대역제한 AWGN 채널에서의 디지털 변조

4주차

대역제한 AWGN 채널에서의 디지털 변조

(1) Detection of NRZ Signals (2) 정합 필터 (Matched Filter) (3) 최적 수신기 (for BER)

Optimum Filter +

Optimal Detector

(4) 최적 수신기의 BER

Optimal Receiver

q Problems

Ÿ 일반적인 2진 통신시스템에 대해

1) BER를 최소화하는 수신기의 구조는?

2) 최적 수신필터 = 정합 필터?

3) 최적필터의 impulse 응답은?

4) Decision threshold?

5) BER performance?

6) 우수한 성능을 얻도록 하는 signaling format?

q System model

Ÿ Binary communication system with tx signal _ or _

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

ˆi

b ( ) ( ) ( )

o o

r t = a t +n t

1 2

o o

o o

r a n

a n a n

= +

ì +

= íî +

< h>

1

0 o( )

h t

optimal filter T^b

1 2

( ) for '1' ( ) ( ) for '0' s t s t

s t

= í ì

î

Sampler Output Distribution - 1

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

ˆi

b ( ) ( ) ( )

o o

r t = a t +n t

1 2

o o

o o

r a n

a n

a n

= +

ì +

= íî +

< h>

1 0 o( )

h t

optimal filter

1 2

( ) for binary 1 ( ) ( ) for binary 0 s t s t

s t

= íì î

Tb

q Rx signal

1 2

( ) ( ) for binary 1, 0 ( ) ( ) ( )

( ) ( ) for binary 0, 0

b b

s t n t t T

r t s t n t

s t n t t T

+ £ £

= + = íì

+ £ £

î

q Rx filter output

( ) ( ) ( )

o o

r t = a t +n t Ÿ Signal component

1 2

( ) ( ) for binary 1 ( ) ( ) ( )

( ) ( ) for binary 0

o o

o

h t s t a t h t s t

h t s t ì *

= * = í

î * Ÿ Noise component

( ) ( ) ( )

o o

n t = h t *n t

Sampler Output Distribution - 2

q Sampler output at



o o( )b o

r @ r T = a +n

1 1 0 1

2 2 0 2

( ) ( ) ( ) ( ) for binary 1 ( )

( ) ( ) ( ) ( ) for binary 0

b b

b b

T

o t T o b

b T

o t T o b

a s t h t s h T d

a a T

a s t h t s h T d

t t t

t t t

=

=

ì = * = -

= íï

ï = * = -

î

ò ò

@

( ) ( ) ( ) 0^b ( ) ( )

b

T

o o b o t T o b

n ^@ n T = n t *h t ₌ =

ò

t

t

t q Gaussian random variable 

mean:

[ ]

[

]

E n = E^é n

t

t

t

t

t

t

ê ú

ë

ò

ò

varinace:

[ ]

2 2

0 0

0 0 2

0 0 0

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2 2

b b

b b b

o

T T

o b o b

T T T

o b o b o

E n

E n n h T h T d d

N N

h T h T d d h t dt

s

t l t l t l

d t l t l t l

é ù

= ë û

= - -

= - - - =

ò ò

ò ò ò

Sampler Output Distribution - 3

q Sampler output 

의 분포

o o

r = a+n à Gaussian

If _ is transmitted, _  _ _ à mean = _, var = ^ If _ is transmitted, _  _ _ à mean = _, var = ^

1 0

0 1

( )

( | sent) f r

f r s

=

2 0

0 2

( )

( | sent) f r

f r s

=

s2

s2

h a1

a2

r0

(error | 1 sent)

P s P(error |s₂ sent)

1

2 2

1 2 2

2

2 2 2

( )

( ) 1 exp

2 2

( )

( ) 1 exp

2 2

o o

o o

f r r

f r

a

r

ps s

ps s

ì - ü

= í - ý

î þ

ì - ü

= í - ý

î þ

q Optimal decision threshold :

2 a a h = ⁺

Sampler Output Distribution - 4

q Bit error probability

Ÿ Error rate when _ is sent

(

)

2

1 2

e ( ) sent ( ) 1 exp

2 2

2

o o a a o o

P s t f r dr r dr

a a Q

h

s

ps s

¥ ¥

-

ì ü

= = í - ý

î þ

æ - ö

ç ÷

è

=

ø

ò ò

Ÿ BER

( ) ( ) ( ) ( )

( ) ( )

1 1 2 2

1 2

e sent sent e sent sent

1 1

2 2

P

P s P s P s P s

Q

a a Q

Q

s

= × + ×

= +

æ - ö

ç ÷

è ø

=

g g

1 0

0 1

( )

( | sent) f r

f r s

=

2 0

0 2

( )

( | sent) f r

f r s

=

s2

s2

h a1

a2

r0

(error | 1 sent)

P s P(error |s₂ sent)

Optimal Receiver (Receive Filter)

100

10^-1

10^-2

10^-3

x

( )

Q x

q Optimal receive filter

Ÿ _  _ 또는 _  _^^ 이 최대일 때, BER 최소

Ÿ 최적 필터는, sampled output이 _  _일 때 SNR이 최대가 되도록 하는 필터와 동일

{ }

1 2 1( ) 2( ) ( )

o t Tb

a a s t s t h t

- = - * = _로부터

à matched to _  _

à

h ( t ) = K [

]

일반적인 2진 시스템용 수신기 - MF Receiver

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

ˆi

b Tb

r1

< h>

1 0 1( ) 1( _b )

h t = s T -t

Tb

r2

2( ) 2( _b ) h t = s T -t

S

r =r -r +

-

Or

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

ˆi

> b

< h

1 0

( ) _d( _b )

h t = s T -t

Tb

1 2

r = r -r

1 2

( ) ( ) ( ) s td = s t -s t

1 2

2 a a h = ⁺

일반적인 2진 시스템용 수신기 - Correlation Receiver

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

ˆi

b Tb

y1

< h>

1 0

Tb

y2

S

y = y - y

ò

1( ) Ks t

ò

2( ) Ks t

+ -

Or

ˆi

b Tb

< h>

1

ò

d( ) Ks t

( ) ( ) ( )

r t = s t +n t

1 2

( ) ( ) ( ) s td @ s t -s t

1 2

2 a a h = ⁺

BER & Optimal Signaling Format

Let s t_d( ) = s t₁( )-s t₂( )

Ÿ Signal component in sampled filter output

( )

( )

1 0 1 1

2 0 2 2

(

( ) )

( )

( )

b

Tb

d T

a s t d dt K E

a

Ks t K

s t s t dt K E

g

g

= = -

= = - +

ò ò

Ÿ Noise component in sampled filter output

[ ]

2 0 2 0 2 2 0 2

1 2

0 ( ) 0 ( ) 2

2 2 2

b b

T T

o d

N N N

h t dt K s t dt K E E

s

ò

ò

g

q BER

1 2

0

1 2

2

2

b 2

E E

Q N

a a

P Q g

s

æ - ö

= =

ç è ø

÷

ø

q Optimal signaling format

Ÿ Maximize





Ÿ Minimize  (cross-correlation of two signals)

2

1 0 1

2

2 0 2

1 2

0

( ) ( ) ( ) ( )

b

b

b

T

T

T

E s t dt E s t dt

s t s t dt

g

=

=

=

ò

ò

ò

Summary

q Problems & solutions

Ÿ 일반적인 2진 통신시스템에 대해

1) BER를 최소화하는 수신기의 구조는? à MF + Sampler + Detector 2) 최적 수신 필터 = 정합 필터? à OK

3) 최적필터의 임펄스 응답은? à h t( ) = Ks T_d( _b -t), s t_d ( ) = s t₁( )-s t₂( )

4) Decision threshold? à ^{1 2}

2 a a h = ⁺

5) BER performance? à ^{1 2}

b 2

a a

P Q

s

æ - ö

= ç ÷

è ø

6) 우수한 성능을 얻도록 하는 signaling format?

- Maximize signal energy

- Minimize cross-correlation of two signals