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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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Probability of Bit Error (BER)
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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Probability of Bit Error (BER)
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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Probability of Bit Error (BER)
/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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Error Performance for Line Codes
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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Error Performance for Line Codes
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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Error Performance for Line Codes
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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