(1) detection of nrz signals (2) 정합 필터 (matched filter) (3)...

디지털통신 1 충북대학교

5주차

대역제한 AWGN 채널에서의 디지털 변조

(1) Detection of NRZ Signals

(2) 정합 필터 (Matched Filter)

(3) 최적 수신기

(4) 최적 수신기의 BER

- BER By Optimal Receivers- BER By Signal Space Representation


(1) Optimal Receiver

1 0

0 1

( )( | sent)

f rf r s=

2 0

0 2

( )( | sent)

f rf r s=

2s2s

h1a2a

0r

1(error | sent)P s 2(error | sent)P s

( ) ( ) ( )r t s t n t= +

ibbT

1y

>< h1

0

bT

2y

S1 2y y y= -

òbT dt

0

1( )Ks t

òbT dt

0

2 ( )Ks t

+

-

{ }1 2( ) ( ) ( ) ( )dh t K s t s t Ks t= - =

( )

( )

1 1 1 10

2 2 2 20

( ) ( )

( ) ( )

b

b

Td

Td

a y s t Ks t dt K E

a y s t Ks t dt K E

g

g

= = = -

= = = - +

ò

ò

q Optimal threshold

1 22

a ah +=


BER By Eb/N0

1 0

0 1

( )( | sent)

f rf r s=

2 0

0 2

( )( | sent)

f rf r s=

2s2s

h1a2a

0r

1(error | sent)P s 2(error | sent)P s

q Average bit energy

1 21 1 2 2Pr( sent) + Pr( sent)

2bE EE E s E s +

= × × =

q BER by 0/bE N

1 2 1 2

0

0

2 2 2

b

b

EQ

a a EQ

N

EP QN

g

g

sæ ö- + -æ ö= = ç ÷ç ÷

è ø è ø

=æ ö-ç ÷è ø


What is Eb / No ?q vs. SNR

: average bit energy

: noise power in 1 Hz bandwidth

b

o

EN : Average energy required to transmit one bit over an AWGN channel

0 0

/b b b bE T E RSNRN B N B

æ öæ ö= = ç ÷ç ÷è øè ø

where Bit rate

BandwidthbRB= : Bandwidth efficiency (대역효율)

Eg) roll-off factor 0a = à / 2bR B =

1a = à / 1bR B =

※ Digital modulation에서 BER을 결정하는 요소는 1비트당 전력이 아니라, 1비트당 에너지이다.

bT0t

2A

bT0t

A


Examples : 예제 9.2 (p.629) à Find BER Pb

(a) A

1( )s t

A

2( )s t

t tbT

bT

2 21 1 20

21 2

1 20

1 2

0 0

( )

( ) ( ) 0

2

2

b

b

T

b

b b

T

bb

E s t dt AT E

E E E AT

s t s t dt

E E EP Q Q

N N

g

g

= = =

= = =

= =

æ ö æ ö+ - ÷ ÷ç ç÷ ÷ç ç= =÷ ÷ç ç÷ ÷÷ ÷ç çè ø è ø

ò

ò

(b) A

1( )s t

A

2( )s t

t tbTbT

( )

2 21 10

2 22 20

21 2

1 20

1 2

0 0

2( )

31 1

( )2 3

1 32 4

( ) ( ) 0

2

2

b

b

b

T

b b

T

b b

b b

T

bb

E s t dt AT E

E s t dt AT E

E E E AT

s t s t dt

E E EP Q Q

N N

g

g

= = =

= = =

= + =

= =

æ ö æ ö+ - ÷ ÷ç ç÷ ÷ç ç= =÷ ÷ç ç÷ ÷÷ ÷ç çè ø è ø

ò

ò

ò


Examples :

(c) A

1( )s t

A

2( )s t

t t

bTbT

21

22

2

/2

1 20 0

2

0 0

43

1 22 334

( ) ( ) 2 sin(2 / )

2 2 4 0.85

32 2 0.85

0.152

b b

b b

b b

b

T T

b

b b b

b b bb

E AT E

E AT E

E AT

s t s t dt t T dt

AT E E

E E EP Q Q

N N

g p

p p

= =

= =

=

= =

= = »

æ ö æ ö- × ÷ ÷ç ç÷ ÷ç ç= =÷ ÷ç ç÷ ÷÷ ÷ç çè ø è ø

ò ò

(d) A

1( )s t

A

2( )s t

t tbT bT

2 21 10

2

0

( ) 2

0

0

bT

b b

bb

E s t dt AT E

E

EP Q

N

g

= = =

=

=

æ ö÷ç ÷ç= ÷ç ÷÷çè ø

ò


Examples :

(e) A

1( )s t

A

2( )s t

t tbT

bT

2 21 1 20

0

1( )

2

2

bT

b b

b

bb

E s t dt AT E E

E

EP Q

N

g

= = = =

= -


ò

(f) A

1( )s t

A

2( )s t

t t

bT

bT

2 21 1 20

0

1( )

2

2

bT

b b

b

bb

E s t dt AT E E

E

EP Q

N

g

= = = =

= -


ò


Examples :

0 [dB]

bE N

010

110-

210-

310-

410-

BER, bP

(a) (b) (d)

(c)

(e) (f)

3dB 8dB


(2) Signal Space Representation

※ 는 유일하지

않지만, 1 2, s s 상호간의

위치관계는 동일

à 동일한 BER

q Signal space representation of binary signalsLet 1 2( ), ( )t tf f : basis functions with energy 1

Then, 1 1 1 2 2

2 1 1 2 2

( ) ( ) ( )( ) ( ) ( )

s t c t c ts t d t d t

f ff f

= +ìí = +î

à represented by 1 1 2

2 1 2

( , )( , )c cd d

=ìí =î

ss à signal space

q Signal constellation

1( )tf

2 ( )tf

1 1 2( , )c cs

2 1 2( , d )ds

1( )tf

2 ( )tf

1 1 2( , )c cs2 1 2( , d )ds

Signal point

orthogonal signaling antipodal signaling


Signaling Examples - 1

0bE

bE

1 0

bT

1f 2f1

bT

b

b

ET

(1) Orthogonal code

2 21 1 2 2 1 2 , , 0

b b bT T TE s dt E s dt s s dtg= = = =ò ò ò

Let 1 2 bE E E= =

1 1 2

2 2 2

0

0 b

b

s E

s E

f f

f f

= +

= +à

( )( )

1

2

, 0

0,

b

b

E

E

=

=

s

s



1f 2f2

bT

1 0

bT

b

b

ET

￭ Basis function을 다르게 정한 경우

1 1 2

2 2 2

2 2

2 2

b b

b b

E Es

E Es

f f

f f

= +

= - +à

1

2

, 2 2

, 2 2

b b

b b

E E

E E

æ ö= ç ÷è øæ ö

= -ç ÷è ø

s

s

0 1f

2f

2bE

2bE-



0bEbE-

(2) Polar NRZ (antipodal)

2 21 1 2 1 1 1 2 1 , ( ) ,

b b bT T TE s dt E s dt E s s dt Eg= = - = = = -ò ò ò

Let 1 2 bE E E= =

1 1

2 1

b

b

s E

s E

f

f

=

= -à

( )( )

1

2

b

b

E

E

=

= -

s

s 1f 2f

1 0


BER By Signal Space Representation

0bE

bE

0bEbE-

q BER in signal space diagramŸ Idea: BER은 전송신호에 의해 결정되며, 수신필터는 단순한 신호의 변환이다

à 송신신호의 constellation이 정해지면 BER을 구할 수 있다

(signal distance) / 2(noise variance)b QP

æ öççè

= ÷÷ø, noise variance =

q Orthogonal code (signal distance)/2 =

à 0

bb

EP QN

æ ö= ç ÷

è ø

q Polar NRZ (antipodal)(signal distance)/2 =

à 0

2 bb

EP QN

æ ö= ç ÷

è ø


Examples:

Unipolar RZ

Manchester

Unipolar NRZ

Polar RZ

Polar NRZ

1 0

bT

(1) Antipodal signaling: polar NRZ, polar RZ, Manchester code

bEg = - à0

2 bb

EP QN

æ ö= ç ÷ç ÷

è ø

(2) Orthogonal signaling: unipolar NRZ

0g = à0

bb

EP QN

æ ö= ç ÷ç ÷

è ø

2 4 6 8 10 12

bP

1

0.5110-

410-

310-

210-

510-

0/ [dB]E N

1r =

1r = -

0r =

3 dB


(참고) Gram-Schmidt Orthogonalization

출처: Essentials of Communication Systems Engineering, pp. 380-384, J.G. Proakis, M. Salehi

q Review of orthogonal expansion

1 1 2 2( ) ( ) ( ) ( )N Nf t c t c t c tf f f= + + +L

where ( ) ( ) ( )i jt t dt i jf f d¥

-¥= -ò

{ }( )i tf : orthonormal basis function set

Then, = ( ), ( ) ( ) ( )i i ic f t t f t t dtf f¥

-¥á ñ = ò


Gram-Schmidt Orthogonalization - 2q Gram-Schmidt orthogonalization목적: Find a complete set of orthonormal basis functions from a set of energy signals,

⋯

결과: 개의 signal waveform을 ≦개의 basis function으로 나타낼 수 있다

방법:

① 1

12

1 11

( )( ) ( )E sE

ts tt t df¥

-¥= = ò : normalization

à 1 11 1 1 1( ) ( ) ( )s t c t E tf f= =

② 2 21 1 2(( ) ) )(s t c t d tf= + , 21 2 1 2 1,c s s dtf f¥

-¥= á ñ = ò

22

22 2 21

21 2 2( ) ( ) ( ) , ( )( ) ( )d ttd t s t s t E d t

Edtf f

¥

-¥= - ® == ò

à 2 21 1 22 2 21 1 2 2( ) ( ) ( ) ( ) ( )s t c t c t c t E tf f f f= + = +


Gram-Schmidt Orthogonalization - 3

③ For -th function

1 1 2 2 , 1 1( ) ( ) ( ) ) ( )(k k k k k k ks t c t d tc t c tf f f- -= + + + +L

,ki k i k ic s s dtf f¥

-¥= á ñ = ò

12

1( ) ( ) ( ) , )( (( ))

k

k k ki i kk

kk

ki

dd t s t c t Ett d t dtE

ff- ¥

-¥=

= - ==®å ò

à1

( ) ( )k

k ki ii

s t c tf=

=å

④ Repeat ③ for all waveforms


Example (G.S. Orthogonalization)

2s

3s

4s

1 2 3

1s1

1f

2f

3f

1/ 2

1/ 2

i) 1 2E = à 1 11 ( )2s tf = à 1 1( ) 2 ( )s t tf=

ii) 2 21 1 2s c df= +

21 2 1 0c s dtf= =ò2

2 2 2 2 22

1, 2 2

dd s E sE

f= = ® = =

à 2 2( ) 2 ( )s t tf=

iii) 3 31 1 32 2 3s c c df f= + +

31 3 1 32 3 20, 2c s dt c s dtf f= = = = -ò ò3

3 3 2 3 3 3 23

2 , 1 2dd s E sE

f f f= + = ® = = +

à 3 2 3( ) 2 ( ) ( )s t t tf f= - +

iv) 4 41 1 42 2 4 43 3s c dc cf f f= + + +

41 4 1 42 4 2 43 4 32, 0, 1c s dt c s dt c s dtf f f= = = = = =ò ò ò4 4 1 32d s f f= - -

à 4 1 32s f f= +


※ Note! (1) ① The set of orthonormal basis functions is not unique for a given function set.

Eg) Alternative set for Example

1 2 3

1f

1 2 3

2f

1 2 3

3f

② Once a set of orthonormal basis functions is obtained, signals can be represented as a linear combination of the basis functions.

1( ) ( ), 1,2, ,

K

m mk kk

s t s t m Mf=

= =å L

( ) ( )mk m ks s t t dtf¥

-¥= ò


※ Note! (2) ③ can alternatively be represented by a vector

1 2( , , , )m m m mKs s s=s L

à a point in -dimensional space à called ‘signal space representation’(signal point) (signal space)

④2 2

1

K

m m mkk

E s dt s¥

-¥=

= =åò à Parseval's theorem

à The square of the Euclidean distance

1f

2f

1ms

2ms

ms

ms 2 : signal energyms


※ Note! (3)

⑤ The change in the basis functions does not change the dimensionality of the space, the lengths (energies) of the signal vectors, or the inner product of any two vectors.à Essentially rotate the signal points around the origin

Example)

Signal space representationBases of p. 9-46 Bases of p. 9-47

1 12s f=

2 22s f=

3 2 32s f f= - +

4 1 32s f f= +

1 ( 2,0,0)=s

2 (0, 2,0)=s

3 (0, 2,1)= -s

4 ( 2,0,1)=s

1 (1,1,0)=s

2 (1, 1,0)= -s

3 ( 1,1,1)= -s

4 (1,1,1)=s

à vector space에 나타내 볼 것

(1) detection of nrz signals (2) 정합 필터 (matched filter) (3)...

Documents