Digital Communication I: Modulation and Coding Course-Lecture 5 potx

Digital Communications I:
Modulation and Coding Course
Term 3 - 2008
Catharina Logothetis
Lecture 5
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.
m
ˆ
i
m
Modulator Decision rule
m
ˆ
i
m
z
i
s
n
z
z
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
aaa=s
), ,,(
21 N
zzz=z
), ,,(
21 N
nnn=n
nsz +=
i
2/
0
N
( )









−=
0
2
2/
0
exp
1
)(
N
N
p
N
n
n
n
π
( )









−
−=
0
2
2/
0
exp
1
)|(
N
N
p
i
N
i
sz
sz
z
π
Lecture 5 6
Detection

Optimum decision rule (maximum a posteriori
probability):

Applying Bayes’ rule gives:
., ,1 where
allfor ,)|sent Pr()|sent Pr(
if
ˆ

Set
Mk
ikmm
mm
ki
i
=
≠≥
=
zz
ik
p
mp
p
mm
k
k
i
=
=
allfor maximum is ,
)(
)|(
if
ˆ
Set
z
z
z
z

Lecture 5 7
Detection …

Partition the signal space into M decision regions,
such that
M
ZZ , ,
1
i
k
k
i
mm
ik
p
mp
p
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
ˆ
Set
z
z
ikmp
mm
k
i

=
=
allfor maximum is )],|(ln[
if
ˆ
Set
z
z
Lecture 5 9
Detection (ML)…

Partition the signal space into M decision regions,
.

Restate the maximum likelihood decision rule as
follows:
M
ZZ , ,
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

j
kjj
i
=−
∑
=
r
Lecture 5 11
Maximum likelihood detector block
diagram
〉〈⋅
1
,s
〉〈⋅
M
s,
1
2
1
E−
M
E
2
1
−
Choose
the largest
z
m
ˆ

Lecture 5 12
Schematic example of the ML decision regions
)(
1
t
ψ
)(
2
t
ψ
1
s
3
s
2
s
4
s
1
Z
2
Z
3
Z
4
Z
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()
ˆ
Pr(
iiii
mZmmm z=≠
sent) inside liessent)Pr( Pr()
ˆ
Pr(
iiii
mZmmm z==
∫
==
i
Z
iiiic
dmpmZmP zz z
z
)|(sent) inside liesPr()(
)(1)(
icie
mPmP −=
i
m

i
s
z
i
Z
sent) and
ˆ
Pr()(
iiie
mmmmP ≠=
Lecture 5 14
Av. prob. of symbol error …

Average probability of symbol error :

For equally probable symbols:
)
ˆ
(Pr)(
1
i
M
i
E
mmMP ≠=
∑
=
)|(
1
1

)(
1
1)(
1
)(
1
11
∑
∫
∑∑
=
==
−=
−==
M
i
Z
i
M
i
ic
M
i
ieE
i
dmp
M
mP
M
mP

M
MP
zz
z
Lecture 5 15
Example for binary PAM
)(
1
t
ψ
b
E
b
E−
0
1
s
2
s
)|(
1
mp z
z
)|(
2
mp z
z

2
)2(

0








==
N
E
QPP
b
EB

2/
2/
)()(
0
21
21









−
==
N
QmPmP
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
z
i
s
k
s
i
s
ki
A
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 ss=

i
s
k
s
∑
≠
=
≤
M
ik
k
ikie
PmP
1
2
),()( ss
∑∑
=
≠
=
≤
M
i
M
ik
k
ikE
P
M
MP

1 1
2
),(
1
)( ss
Lecture 5 17
Union bound:
Example of union bound
1
ψ
1
s
4
s
2
s
3
s
1
Z
4
Z
3
Z
2
Z
r
2
ψ
1

ψ
1
s
4
s
2
s
3
s
2
A
r
1
ψ
1
s
4
s
2
s
3
s
3
A
r
1
ψ
1
s
4

s
2
s
3
s
4
A
r
2
ψ
2
ψ
2
ψ
∫
∪∪
=
432
)|()(
11
ZZZ
e
dmpmP rr
r
∫
=
2
)|(),(
1122
A

dmpP rrss
r
∫
=
3
)|(),(
1132
A
dmpP rrss
r
∫
=
4
)|(),(
1142
A
dmpP rrss
r
∑
=
≤
4
2
121
),()(
k
ke
PmP ss
Lecture 5 18
Upper bound based on minimum distance









=−=
=
∫
∞

2/
2/
)exp(
1
sent) is when , than closer to is Pr(),(
0
0
2
0
2
N
d
Qdu
N
u
N

P
ik
d
iikik
ik
π
ssszss








−≤≤
∑∑
=
≠
=
2/
2/
)1(),(
1
)(
0
min
1 1
2
N

d
QMP
M
MP
M
i
M
ik
k
ikE
ss
kiik
d ss −=
ik
ki
ki
dd
≠
=
,
min
min
Minimum distance in the signal space:
Lecture 5 19
Example of upper bound on av. Symbol
error prob. based on union bound
)(
1
t
ψ

)(
2
t
ψ
1
s
3
s
2
s
4
s
2,1
d
4,3
d
3,2
d
4,1
d
s
E
s
E−
s
E−
s
E
4, ,1 , === iEE
sii

s
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
b
R
W
: Bit rate
: Bandwidth
Lecture 5 21
Example of Symbol error prob. For PAM
signals
)(
1
t
ψ
0
1
s
2
s
b
E
b

E−
Binary PAM
)(
1
t
ψ
0
2
s
3
s
5
2
b
E
5
6
b
E
5
6
b
E
−
5
2
b
E
−
4

s
1
s
4-ary PAM
T
t
)(
1
t
ψ
T
1
0