Correlative level coding

Baseband Pulse Transmission
Correlative-Level Coding
Baseband M-ary PAM Transmission
Tapped-Delay-Line Equalization
Eye Pattern
Hyeong-Seok Yu
Vada Lab.
gargoyle@vada1.skku.ac.kr

1

 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

2

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

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

+1
ak = 
if symbol bk is 1 ck = ak + ak −1
−1 if symbol bk is 0

3

Duobinary Signaling
 Ideal Nyquist channel of bandwidth
W=1/2Tb
H I ( f ) = H Nyquist ( f )[1 + exp(− j 2πfTb )]
= H Nyquist ( f )[exp( jπfTb ) + exp(− jπfTb )] exp(− jπfTb )
= 2 H Nyquist ( f ) cos(πfTb ) exp(− jπfTb )
1, | f |≤ 1 / 2Tb
H Nyquist ( f ) = 
0, otherwise
2 cos(π fTb ) exp(− jπ fTb ), | f |≤ 1/ 2Tb
HI ( f ) = 
 0, otherwise
sin(πt / Tb ) sin[π (t − Tb ) / Tb ]
hI (t ) = +
πt / Tb π (t − Tb ) / Tb
Tb2 sin(πt / Tb )
=
πt (Tb − t )

4

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 a represent the estimate of the original pulse ak as
k

conceived by the receiver at time t=kTb
^ ^
ak = ck − ak −1

 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

5

Duobinary Signaling
d k = bk ⊕ d k −1
 symbol 1 if either symbol bk or d k −1 is 1
dk = 
 symbol 0 otherwise

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

ck = ak + ak −1

 0 if data symbol bk is 1
ck = 
±2 if data symbol bk is 0

6

Duobinary Signaling
 |ck|=1 : random guess in favor of symbol 1 or 0
If | ck |< 1, say symbol bk is 1
If | ck |> 1, say symbol bk is 0

7

Modified Duobinary Signaling
 Nonzero at the origin : undesirable
 Subtracting amplitude-modulated pulses spaced 2Tb second
ck = ak + ak −1
H IV ( f ) = H Nyquist ( f )[1 − exp(− j 4π fTb )]
= 2 jH Nyquist ( f ) sin(2π fTb ) exp(− j 2π fTb )

 2 j sin(2π fTb ) exp(− j 2π fTb ), | f |≤ 1/ 2Tb
H IV ( f ) = 
 0, elsewhere
sin(π t / Tb ) sin[π (t − 2Tb ) / Tb ]
hIV (t ) = −
π t / Tb π (t − 2Tb ) / Tb
2Tb2 sin(π t / Tb )
=
π t (2Tb − t )
8


 precoding
d k = bk ⊕ d k − 2
 symbol 1 if either symbol bk or d k − 2 is 1
=
 symbol 0 otherwise

9


If | ck |> 1, say symbol bk is 1
If | ck |< 1, say symbol bk is 0

10

Generalized form of correlative-level coding
Type of N w0 w1 w2 w3 w4 comments
class
I 2 1 1 Duobinary
II 3 1 2 1
III 3 2 1 –1
IV 3 1 0 –1 Modified
V 5 -1 0 2 0 -1

N −1
 t 
h(t ) = ∑ wn sin c
 − n

n  Tb 

11

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 : T = Tb log 2 M
 To realize the same average
probability of symbol error,
transmitted power must be
increased by a factor of M2/log2M
compared to binary PAM

12

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

13


 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
h(t ) = ∑ w δ (t − kT )
k =− N
k

14


 P(t) is equal to the convolution of c(t) and h(t)
N
p(t ) = c(t ) ∗ h(t ) = c(t ) ∗ ∑ w δ (t − kT )
k =− N
k

N N
= ∑ w c(t ) ∗ δ (t − kT ) = ∑ w c(t − kT )
k =− N
k
k =− N
k

 nT=t sampling time, discrete convolution sum

N
p (nT ) = ∑ w c((n − k )T )
k =− N
k

15

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

1, n = 0 1, n=0
p (nT ) =  =
0, n ≠ 0 0, n = ±1, ± 2, .....,± N
 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

16

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

17

Adaptive Equalizer
Least-Mean-Square Algorithm
 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
 an : desired response, en : error signal, yn : actual response
 Mean-square error is defined by cost fuction
ε = E en 
2
 

18

Adaptive Equalizer
 Ensemble-averaged cross-correlation

∂ε  ∂e   ∂y 
= 2 E en n  = −2 E en n  = −2 E [ en xn − k ] = −2 Rex (k )
∂wk  ∂wk   ∂wk 

Rex (k ) = E [ en xn − k ]

 Optimality condition for minimum mean-square error
∂ε
=0 for k = 0, ± 1,...., ± N
∂wk

19

Adaptive Equalizer
 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 ε min )
 Steepest descent algorithm
 The successive adjustments to the tap-weight in direction opposite to
the vector of gradient ∂ε / ∂wk )
 Recursive formular (µ : step size parameter)

1 ∂ε
wk (n + 1) = wk (n) − µ , k = 0, ± 1,...., ± N
2 ∂wk
= wk (n) − µ Rex (k ), k = 0, ± 1,...., ± N

20

Adaptive Equalizer
 Least-Mean-Square Algorithm
 Steepest-descent algorithm is not available in an unknown environment
 Approximation to the steepest descent algorithm using instantaneous
estimate
)
Rex (k ) = en xn − k
) )
wk (n + 1) = wk (n) + µ en xn − k
 LMS is a feedback system
 In the case of small µ,
roughly similar to steepest
descent algorithm

21

Adaptive Equalizer
Operation of the equalizer
 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

22

Adaptive Equalizer
Implementation Approaches
 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

23

Adaptive Equalizer
Decision-Feed back equalization

 Baseband channel impulse response : {hn}, input : {xn}
yn = ∑hk xn −k
k

= h0 xn + ∑hk xn −k + ∑hk xn −k
k <0 k >0

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

24

Adaptive Equalizer
Decision-Feed back equalization
 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
) (1) ) (1) ) (1)
 wn   xn  wn +1 = wn +1 − µ1en xn
cn =  ) (2)  vn =  )  en = an − cn vn
T
) (2) ) (2) )
 wn   an  wn +1 = wn +1 − µ1en an

25

Eye Pattern
 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

26

Correlative level coding

More Related Content

What's hot

Viewers also liked

Similar to Correlative level coding

Recently uploaded

Correlative level coding