Modul #11

TE3113 TE3113 SISTEM KOMUNIKASI 1SISTEM KOMUNIKASI 1

TRANSMISIBASE-BAND

Program Studi S1 Teknik TelekomunikasiDepartemen Teknik Elektro - Sekolah Tinggi Teknologi Telkomp gg g

Bandung – 2007

Review Probabilitas dan StatistikG i /N l

Normal Distribution: Completely characterized by mean (μ) and variance (σ2)

Gaussian/Normal

⎤⎡ 21(μ) ( )

0.8

pN(y)( ) ⎥

⎦

⎤⎢⎣

⎡−= 2

2

2exp

21

σπσyypN

0.6

0.7 N1(μ=0,σ1)

0.3

0.4

0.5

σ2 > σ1

0.1

0.2

N2(μ=0,σ2)

2Modul 11 - Siskom I - Base Band Transmission

-3 -2 -1 0 1 2 30 y

Gaussian: Rapidly Dropping Tail Probability!

P

R

dyyP ∫∞

⎥⎦

⎤⎢⎣

⎡ −−=

ση

πσ 2

2

2)(exp

21

+ ⎦⎣ση σπσ 22

dyyR ∫∞

+⎥⎦

⎤⎢⎣

⎡ −−=

ση ση

πσ 22

2

2)(exp

21

3Modul 11 - Siskom I - Base Band Transmission

+ ⎦⎣σηπσ 22

Review Probabilitas dan StatistikG i /N l

pN(y)0.4

Normal Distribution: Completely characterized by mean (μ=0) and

i ( 2 1)

Gaussian/Normal

0.25

0.3

0.35

N(μ=0,σ=1)

variance (σ2=1)

Q-function: one-sided tail of normal pdf 0 05

0.1

0.15

0.2

pdf-4 -3 -2 -1 0 1 2 3 4

0

0.05

z

p ( y > z ) =

erfc(): two-sided tail. So:

4Modul 11 - Siskom I - Base Band Transmission

Standardize the Normal Distribution

Z X=

− μNormal DistributionNormal Distribution

Z =σ Standardized Normal

DistributionStandardized Normal

Distribution

σ σ = 1

Xμ

One table!One table!

μ = 0 ZOne table!One table!

5Modul 11 - Siskom I - Base Band Transmission

Example P(X ≥ 8)N lN l St d di d N lSt d di d N lSt d di d N lSt d di d N lX 8 5

10

Normal DistributionNormal Distribution

Standardized Normal Standardized Normal DistributionDistribution

Standardized Normal Standardized Normal DistributionDistributionZ X

=−

=−

=μ

σ8 510

30.

1σ = 10 σ = 1

.5000.50003821382138213821

Z0 30

.1179.1179.1179.1179

.5000.5000.3821.3821.3821.3821

Xμ = 5 8 Zμ = 0 .30Shaded area exaggeratedShaded area exaggeratedShaded area exaggeratedShaded area exaggerated

PERTANYAAN ! Luas daerah yang diarsir = 0.3821 = Q(??); ?? = 0.3

6Modul 11 - Siskom I - Base Band Transmission

Q function:Q-function: Tail of Normal DistributionDistribution

Q(z) = P(Z > z) = 1 – P[Z < z]

7Modul 11 - Siskom I - Base Band Transmission

Baseband Digital Transmission Link

8Modul 11 - Siskom I - Base Band Transmission

Sinyal Terima + AWGN

original messageoriginal messaged(t)

received wave y(t)=d(t)+n(t)

Modul 11 - Siskom I - Base Band Transmission 9

AWGN

n(t) = gaussian noise dengan zero-mean

σ

)(vpN

( ) ⎥⎤

⎢⎡−=

2

exp1 vvp

σ = standar deviasi = tegangan effektive noise

( ) ⎥⎦

⎢⎣−= 22

exp2 σπσ

vpN

σ standar deviasi tegangan effektive noise

Modul 11 - Siskom I - Base Band Transmission 10

Gangguan Noise Terhadap Sinyal Digital

2 Si l ki i

2

3

1

1.5

Signal kirim

Sinyal dengan AWGN

1

-0.5

0

0.5

0

-1.5

-1 -1

0 1 2 3 4 5 6 7 8 9 10-2

-3

-2

Modul 11 - Siskom I - Base Band Transmission 11

-30 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Determining Decision Threshold

)()/(,0:

0

0

AypHypnAYaH

NY

k

+=

+−==

1

1

: 1,( | ) ( )

= = += −

k

Y N

H a Y A np y H p y A

Choose Ho (ak=0) if Y<VChoose H1 (ak=1) if Y>V

The comparator implements decision rule:

( | ) ( | )Vp P Y V H p y H dy∫≡ < =

-A V =Theshold

( k )1 1 1

0 0

( | ) ( | )( | ) ( | )

e Y

Veo Y

p P Y V H p y H dyp P Y V H p y H dy

−∞

∞

∫

∫

≡ < =

≡ > =

Average error error probability: = +P PP PP Transmitted ‘0’Average error error probability: 0 0 1 1+e e eP PP PP1

20 1 0 11/ 2 ( )= = ⇒ = +e e eP P P P P

Ch l i i G i ith th df

Transmitted 0but detected as ‘1’

( ) ⎥⎤

⎢⎡ 2

exp1 yypChannel noise is Gaussian with the pdf:

12Modul 11 - Siskom I - Base Band Transmission

( ) ⎥⎦

⎢⎣−= 22

exp2 σπσ

yypN

Error rate and Q-function

120 1 0 11/ 2 ( )= = ⇒ = +e e eP P P P P

0 ( )∞

∫= Ve Np p y dy-A V =Theshold

V threshold = 0

dyAyppV

ee ∫∞

=⎥⎦

⎤⎢⎣

⎡ +−==

02

2

0 2)(exp

21

σπσThis can be expressed by using the Q-function

( ) Vd Q∞

∫⎛ ⎞⎜ ⎟

by⎟⎞

⎜⎛

⎟⎞

⎜⎛ 2AQAQ

0 ( )Ve Np p y dy Qσ

∞

∫⎛ ⎞= = ⎜ ⎟⎝ ⎠

13Modul 11 - Siskom I - Base Band Transmission

⎟⎟⎠

⎜⎜⎝

=⎟⎠⎞

⎜⎝⎛= 2σσ

QQpe

Baseband Binary Error Rate in Terms of Pulse Shape and S/Np

12 0 1 0 1( ) Ap p p p p p Q⎛ ⎞= + = = ⇒ = ⎜ ⎟

⎝ ⎠

setting V=0 yields then

⎟⎟⎞

⎜⎜⎛

=⎟⎟⎞

⎜⎜⎛

=⎟⎟⎞

⎜⎜⎛

=⎟⎠⎞

⎜⎝⎛= 2

2 2EQSQAQAQp be2 0 1 0 1( )

2e e e e e ep p p p p p Qσ

⇒ ⎜ ⎟⎝ ⎠

for polar, rectangular NRZ [-A,A] bits

Si lProbability of occurrence

⎟⎠

⎜⎝

⎟⎠

⎜⎝⎟

⎠⎜⎝

⎟⎠

⎜⎝ 0

2 NQ

NQQQpe σσ

( ) 222 11 AAASSignal power: ( ) 222

21

21 AAAS =−+=

Noise power: bN T

NRNBWN21.

2.. 00

2 ==== ησb

N T22 00η

Energy Bit to Noise Spectral Density Ratio

NS

TNTSR

NTS

RbN

TS

BWN

TSNE

b

bbbb

N

bb .21

.21..

2..

2

..

0

=====

Note that (BW pulse shapping filter)14Modul 11 - Siskom I - Base Band Transmission

2b

NRBW =

When p0 = p1 =1/2, the value of V that minimizes the probability of error is V = 0.

⎞⎛⎞⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟

⎟⎠

⎞⎜⎜⎝

⎛=

NSQ

NEQp b

e0

2

15Modul 11 - Siskom I - Base Band Transmission

TabelQ-functionQ function

Modul 11 - Siskom I - Base Band Transmission 16