Digital Communications I:
Modulation and Coding Course
Spring - 2015
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
mapping from z to an estimate m̂ of the
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
m̂
Statistics of the observation Vector

AWGN channel model: z  si  n


Signal vector si  (ai1, ai 2 ,...,aiN ) is deterministic.
Elements of noise vector n  (n1, n2 ,...,nN ) are i.i.d
Gaussian random variables with zero-mean and
variance N0 / 2 . The noise vector pdf is
pn (n) 

1
N 0 N / 2
2

n

exp 
N0







The elements of observed vector z  ( z1, z2 ,..., z N ) are
independent Gaussian random variables. Its pdf is
pz ( z | s i ) 
1
N 0 N / 2
Lecture 5

z  si

exp 
N0


2




5
Detection

Optimum decision rule (maximum a
posteriori probability):
Set mˆ  mi if
Pr(mi sent | z )  Pr(mk sent | z ), for all k  i
where k  1,...,M .

Applying Bayes’ rule gives:
Set mˆ  mi if
pk
pz (z | mk )
, is maximumfor all k  i
pz (z)
Lecture 5
6
Detection …

Partition the signal space into M decision
regions, Z1,...,Z M such that
Vector z lies inside region Zi if
pz (z | mk )
ln[ pk
], is maximumfor all k  i.
pz ( z )
T hatmeans
mˆ  mi
Lecture 5
7
Detection (ML rule)

For equal probable symbols, the optimum
decision rule (maximum posteriori probability)
is simplified to:
ˆ  mi if
Set m
pz (z | mk ), is maximumfor all k  i
or equivalently:
ˆ  mi if
Set m
ln[ pz (z | mk )], is maximumfor all k  i
which is known as maximum likelihood.
Lecture 5
8
Detection (ML)…

Partition the signal space into M decision
regions, Z1,...,Z M .

Restate the maximum likelihood decision
rule as follows:
Vector z lies inside region Zi if
ln[ pz (z | mk )], is maximumfor all k  i
T hatmeans
mˆ  mi
Lecture 5
9
Detection rule (ML)…

It can be simplified to:
Vect or z lies inside region Z i if
z  s k , is minimumfor all k  i
or equivalently:
Vect or r lies inside region Z i if
N
1
z
a

Ek , is maximumfor all k  i
 j kj
2
j 1
where Ek is t heenergy of sk (t ).
Lecture 5
10
Maximum likelihood detector block
diagram
,s1 
1
 E1
2
z
Choose
the largest
, s M 
1
 EM
2
Lecture 5
11
m̂
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  mi and mi sent)
Pe (mi )  Pr(m
ˆ  mi )  P r(mi sent )P r(z does not lie inside Z i mi sent )
P r(m

Probability of correct decision for a transmitted symbol
Pr(mˆ  mi )  Pr(mi sent)Pr(z lies inside Z i mi sent)
Pc (mi )  P r(z lies inside Z i mi sent )   pz (z | mi )dz
Pe (mi )  1  Pc (mi )
Lecture 5
Zi
13
Av. prob. of symbol error …

Average probability of symbol error :
M
ˆ  mi )
PE ( M )   Pr(m
i 1

For equally probable symbols:
1 M
1 M
PE ( M ) 
 Pe (mi )  1 
 Pc (mi )
M i 1
M i 1
1 M
1
  pz (z | mi )dz
M i 1Z i
Lecture 5
14
Example for binary PAM
This is a poor “artist’s conception”
of Gaussian curves
pz (z | m2)
pz (z | m1)
s2
 Eb
s1
0
 1 (t )
Eb
 s s /2 
1
2

Pe (m1 )  Pe (m2 )  Q


N
/
2
0


 2E
b
PB  PE (2)  Q
 N0

Lecture 5




15
Erfc / Q(x) Table
•
•
•
•
This table gives your erfc(x) and is
normalized for  = 1 and mean = 0
Note that it calculates the area under the
Gaussian function from x to ∞ (the tail)
where x is a positive number.
Draw a picture to see which portion of the
area under the curve is your interest (eg. The
area from 1 to x) and use the table to give
you the required area (eg. 0.5 – Q(x)).
The table can give you 5 digits (the 5th digit
is obtained by linear interpolation) to the
right of the decimal point. It is a twodimensional lookup.
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(Aki )  P2 (sk , si ) depends only on s i and s k .

Applying Union bounds yields

M
Pe ( mi )   P2 (s k , s i )
k 1
k i
1 M M
PE ( M ) 
  P2 (s k , s i )
M i 1k 1
k i
Lecture 5
17
Example of union bound
Pe (m1 ) 
r
Z2
 pr (r | m1 )dr
Z 2 Z3 Z 4
2
Z1
s2
s1
1
Union bound:
s4
Z4
s3
Z3
4
Pe (m1 )   P2 (s k , s1 )
k 2
2
A2 r
s2
r
2
s2
s1
r
s3
s4
P2 (s 2 , s1 )   pr (r | m1 )dr
s2
s1
1
2
s1
1
s3
A3
s4
P2 (s3 , s1 )   pr (r | m1 )dr
A3
A2
Lecture 5
1
s3
A4
s4
P2 (s 4 , s1 )   pr (r | m1 )dr
A4
18
Upper bound based on minimum distance
P2 (s k , si )  Pr(z is closer tos k thansi , when si is sent)

 
d ik
 d /2
u2
exp(
)du Q ik
 N /2
N0
N0
0

1




d ik  s i  s k
d

1 M M

min / 2 
PE ( M ) 
  P2 (s k , si )  ( M  1)Q

M i 1k 1
N
/
2
0


k i
Minimum distance in the signal space:
Lecture 5
d m in  m in d ik
i,k
ik
19
Example of upper bound on av. Symbol
error prob. based on union bound
si

Ei 
Es ,
i  1,...,4
 2 (t )
d i , k  2 Es
ik
Es
d min  2 Es
s2
d1, 2
d 2,3
s3
 Es
s1
d 3, 4
d1, 4
Es
s4
 Es
Lecture 5
20
 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.
Eb
STb
S W


N0
N /W
N Rb
Lecture 5
Rb
W
: Bit rate
: Bandwidth
21
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
22
T t
 1 (t )