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
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
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
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
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
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
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
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
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
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
Lecture 5 11
Maximum likelihood detector block diagram
1,s
Ms,
12
1E
ME2
1
Choose
the largestz m̂
Lecture 5 12
Schematic example of the ML decision regions
)(1 t
)(2 t
1s3s
2s
4s
1Z
2Z
3Z
4Z
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
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
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
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
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
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:
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
,
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
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
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