noise tsks01 digital communication lecture 4 · digital to analog analog to digital medium digital...

TSKS01 Digital Communication

Lecture 4

Digital Modulation � Detection

Mikael Olofsson

Department of EE (ISY)

Div. of Communication Systems

2013-09-24 TSKS01 Digital Communication - Lecture 4 2

A One-way Telecommunication System

Channel

Source

encoder

Source

decoder

Source

Destination

Channel

encoderModulator

Channel

decoder

De-

modulator

Source

coding

Channel

coding

Packing

Unpacking

Error control

Error correction

Digital to analog

Analog to digital

Medium

Digital

modulation


Situation

ChannelSender ReceiverSource Destination

Noise

ChannelVector

sender

Vector

detectorSource DestinationModulator

De-

modulator

Noise

Vector channelVector

sender

Vector

detectorSource Destination

Vector noise


Last Time � Signals as Vectors

Signals:

Tt <≤0

( ) 1,,1,0, −= Mitsi �

ON basis:

( ) ( )��

��

≠

===� ji

jidttt ji

T

ji,0

,1,

0

δφφ

( ) 1,,1,0, −= Njtj �φ

Linear combinations:

( ) ( )�−

=

=1

0

,

N

j

jjii tsts φ

Vectors:

�

��

�

=

−1,

0,

Ni

i

i

s

s

s �

Inner products (correlation):

( ) ( ) ( ) ,,0

dttbtaba

T

�= �−

=

=1

0

N

j

jjbaba �

ON basis � ( ) baba �=,

Norms: ( )aaa ,2

=

aaa �=2

Euclidean distance:

( ) 222 , bababad −=−=

Angles: ( ) ( )ba

ba

ba

ba

⋅=

⋅=

,cos

�α

Equal!


Last Time � Impact of White Gaussian Noise

Received: ( ) ( ) ( )tWtstX i +=

( ) ( ) ( )dtttWWW

T

jjj �==0

, φφ

Statistical properties:Noise: ( ) ( ) ( )tWtWtWN

j

jj′+=�

−

=

1

0

φ

( ) ( )tWtXN

j

jj′+=�

−

=

1

0

φ

( ) ( ) ( )dtttXXX

T

jjj �==0

, φφ jji Ws += ,

Vectors: WsX i +=

�

��

�

=

−1

0

NX

X

X �

�

��

�

=

−1

0

NW

W

W �

Wj Gaussian with mean 0.

Xj Gaussian with mean si,j.

Wj and Wk independent for j ≠ k.

Xj and Xk independent for j ≠ k.

Wj and W�(t) independent.

( )22

020 NNfR

jWW =�= σ


AWGN � Additive White Gaussian Noise

Orthogonal noise components

are statistically independent.

X1 & X2 independent

Y1 & Y2 independent

PSD = Variance

Y2

Y1

X1

X2

σXi= σYi

= RW ( f ) = N0/22 2


Demodulation

ChannelModulatorDe-

modulator

Noise

Projections:

Do this!

Signals are

projected on

basis functionsNoise is

projected on

basis functions


Correlation Receiver


Matched Filters

A filter with impulse response hj(t) = φj(T � t) is matched to φj(t).

Example: Usage (φj�hj)(t):

T 2T

Maximum at t = T.

Orthogonal signal φi(t):

Zero at t = T.

T 2T

T

φj(t)hj(t)

Then (φi�hj)(t):

(φi(t),φj(t)) = 0


Demodulation Using Matched Filters

ChannelModulatorDe-

modulator

Noise

Do this!

Filter: Matched to

Output:

Sample at t = T : And this!


Matched Filter Receiver


The Vector Detector

Knowledge that can be used:

All entities are realizations of stochastic variables.

Use statistical descriptions of those stochastic variables.

The task of the vector detector:

Observe and output according to a well chosen decision rule,

designed to minimize the error probability.

x a�

Vector channelVector

sender

Vector

detectorSource Destination

Vector noise

x a�isia


1 3

3

1

�1

φ0

φ1

0s

1s

2s

Vector Detection

?

Received vector: x

x

Goal: Minimize the error probability

{ }xXAA =≠ |�Pr

Estimated

symbol

Sent

symbol

Received

vector

Observed

realization

Equivalent decision rule:

Set if

is maximized for k = i.

{ }xXaA k == |Priaa =�


Independent of ak.

"4"2

02

46

"20

24

680

0.01

0.02

0.03

Detection

Bayes rule: { }( )

( ){ }k

X

kAX

k aAxf

axfxXaA ==== Pr

||Pr

|

New decision rule:

Set if

is maximized for k = i.

iaa =�

( ) { }kkAXaAaxf =⋅Pr|

|


Maximum Likelihood (ML) Detection

Assumption: { } { }1,,1,01

Pr −∈∀== MkM

aA k �

ML decision rule: Set if is maximized for k = i. iaa =� ( )kAX

axf ||

( ) ( )∏−

=

=1

0

||||

N

j

kjAXkAXaxfaxf

j

Independent variables:

( ) 02

,

1

0 0

1 NsxN

j

jkjeN

−−−

=

∏ ⋅=π

( )( )�

⋅=

−

=

−−−

1

0

2,

0

1

2

0

N

j

jkj sxNN

eNπ

Natural logarithm (strictly increasing function):

( )( ) ( ) ( )� −−−=−

=

1

0

2

,

0

0|

1ln

2|ln

N

jjkjkAX

sxN

NN

axf π ( ) ( )ksxdN

NN

,1

ln2

2

0

0 −−= π

Constant

Equivalent ML rule: Set if is minimized for k = i. iaa =� ( )ksxd ,


1 3

3

1

�1

φ0

φ1

2s

0s

1s

ML Decision Regions

?

B2

B1

B0

x

Interprete as the nearest signal.x

!

Result:

Decision regions consisting

of all points closest to a

signal point.

Notation:

Bi is the decision region of

the signal vector si.

Thus also of the signal si (t)

and of the message ai.

The borders are orthogonal to

straight lines between signals:

In 2 dimensions: Lines.

In 3 dimensions: Planes.

In more dim: Hyperplanes.

Borders cut the lines mid-way.


Error Probability

Symbol error probability:

{ } { } { }�−

=

=≠⋅==≠=1

0

e |�PrPr�PrM

i

iii aAaAaAAAP

{ } { }�−

=

=∉⋅==1

0

|PrPrM

i

iii aABXaA

ML detection again { } { }

��

�−∈∀== 1,,1,0

1Pr Mi

MaA i � :

{ }�−

=

=∉=1

0

e |Pr1 M

i

ii aABXM

P ( ) N

M

i

iAXdxdxaxf

M�� 1

1

0|

|1��

−

=

=

iBx ∉

This is generally hard to calculate.


A Special Case � Two signals in One Dim

( )10 , ssd

B1B0

fX|A(x|a0) fX|A(x|a1)

{ } { }0100 |Pr|Pr aABXaABX =∈==∉

s0 s1

φ0

��

��

=+

>= 0

0,10,0

0 |2

Pr aAss

X

��

��

=−+

>= 00,0

0,10,0

0 |2

Pr aAsss

W��

�� −

>=2

Pr0,00,1

0

ssW

( )��

��

>=2

,Pr 10

0

ssdW

( )

�

��

�=

2

2,

0

10

N

ssdQ

( )

�

��

�=

0

10

2

,

N

ssdQ

Similarily for .{ }11 |Pr aABX =∉

This works in all

directions.

B1B0

s0

s1( )�QP =� e

2

10 ss +


Back to M Signals in N Dimensions

1 3

3

1

�1

φ0

φ1

2s

0s

1s

B2

B1

B0


We had:

{ }�−

=

=∉=1

0

e |Pr1 M

i

ii aABXM

P

{ }��−

= ≠

=∈=1

0

|Pr1 M

i ij

ij aABXM

Hard to calculate!!

Could we take this to

the more simple case

in one dimension?


The Union Bound

1 3

3

1

�1

φ0

φ1

2s

0s

1s

B2

B1

B0


Define overestimated regions:

( ) ( ){ }ijji sxdsxdxB ,,:, <=

B0,1B0,2

An upper bound based on

overestimating the decision

regions. We had:

{ }��−

= ≠

=∈=1

0

e |Pr1 M

i ij

ij aABXM

P

{ }��−

= ≠

=∈≤1

0

,e |Pr1 M

i ij

iji aABXM

P

Overestimated error probability:

��−

= ≠

�

��

�≤

1

0 0

,

e2

1 M

i ij

ji

N

dQ

MP

Back to the one-dim case:

( )jiji ssdd ,, =Distances:


The Nearest Neighbour Approximation

1 3

3

1

�1

φ0

φ1

2s

0s

1s


��−

= ≠

�

��

�≤

1

0 0

,

e2

1 M

i ij

ji

N

dQ

MP

We had the union bound:

0,11,0 dd =

1,22,1 dd =

0,22,0 dd =

� �−

= =

�

��

�≈

1

0 : 0

mine

min,2

1 M

i ddj jiN

dQ

MP

Nearest neighbour approximation:

Dominated by the smallest distance.

jiji

dd ,min min≠

=

min,:# ddjn jii ==

�−

=

�

��

�=

1

0 0

min

2

1 M

i

iN

dQn

M

mind=


Comments

Alternative upper bound:

As the union bound, but only consider pairs of points whose

decision regions share a common border.

At high SNR Eavg/N0:

Both the union bound on � and the nearest neighbour of �

the error probability are close to the real error probability.

Alternative approximation:

As the nearest neighbour approximation, but consider the two or

three smallest distances.

Very simple approximation:

�

��

�≈

0

mine

2N

dQP

Very simple upper bound:

( )

�

��

�⋅−≤

0

mine

21

N

dQMP


Maximum á-Posteriori (MAP) Detection 1(2)

B1B0

s0 s1

φ0

fX|A(x|a0)Pr{A=a0}�

fX|A(x|a1)Pr{A=a1}�

Back to this rule (MAP):

Set if is maximized for k = i. iaa =� ( ) { }kkAXaAaxf =⋅Pr|

|

( ) { }( )kk aANsxd =⋅− Prln, 0

2

Equivalent reasoning leads to this rule:

Set if is minimized for k = i. iaa =�


Maximum á-Posteriori (MAP) Detection 2(2)

The borders are still:

Straight lines (hyperplanes)

Orthogonal to straight lines

between signal points.

1 3

3

1

�1

φ0

φ1

2s

0s

1s

B2

B1

B0

But:

Borders cut the lines mid-way

only if the two probabilities are

equal.

Borders depend on N0.

Note:

A signal point may in extreme

cases not even be in its own

detection region.


Special Case: Orthogonal Decision Borders

2s

0s1s

B2

B1B0

3sB3

1,0d

2,1d 2,0d

UB: 0,21,20,1e qqqP ++≤1,20,1e qqP +≤Alternative bound:

NN: 1,2e qP ≈ 1,20,1e qqP +≈Alternative approx.:

Exact: 1,20,11,20,1e qqqqP −+=

(orthogonal noise components are independent)

�

��

�=

0

,

ji,2N

dQq

jiDefine notation:

2

2,1

2

1,0

2

2,0 ddd +=Pythagoras:

1,2e qP >Lower bound:


0θ

α

Cost of Non-Optimal Detection

Detected signal points: ( ) ( ) ( ) ( )ααθφθφθ coscos,, 000000 ⋅±=⋅⋅⋅±=⋅±=± aaas

d′Signals: )()( 00 tats φ⋅=

)()( 01 tats φ⋅−=

Wrong basis function in receiver:

)(0 tθ

What is the cost of this non-ideal

situation?

0φ0s

1s

ad 2=

( )αcos⋅=′ ddEffective distance:

�

��

� ′=

0

e2N

dQPResulting error probability:

Signals are

projected on

basis functions


Example of Non-Optimal Detection

( ) ( ) 2

0

e 1025.124.252

−⋅≈≈=

�

��

� ′= QQ

N

dQPAt α = π /4:

Ideal detection: ( ) ( ) 4

0

e 109.716.3102

−⋅≈≈=

�

��

�= QQ

N

dQP

d′0θ

0φ0s

1s

ad 2=

α

SNR: 50

2

=N

a

( )αcos⋅=′ dd

Distances: ad 2=

Signal points: a±

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