duobinary_signaling.pdf

7/27/2019 duobinary_signaling.pdf

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Duobinary Signaling

From: Sklar, Ch. 2, Section 2.9 Correlative Coding = Partial Response Signaling

Special Case: Duobinary Signaling = Class 1 Partial ResponseSignaling

Recall Nyquists claim from 1928: It is possible to tx Rs symbols/secwith no ISI in (minimum) bandwidth of W = Rs/2.

But: it requires a brick-wall filter to obtain the Nyquist pulse shape.

Adam Lender, 1963: It is possible to tx Rs symbols/sec with no ISI in

(minimum) bandwidth of W = Rs/2. And: infinitely-sharp pulse-shaping filters are not required.

Technique - Duobinary signaling: use controlled ISI to youradvantage (to cancel out the ISI at the detector)

ECE 561 1D. van Alphen


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Block Diagram: Duobinary Signaling

Delay T

T

1/(2T)

H(f)noise

Decoder[xk]

Digital filter

yk = xk + xk-1

Brick-wall

Nyquist filter t = kT

channel

Say binary sequence [xk] is txd at rate Rs symbols/sec, over a system

with bandwidth W = Rs/2 = 1/(2T)

Note: the brick-wall filter is just a model; it wont really be used inour final actual system (recall: not realizable)

Also note: symbols ykare not independent (each carries memory of

the previous symbol) in other words, the yksare correlated

yk

xk



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Block Diagram: Duobinary Signaling

Delay T

T

1/(2T)

H(f)noise

Decoder[xk]

Digital filter

yk = xk + xk-1

Brick-wall

Nyquist filter t = kT

channel

If we could realize the brick-wall filter, there would be no ISI, so:

in the absence of noise,

yk = yk

Duobinary decoding: since yk = xk + xk-1 : xk = yk xk-1

Assume xk { 1}, so yk {0, 2}, a 3-ary alphabet

Decoding: xk = yk xk-1

bipolar

ykxk



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Duobinary Signaling An Example

xk (binary) 0 0 1 0 1 1 0

xk (bipolar) -1 -1 1 -1 1 1 -1yk ___ ___ ___ ___ ___ ___

xk (bipolar) ___ ___ ___ ___ ___ ___

xk (binary) ___ ___ ___ ___ ___ ___

Ref data data

Note: because the encoding is differential (like difference equations & differential

equation) we need an initial condition, or an assumed value (or referencevalue) for the first xk.

Given ref and data values;

Binary BipolarEncode: yk = xk + xk-1

Decode: xk = yk xk-1

Bipolar Binary

Observe alternative decision rule: yk = 2 xk = _____

-2 xk = _____

0 xk

= negative of previous decisionECE 561 4D. van Alphen


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Precoding of Duobinary Signals

Bad Property of Duobinary Signaling (as presented so far) Error Propagation: A single error in yk-hat will (often) result in 2 errors in xk-

hat

Error Propagation can be avoided by pre-coding of the symbols

We change the xks to coded values: wks

wk = xk

wk-1

where = mod 2 addition = exclusive-or operation Then do binary bipolar (on the wks);

Then do the correlative coding: yk = wk + wk-1;

Pass the yksthrough the brick-wall Nyquist filter;

At the decoder: if yk = 2 xk = binary 0

0 xk = binary 1

For decoding, eachdecision is independent

of previous decisions

no error propagation



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Example with Pre-coding

xk (binary) 0 1 0 1 1 0

wk (binary) 0 ___ ___ ___ ___ ___ ___

wk (bipolar) ____ ___ ___ ___ ___ ___ ___

yk ___ ___ ___ ___ ___ ___

xk (binary) ___ ___ ___ ___ ___ ___

Ref data data

Given ref and data values;Pre-code: wk = xk wk-1

Binary Bipolar

Encode: yk = wk + wk-1

Decode: 2 0, 0 1

Block Diagram with Pre-coding


Delay T

T

1/(2T)

H(f)noise

Decoder[xk]

yk = wk + wk-1

Brick-wall

Nyquist filtert = kT

channel

binary mod-2 add

yk

xkwk

wk-1wk-1


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Duobinary Equivalent Transfer Function

(equiv. to block diagram of pp. 2-3)

Problem: we need to get rid of that brick-wall filter in the model!

Looking at the first part of our (not pre-coded) system, using an analog model:

The second part of our (not pre-coded) system was the Nyquist brick-wall filter:

H2(f) = T, |f| < 1/(2T)

0, else

Delay T

x(t) y(t)= x(t) + x(t-T)

Y(f) = _____ + ____ e-j2 fT

= X(f) [ ___ + _______]

H1(f) = ____ / _____ = _____________



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Duobinary Equivalent Transfer Function

(equiv. to block diagram of pp. 2-3)

Thus, the over-all equivalent transfer function for the duobinary sytem(without pre-coding) is:

Heq(f) = H1(f) H2(f) = T [1 + e-j2 fT], |f| < 1/(2T)

= T [ej fT + e-j fT] e-j fT, |f| < 1/(2T)

= 2T cos( fT) e-j fT |f| < 1/(2T)

So: |Heq(f)| = 2T cos(2 fT), |f| < 1/(2T)

Phase response: corresponds to time delay T/2

ECE 561 D. van Alphen 8

|Heq(f)|

-1/(2T) 1/(2T)


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|Heq(f)| = = 2T cos( fT) e-j fT , |f| < 1/(2T)

Duobinary Equivalent Filter is called: a cosine filter Not the same as the raised cosine or RC filter of Ch. 3

Implementation: xk precode wk Heq(f)


T

Ttsinc

T

tsinc)t(heq

-2 -1 0 1 2 3-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4Impulse Response, Duobinary Equivalent Filter

t/T

heq

(t)

-1 -0.5 0 0.5 10

0.5

1

1.5

2Magnitude Response, Duobinary Equivalent Transfer Function

f/T

|He

q(f)|


10/11

Comparison: Binary vs. Duobinary Signaling

Duobinary: 0 ISI with minimum bandwidth No error propagation if precoding is used

Problem of duobinary: because there are 3 amplitude levels (as opposed to 2

for traditional binary signaling), the receiver will have more difficulty

distinguishing the levels

2.5 dB more Eb/N0 required for performance identical to BPSK without

ISI (even with precoding for the duobinary)

Depending on the severity of the ISI, duobinary may perform better than

BPSK



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The Larger Class of Partial Response Signaling

Class 1 PR Signaling = Duobinary Signaling: yk = xk + xk-1

Class 2 PR Signaling: yk = xk +2 xk-1 + xk-2

Class 4 PR Signaling: yk = xk + xk-2

See Charan Langtons Tutorial 16: Partial Response Signaling, for a discussion

of all the other variations (Classes 1 5, pros and cons)


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duobinary_signaling.pdf