WMCS Lab Digi Communication lecture note [Digi Comm]

1. 8장 펄스부호 변조(PCM)
디지털통신(Digital Comm.)

2. Contents
 PCM (pulse code modulation)
 Sampling
 Quantization
 Encoding
 PCM waves
 TDM
 Delta modulation
 정합필터 (matched filter)
 MAP (maximum A-Posteriori criterion) detection
 Error probability
디지털통신(Digital Comm.)
2

3. PCM (Pulse Code Modulation)
<< Advantages of PCM System >>
■ PCM system can use repeater
펄스 파형이 두 가지뿐이므로 펄스 재생이 용이하다.
재생 중계기로 잡음의 영향이 누적되는 것을 막을 수 있다.
■ stability of digital system
신뢰도와 안정성이 높은 디지털 시스템을 이용할 수 있다.
논리 회로는 IC화에 적합하다.
■ PCM system can memory device
IC 메모리 소자의 발전으로 많은 양의 데이터를 저장, 전송할 수 있다.
■ redundancy removal
특수한 부호화에 의해 신호의 용장을 크게 줄일 수 있다
(Data compression)
■ error check & correction
신호의 용장을 늘려 수신기의 비트 오판 확률을 줄일 수 있다
(Channel coding)
디지털통신(Digital Comm.)
3

4. PCM (Pulse Code Modulation)
펄스의 부호화
Quantization
L levels
디지털통신(Digital Comm.)
4

5. PCM (Pulse Code Modulation)
■ Sampling
■ Quantization
■ binary encoding
L  2n or n  log2 L
L=quantization levels,
n=bits
■ 양자화 잡음
(Quantization noise)
디지털통신(Digital Comm.)
5

6. PCM
- Quantization
Example)
8 levels  23bits
Binary encoding
Quantization noise
디지털통신(Digital Comm.)
6

7. PCM
- Quantization
Quantization levels : 256 = 28
bits
디지털통신(Digital Comm.)
levels
7

8. SNR for quantized pulses
 Quantization noise
Distortion introduced by the need to approximate the analog waveform
with quantized samples
 SNR for quantized pulses
Vp
Vp -q/2
Vp -3q/2
MSE  
↕ q volts
Quantized
values
q / 2 2
e
q / 2

-Vp+3q/2
-Vp+q/2
-Vp
디지털통신(Digital Comm.)

L levels Vpp
-q/2
-3q/2
-5q/2
q / 2
q / 2

5q/2
3q/2
q/2
e  x(nT )  xq (nT )
e 2 p(e) de
q2
1
de 
→ 양자화잡음
q
12
2
2
 V pp 
Lq 
L2 q 2

2
Vp  
 2   2   4

 


L2 q 2 4
S
 3L2
   2
 N q
q 12
8

9. Quantization
 균일 양자화 (uniform quantization)
Quantization steps are uniform in size
SNR is worse for low-level signals than high-level signals
 불균일 양자화 (non-uniform quantization)
Fine quantization of the weak signals
Coarse quantization of the strong signals
Quantization noise can be made proportional to signal noise
압신 (companding) = 압축 (compression) + 신장 (expanding)
디지털통신(Digital Comm.)
9

10. Quantization
Quantizing levels
15
15
14
14
13
Strong signal
13
12
12
11
11
10
9
Week signal
10
9
8
8
7
6
5
4
3
2
6
7
5
4
3
2
1
1
0
Uniform quantization
디지털통신(Digital Comm.)
0
Non-uniform quantization
10

11. Quantization
y  ymax
 압신 (companding)
-  -law 압축 : 북미에서 사용 (표준값 255)
- A -law 압축 : 유럽 표준으로 사용 (표준값 87.6)
 -law charateristic curves
1
A-law charateristic curves
A  100
0.9
0.8
0.8
0.7
0.7
  100
0.6
Output,|y|/y max
Output,|y|/y max
 1 for x  0
sgn x  
1 for x  0
1
  255
0.9
log e [1   ( x / xmax )]
sgn x
log e (1   )
 5
0.5
0.4
 1
0.5
A5
0.4
A2
0.3
0.3
 0
0.2
0.2
A 1
0.1
0.1
0
A  87.6
0.6
0
0.2
0.4
0.6
Input,|x|/x max
디지털통신(Digital Comm.)
0.8
1
0
0
0.2
0.4
0.6
Input, |x|/x max
0.8
1
11

12. Non-uniform quantization
 비선형 양자기와 비선형 복호기에 의한 비선형 양자화 방식

아날로그 부품을 이용, 정의된 레벨의 임계치를 정확하게 유지하는 것이 다소 어려움.
하지만 설계 기술의 발전으로 문제점이 극복되고 있다.
 Companding에 의한 비직선 양자화 방식

Companding (Compressing + expanding) 방식
: 송신기의 압축기능과 수신기의 신장기능을 합친 복합어

현재 대부분의 PCM 시스템에서 적용
디지털통신(Digital Comm.)
12

13. A/D & D/A system
PCM System
(Channel)
디지털통신(Digital Comm.)
13

14. 기저대역 디지털 전송
PCM
 펄스 부호 변조 (PCM)
sampling → quantization → encoding
(a) PCM sequence (b) PCM pulse (c) pulse waves
디지털통신(Digital Comm.)
14

15. 다중화(Multiplexing)
 다중화 개념
 다중화 종류





주파수분할 다중화(FDM)
시분할 다중화(TDM)
통계적 시분할 다중화(Statistical TDM)
코드분할 다중화(CDM)
파장분할 다중화(WDM)
정보통신경영연구실디지털통신(Digital
Comm.)
15

16. Multiplexing
 FDM (Frequency Division Multiplexing)
 TDM (Time Division Multiplexing)
디지털통신(Digital Comm.)
16

17. 시분할다중화(Time Division Multiplexing; TDM)
Ex) phone voice 0~3.4KHz
f m  4 Hz
f s  2  f m  8 Hz
N개 채널을 TDM할경우 각 채
널 sampling pulse의 폭 (τ) :
PAM signals

1
N  fs
Total BW of TDM signal:

BT 
1

 2 Nf m  Nf s Hz
(in PAM signal transmission)
디지털통신(Digital Comm.)
17

18. 시분할다중화(Time Division Multiplexing; TDM)
8 KHz  (24 ch  8 bits  1 sync bit)  1.544 Mbps
■ PCM T1 frame
fs
= 193 bits
193bits/frame
CH 1
CH 2
CH 24
Frame sync bit
디지털통신(Digital Comm.)
18

19. Digital hierarchy (TDM)
Service
DS-1
Line Rate(Mbps)
1.544
T-1
Voice Ch.
24
DS-2
디지털통신(Digital Comm.)
6.312
96
DS-3
T-3
44.736
672
DS-4
 DS(Digital Signal) service in NAS
T-2
T-4
274.176
4032
19

20. Digital hierarchy (TDM)
 Digital hierarchy (TDM) - 비동기식 디지털 다중화 계위
ITU-T
NAS
Korea
Europe
3.2kbps
6.4kbps
12.8kbps
Bearer군
전송속도
5차군
1
1
1
1
1.544Mbps
1.544Mbps
2.048Mbps
2.048Mbps
CH수
24
24
30
30
6.312Mbps
6.312Mbps
6.312Mbps
8.448Mbps
CH수
24 x 4 = 96
24 x 4 = 96
30 x 3 = 90
30 x 4 =120
32.064Mbps
44.736Mbps
44.736Mbps
34.368Mbps
CH수
96 x 5 = 480
96 x 7 = 672
90 x 7 =630
120 x 4 =480
97.728Mbps
274.176Mbps
139.264Mbps
139.264Mbps
CH수
480 x 3 = 1,440
672 x 6 = 4,032
630 x 3 = 1,890
480 x 4 =1,920
전송속도
4차군
CH수
전송속도
3차군
64Kbps
전송속도
2차군
64Kbps
전송속도
1차군
64Kbps
전송속도
0차군
64Kbps
397.2Mbps
564.992Mbps
139.264Mbps
CH수
1,440 x 4 = 5,760
1,890 x 4 =7,560
1,920 x 4 =7,680
계위 구조
∙ITU-T:4×5×3×4, NAS:4×7×6, Europe:4×4×4×4, Korea:3×7×3×4
디지털통신(Digital Comm.)
20

21. Pulse distortion in filter
HPF
NRZ
(NonReturn to Zero)
Distorted signal
디지털통신(Digital Comm.)
21

22. PCM waves for baseband transmission
 PCM waves
 NRZ (nonreturn-to-zero)
 RZ (return-to-zero)
 위상 부호화 (phase encoded)
 다치 2진수 (multilevel binary)
디지털통신(Digital Comm.)
22

23. PCM waves for baseband transmission
0
PCM waves
디지털통신(Digital Comm.)
23

24. Synchronization Problem
Self-clocking
• Manchester code has a transition in the middle of every bit interval
whether a one or a zero is being sent
• This guaranteed transition provides a clocking signal
디지털통신(Digital Comm.)
24

25. NRZ coding
 T 
S ( )  A 2Tb sinc 2  b 
 2 
대역폭 면에서
유리하다
디지털통신(Digital Comm.)
25

26. RZ coding
 A, 0  t  Tb 2
x1 (t )  
0, Tb 2  t  Tb
x 0 (t )  0
ATb
 T 
sinc b  exp(  j Tb 4), X 0 ( )  0
2
 4 
1
 
2
2
X 1 ( k b )  (  k b )
S ( ) 
X 1 ( ) 
4Tb
2Tb2 k  
X 1 ( ) 

A 2Tb
A 2
2  Tb 
sinc 


16
4 
8

디지털통신(Digital Comm.)

k
sinc 2   (  k b ).
2
k  

26

27. AMI 방식
A 2Tb
2  Tb 
2  Tb 
sinc 
S ( ) 

 sin 
4
 2 
 4 
DC 성분=0
디지털통신(Digital Comm.)
27

28. Manchestor coding
 A, 0  t  Tb 2
x1 (t )   x0 (t )  
Tb 2  t  Tb
 A,
X 1 ( )   X 0 ( ) 
ATb
 T 
sinc b [exp( j 3 Tb 4)  exp( j Tb 4)]
2
 4 
 T 
 T 
S ( )  A 2Tb sinc 2  b  sin 2  b 
 4 
 4 
DC 성분=0:
전송시 AC coupling이 가능
디지털통신(Digital Comm.)
28

29. 델타 변조 (Delta modulation; DM)
근사적인 양자화
(1 bit PCM)
Quantization step size
디지털통신(Digital Comm.)
29

30. 델타 변조 (DM)
 V , x(t )  xq (t )

(t )  
 V , x(t )  xq (t ).

근사적인 양자화
(Ts와 S에 의존)
1 bit PCM
디지털통신(Digital Comm.)
30

31. 델타 변조 (DM)

Quantization noise depends on step size
Quantization noise
 (nTs )  x(nTs )  xq (nTs )
(a) Small step size
(b) big step size
To resolve the unmatched signal magnitude problem
• increase f s → bandwidth increased
• enlarge step size → quantization noise increased
디지털통신(Digital Comm.)
31

32. Adaptive delta modulation (ADM)
신호가 급격히 변하는
부분에 감쇠 진동 발생
If) current bit ≠ previous bit
→ decrease quantization
step size
If) current bit =
previous bit
→ Increase
quantization
step size
디지털통신(Digital Comm.)
32

33. 차동 펄스부호변조 (Differential PCM)
Encoding this
prediction error
■ linear prediction
ˆ
x ( nTs )  a1 x[( n  1)Ts ]  a2 x[(n  2)Ts ]    a N x[( n  N )Ts ]
■ prediction error
ˆ
e( nTs )  x ( nTs )  x ( nTs )
디지털통신(Digital Comm.)
33

34. Quantization noise
■ DM
2
nq (t )
M: quantization levels
S : step size
■ PCM
S2  B 
 
 
3  fs 
 
p( x) 
2
n
S o x (t ) 3  f s  2


  x (t )
2
N o nq (t ) S 2  B 
(최악의 상황)
x 2 (t )  0.5 A2
 2A 
 2A 
f s  f m 
  B

 S 
 S 
MS
MS
x
2
2

MS 2

2
MS 2
x 2 (t ) ( MS ) 2
So  2 
Ts
12Ts2
S o ( MS ) 2 12Ts2

 2  M 2  (2 N ) 2  22 N .
2
N o 12Ts
S
BT  2 NB
(경사과부하가 없을 조건)
x 2 (t ) 

x2
( MS ) 2
x (t ) 
x p ( x) dx 
dx 
.
 MS 2
 MS 2 MS
12
2
2
x(t )  A sin 2Bt
1
,
MS
(B=x(t)의 최대 주파수, N=비트수, BT=전송 대역폭)
2
f s2
2 2
S
8 B
3
So
3 f 
3 B 
 2 s  2 T
N o 8  B  8  B 
(BT=전송 대역폭)
디지털통신(Digital Comm.)
3
So
 2 2 N  2( BT
No
B)
.
If) 8bits/sample → BT/B = 2N = 16
PCM : 216=48 dB (성능 우수)
DM : 155.6 = 22 dB
34

35. Quantization noise
2
신호 대 잡음 비
BT의 자승에 비례
신호 대 잡음 비
BT의 자승에 비례
신호 대 양자화 잡음 비
BT에 지수적으로 비례
신호 대 양자화 잡음 비
BT의 3승에 비례
디지털통신(Digital Comm.)
So  BT  Si
 
No  B  N B
2
So  BT 
 
No  B 
So
 2( BT B )
No
(광대역 FM)
 Si 
  (PPM)
N 
 i
(PCM)
So
3  BT 
 2 
N o 8  B 
3
(DM)
35

36. 기저대역 디지털 전송
Digital receiver
 Signal demodulation & detection
1 단계
파형에서 샘플로의
변환
사전검출점
(Predetection point)
detect
demodulation & sample
AWGN
si t 

2 단계
결정
Sample at
t T
r t 
Freq. down
conversion
Receiving
filter
Threshold
comparision
Equalizing filter
Transmitted
waveform
z T 
z t 
For bandpass signal
Received waveform
Baseband pulse
r t   si t   hc t   nt 
optional
Essential
디지털통신(Digital Comm.)
z T 
H1
>
<
ˆ
mi

H2
Compensation for
channel-induced ISI
Baseband pulse
z t  
a i t   n 0 t 
sample
(test statistic)
Message Symbol
ˆ
mi
z T  
ai T   n0 T 
36

37. 기저대역 디지털 전송
Digital receiver
 주파수 하향변환 (down-conversion)
 수신기 필터 (receiving filter)
- 정합필터(matched filter)
- 상관기(correlator)
 등화 필터 (equalizing filter)
- 등화기(equalizer)
디지털통신(Digital Comm.)
37

38. Digital signal distorted by channel noise
Digital receiver
Rectangular
pulse with noise
Band-limited
rectangular pulse
Heavily band-limited
rectangular pulse
•SNR is max, when t=Tb.
•Sampling time in receiver
디지털통신(Digital Comm.)
38

39. Eye pattern
•Band-limited
•No additive noise
•No bandlimit
•No additive noise
최적 표본화 시간
시간 오류에
대한 민감도
•Band-limited
•Noise added
시간 지터
(timing jitter)의
측정값
잡음 마진
(noise margin)
심볼 간 간섭에
의한 왜곡
심볼 간 간섭에 의한 아이 패턴
디지털통신(Digital Comm.)
39

40. Linear filter designed to provide the maximum SNR at its
output for a given transmitted symbol waveform
Matched filter (MF)
1
so (t ) 
2

1
so (Tb ) 
2
SNR

H ( ) S ( ) exp( jt )d




S ( ) : FT of s (t )
H ( ) S ( ) exp( jTb )d

2
so (Tb )
2
o
n (Tb )
디지털통신(Digital Comm.)




S n ( ) : power spectral density of n(t )
2
S n ( ) H ( ) : power spectral density of no (t )
H ( ) S ( ) exp( jTb )d 
2 


2
2
H ( ) S n ( )d 

 [watts/Hz]
2
40

41. Matched filter (MF)
2


2
2
X ( )Y ( )d     X ( ) d    Y ( ) d 
 
 
  




X ( )  KY *( )

Schwartz’s inequality
SNR

2
so (Tb )
2
no (Tb )


f (t )  F ( w)
f (t )  F* ( w)
f (t  Tb ) exp( jwTb )F( w)


H ( ) S ( ) exp( jTb )d 
 


2
2
H ( ) d 




X ( )Y ( )d 
 


2
X ( ) d 
X ( )  H ( ), Y ( )  S ( ) exp( jTb )
 max 
1
 


2
S ( ) d 
2E


1
1

  

  
2
Y ( ) d 
2
S ( ) d 
Signal energy

 H ( )  KS  ( ) exp( jTb )


h(t )  Ks (Tb  t )

디지털통신(Digital Comm.)
2
Mirror image of the message signal s(t),
delayed by the symbol time duration Tb.
41

42. Matched filter (MF)
Impulse response of matched filter
디지털통신(Digital Comm.)
causality
•SNR is max, when t=Tb.
•Sampling time in receiver
42

43. Matched filter (MF)
 The mathematical operation of MF is convolution
so (t )  no (t )  [ s (t )  n(t )]  h(t )
• The process of convolving two signals reverses one of them in time
• The impulse response of an MF is defined in terms of a signal
that is reversed in time
Convolution in the MF results in a second time-reversal (correlation)
디지털통신(Digital Comm.)
43

44. 상관기 (correlator)
• MF를 구현하는 것보다 correlator가 용이
• MF : convolution을 이용, correlator : correlation 이용
so (t )  no (t )  [ s (t )  n(t )]  h(t ) 
 [s( )  n( )]s(T
t
0
b
 t   )d .
Output of MF
Tb
  [ s ( )  n( )]s ( ) d
0
디지털통신(Digital Comm.)
at sampling time Tb
44

45. 2진 정합필터 수신기(Binary MF receiver)
-V
+V
디지털통신(Digital Comm.)
Antipodal signals, s0(t)=-s1(t)
45

46. Performance of digital receiver
기저대역 디지털 통신
디지털통신(Digital Comm.)
46

47. Example of matched filter in digital communications
 Imagine we want to send the sequence "0101100100" coded in
non polar NRZ through a certain channel.
 If we model our noisy channel as an AWGN channel, white Gauss
ian noise is added to the signal
 To increase our signal-to-noise ratio, we pass the received sign
al through a matched filter.
디지털통신(Digital Comm.)
47

48. Equalizer
 등화기(equalizer)
Input
data
Transmitting
filter
Ht  f 
Channel
Hc  f 
+
Receiving
filter
Hr  f 
Equalizer
T0 detector
He  f 
Noise
n(t )
[System block diagram with equalizer]
 Channel = band-limited filter
H c ( f )  H c ( f ) e j ( f )
 In ideal channel,
H c  f  : constant value
  f  : linear value according to f
디지털통신(Digital Comm.)
48

49. Equalizer
 Total system frequency response without equalizer
H ( f )  Ht ( f )Hc ( f )H r ( f )
여기서, H ( f ) 는 송수신간 equivalent transfer function
 In real, H c  f  , Intersymbol interference (ISI) is generated by the variable   f
0.9
0
0
-3T
0.1
0.2
-2T
0
-0.3
0
2T
3T
t
[Received pulse exhibiting distortion]
 Total system frequency response with equalizer
H ( f )  Ht ( f )Hc ( f )Hr ( f )He ( f )
He ( f ) 
디지털통신(Digital Comm.)
1
1

e j ( 
Hc ( f ) | Hc ( f ) |
C
( f ))
49


50. Equalizer
Transversal filter
(linear equalizer)
 Equalizer compensates for channel-induced ISI
 Inverse transfer function of channel
N
h(t ) 
 a  (t  nT )
n
n 0
H ( ) 
N
a
n
exp( jnT ).
n 0
Tap coefficient
디지털통신(Digital Comm.)
50

51. Equalizer
Transversal filter
xk
T
c N
T

T
T
c N 1
c N 1
cN


zk
계수조정을 위한 알고리즘
트랜스버설 필터
z (k ) 
N
 x ( k  n )c
n N
n
k  2 N , , 2 N , n   N , , N
z  xc
[Tap coefficient]
1. 제로 포싱 해법
(zero-forcing solution)
2. 최소 평균 자승 오차
(minimum mean-square error : MMSE)
c  x 1 z
디지털통신(Digital Comm.)
51

52. ZF equalizer was developed by Lucky, 1965, for wireline
communication. Not often used for wireless links. However, it
performs well for static channels with high SNR, such as local
wired telephone lines.
Equalizer
 제로 포싱 등화기 (zero-forcing (ZF) equalizer)
0.9
1.0
0.2
0
-3T
0.1
0
-2T
0
0
-3T
0
-2T
Time
0
2T
3T
0
0
Time
2T
3T
-0.3
제로 포싱에 의한 등화기를 사용한 경우
c  x 1 z
디지털통신(Digital Comm.)
52

53. Zero-forcing equalizer
 Use a zero-forcing solution to find the weights (c-1, c0, c1)
 3 taps(c-1, c0, c1)
 received distorted set of pulse samples {x(k)}, with voltage values
0.0, 0.2, 0.9 -0.3, 0.1.
 Calculate the ISI values of the equalized pulse at the sample times
k=-3,-2,-1,0,1,2,3
Z=XC
Z=XC
0   x(0) x(1)
1    x(1) x(0)
  
0   x(2) x(1)
  
0   0.9 0.2
1    0.3 0.9
  
0   0.1 0.3
  
C1   0.2140 
 C    0.9631 
 0 

 C1   0.3448 
  

디지털통신(Digital Comm.)
x(2)  C1 
x(1)   C0 
 
x(0)   C1 
 
0  C1 
0.2   C0 
 
0.9   C1 
 
 Z 3   0.0 0.0 0.0 
 Z   0.2 0.0 0.0 
 2  

 Z 1   0.9 0.2 0.0   0.2140 
  


 Z 0    0.3 0.9 0.2   0.9631 
 Z1   0.1 0.3 0.9   0.3448 


  

 Z 2   0.0 0.1 0.3
 Z   0.0 0.0 0.1 

 3 
0.0, -0.0428, 0.0, 1.0, 0.0 -0.0071, 0.0345
53

54. Equalizer
 사전 설정(preset) 등화기/적응(adaptive) 등화기
: 계산 자동 성질(the automatic nature of operation)에 따른 분류.
사전 설정(preset) equalizer
적응(adaptive) equalizer
channel
environment
• Known channel frequency
response
• time-invariant channel
• slow time-varying channel
Characteristics
• fixed tap coefficient duration
data transmission
• Tap coefficients are adaptive to
the channel response
method
• Tap coefficients is determined
by using training sequence(학습
스퀀스)
• 보편적으로 알고 있는 채널에 대하
여 탭 가중치 설정
• Preamble
• 학습 시퀀스를 사용 주기적으로 탭
가중치 값을 바꿈
operation
대부분의 경우 데이터 전송 시작 전
에 끝냄
데이터 전송과 동시에 동작
디지털통신(Digital Comm.)
54

55. Equalizer
 등화기의 종류
기준
기준에 따라 분류
설명
트랜스버설 등화기
선형성
제로 포싱 등화기
최소 평균 자승 오차 등화기
결정 궤환 등화기
사전 설정(preset) 등화기
계산 자동 성질
(the automatic
nature of operation) 적응(adaptive) 등화기
필터의 업데이트 속
도(rate)
디지털통신(Digital Comm.)
순방향 원소들로 구성.
궤환(feedback)
원소도 포함.
계수값을 처음에 결정함.
상황에 따라 적응적으로
계수값을 바꿈.
심볼 간격(symbol-spaced) 등화기
1 심볼당 1 샘플
부분 간격(fractionally spaced) 등화기
1 심볼당 다수 샘플
55

56. 검출 과정 (Detection process)
판정 이론 : (Maximum A-Posteriori) MAP criterion
Hypothesis H0 and H1 :
The signal source at the transmitter consists of s0(t)and s1(t) waveforms
사전확률(A-priori probability ) : P( H 0 ), P( H1 )
사후확률(A-posteriori probability ) : P( H 0 y ), P( H1 y )
MAP criterion
If)
(최소에러조건)
P ( H1 y )  P ( H 0 y )
or
P ( H1 y )
1 ,
P( H 0 y )
→ choose hypothesis H1, otherwise choose hypothesis H0
디지털통신(Digital Comm.)
56

57. 검출 과정 (Detection process)
Likelihood (conditional probability )
p ( y ) P( H 0 )
P( H 0 y )  0
,
p( y )
p ( y ) P ( H1 )
P ( H1 y )  1
.
p( y )
If)
가능성비
(likelihood
ratio)
p0 ( y )  P( y H 0 ) : 0이 전송될때 y의 확률밀도
p1 ( y )  P( y H1 ) : 1이 전송될때 y의 확률밀도
(likelihood ratio test)
Likelihood of s0
Likelihood of s1
p1 ( y ) P( H 0 )
 ( y) 

p0 ( y ) P( H1 )
p1 ( y ) P( H1 )  p0 ( y ) P( H 0 )
MAP chooses H1.
If P( H 0 ) and P( H1 ) are equally likely, the MAP criterion is known as
the maximum likelihood criterion.
 ( y) 
디지털통신(Digital Comm.)
p1 ( y )
1
p0 ( y )
57

58. 검출 과정 (Detection process)
가능성비(likelihood ratio)
 ( y) 
p1 ( y )

p0 ( y )
Likelihood of s0
Likelihood of s1
 ( y  S1 ) 2 
exp 

2
2
2
 2

2
 ( y  S 0 ) 
1
exp 

2
2 2
 2

1
 S12 
 y2 
 2 yS1 
exp   2  exp   2  exp 
2 
 y ( S1  S0 ) S12  S0 2  P( H 0 )
 2 
 2 
 2 

 exp 


2
2 2  P( H1 )
 S0 2 
 y2 
2 yS0 


exp   2  exp   2  exp 
2 
 2 
 2 
 2 
if ) P( H 0 ) and P( H1 )are equally likey
y ( S1  S0 )
S12  S0 2

2

2 2
S S
y 1 0
2
if ) S0  V , S1  V (antipodal signals)
maximum likelihood criterion
Threshold =0
 y  0, S1 (V ) is chosen
y  0, S0 (V ) is chosen
디지털통신(Digital Comm.)
58

59. 에러 확률 (Error probability)
Simple receiver without MF
sin(t Tb ) B  1 (transmission BW)
s (t )  V
2Tb
t T
b
Pe 
1
2 B


0
Likelihood of s0
Likelihood of s1
 ( y  V ) 2 
exp 
dy, ( PFA  PM )
2 B 


( y의 신호성분은 s (0)  V )
y V
dy


 x  2 B , y  2 Bx  V , dx  2 B 




2 
erfc( x ) 
exp(u 2 )du


 x



2 B
2
2 2 B 
Pe 


V
2 B
 V
1
erfc 
 2 B
2

exp( x) 2 dx
 1
 S
 erfc 

 2N
 2


디지털통신(Digital Comm.)




59

60. Error probability of receiver without MF
Co-error function
 S
1
Pe  erfc
 2N
2





x
erfc(x)
x
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
1.000
0.777
0.572
0.396
0.258
0.157
8.97102
4.77102
2.37102
1.09102
4.38103
2.2
2.4
2.6
2.8
3.0
3.3
3.7
4.0
5.0
erfc(x)
1.86103
6.9104
2.4104
7.5105
2.2105
3.06106
1.67107
1.54108
1.541012
erfc(0)  1
erfc()  0
디지털통신(Digital Comm.)
60

61. 에러 확률 (Error probability)
정합필터 수신기(receiver with MF)
s  V 2Tb
n(t)의 분산
n

2
 E 


 E V 2



Vn ( t ) dt 


Tb
0
Tb
Tb
0
0
 
 
V
2
Tb
Tb
0
V
2
0
V 2

2
Tb

0
0
2

Tb
0
0
Vn(t )dt
(출력 잡음)
2
n ( t ) n ( ) dtd  .


R n ( t   ) dtd 
Tb
 

Tb
(출력 신호)
 ( t   ) dtd 
(1) dt
V 2 Tb

.
2
V 4Tb2
2V 2Tb
S


N V 2Tb 2

디지털통신(Digital Comm.)
[참고] F
1
[ S n ( )]  R n ( t )
R n (t )  F
 F
1
[ S n ( )]
1  
 
 2   2  ( t ).
 
61

62. 에러 확률 (Error probability)
Pe  

0
 ( y  V 2Tb ) 2 
1
exp 
 dy, ( PM  PFA )
2
2
2




y  V 2Tb
dy
 2 
, y  2 x  V 2Tb,
x
dx
2





2
2
erfc( x) 


 x exp(u )du



Pe 
1



V 2Tb
2
 V 2Tb
1
 erfc 
2
 2
exp( x 2 )dx
 2V 2T
 1
b
  erfc 


 2





 S 
1
 erfc 
 N

2


※error probability of receiver w/o MF
디지털통신(Digital Comm.)
Pe 
 S
1
erfc 
 2N
2





62

63. 에러 확률 (Error probability)
2진 정합필터 수신기
디지털통신(Digital Comm.)
63

64. 에러 확률 (Error probability)

E[ y ]  E 






f (t ){s1 (t )  s0 (t )}dt   E 


Tb
0
Tb
0

Tb
0

{si (t )  n(t )}{s1 (t )  s0 (t )}dt 

si (t ){s1 (t )  s0 (t )}dt  mi
y의 분산 : f (t )  n(t )

Tb
 E


Tb
Tb
0
0
E[n(t )]  0

 2  E   f (t ){s1 (t )  s0 (t )}dt  E  f (t ){s1 (t )  s0 (t )}dt 
 0



 0


 
Tb
Tb
0
0
 

2
Tb

Tb
0
2
n(t )n( ){s1 (t )  s0 (t )}{s1 ( )  s0 ( )}dtd 




 (t   ){s1 (t )  s0 (t )}{s1 ( )  s0 ( )} dtd
2


{s1 (t )  s0 (t )}2 dt.
디지털통신(Digital Comm.)
64

65. 에러 확률 (Error probability)
 ( y) 
p1 ( y )

p0 ( y )
 ( y  m1 ) 2 
1
exp 

 ( y  m1 ) 2 ( y  m0 ) 2 
2 2 
2 

 exp 


2
2
 ( y  m0 ) 
2
2 2 
1

exp 

2
2
2 


다음의 경우에 가정 H1을 선택한다.
 ( y  m1 )
( y  m0 ) 

  o
2
2
2
2


 ( y )  exp 
또는
2
2
2
 y (m1  m0 ) 2 m0  m12 
 ( y )  exp 

  o
2
2

2


만일
m1  m0  2 ln o

y
2
m1  m0
가정 H1을 선택하고
그렇지 않으면 H0을 선택
if ) o  1
Threshold
(문턱값)
디지털통신(Digital Comm.)
이면
y
m1  m 0
2
65

66. 에러 확률 (Error probability)
m1  m0  2 ln o m1  m0


y
m1  m0
2
2
디지털통신(Digital Comm.)
(o  1)
66

67. 에러 확률 (Error probability)
Average energy of S0(t) and S1(t)
1
E
2



Tb
Tb
0
0
Pe  PFA  P( z  0 H 0 )
2
{s0 (t )  s12 (t )}dt
 {z  E[ z ]}2 
1
exp  

 dz
2
0
2
2


 E (1   ) 
1
 erfc 

2
2  E (1   ) 



 E (1   ) 
1
 erfc 
.
2
2 


s0 (t ) s1 (t )dt
E
Correlation
coefficient
m1  m0
2
( z  0이면 가정 H1 )
z  y
m  m0
E[ z ]   1
  E (1   )
2
Z  y 
2
2


2
Tb
0

2
Tb
0
  0, E[ z ]   E
  1, E[ z ]  2 E
{s1 (t )  s0 (t )} dt
2
2
{s12 (t )  s0 (t )  2 s1 (t ) s0 (t )}dt
  E (1   ).
디지털통신(Digital Comm.)
m1  m0
y의 threshold
2
1 Tb

[ s1 (t )  s0 (t )][ s1 (t )  s0 (t )]dt
2 0
E  E0
1 Tb 2
2

[ s1 (t )  s0 (t )]dt  1
2 0
2
yo 


67

68. 에러 확률 (Error probability)
 E (1   ) 
1
Pe  erfc

2
2 




1
Pe  erfc 
2


,E 
, 
E (1   ) 

2


E 0  E1
1

2
2

Tb
0
0
2
{ s 0 ( t )  s12 ( t )} dt
s 0 ( t ) s1 ( t ) dt
E
E  E0
, yo  1
2
디지털통신(Digital Comm.)

Tb
(  o  1)
68