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

2014-09-23 TSKS01 Digital Communication - Lecture 4 2

A One-way Telecommunication System

Channel

Source encoder

Source decoder

Source

Destination

Channel encoder

Modulator

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 detector

Source DestinationModulatorDe-

modulator

Noise

Vector channelVector sender

Vector detector

Source Destination

Vector noise


Last Time – Signals as Vectors

Signals:

Tt <≤0( ) 1,,1,0, −= Mitsi K

ON basis:

( ) ( )

≠=

==∫ ji

jidttt ji

T

ji ,0

,1,

0

δφφ

( ) 1,,1,0, −= Njtj Kφ

Linear combinations:

( ) ( )∑−

=

=1

0,

N

jjjii tsts φ

Vectors:

=

−1,

0,

Ni

i

i

s

s

s M

Inner products (correlation):

( ) ( ) ( ) ,,0

dttbtabaT

∫= ∑−

=

=1

0

N

jjjbaba o

ON basis ⇒ ( ) baba o=,

Norms: ( )aaa ,2 =

aaa o=2

Euclidean distance:

( ) 222 , bababad −=−=

Angles: ( ) ( )ba

ba

ba

ba

⋅=

⋅= ,

cosoα

Equal!


Last Time – Impact of White Gaussian Noise

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

( ) ( ) ( )dtttWWWT

jjj ∫==0

, φφ

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

jjj ′+=∑

−

=

1

0

φ

( ) ( )tWtXN

jjj ′+=∑

−

=

1

0

φ

( ) ( ) ( )dtttXXXT

jjj ∫==0

, φφ jji Ws += ,

Vectors: WsX i +=

=

−1

0

NX

X

X M

=

−1

0

NW

W

W M

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 detector

Source Destination

Vector noise

x aisia


1 3

3

1

–1

φ0

φ1

0s

1s

2s

Vector Detection

?

Received vector: x

x

Goal: Minimize the error probability

{ }xXAA =≠ |ˆPr

Estimated symbol

Sent symbol

Received vector

Observed realization

Equivalent decision rule:

Set if is maximized for k = i.

{ }xXaA k == |Priaa =ˆ


Independent of ak.

−4−2

02

46

−20

24

680

0.01

0.02

0.03

Detection

Bayes rule: { } ( )( ) { }k

X

kAXk aA

xf

axfxXaA ==== Pr

||Pr |

New decision rule:

Set if

is maximized for k = i.

iaa =ˆ

( ) { }kkAX aAaxf =⋅Pr||


Maximum Likelihood (ML) Detection

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

Pr −∈∀== MkM

aA k K

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

( ) ( )∏−

=

=1

0|| ||

N

jkjAXkAX axfaxf

j

Independent variables:

( ) 02

,

1

0 0

1 NsxN

j

jkjeN

−−−

=∏ ⋅=

π( )

( )∑⋅=

−

=

−−−

1

0

2,

0

1

20

N

jjkj sx

NN eNπ

Natural logarithm (strictly increasing function):

( )( ) ( ) ( )∑ −−−=−

=

1

0

2,

00|

1ln

2|ln

N

jjkjkAX sx

NN

Naxf π ( ) ( )ksxd

NN

N,

1ln

22

00 −−= π

Constant

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

0e |ˆPrPrˆPr

M

iiii aAaAaAAAP

{ } { }∑−

=

=∉⋅==1

0

|PrPrM

iiii aABXaA

ML detection again { } { }

−∈∀== 1,,1,01

Pr MiM

aA i K :

{ }∑−

=

=∉=1

0e |Pr

1 M

iii aABX

MP ( ) N

M

iiAX dxdxaxf

MLL 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

0,10,00 |

2Pr aA

ssX

=−+

>= 00,00,10,0

0 |2

Pr aAsss

W

−

>=2

Pr 0,00,10

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( )LQP =⇒ e

210 ss +


Back to M Signals in N Dimensions

1 3

3

1

–1

φ0

φ1

2s

0s

1s

B2

B1

B0


We had:

{ }∑−

=

=∉=1

0e |Pr

1 M

iii aABX

MP

{ }∑∑−

= ≠

=∈=1

0

|Pr1 M

i ijij aABX

M

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

0e |Pr

1 M

i ijij aABX

MP

{ }∑∑−

= ≠

=∈≤1

0,e |Pr

1 M

i ijiji aABX

MP

Overestimated error probability:

∑∑−

= ≠

≤

1

0 0

,e

2

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

,e

2

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

ii

N

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 =ˆ ( ) { }kkAX aAaxf =⋅Pr||

( ) { }( )kk aANsxd =⋅− Prln, 02

Equivalent reasoning leads to this rule:

Set if is minimized for k = i. iaa =ˆ


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-wayonly 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:

22,1

21,0

22,0 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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tsks01 digital communication lecture 4 · digital to analog analog to digital medium digital ......

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