Digital Communications I: Modulation and Coding Course
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Digital Communications I: Modulation and Coding Course
Spring - 2013Jeffrey N. Denenberg
Lecture 3c: Signal Detection in AWGN
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Lecture 5 2
Last time we talked about:
Receiver structure Impact of AWGN and ISI on the
transmitted signal Optimum filter to maximize SNR
Matched filter and correlator receiver Signal space used for detection
Orthogonal N-dimensional space Signal to waveform transformation and
vice versa
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Lecture 5 3
Today we are going to talk about:
Signal detection in AWGN channels Minimum distance detector Maximum likelihood
Average probability of symbol error Union bound on error probability Upper bound on error probability
based on the minimum distance
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Lecture 5 4
Detection of signal in AWGN
Detection problem: Given the observation vector ,
perform a mapping from to an estimate of the transmitted symbol, , such that the average probability of error in the decision is minimized.
m̂im
Modulator Decision rule m̂imzis
n
zz
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Lecture 5 5
Statistics of the observation Vector
AWGN channel model: Signal vector is deterministic. Elements of noise vector are i.i.d
Gaussian random variables with zero-mean and variance . The noise vector pdf is
The elements of observed vector are independent Gaussian random variables. Its pdf is
),...,,( 21 iNiii aaas
),...,,( 21 Nzzzz
),...,,( 21 Nnnnn
nsz i
2/0N
0
2
2/0
exp1
)(NN
p N
nnn
0
2
2/0
exp1
)|(NN
p iNi
szszz
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Lecture 5 6
Detection
Optimum decision rule (maximum a posteriori probability):
Applying Bayes’ rule gives:
.,...,1 where
allfor ,)|sent Pr()|sent Pr(
if ˆSet
Mk
ikmm
mm
ki
i
zz
ikp
mpp
mm
kk
i
allfor maximum is ,)(
)|(
if ˆSet
z
z
z
z
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Lecture 5 7
Detection …
Partition the signal space into M decision regions, such that MZZ ,...,1
i
kk
i
mm
ikp
mpp
Z
ˆ
meansThat
. allfor maximum is ],)(
)|(ln[
if region inside lies Vector
z
z
z
z
z
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Lecture 5 8
Detection (ML rule)
For equal probable symbols, the optimum decision rule (maximum posteriori probability) is simplified to:
or equivalently:
which is known as maximum likelihood.
ikmp
mm
k
i
allfor maximum is ),|(
if ˆSet
zz
ikmp
mm
k
i
allfor maximum is )],|(ln[
if ˆSet
zz
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Lecture 5 9
Detection (ML)…
Partition the signal space into M decision regions, .
Restate the maximum likelihood decision rule as follows:
MZZ ,...,1
i
k
i
mm
ikmp
Z
ˆ
meansThat
allfor maximum is )],|(ln[
if region inside lies Vector
z
z
z
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Lecture 5 10
Detection rule (ML)…
It can be simplified to:
or equivalently:
ik
Z
k
i
allfor minimum is ,
if region inside lies Vector
sz
z
).( ofenergy theis where
allfor maximum is ,2
1
if region inside lies Vector
1
tsE
ikEaz
Z
kk
k
N
jkjj
i
r
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Lecture 5 11
Maximum likelihood detector block diagram
1,s
Ms,
12
1E
ME2
1
Choose
the largestz m̂
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Lecture 5 12
Schematic example of the ML decision regions
)(1 t
)(2 t
1s3s
2s
4s
1Z
2Z
3Z
4Z
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Lecture 5 13
Average probability of symbol error
Erroneous decision: For the transmitted symbol or equivalently signal vector , an error in decision occurs if the observation vector does not fall inside region . Probability of erroneous decision for a transmitted symbol
or equivalently
Probability of correct decision for a transmitted symbol
sent) inside lienot does sent)Pr( Pr()ˆPr( iiii mZmmm z
sent) inside liessent)Pr( Pr()ˆPr( iiii mZmmm z
iZ
iiiic dmpmZmP zz z z )|(sent) inside liesPr()(
)(1)( icie mPmP
imis
z iZ
sent) and ˆPr()( iiie mmmmP
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Lecture 5 14
Av. prob. of symbol error …
Average probability of symbol error :
For equally probable symbols:
)ˆ(Pr)(1
i
M
iE mmMP
)|(1
1
)(1
1)(1
)(
1
11
M
i Z
i
M
iic
M
iieE
i
dmpM
mPM
mPM
MP
zzz
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Lecture 5 15
Example for binary PAM
)(1 tbEbE 0
1s2s
)|( 1mp zz)|( 2mp zz
2
)2(0
N
EQPP b
EB
2/
2/)()(
0
2121
NQmPmP ee
ss
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Lecture 5 16
Union bound
Let denote that the observation vector is closer to the symbol vector than , when is transmitted.
depends only on and . Applying Union bounds yields
zisks
is
kiA
Union bound
The probability of a finite union of events is upper bounded by the sum of the probabilities of the individual events.
),()Pr( 2 ikki PA ssis
ks
M
ikk
ikie PmP1
2 ),()( ss
M
i
M
ikk
ikE PM
MP1 1
2 ),(1
)( ss
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Lecture 5 17
Union bound:
Example of union bound
1
1s
4s
2s
3s
1Z
4Z3Z
2Z r 2
1
1s
4s
2s
3s
2A r
1
1s
4s
2s
3s
3A
r
1
1s
4s
2s
3s
4A
r2 22
432
)|()( 11
ZZZ
e dmpmP rrr
2
)|(),( 1122
A
dmpP rrss r 3
)|(),( 1132
A
dmpP rrss r 4
)|(),( 1142
A
dmpP rrss r
4
2121 ),()(
kke PmP ss
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Lecture 5 18
Upper bound based on minimum distance
2/
2/)exp(
1
sent) is when , than closer to is Pr(),(
00
2
0
2
N
dQdu
N
u
N
P
ik
d
iikik
ik
ssszss
2/
2/)1(),(
1)(
0
min
1 12
N
dQMP
MMP
M
i
M
ikk
ikE ss
kiikd ss
ik
kikidd
,
min minMinimum distance in the signal space:
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Lecture 5 19
Example of upper bound on av. Symbol error prob. based on union
bound
)(1 t
)(2 t
1s3s
2s
4s
2,1d
4,3d
3,2d
4,1dsEsE
sE
sE
4,...,1 , iEE siis
s
ski
Ed
ki
Ed
2
2
min
,
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Lecture 5 20
Eb/No figure of merit in digital communications
SNR or S/N is the average signal power to the average noise power. SNR should be modified in terms of bit-energy in DCS, because: Signals are transmitted within a symbol
duration and hence, are energy signal (zero power).
A merit at bit-level facilitates comparison of different DCSs transmitting different number of bits per symbol.
b
bb
R
W
N
S
WN
ST
N
E
/0
bR
W: Bit rate
: Bandwidth
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Lecture 5 21
Example of Symbol error prob. For PAM signals
)(1 t0
1s2s
bEbE
Binary PAM
)(1 t0
2s3s
52 bE
56 bE
56 bE
52 bE
4s 1s4-ary PAM
T t
)(1 t
T1
0