lecture 06 - optimal receiver design

7/29/2019 Lecture 06 - Optimal Receiver Design

1/21

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


2/21

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


3/21

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 = +,


4/21

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


5/21

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 ,


6/21

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

,

,


7/21

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 ,


8/21

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


9/21

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, = =

=


10/21

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


11/21

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


12/21

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


13/21

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= = = =


14/21

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


15/21

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


16/21

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


17/21

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


18/21

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


19/21

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


20/21

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


21/21

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

lecture 06 - optimal receiver design

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