Lec9 Error Probability

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TE312: Introduction to

Digital Telecommunications

PART IIBASEBAND DIGITAL

TRANSMISSION

Lecture #9Optimum Digital Receivers

Error Performance


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Introduction

Reading Assignment

Simon Haykin, Digital Communications,John Wiley & Sons, Inc., 1988, Chapter 6,Sec. 6.2.

Simon Haykin, Communication Systems, 4th

Ed., John Wiley & Sons, Inc., 2001, Chapter

3, Sec. 3.7.


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Probability of Bit Error (BER)

One-Dimensional Signal Set

Recall that

( ) ( ) ( )

( ) ( ) ( )

1 11 1 12 2

2 21 1 22 2

s t s t s t

s t s t s t

= +

= + 0 bt T

A signal set is one-dimensional when ( )2 0t = ,

12 22 0s s= = .

2 21s=s1 11s=s( )1 t


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Probability of Bit Error (BER)One-Dimensional Signal Set

The MAP decision rule reduces to

( )

( )( )

12 1

2 12 1 1

1ln

2o

pE E N T

p s s

+

11 2 1 11 21

2

1

2 lnop

E E r s s Np

r

< +

Defining a new Gaussian random variable L withsample value 2l r r1= , the decision rule becomes

1

1 2

set if

0

b b

l r r

=

= >


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Two-Dimensional Signal Set

( )

( )

[ ]

1 11

2 22

|

|

b

b

o

E L s t s E

E L s t s E

Var L N

= =

= =

=

( )( ) ( )2

1

1| exp

22

b

L

oo

l Ef l s t

NN

=

( )( ) ( )2

2

1| exp

22

b

L

oo

l Ef l s t

NN

+ =


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Two-Dimensional Signal Set

Probabil ity of making an error is given by

22s 11s0T=

( )( )1|Lf l s t( )( )2|Lf l s t

l

( ) ( )

( )( ) ( )( )

1 2

0

1 20

1 10 | 0 |2 2

1 1 | |

2 2

e

L L

p P l s t P l s t

f l s t dl f l s t dl

= < + >

= +


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/b oE N is the standard quality measure for digitalcommunication system performance.

[ ]Q x is a monotonically decreasing function of.Hencex ep decreases with the increase in /b oE N .

For fixed channel noise psd, o/bE N is increased byincreasing bE corresponding to an increase in the

Euclidean distance between signals ( )1s t and ( )2s t .

Different modulation methods can be comparedbased on the required / obE N for the fixed ep .


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Error Performance for Line Codes

Error performance for the line codes is determinedby probabil ity of bit error for an AWGN channel.

The signal is corrupted by additive white Gaussiannoise of zero-mean and power spectral density(psd) / 2.

oN

The two bits are assumed to be equally likely, i.e.,

1 2p p= .

For comparison purpose, the amplitude of eachpulse is the same for all line codes.


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Signal vectors and1s

2s

1 1E = s , [ ]2

2 10 where bE A T= =s

Signal constellation diagram for the unipolar NZRline code is one-dimensional.

2 0=s 1 1E=s

/ 2T E=

1Z

( )1 t

2Z


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Signal vectors and1s

2s

1 1E = s ,2

2 2 1 2where bE E E A T = = = s

Signal constellation diagram for the polar NZR linecode is one-dimensional.

2 2E= s 1 1E=s

0T=

1Z

( )1 t

2Z


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For an AWGN channel, the probability of bit error ep is expressed as

avb1 22e

o o

EEp Q QN N

= =

The expression for ep applies for the polar RZ andManchester line codes.

bt T=

( )1 t

1r

Threshold T=0

Decisiondevice 1

1

bit 1 ifbit 0 if

r Tb

r T

>=


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Lec9 Error Probability

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