Post on 14-Apr-2018
7/29/2019 Lecture 06 - Optimal Receiver Design
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Instructor: Dr. Phan Van Ca
Lecture 06 : Optimal Receiver Design
Digital Communications
Modulation
We want to modulate digital data using signal sets which are :
z
bandwidth efficientz energy efficient
A signal space representation is a convenient form for
viewing modulation which allows us to:
z design energy and bandwidth efficient signal constellations
z determine the form of the optimal receiver for a given
constellation
z evaluate the performance of a modulation type
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Problem Statement
We transmit a signal , where
is nonzero only on .
Let the various signals be transmitted with probability:
The received signal is corrupted by noise:
Given , the receiver forms an estimate of the signalwith the goal of minimizing symbol error probability
( ) ( ){ }s t s t s t s t M 1 2, ( ), , ( ) ( )s t[ ]t T 0,
( )[ ] ( )[ ]p s t p s tM M1 1= =Pr , , Pr
( ) ( ) ( )r t s t n t = +
r t( )( )s t
( )s t
( ) ( )[ ]P s t s t s = Pr
Noise Model
The signal is corrupted by Additive White Gaussian Noise
(AWGN)
The noise has autocorrelation and
power spectral density
Any linear function of will be a Gaussian random
variable
n t( )
n t( ) ( ) ( ) nnN
= 02
( )nn f N= 0 2
n t( )
s t( )
n t( )
r t( )
Channel
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Signal Space Representation
The transmitted signal can be represented as:
,
where .
The noise can be respresented as :
where
and
( )s t s f t m m k k k
K
( ) ,= =1
( )s s t f t dt m k m k T
, ( )= 0
( ) ( )n t n t n f t k kk
K= +
=( )
1
( ) ( )n n t f t dt k k
T
= 0
( ) ( ) = =
n t n t n f t k kk
K( )
1
Signal Space Representation (continued)
The received signal can be represented as :
where
( ) ( ) ( ) ( )r t s f t n f t n t r f t n t m k kk
K
k k
k
K
k k
k
K= + + = +
= = =, ( ) ( )
1 1 1
r s nk m k k = +,
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The Orthogonal Noise:
The noise can be disregarded by the receiver
( ) ( )
( ) ( )
( ) ( )
s t n t dt s t n t n f t dt
s f t n t n f t dt
s f t n t dt s n f t dt
s n s n
m
T
m k kk
KT
m k kk
K
k kk
KT
m kk
K
k
T
m k kk
K
k
T
m k kk
Km k k
k
K
( ) ( ) ( )
( )
( )
,
, ,
, ,
=
=
=
= =
=
= =
= =
= =
0 10
1 10
1 0 1
2
0
1 1
0
( )n t
( )n t
We can reduce the decision to a finite
dimensional space!
We transmit a Kdimensional signal vector:
We receive a vector which is the sum of
the signal vector and noise vector
Given , we wish to form an estimate of the transmitted
signal vector which minimizes
[ ] { }s s s= s s sK M1 2 1, , , , ,
[ ]r s n= = +r rK1, ,[ ]n = n nK1, ,
r s[ ]Ps = Pr s s
s
Channel
n
rReceiver
s
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MAP (Maximum a posteriori probability)
Decision Rule
Suppose that signal vectors are transmitted
with probabilities respectively, and the signal
vector is received
We minimize symbol error probability by choosing the
signal which satisfies :
Equivalently :
or
{ }s s1 , , M{ }p pm1, ,
r
sm ( ) ( )Pr Pr ,s r s rm i m i
( ) ( )( )
( ) ( )( )
p
p
p
pm im m i i
r s s
r
r s s
r
Pr Pr ,
( ) ( ) ( ) ( )p p m im m i ir s s r s sPr Pr ,
Maximum Likelihood (ML) Decision Rule
If or the a priori probabilities are unknown,
then the MAP rule simplifies to the ML Rule
We minimize symbol error probability by choosing the
signal which satisfies
p pm1 = =
sm ( ) ( )p p m im irs rs ,
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Evaluation of Probabilities
In order to apply either the MAP or ML rules, we need to
evaluate:
Since where is constant, it is equivalent to
evaluate :
is a Gaussian random process
zTherefore is a Gaussian random variable
z Therefore will be a Gaussian p.d.f.
( )p mrs
r s n= +m sm( ) ( )p p n nkn = 1, ,
( ) ( )n n t f t dt k k
T
= 0
n t( )
( )p n nK1, ,
The Noise p.d.f
[ ] ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
E n n E n t f t dt n s f s ds
E n t n s f t f s dsdt E n t n s f t f s dsdt
t s f t f s dsdt N
t s f t f s dsdt
Nf t f t dt
N i k
i k
i k i
T
k
T
i k
TT
i k
TT
nn i k
TT
i k
TT
i k
T
=
=
=
= =
= ==
0 0
00 00
00
0
00
00
0
2
2
2
0
,
,
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The Noise p.d.f (continued)
Since , individual noise components are
uncorrelated (and therefore independent)
Since , each noise component has a variance of
.
[ ]E n n i ki k = 0,
[ ]E n Nk2 0 2=
( ) ( ) ( )
( )
( )
p n n p n p n
Nn N
N n N
K K
k
K
k
Kk
k
K
1 1
01
20
02 2
10
1
, ,
exp
exp
=
=
=
=
=
N0 2
Conditional pdf of Received Signal
Transmitted signal values in each dimension represent the
mean values for each signal
( ) ( ) ( )p N r s NmK
k m kk
Kr s =
= 0
2 2
10exp ,
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Structure of Optimum Receiver
MAP rule :
{ }( ) arg max
, ,
s r s
s s
= 1 M
p pm m
{ }( ) ( ) arg max exp
, ,,s
s s
=
=10
2 2
10
M
p N r s NmK
k m kk
K
{ }( ) ( ) arg max ln exp
, ,,s
s s
=
=10
2 2
10
M
p N r s NmK
k m kk
K
{ }[ ] [ ] ( ) arg max ln ln
, ,,s
s s
= =1
2
10
0
2
1 M
pK
NN
r sm k m k k
K
Structure of Optimum Receiver (continued)
Eliminating terms which are identical for all choices:
{ }[ ] [ ] arg max ln ln
, ,
, ,
s
s s
=
+
= ==
12
1 2
0
0
2
1
2
11
M
pK
N
Nr r s s
m
k k m k k
K
m kk
K
k
K
{ }
[ ] arg max ln, ,
, ,s
s s
= +
= =1
2 1
0 1 0
2
1 M
p
N
r s
N
sm k m k k
K
m k
k
K
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Final Form of MAP Receiver
Multiplying through by the constant :N0 2
{ }[ ] argmax ln
, ,, ,s
s s
= + = =1
0
1
2
12
1
2 M
Np r s sm k m k
k
Km k
k
K
Interpreting This Result
weights the a priori probabilities
z If the noise is large, counts a lot
z
If the noise is small, our received signal will be an accurateestimate and counts less
represents the correlation with
the received signal
represents signal energy
pm
[ ]N
pm0
2ln
pm
( )r s s t r t dt k m kk
K
m
T
, ( )= =
1 0
( )12
12 2
2 2
01
s s t dt Em k mT
k
K m, = =
=
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An Implementation of the Optimal Receiver -
Correlation Receiver
Choose
Largest
( )r t
( )s t1
dtT
0
E1 2 ( )p0 12
ln
( )r t
( )s tM
dtT0
EM 2 ( )N
pM0
2ln
Simplifications for Special Cases
ML case: All signals are equally likely ( ). A
priori probabilities can be ignored.
All signals have equal energy ( ). Energy
terms can be ignored.
We can reduce the number of correlations by directly
implementing:
p pM1= =
E EM1= =
{ } [ ]arg max ln
, ,, ,s
s s= + = =1
0
1
2
12
1
2 M
N
p r s sm k m k k
K
m kk
K
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Reduced Complexity Implementation:
Correlation Stage
( )r t
( )f t1
dtT0 r1
[ ]r = r rK1
( )r t
( )f tK
dtT0
rK
Reduced Complexity Implementation -
Processing Stage
r
s s
s s
M
K M K
11 1
1
, ,
, ,
Choose
Largest
EM 2 ( )N
pM0
2ln
s rM k kk
K
,
=
1
E1 2
s rk k
k
K
1
1
,
=
( )N
p0 12
ln
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Matched Filter Implementation
Assume is time-limited to , and let
Then
where denotes the convolution of the signals
and evaluated at time
We can implement each correlation by passing through
a filter with impulse response
( )f tk [ ]t T 0, ( ) ( )h t f T t k k=
( ) ( )
( ) ( ) ( )
r r t f t dt r t f T T t dt
r t h T t dt r t h t
k k
T
k
T
k
T
k t T
= =
= = =
( ) ( ) ( )
( )
0 0
0
( ) ( )r t h t k t T = ( )r t
( )h tk T
( )r t( )h tk
Matched Filter Implementation of
Correlation
[ ]r = r rK1
( )r t ( )h t1
( )r th t
K( )
t T=
t T=
r1
rK
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Example of Optimal Receiver Design
Consider the signal set:
-1
+1t
s t1( )
-1
+1t
s t3( )
-1
+1t
s t2( )
-1
+1t
s t4( )
1 2
1 2
1 2
1 2
Example of Optimal Receiver Design
(continued)
Suppose we use the basis functions:
-1
+1t
f t1( )
1 2 -1
+1t
f t2( )
1 2
s t f t f t 1 1 21 1( ) ( ) ( )= + s t f t f t 2 1 21 1( ) ( ) ( )=
s t f t f t 3 1 21 1( ) ( ) ( )= + s t f t f t 4 1 21 1( ) ( ) ( )=
T= 2E E E E1 2 3 4 2= = = =
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1st Implementation of Correlation Receiver
( )r t
( )s t1
dt0
2
( )N
p0 12
ln ChooseLargest
( )r t
( )s t4
dt0
2
( )N
p0 42
ln
Reduced Complexity Correlation Receiver -
Correlation Stage
( )r t
( )f t2
dt02 r2
[ ]r = r r1 2
( )r t
( )f t1
dt02 r1
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Reduced Complexity Correlation Receiver -
Processing Stage
( )N p0 4 2ln
Choose
Largest
1 11 2 + r r
( )N p0 1 2ln
( )N p0 2 2ln
( )N p0 3
2ln
1 11 2 r r
+ 1 11 2r r
1 11 2r r
Matched Filter Implementation of
Correlations
( ) ( )h t f t k k= 2
+1 t
h t1( )
1 2
+1 t
h t2( )
1 2
( )r t ( )h t1
( )r t h t2( )
r1
r2
[ ]r = r r1 2
t= 2
t= 2
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Summary of Optimal Receiver Design
Optimal coherent receiver for AWGN has three parts:
z Correlates the received signal with each possible transmitted
signal signalz Normalizes the correlation to account for energy
z Weights the a priori probabilities according to noise power
This receiver is completely general for any signal set
Simplifications are possible under many circumstances
Decision Regions
Optimal Decision Rule:
Let be the region in which
Then is the ith Decision Region
{ }
[ ] arg max ln
, ,
, ,s
s s
= +
= =1
0
1
2
12
1
2
M
Np r s sm k m k
k
K
m k
k
K
RiK
[ ]
[ ]
Np r s s
Np r s s i j
i k i k k
K
i kk
K
j k j k
k
K
j k
k
K
0
1
2
1
0
1
2
1
2
1
2
2
1
2
ln
ln ,
, ,
, ,
+
+
= =
= =Ri
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A Matlab Function for
Visualizing Decision Regions
The Matlab Script File sigspace.m (on course web page)
can be used to visualize two dimensional signal spaces and
decision regions
The function is called with the following syntax:
sigspace( , )
z and are the coordinates of the ith signal point
z is the probability of the ith signal (omitting gives ML)
z is the signal to noise ratio of digital system in dB
E Nb 0[ ]x y p x y p y pM M M1 1 1 2 2 2; ; ;
xi yi
piE Nb 0
Average Energy Per Bit:
is the energy of the ith signal
is the average energy per symbol
is the number of bits transmitted per symbol
is the average energy per bit
z allows fair comparisons of the energy requirements of
different sized signal constellations
E si i kk
K=
=,
2
1
EM
Es ii
M= =
1
1
log2 M
EE
Mb
s=log2
Eb
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Signal to Noise Ratio for Digital Systems
is the (two-sided) power spectral density of the
background noise
The ratio measures the relative strength of signal
and noise at the receiver
has units of Joules = Watts *sec
has units of Watts/Hz = Watts*sec
The unitless quantity is frequently expressed in dB
N0 2
E Nb 0
Eb
N0
E Nb 0
Examples of Decision Regions - QPSK
sigspace( [1 0; 0 1; -1 0; 0 -1], 20)
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
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QPSK with Unequal Signal Probabilities
sigspace( [1 0 0.4; 0 1 0.1; -1 0 0.4; 0 -1 0.1], 5)
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
QPSK with Unequal Signal Probabilities -
Extreme Case
sigspace([0.5 0 0.4; 0 0.5 0.1; -0.5 0 0.4; 0 -0.5 0.1],-6)
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
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Unequal Signal Powers
sigspace( [1 1 ; 2 2; 3 3; 4 4], 10)
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.5
1
1.5
2
2.5
3
3.5
4
Signal Constellation for 16-ary QAM
sigspace( [1.5 -1.5; 1.5 -0.5; 1.5 0.5; 1.5 1.5; 0.5 -1.5; 0.5 -
0.5; 0.5 0.5; 0.5 1.5;-1.5 -1.5; -1.5 -0.5; -1.5 0.5; -1.5 1.5; -
0.5 -1.5; -0.5 -0.5; -0.5 0.5; -0.5 1.5],10)
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
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Notes on Decision Regions
Boundaries are perpendicular to a line drawn between two
signal points
If signal probabilities are equal, decision boundaries lie
exactly halfway in between signal points
If signal probabilities are unequal, the region of the less
probable signal will shrink.
Signal points need not lie within their decision regions for
case of low and unequal probabilities.E Nb 0