Entropy Entropy Channel Capacity Information Source Channel coding

정보이론 • 정보량과 Entropy • Entropy 계산의 예 • Channel Capacity 잡음 Information Source 정보량 Channel coding 송신 전력 Communication Channel Detection 채널 용량 오류 확률 4

Channel Capacity(채널 용량) • Shannon-Hartley Capacity Theorem C : Channel Capacity S : Average Signal Power N : Average Noise Power W : Bandwidth In AWGN channel C/W(bits/sec/Hz) 32 16 8 4 2 1 -10 10 20 30 40 SNR(d. B) 1/2 Ex) Telephone line : 3000 Hz, SNR=30 d. B=1000 C=3000 log 2(1+1000) =3000 x 9. 967 = 29. 9 Kbps 30 Kbps 1/4 1/8 9

Shannon Limit 10

채널 부호화의 종류 • Waveform Coding 심볼 파형을 서로 구별하기 용이하도록 변형 – 직교 부호(Orthogonal Code) – 쌍직교 부호(Bi-orthogonal Code) – Transorthogonal Code • Structured Sequence – – – 오류 비트를 검출하고 정정할 수 있도록 Cyclic Code 블록 부호화(Block Coding) Reed-Solomon Code 길쌈 부호화(Convolutional Coding) 터보 부호화(Turbo Coding) 12

Orthogonal Signaling Information k bit Orthogonal Waveform coding 2 k symbol MFSK Modulation Waveform MPSK • Orthogonal : MFSK • Non-Orthogonal : MPSK • Orthogonal Waveform Code : – Ex) Hadamard code • Biorthogonal code • Transorthogonal code = simplex code – Delete 1 st digit of Orthogonal code 14

직교 부호 (Hadamard code) In general, 18

Biorthogonal code In general, 20

오류제어와 채널 부호화 • 오류 제어방식 – 무시, 루프 비교, 재전송, 오류 수정 • 오류 검출 부호 – Parity Check, CRC, FCS • 오류 정정 부호 – linear block code, Hamming code, extended Golay code, BCH code, Reed. Solomon code – Convolutional code, turbo code 22

ARQ 시스템 Stop & Wait 정지대기 Goback-N N-후진 Selective repeat 선별 반복 23

오류 검출부호 • Single Parity Check Code – Even parity, Odd Parity, no parity – Data : 4~8, Stop : 1/1. 5/2, 속도: 110 ~ 11520 • Block Sum Check 0 0 1 1 1 0 0 0 1 1 1 0 1 0 1 1 1 0 0 0 1 24

CRC Generation 26

CRC 발생회로 Switch 1 G(x) … … 1) 2) 3) 4) 5) register 초기화 SW 1 close, SW 2 down Information k bit shift out and back SW 1 open , SW 2 up Parity p bit shift out M(x) R(x) T(x) Switch 2 27

Standard CRC 28

Linear Block Code k bits data n-k parity symbol (n, k) code n bits channel symbol • k : data block size (bits) • n : channel symbol size (bits) • k/n : code rate – Ex) parity code : k/k+1 – Product code : k/n = MN/(M+1)(N+1) • Linearity – Include all zero vector “ 0000” – Closed in modulo-2 addition 29

Extended Golay code • • (23, 12) : Golay Code (24, 12) : Golay Code + parity bit Minimum distance 8 오류 정정 : all triple error, some four-error(19%) 33

Reed-Solomon Code • • One Special subclass of BCH code largest possible code minimum distance n = 2 m-1 k = n – 2 t dmin = n-k+1 = 2 t+1 t=(dmin -1)/2 = (n-k)/2 useful for burst-error correction 35

Bit Error Probability PE PB 10 -2 10 -3 Hamming(31, 26), t=1 Hamming(15, 11), t=1 Hamming(7, 4), t=1 10 -4 10 -5 Extended Golay(24, 12), t=3 BCH(127, 64), t=10 -5 BCH(127, 36), t=15 10 10 -6 p 10 -7 10 -1 10 -2 10 -3 10 -4 10 -7 Uncoded Eb/N 0 4 5 6 7 8 9 10 36 11

Convolutional Code Data Sequence 1 2 k Linear Sequential Circuit: Memory M 1 2 n Code Sequence (n, k, M) 길쌈부호 k = 입력 열, n = 출력 열 M = 메모리 크기 K=M+1 : 구속장 길이(constraint length) R=k/n : 부호율 (일반적으로: K=7, 9; R=1/2, 1/3 ; M=6, 8 인 부 호 사용) 37

길쌈부호의 예 • (2, 1, 2) : g 1=101, g 2 = 111, m = 101 + Input Data m 1 0 0 0 r 2 0 1 0 0 0 r 1 0 0 r 0 0 1 0 u 1 0 0 0 1 0 r 2 u 2 0 1 1 0 r 1 r 0 u 1 Output Code word u 2 + 38

길쌈부호의 예 • (2, 1, 2) : g 1=101, g 2 = 111 , m = 101 m(X) = 1 + X 2 g 1(X) = 1 + X 2 g 2(X) = 1 + X 2 m(X) g 1(X)=1 + X 4 m(X) g 2(X)=1 + X 3 + X 4 m(X) g 1(X)=1 + X 4 3 + X 4 m(X) g (X)=1 + X 2 + 11 01 00 01 11 39

상태표 Time ti+1 Input m r 1 r 0 u 1 u 2 - 0 0 - - 0 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 1 1 1 Output 40

상태 천이도 입력/출력 0/11 01 0/00 00 입력 출력 1 0 1/11 0/01 10 1/00 0/10 11 1/01 1/10 레지스터 상태 11 01 00 01 11 00 00 41

Trellis Diagram 출력 u 1 u 2 상태 t 1 a=00 t 2 00 11 t 3 00 11 01 10 00 t 6 00 11 11 11 00 00 00 01 10 0 10 1 10 d=11 t 5 11 11 b=10 c=01 t 4 01 10 01 01 입력 42

자유 거리(dfree) 상태 t 1 t 2 t 3 0 a=00 2 t 4 0 0 2 2 1 1 1 d=11 0 2 1 1 t 6 0 2 b=10 c=01 t 5 dfree=5 2 2 2 0 0 0 1 1 1 1 1 43

Viterbi Decoding(1) U = 00 Z = 00 t 1 a=00 0 00 2 11 00 01 t 2 1 00 1 11 00 10 t 3 b=10 c=01 d=11 0 01 2 10 00 00 t 4 00 00 t 5 1 00 2 1 3 11 0 00 2 2 11 5 1 11 2 13 00 2 01 0 3 10 4 0 10 2 1 01 6 2 11 0 00 1 01 1 10 1 01 0 00 t 6 2 11 0 00 1 01 1 10 1 01 44

Viterbi Decoding(2) U = 00 Z = 00 t 1 a=00 00 01 t 2 1 00 1 11 00 10 t 3 00 00 t 4 1 00 2 1 3 00 00 t 5 0 00 2 2 5 b=10 2 3 4 0 3 c=01 2 01 0 3 10 4 1 01 1 3 10 2 1 10 1 3 01 2 d=11 1 6 2 0 00 t 6 5 2 11 0 00 1 01 1 2 1 10 1 01 45

Viterbi Decoding(3) U = 00 Z = 00 t 1 a=00 00 01 t 2 1 00 1 11 00 10 t 3 00 00 t 4 1 00 2 1 00 00 t 5 0 00 2 5 0 10 1 10 d=11 0 00 2 2 5 2 11 4 0 2 00 b=10 c=01 t 6 1 6 3 2 1 3 01 2 1 10 1 4 3 0 00 2 2 5 2 4 11 0 3 00 1 01 1 3 4 46

Viterbi Decoding(4) U = 00 Z = 00 t 1 a=00 00 01 t 2 1 00 1 11 00 10 t 3 00 00 t 4 1 00 2 1 00 00 t 5 0 00 2 5 0 10 1 10 d=11 0 00 2 2 5 4 0 2 00 b=10 c=01 t 6 1 6 3 2 1 3 01 2 1 10 1 4 3 t 7 0 00 2 2 5 2 4 11 0 3 00 1 01 1 3 4 47

Viterbi 복호 연습 Z = 10 t 1 a=00 00 t 2 00 11 00 t 3 00 11 01 10 00 11 00 t 5 00 11 t 6 00 11 11 t 7 00 11 11 11 00 00 01 10 10 10 10 d=11 10 t 4 11 b=10 c=01 10 01 01 01 48

Viterbi 복호 연습 Z = 10 t 1 a=00 00 t 2 00(1) 00 10 t 3 00(0) 10 t 4 00(0) 1 00 t 5 00(1) t 6 00(1) t 7 00(0) 4 11(1) 11(2) 3 11(2) b=10 01(1) c=01 11(1) 11(2) 10(1) 00(0) 00(1) 00(0) 01(1) 01(2) 01(1) 10(1) d=11 11(1) 2 01(1) 10(0) 01(2) 10(1) 01(1) 49

길쌈부호의 응용예 구속장 (K) 생성다항식 dfree 수정가능 코딩이 비트수(t) 득(d. B) 사용 분야 5 23=10011 35=11101 7 3 4. 3 GSM 6 65=110101 57=101111 8 3 4. 6 IS-54 미국이동통신 7 133=1011011 171=1111001 10 4 5. 2 위성통신 9 753=111101011 561=101110001 12 5 7. 8 IS-95 CDMA Forward 채널 9 557=101101111 663=110110011 711=111001001 18 8 IS-95 CDMA Reverse 채널 51

인터리빙(Interleaving) 53

Turbo Code • RSC-인터리빙-RSC의 이중 구조의 encoding 방식을 적용 • 복조시 반복 알고리즘을 사용 RSC: Recursive Systematic Convolutional Code 54

RSC (Recursive Systematic Convolutional Code ) uk dk 출력 uv 상태 a=00 00 11 + ak ak-1 11 ak-2 b=10 + vk c=01 0 1 입력 d=11 00 10 01 01 10 55

Turbo Encode uk dk + ak ak-2 + Interleaver + ak-1 ak-2 vk + 56

Turbo Decoder zk Decoder DEC 1 xk y 1 k Interleaving Decoder DEC 1 y 2 k Deinterleaving 0 yk Bit slice 0 dk 57