Universität Bremen Digitale Übertragung 1 Grundlagen der Nachrichtentechnik I. Kontinuierliche...

Digitale Übertragung 1UniversitätBremen

Grundlagen der NachrichtentechnikGrundlagen der Nachrichtentechnik

I. Kontinuierliche Signale und Systeme

1. Fouriertransformation

2. Tiefpass-Darstellung v. Bandpass-Signalen

3. Eigenschaften von Übertragungskanälen

III. Diskretisierung v. Quellensignalen

1. Abtasttheorem

2. Pulsamplitudenmodulation

3. Pulsdauer- und Pulsphasenmodulation

4. Pulscodemodulation

5. Prinzip des Zeitmultiplex

II. Analoge Übertragung

1. Analoge Modulationsverfahren

2. Empfängerstrukturen

3. Einfluss von Rauschen

IV. Digitale Übertragung

1. Struktur e. Datenübertragungssystems

2. Erste u. Zweite Nyquist-Bedingung

3. Rauschangepasstes Empfangsfilter

4. Bitfehlerwahrscheinlichkeit

5. Digitale Modulationsverfahren

__________________________________________________________________


1. Structure of Data Transmission Systems1. Structure of Data Transmission Systems

objective: transmitting discrete values d(i) across an analog channel

weighting time-shifted analog impulses gTx(t-iT) with d(i)

( )s t

i

iTtT )(0)(id )(tgTx

i

T )(id )( iTtgTx


g(t): impulse response of a transmission system (infinite length)

equally spaced zeros,

interval Tfn

2

1

2. 1st Nyquist Criterion: Time domain2. 1st Nyquist Criterion: Time domain

TfN

2

1

02t0t

t0

1g(t)

-1

shaping function

no ISI !


limitation of length )20( 0tt by multiplying with a shaping function

and sampling (rate Na fT

f 21

), Tit 00

1st Nyquist Criterion in time domain

0

0

for 0

for 1)(

ii

iiiTgT

1st Nyquist Criterion: Time domain1st Nyquist Criterion: Time domain


1st Nyquist Criterion: Frequency domain1st Nyquist Criterion: Frequency domain

0-1

-

- 2 j i TT

i

G j i e

2a Nf f 4 Nff

1

0j i TG j e

0(limited bandwidth)


symmetry to

Nf : Nyquist rolloff

1st Nyquist Criterion: Frequency domain1st Nyquist Criterion: Frequency domain

1

0,5

TfN 2

1 f

a

a

b

b

f f

)(0 jG

TijejGjG 0)()( 0 )( jG with linear phase: ))(( 0 jG


Cosine rolloff filterCosine rolloff filter

20 )2(1

)cos()sin()(

Tt

Tt

Tt

Tt

rc tg

r

r

r : rolloff factor 10 r

)1()1( 21

21 rfr TT

Trf 21)1(

)1(21 rf T

)2(0 fjGrc

1

0

ifrr ))1(cos(1 221 T

f


Cosine rolloff filter: Examples (w=4)Cosine rolloff filter: Examples (w=4)


Demonstration: Eye pattern (r=0,5)Demonstration: Eye pattern (r=0,5)


Cosine rolloff filter: Eye patternCosine rolloff filter: Eye pattern

2nd Nyquist

1st Nyquist

2nd Nyquist:

1st Nyquist:

2nd Nyquist:

1st Nyquist:

2nd Nyquist:

1st Nyquist:

2nd Nyquist:

1st Nyquist:


Cosine rolloff filter: Bandwidth efficiency Cosine rolloff filter: Bandwidth efficiency

data rate 1/ 2 bit/s

bandwidth (1 ) / 2 1 Hzrc

T

r T r

Hz

bit/s2

)1(

2

Hz

bit/s1

r

2nd Nyquist (r=1) r=0


d(i) gTx(t)

Noise na(t)

?

)()()( 0 iTniTriTr

maxN

S

task: design a gRx(t) that maximizes the -RatioS

N

3. Matched Filter3. Matched Filter

)(

)(

)()()(

0

00 Tin

Tir

iidTgTTir

Ti

gRx(t)


Matched FilterMatched Filter

0 0 0

0

( )( )

0 ;

g T t T i Tg t

t i T i i

2

2 20 0E ( ) ( ) ( )

D Rx TxR iT g g T d

i.e.

with ( ) ( ) ( )Tx Rxg t g t g t ( )g tand meets 1.Nyquist criterion

2

2 2 22 20 0 0 0E ( ) E ( ) ( ) ( )

D

DR iT D i i g T g T

ˆ( )d ipower of discrete signal

2

2

0E ( ) E

l

R iT D l g l iT


Matched filterMatched filter

2 2 20 1E ( ) E ( ) ( )

2 2a a Rx

NN t N iT G j d

f

20N

fN

2

0 fN

2

0

fN 0 noise power on channel is

noise power at output of receive filter gRx(t) :

20N

channel noise na(t) is white with spectral power density

:( )n ipower of interference


Matched filterMatched filter

2 2 20 01E ( ) ( ) ( )

2 2 2Rx Rx

N NN iT G j d g d

noise power at output of receive filter gRx(t) :

20N

channel noise na(t) is white with spectral power density

power of interference )(in :

20N

Parseval‘s theorem

2( )RxG j


determine signal-to-noise-ratio

N

S

2

20( ) ( )D Rx Txg g T d

20 ( )2 Rx

Ng d

max!

by defining the mean energy of a single transmitted symbol:

SE

dgd Tx

22 )(2D


2

0 2

0

2

( )

(( ))2

S

TR

Rx Tx

xx

S

N Ng

Eg T

g

d

d d

g


since

dTgdg TxTx

2

0

2)()(

N

S- Ratio can be estimated by Schwartz´s inequality

2

0( )Rx Txg g T d

22( ) ( )x TR xgg d d

this implies

2/0N

E

N

S S


2

0 2

0

2

( )

(( ))2

S

TR

Rx Tx

xx

S

N Ng

Eg T

g

d

d d

g


When does equality apply (maximum )?N

S

0 Rx Txg ( t ) g (T t )


0( ) Rx Txg t g T t

2

2 2

Rx Rx Rx Rxg g d g d g d

2 2

2 2

Rx Rxg d g d


0T

example:

transmit filter receive filter


t

)()( 0 tgtTg RxTx

0T t

)( tgTx

0T t

)(tgTx

Matched Filter: optimal receive filter for maximizedN

S

(matched)



1

2T

1

2T

1

2T

1

2T

0.5

1/ 2

11

ff

G j

Tx Rx Tx Tx Rx RxG j G j G j G j G j G j G j

Tx RxG j G j G j

Nyquist slope Nyquist slope

Raised cosine design Root raised cosine design


total system impulse response:

( ) ( ) ( )Rx Txg t g g t d

)( 0 TgTx ( )Txg t d with 0T

0( ) ( ) ( )

Tx Txg t g g t T d ( ) ( )Eggr g g d

reminder:


implemented as matched filter!)(tgRx

0 Tx Tx

Eg gg t r t T

The impulse response of the total system

is the shifted Energy-ACF of the transmit filter.








1. Abtasttheorem















__________________________________________________________________


4. Bit Error Probability4. Bit Error Probability

We assume: • Binary transmission with

• transmission system fulfills 1. Nyquist criterion

• noise , independent of data sourcen(iT)

Probability density function (pdf) of

},{)( id 0d 1d

)(npN

n

n(iT)

d(i) gTx(t)

Noise na(t)

0 ( ) ( )r nT iTi

Ti

gRx(t)


Conditional pdfsConditional pdfs

The transmission system induces two conditional pdfs depending on )(id

• if )(id 0d • if )(id 1d

0d 1dx x

)(0p x )( xpN 0d )(1p x )( xpN 1d

)(0p x )(1p x


Probability of wrong decisionsProbability of wrong decisions

11

0 2( ) P P

0

1 10 0 0 02 2

1 12

1 (

12

)

1 1 1 12( ) ( ) ( ) ( )

S

S S

S

xp

b

dx

P x dx x dx x xP Q pP pQ p p dx

Placing a threshold S

Probability of wrong decision

)(0p x )(1p x

0d 1dS S

When we define and as equal a-priori probabilities of and

we will get the bit error probability

0d 1d1P0P

xd

xd0Q 1Q )(1p x)(0p x

x x

S

S


22

Conditions for „illustrative“ solutionConditions for „illustrative“ solution

10

2

S

d d

211P 0P )()( xpxp NN

substituting

S

xd)(1p x

xxpN d)(2S 0d 1d

S

1d

for2x S 0d 1d

0d 1d1d 1d0d

x 1d x

equivalently

with

00

d)(21

d)(21

NN pp20d 1d

xx2 0d1d

x x

0

d)(2121

xxpP Nb

2 0d1d

xxPb dd121

S

)(1p xS

)(0p x

xd)(1p x

d)(Np20d 1d

S

xx

S

01

2

0

0 d

' '1

d2

S

N

d d

p

x

x x

xp

With and


Special Case: Gaussian distributed noiseSpecial Case: Gaussian distributed noise

• many independent interferers

• central limit theorem

• Gaussian distribution

Motivation:

Definition of „Error Function“ and „Error Function Complement“

)erfc(x 1 )erf( x)erf(

x

0

de2 2

x x

x

2

2

2e2

1)( N

NNp

n

n

no closed solution

xx

2 0d1d

0

2 de2

21

21 2

2

N

NbP


Error function and its complementError function and its complement

-3 -2 -1 0 1 2 3-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

x

erf

(x),

erf

c(x)

erf(x) erfc(x)


Bit error rate with error function complementBit error rate with error function complement

Expressions with andSE 0N

antipodal: unipolar1d 0d;d d

N

bP22

erfc21 0d1d

1d 0d;d 0

0

erfc21

NE

P Sb

0

1erfc

2 2S

b

EP

N

1 0

2

2

1 1erfc erfc

2 22 2 2

1 1 SNRerfc erfc

2 2 2 2

b

N N

N

d d dP

d

2

2 matched0

SNR/ 2S

N

Ed

N

2

2 matched0

/ 2SNR

/ 2S

N

Ed

N

2

2

2

2

1 1erfc erfc

2 2 82 2

1 / 2 1 SNRerfc erfc

2 4 2 4

bNN

N

d dP

d

1 2

2

0

22

0

1 2 11 e d

2 2N

b

N

dx

d

xP


Bit error rate for unipolar and antipodal transmissionBit error rate for unipolar and antipodal transmission

-2 0 2 4 6 8 1010

-4

10-3

10-2

10-1

dBinNE

0

S

BE

R

theoretical

simulationunipolar

antipodal

Digitale Übertragung 2-31UniversitätBremen

Signal-Störverhältnis bei PCM-ÜbertragungSignal-Störverhältnis bei PCM-Übertragung

Annahme:

Bei Fehlentscheidungen eines PCM-Wortes ist nur ein Bit verfälscht Dann sieht die Amplitudenfehlerverteilung nach der Dekodierung

folgendermaßen aus:

( ) 2 vv k 1- P

v-0.5-1 0.5 1

P /2

p ( v)V

b

b

l


Berechnung des S/N bei PCMBerechnung des S/N bei PCM

Die Leistung des PCM-Fehlers berechnet sich zu:

Mit

folgt

1 12 2

0 0

2V 2 2 2

2

l lb

b

PP

2-2

2

1 2 4für 2 1

1 2 3

ll

b bP P

0

1erfc

2 2S

b

EP

N

2V

0

2erfc

3SE

N


Berechnung des S/N bei PCMBerechnung des S/N bei PCM

Zusätzlich tritt noch der Quantisierungsfehler auf

und sind unabhängig voneinander

Sinusförmiges Signal

2Q

22 21

212 3

lQ

Q

2Q2

V

2 22 2

0

2 erf3

21

c lQC

SVP M

E

N

2V 1/ 2

0

2V

2 2PCM

3/ 2

2erfc 2SE lPCM

N

S

N


Darstellung des Schwellwert-EffektesDarstellung des Schwellwert-Effektes

0 2 4 6 8 10 12 14 160

10

20

30

40

50

60

70

80

90

100

(Es/N

0) in dB

(S/N

) PC

M

16 bit

12 bit

8 bit

PCM-Schwelle

PCM-Schwelle:

Beipiel:

22QV

0

22 erfc 2S

Schwelle

lE

N

12l

0

12 dBS

Schwelle

E

N








1. Abtasttheorem















__________________________________________________________________


5. Digital Modulation Methods5. Digital Modulation Methods

up to now: real data

i

I Tts )( )(id I )( iTtgTx

)(id I

i

Tts )( )()( idjid QI )( iTtgTx

complex data )()()( idjidid QI

)(ts : ‘ complex envelope ‘

bandpass transmission allows the application of complex baseband signal )(ts


Transmitter ConfigurationTransmitter Configuration

RF-signal (carrier frequency 0):

tjets 0)(

is real ! )(tx

)sin()()()cos()()( 00 tiTtgidTtiTtgidTi

TxQi

TxI

block diagram of Quadrature-Amplitude-Modulation (QAM) transmitter:)cos( 0t

)sin( 0t

)(tgTx

)(tgTx

)(id I

)(idQ

signalmapping

(ROM)

)(tx

0

1)( Mld

}1,0{)( jb

S/ P

co

nv.

impulsegenerator

impulsegenerator

+

-

Re)( tx

sourcebits:


Examples for Signal Space ConstellationsExamples for Signal Space Constellations

)(id I

)(idQ

)(id I

)(idQ

)(id I

)(idQ

)(id I

)(idQ

)(id I

)(idQ

)(id I

)(idQ

4 ASK (M=4)

8 PSK (M=8)

QPSK, =0(M=4)

QPSK, =/4 (M=4)

16 PSK/ASK(M=16)

16 QAM (M=16)

M: number of signal points every signal point represents ld(M) bits


signal in lowpass domain:

mean symbol energy:

signal in bandpass domain:

mean symbol energy:

( ((( )) ))T Txpl

x g tdt lll TjdT

02

0 0 (

( ) 2 Re ( )

2 cos(2 ) sin(2( ) )( ))) (

j f t

T

B

x

p Tp

l lTxg t l

x t x t e

T gT f d l t lTf d l

222 ( )TS xE T D dt tg

2

2 2

22 2 ( )2 2X S

D

TxE T dtD

g ED

t


Linear Modulation with Nyquist Impulse ShapingLinear Modulation with Nyquist Impulse Shaping

QPSK diagram under limited bandwidth conditions

if system (tx and rx filter) meets 1st Nyquist : 4 sharp signal points (right diagram)


Linear Modulation with Nyquist Impulse ShapingLinear Modulation with Nyquist Impulse Shaping

QPSK diagram under limited bandwidth conditions

if system (tx and rx filter) meets 1st Nyquist : 4 sharp signal points (right diagram)

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Universität Bremen Digitale Übertragung 1 Grundlagen der Nachrichtentechnik I. Kontinuierliche...

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