Modulation, Demodulation and Coding Course
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Transcript Modulation, Demodulation and Coding Course
Digital Communications I:
Modulation and Coding Course
Spring - 2013
Jeffrey N. Denenberg
Lecture 3c: Signal Detection in AWGN
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
2
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
3
Detection of signal in AWGN
Detection problem:
Given the observation vector z , perform a
ˆ of the
mapping from z to an estimate m
transmitted symbol, mi , such that the
average probability of error in the decision
is minimized.
n
mi
Modulator
si
Lecture 5
z
Decision rule
4
mˆ
Statistics of the observation Vector
AWGN channel model: z si n
(a
,a
,...,
a
) is deterministic.
Signal vector s
i
i1
i2
iN
,n
n
Elements of noise vector n(n
are i.i.d
1
2,...,
N)
Gaussian random variables with zero-mean and
variance N 0 / 2 . The noise vector pdf is
2
n
1
p
(
n
)
exp
n
N
/2
N
N
0
0
The elements of observed vector z(z1,z2,...,
zN)are
independent Gaussian random variables. Its pdf is
2
z
s
1
i
p
(
z
|s
)
N
exp
z
i
/
2
N
N
0
0
Lecture 5
5
Detection
Optimum decision rule (maximum a
posteriori probability):
ˆ
Set
m
m
iif
Pr(
m
|z
)
Pr(
m
sent
|z
)
,
for
all
k
i
isent
k
where
k
1
,...,
M
.
Applying Bayes’ rule gives:
ˆ
Set
m
m
iif
p
(
z
|m
)
z
k
p
,is
maximum
for
all
k
i
k
p
(
z
)
z
Lecture 5
6
Detection …
Partition the signal space into M decision
regions, Z1,...,ZM such that
Vector
zlies
inside
region
Z
i if
p
z|m
z(
k)
ln[
p
],is
maximum
for
all
k
i.
k
p
z
)
z(
That
means
ˆ
m
m
i
Lecture 5
7
Detection (ML rule)
For equal probable symbols, the optimum
decision rule (maximum posteriori probability)
is simplified to:
ˆ
Set
m
m
iif
p
(
z
|m
),
is
maximum
for
all
k
i
z
k
or equivalently:
ˆ
Set
m
m
if
i
ln[
p
(
z
|
m
)],
is
maximum
for
all
k
i
z
k
which is known as maximum likelihood.
Lecture 5
8
Detection (ML)…
Partition the signal space into M decision
regions, Z1,...,ZM.
Restate the maximum likelihood decision
rule as follows:
Vector
z
lies
inside
region
Z
i if
ln[
p
z
|m
is
maximum
for
all
k
i
z(
k)],
That
means
ˆ
m
m
i
Lecture 5
9
Detection rule (ML)…
It can be simplified to:
Vector
z
lies
inside
region
Z
iif
z
s
,is
minimum
for
all
k
i
k
or equivalently:
Vector
r
lies
inside
region
Z
iif
N
1
z
E
maximum
for
all
k
i
ja
kj
k,is
2
j
1
where
E
the
energy
of
s
t).
kis
k(
Lecture 5
10
Maximum likelihood detector block
diagram
,s1
1
E1
2
z
Choose
the largest
, s M
1
EM
2
Lecture 5
11
mˆ
Schematic example of the ML decision regions
2 (t )
Z2
s2
s3
s1
Z3
Z1
1 (t )
s4
Z4
Lecture 5
12
Average probability of symbol error
Erroneous decision:
For the transmitted symbol mi
or equivalently signal vector s i , an error in decision occurs
if the observation vector z does not fall inside region Z i.
Probability of erroneous decision for a transmitted symbol
or equivalently
ˆ
P
(
m
)
Pr(
m
m
m
e
i
iand
isent)
ˆ
Pr(
m
m
)
Pr(
m
sent)Pr
z
does
not
lie
insi
Z
m
se
i
i
i
i
Probability of correct decision for a transmitted symbol
ˆ
Pr(
m
m
)
Pr(
m
sent)Pr(
z
lies
inside
Z
m
sen
i
i
i
i
P
(
m
)
Pr(
z
lies
inside
Z
m
sent)
p
(
z
|
m
)
d
z
c
i
i
i
z
i
P
m
1
P
m
e(
i)
c(
i)
Lecture 5
Z
i
13
Av. prob. of symbol error …
Average probability of symbol error :
M
ˆ
P
(
M
)
Pr
(
m
m
E
i)
i
1
For equally probable symbols:
1M
1M
P
(
M
)
P
(
m
1
P
(
m
E
e
i)
c
i)
M
M
i
1
i
1
1M
1
p
(
z
|m
d
z
z
i)
M
i
1Z
i
Lecture 5
14
Example for binary PAM
pz (z | m2)
pz (z | m1)
s2
Eb
s1
0
1 (t )
Eb
s
s
/
2
1
2
P
(
m
)
P
(
m
)
Q
e
1
e
2
N
/
2
0
2
E
b
P
P
(
2
)
Q
B
E
N
0
Lecture 5
15
Union bound
Union bound
The probability of a finite union of events is upper bounded
by the sum of the probabilities of the individual events.
Let Aki denote that the observation vector z is closer to
the symbol vector s k than s i , when s i is transmitted.
Pr(
A
)P
sk,si) depends only on s i and s k .
ki
2(
Applying Union bounds yields
M
P
m
P
(sk,s
e(
i)
2
i)
k
1
k
i
1MM
P
(
M
)
P
(
s
,s
E
2
k
i)
M
i
1k
1
k
i
Lecture 5
16
Example of union bound
r
Z2
P
(
m
)
p
(
r
|m
)
d
r
e
1
r
1
2
Z1
s2
s1
Z
Z
Z
2
3
4
1
Union bound:
s4
Z4
s3
4
Z3
P
(
m
)
P
(
s
s
)
e
1
2
k,
1
k
2
2
A2 r
s2
r
2
s2
s1
r
s3
s4
P
(
s
,s
)
(
r
|m
)
d
r
2
2
1
r
1
p
A
2
s2
s1
1
2
s1
1
s3
A3
s4
P
(
s
,s
)
(
r
|m
)
d
r
2
3
1
r
1
p
A
3
Lecture 5
1
s3
A4
s4
P
(
s
,s
)
(
r
|m
)
d
r
2
4
1
r
1
p
17
A
4
Upper bound based on minimum distance
P
(
s
,
s
)
Pr(
z
is
closer
s
s
, when
s
to
sent)
2
k
i
k than
i
iis
2
d
/
2
1
u
ik
exp(
)
du
Q
N
N
N
/
2
0
d
0
0
ik
dik si sk
M
M
1
d
/
2
min
P
(
M
)
P
(
s
,
s
)
(
M
1
)
Q
E
2
k
i
M
N
/
2
i
1
k
1
0
k
i
Minimum distance in the signal space:
Lecture 5
dminmin
dik
i,k
ik
18
Example of upper bound on av. Symbol
error prob. based on union bound
2 (t )
s
E
E
,
i
1
,...,
4
i
i
s
d i , k 2 Es
ik
Es
d min 2 Es
s2
d1, 2
d 2,3
s3
Es
s1
d 3, 4
d1, 4
Es
s4
Es
Lecture 5
19
1 (t )
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.
E
ST SW
b
b
N
/WN
R
0 N
b
Lecture 5
Rb
W
: Bit rate
: Bandwidth
20
Example of Symbol error prob. For PAM
signals
Binary PAM
s2
s1
Eb
s4
6
Eb
5
0
1 (t )
Eb
4-ary PAM
s3
s2
2
Eb
5
0
2
Eb
5
s1
6
Eb
5
1 (t )
1
T
0
Lecture 5
21
T t
1 (t )