ary PAM TransmissionBaseband M-ary PAM TransmissionTapped-Delay-Line EqualizationTapped-Delay-Line Equalization

Eye PatternEye Pattern

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Correlative-Level CodingCorrelative-Level Coding

Correlative-level coding (partial response signaling) adding ISI to the transmitted signal in a controlled

manner Since ISI introduced into the transmitted signal is

known, its effect can be interpreted at the receiver A practical method of achieving the theoretical

maximum signaling rate of 2W symbol per second in a bandwidth of W Hertz

Using realizable and perturbation-tolerant filters

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Duobinary Signaling Duobinary Signaling Dou : doubling of the transmission capacity of a straight binary

system

Binary input sequence {bk} : uncorrelated binary symbol 1, 0

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01k

kk

if symbol b isa

if symbol b is

+= −

1−+= kkk aac

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Duobinary Signaling Duobinary Signaling Ideal Nyquist channel of bandwidth

W=1/2Tb

)exp()cos()(2

)exp()]exp())[exp((

)]2exp(1)[()(

bbNyquist

bbbNyquist

bNyquistI

fTjfTfH

fTjfTjfTjfH

fTjfHfH

πππππ

π

−=

−−+=

−+=

≤

=otherwise

TffH b

Nyquist

2/1||

,0

,1)(

2cos( )exp( ), | | 1/ 2( )

0,b b b

I

fT j fT f TH f

otherwise

π π− ≤=

)(

)/sin(

/)(

]/)(sin[

/

)/sin()(

2

tTt

TtT

TTt

TTt

Tt

Ttth

b

bb

bb

bb

b

bI

−=

−−+=

ππ

ππ

ππ

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Duobinary Signaling Duobinary Signaling The tails of hI(t) decay as 1/|t|2, which is a faster rate of decay

than 1/|t| encountered in the ideal Nyquist channel. Let represent the estimate of the original pulse ak as

conceived by the receiver at time t=kTb

Decision feedback : technique of using a stored estimate of the previous symbol

Propagate : drawback, once error are made, they tend to propagate through the output

Precoding : practical means of avoiding the error propagation phenomenon before the duobinary coding

^

ka

^

1

^

−−= kkk aca

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Duobinary Signaling Duobinary Signaling

{dk} is applied to a pulse-amplitude modulator, producing a corresponding two-level sequence of short pulse {ak}, where +1 or –1 as before

11 1

0k k

k

symbol if either symbol b or d isd

symbol otherwise−

=

1−+= kkk aac

1−⊕= kkk dbd

10

02k

kk

if data symbol b isc

if data symbol b is

= ±

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Duobinary Signaling Duobinary Signaling |ck|=1 : random guess in favor of symbol 1 or 0

0,1||

1,1||

isbsymbolsaycIf

isbsymbolsaycIf

kk

kk

><

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Modified Duobinary Signaling Modified Duobinary Signaling Nonzero at the origin : undesirable Subtracting amplitude-modulated pulses spaced 2Tb second

1−+= kkk aac( ) ( )[1 exp( 4 )]

2 ( )sin(2 )exp( 2 )

IV Nyquist b

Nyquist b b

H f H f j fT

jH f fT j fT

ππ π

= − −

= −

2 sin(2 )exp( 2 ), | | 1/ 2( )

0,b b b

IV

j fT j fT f TH f

elsewhere

π π− ≤=

2

sin( / ) sin[ ( 2 ) / ]( )

/ ( 2 ) /

2 sin( / )

(2 )

b b bIV

b b b

b b

b

t T t T Th t

t T t T T

T t T

t T t

π ππ π

ππ

−= −−

=−

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Modified Duobinary SignalingModified Duobinary Signaling

precoding

2

21 1

0

k k k

k k

d b d

symbol if either symbol b or d is

symbol otherwise

−

−

= ⊕

=

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Modified Duobinary SignalingModified Duobinary Signaling

|ck|=1 : random guess in favor of symbol 1 or 0

| | 1, 1

| | 1, 0k k

k k

If c say symbol b is

If c say symbol b is

><

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Generalized form of correlative-level codingGeneralized form of correlative-level coding |ck|=1 : random guess in favor of symbol 1 or 0

Type of Type of classclass

NN ww00 w w11 w w22 w w33 w w44 commentscomments

II 22 1 11 1 DuobinaryDuobinary

IIII 33 1 2 11 2 1

IIIIII 33 2 1 2 1 ––11

IVIV 33 1 0 1 0 ––11 ModifiedModified

VV 55 -1 0 2 0 -1-1 0 2 0 -1

∑−

−=

1

sin)(N

n bn n

T

tcwth

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Baseband M-ary PAM Trans.Baseband M-ary PAM Trans.

Produce one of M possible amplitude level

T : symbol duration 1/T: signaling rate, symbol per

second, bauds Equal to log2M bit per second

Tb : bit duration of equivalent binary PAM :

To realize the same average probability of symbol error, transmitted power must be increased by a factor of M2/log2M compared to binary PAM

MTT b 2log=

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Tapped-delay-line equalization Tapped-delay-line equalization

Approach to high speed transmission Combination of two basic signal-processing operation Discrete PAM Linear modulation scheme

The number of detectable amplitude levels is often limited by ISI

Residual distortion for ISI : limiting factor on data rate of the system

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Equalization : to compensate for the residual distortion Equalizer : filter

A device well-suited for the design of a linear equalizer is the tapped-delay-line filter

Total number of taps is chosen to be (2N+1)

∑−=

−=N

Nkk kTtwth )()( δ

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P(t) is equal to the convolution of c(t) and h(t)

nT=t sampling time, discrete convolution sum

∑∑

∑

−=−=

−=

−=−∗=

−∗=∗=

N

Nkk

N

Nkk

N

Nkk

kTtcwkTttcw

kTtwtcthtctp

)()()(

)()()()()(

δ

δ

∑−=

−=N

Nkk TkncwnTp ))(()(

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Nyquist criterion for distortionless transmission, with T used in place of Tb, normalized condition p(0)=1

Zero-forcing equalizer Optimum in the sense that it minimizes the peak distortion(ISI) – worst

case Simple implementation The longer equalizer, the more the ideal condition for distortionless

transmission

±±±==

=

≠=

=Nn

n

n

nnTp

.....,,2,1

0

,0

,1

0,0

0,1)(

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Adaptive Equalizer Adaptive Equalizer

The channel is usually time varying Difference in the transmission characteristics of the individual links that

may be switched together Differences in the number of links in a connection

Adaptive equalization Adjust itself by operating on the the input signal

Training sequence Precall equalization Channel changes little during an average data call

Prechannel equalization Require the feedback channel

Postchannel equalization synchronous

Tap spacing is the same as the symbol duration of transmitted signal

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Adaptive EqualizerAdaptive Equalizer

Adaptation may be achieved By observing the error b/w desired pulse shape and actual pulse

shape Using this error to estimate the direction in which the tap-weight

should be changed Mean-square error criterion

More general in application Less sensitive to timing perturbations

: desired response, : error signal, : actual response Mean-square error is defined by cost fuction

Least-Mean-Square AlgorithmLeast-Mean-Square Algorithm

na ne ny

2nE eε =

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Ensemble-averaged cross-correlation

Optimality condition for minimum mean-square error


[ ]2 2 2 2 ( )n nn n n n k ex

k k k

e yE e E e E e x R k

w w w

ε−

∂ ∂∂ = = − = − = − ∂ ∂ ∂

[ ]( )ex n n kR k E e x −=

0 0, 1,....,k

for k Nw

ε∂ = = ± ±∂

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Mean-square error is a second-order and a parabolic function of tap weights as a multidimentional bowl-shaped surface

Adaptive process is a successive adjustments of tap-weight seeking the bottom of the bowl(minimum value )

Steepest descent algorithm The successive adjustments to the tap-weight in direction opposite to

the vector of gradient ) Recursive formular (µ : step size parameter)


minε

/ kwε∂ ∂

1( 1) ( ) , 0, 1,....,

2

( ) ( ), 0, 1,....,

k kk

k ex

w n w n k Nw

w n R k k N

εµ

µ

∂+ = − = ± ±∂

= − = ± ±

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Least-Mean-Square Algorithm Steepest-descent algorithm is not available in an unknown environment Approximation to the steepest descent algorithm using instantaneous

estimate


( )

( 1) ( )ex n n k

k k n n k

R k e x

w n w n e xµ−

−

=+ = +

)

) )

LMS is a feedback system In the case of small µ,

roughly similar to steepest descent algorithm

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Training mode Known sequence is transmitted and synchorunized version is generated

in the receiver Use the training sequence, so called pseudo-noise(PN) sequence

Decision-directed mode After training sequence is completed Track relatively slow variation in channel characteristic

Large µ : fast tracking, excess mean square error

Operation of the equalizerOperation of the equalizer

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Analog CCD, Tap-weight is stored in digital memory, analog sample and

multiplication Symbol rate is too high

Digital Sample is quantized and stored in shift register Tap weight is stored in shift register, digital multiplication

Programmable digital Microprocessor Flexibility Same H/W may be time shared

Implementation ApproachesImplementation Approaches

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Baseband channel impulse response : {hn}, input : {xn}

Using data decisions made on the basis of precursor to take care of the postcursors The decision would obviously have to be correct

Decision-Feed back equalizationDecision-Feed back equalization

00 0

n k n kk

n k n k k n kk k

y h x

h x h x h x

−

− −< >

=

= + +

∑∑ ∑

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Feedforward section : tapped-delay-line equalizer

Feedback section : the decision is made on previously detected symbols of the input sequence Nonlinear feedback loop by

decision device

Decision-Feed back equalizationDecision-Feed back equalization

(1)

(2)

nn

n

wc

w

=

)

)n

nn

xv

a

=

)

Tn n n ne a c v= −

(1) (1)1 1 1

(2) (2)1 1 1

n n n n

n n n n

w w e x

w w e a

µµ

+ +

+ +

= −

= −

) )

) ) )

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Experimental tool for such an evaluation in an insightful manner Synchronized superposition of all the signal of interest viewed within a

particular signaling interval Eye opening : interior region of the eye pattern

In the case of an M-ary system, the eye pattern contains (M-1) eye opening, where M is the number of discreteamplitude levels

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Technology

Correlative level coding