Digital Communications

Transcription

1 Digital Communications Chapter 8: Trellis and Graph Based Codes Saeedeh Moloudi May 7, 2014

2 Outline 1 Introduction 2 Convolutional Codes 3 Decoding of Convolutional Codes 4 Turbo Codes May 7, 2014 Proakis-Salehi Chapter 8 1 / 25

3 Introduction May 7, 2014 Proakis-Salehi Chapter 8 2 / 25

4 Channel Coding Channel Coding Theorem:For every code rate R < C there exists a code which the probability of error after decoding approaches zero. Shannon Limit: For a given R, Shannon limit is the smallest value of E b /N 0 for which reliable communications is possible. May 7, 2014 Proakis-Salehi Chapter 8 3 / 25

5 Gain of the Coding Coding Gain: At a given probability of error per bit P b (E), the difference between the E b /N 0 required to obtain that P b (E) with coding versus without coding and it is typically measured in db. May 7, 2014 Proakis-Salehi Chapter 8 4 / 25

6 Block Codes Example (Repetition Codes) u = (0) v = (0,0,0) u = (1) v = (1,1,1) K = log(m), K = 1 A codeword length N = 3 Code Rate, R = K N, R = 1/3 v = u G, G = (1,1,1) May 7, 2014 Proakis-Salehi Chapter 8 5 / 25

7 Convolutional Codes May 7, 2014 Proakis-Salehi Chapter 8 6 / 25

8 The Structure of Convolutional Codes Block codes: v t = u t G Convolutional Codes: v t = f (u t,u t 1,...,u t m ) (1) Encoder of Convolutional Codes: 1 3 Memory: In (1), m is called memory. May 7, 2014 Proakis-Salehi Chapter 8 7 / 25

9 Convolutional Codes According to block diageram of encoder in previous slid: Function Generator: For each out put we can define a function generator: g 1 = [100], g 2 = [101], g 3 = [111] Output: c (1) = u g 1, c (2) = u g 2, c (3) = u g 3. c = (c (1) 1,c(2) 1,c(3) 1,c(1) 2,c(2) 2,c(3) 2,...) Encoding in the transform domain: u(d) = i=0 u i D i g1 (D) = 1, g 2 (D) = 1 + D 2, g 3 = 1 + D + D 2 Example c (j) = u(d)g j (D) c(d) = c (1) (D 3 ) + Dc (2) (D 3 ) + D 2 c (3) (D 3 ) For input u(d) = 1 + D 3 + D 4 + D 5 the out put sequence is ( ) May 7, 2014 Proakis-Salehi Chapter 8 8 / 25

10 Convolutional Codes Example (Convolutional Codes: Rate= 2/3) May 7, 2014 Proakis-Salehi Chapter 8 9 / 25

11 State Diagram and Trellis , L ,--1 ', 010 ',, 100 ' ' ' I 111,.,'" 110 / / I I I I I I State: The stored values at the out put of delay elements May 7, 2014 Proakis-Salehi Chapter 8 10 / 25

12 State Diagram and Trellis a b c 000 lll 001 Ill d 101 Steady state May 7, 2014 Proakis-Salehi Chapter 8 11 / 25

13 Decoding of Convolutional Codes May 7, 2014 Proakis-Salehi Chapter 8 12 / 25

14 Viterbi Algorithm (Maximum Likelihood Decoding) Example (Viterbi algorithm) Consider previous Trellis and received sequence r = (101, 000, 100) The initial and terminated states: State a Possible Code sequences: (000, 000, 000) and (111, 001, 011) a b c 000 lll 001 Ill d 101 Steady state May 7, 2014 Proakis-Salehi Chapter 8 13 / 25

15 Viterbi Algorithm (Maximum Likelihood Decoding) Branch Metric: For the the jth branch of the ith path, branch metric is : µ (i) j = logp(r j c (i) j ) Path Metric: Example PM (i) = µ (i) j j For a channel with the probability of a bit error: p PM (0) = 6log(1 p) + 3logp PM (1) = 4log(1 p) + 5logp For p < 1/2 PM (0) > PM (1), Then the decoded sequence is (000, 000, 000) May 7, 2014 Proakis-Salehi Chapter 8 14 / 25

16 Viterbi Algorithm and WAGN Channel General Memoryless Channel For trellis T c (m) = max c T j logp(r j c j ) Soft Decision Decoding- AWGN Channel For trellis T c (m) = min c T j r j c j 2 Hard Decision Decoding- AWGN Channel For trellis T c (m) = min c T j d H (y j,c j ) May 7, 2014 Proakis-Salehi Chapter 8 15 / 25

17 BCJR Algorithm: Maximum A Posteriori Decoding Output of MAP Decoder: û = argmax u P(u r) = argmax u P(r u)p(u) P(r) = argmax P(r u)p(u) u MAP decoder can obtain from Viterbi algorithm by incorporating the a priori probabilities into branch metric values. Symbol by Symbol MAP Decoder: û i = arg max u {0,1} P(u i r) A Posteriori Probability (APP) Decoder: Provide soft outputs L(u i ) = log P(u i = 0 r) P(u i = 1 r) = log u:u i =0 P(r u)p(u) u:ui =1 P(r u)p(u) May 7, 2014 Proakis-Salehi Chapter 8 16 / 25

18 BCJR Algorithm: Maximum A Posteriori Decoding r <t r t r >t t = t 1 = May 7, 2014 Proakis-Salehi Chapter 8 17 / 25

19 BCJR Algorithm: Maximum A Posteriori Decoding 1 1. Forward recursion: t = 1,...,L + m 1 ( a t (s)=â g t (s 0,s) a t 1 (s 0 1 if s = 0 ), a 0 (s)= s 0 0 s 6= 0 with g t (s 0,s)=p(r t s 0,s) P(s s 0 )=p(r t v t ) P(u t ) 2. Backward recursion: t = L + m,...,2 b t 1 (s 0 )=Â s g t (s 0,s) b t (s), 3. Soft output: t = 1,...,L + m L(u (i) t )=log b N (s)= ( 1 if s = 0 0 s 6= 0 Â a t 1 (s 0 ) g t (s 0,s) b t (s) (s 0,s):u (i) t =0 a t 1 (s 0 ) g t (s 0,s) b t (s) Â (s 0,s):u (i) t =1 1 This slide is used from Error Control Coding course May 7, 2014 Proakis-Salehi Chapter 8 18 / 25

20 Â (i) 0 ):u (i)=1 (s (s 0,s,s ):utt =1 a att 11(s (s )) ggtt(s (s,,ss)) bbtt(s (s)) BCJR Algorithm: Maximum A Posteriori Decoding 2 Michael Lentmaier Michael Lentmaier EDI042 EDI042 Error Error Control Control Coding: Coding: Chapter Chapter / /35 35 The Log-Domain BCJR Algorithm Algorithm (Log-APP) (Log-APP) 1. Forward recursion: t = 1, 1,...,,LL+ +m m 11 0 att (s ) = max gtt (s 00, s ) + +a att 11(s (s 0)),, aa00 (s (s)) = = 0 s s0 with K K NN ( ( 00 ifif ss = =00 ss 6= 6=00 (i) (j) (j) gtt (s 00, s ) = Â Laa ut(i) 1/2 1/2 uutt + +Â 1/2 vvtt t ÂLLchch vvtt 1/2 (i) (i) i=1 i=1 (j) (j) j=1 j=1 2. Backward recursion: t = = LL + +m, m,...,,22 btt 11(s 00) = max gtt (s 00, s )) + (s ) = + bbtt (s (s)),, bbl+m L+m (s ) = s s ( ( 00 ifif ss = =00 ss 6= 6=00 3. Soft output: t = 1,..., L + m m (i) L(ut(i) )= t max (i) 0 ):u (i)=0 (s (s 0,s,s ):utt =0 max (i) 0 ):u (i)=1 (s (s 0,s,s ):utt =1 Michael Lentmaier Michael Lentmaier 2 This May 7, a att 11(s (s 0))+ +ggtt (s (s 0,,ss))+ +bbtt (s (s)) 0 0 a att 11(s (s 0))+ +ggtt (s (s 0,,ss))+ +bbtt (s (s)) EDI042 EDI042 Error Error Control Control Coding: Coding: Chapter Chapter / /35 35 slide is used from Error Control Coding course Proakis-Salehi Chapter 8 19 / 25

21 Log-Domain Formulation BCJR Algorithm: Maximum A Posteriori I The probabilities at the input can also be represented as Decoding L-values (log-likelihood ratios): (i) La ut (i) = log P ut = 0 (i) P ut = 1, (j) Lch vt e easily shown that (Problem 8.22) I (j) (j) Digital p rt Communications vt = 0 = log (j) (j) p rt vt = 1 In case of an AWGN channel we have Lch vt = max*{x, y} = max{x, y} + ln (1 + e-lx-yl) (j) 2 s2 (j) rt (8.8-31) Definition max*{x, y, z} =max* {max*{x, y}, z} The max operator is defined as Its maximum occurs are not close. (1 + e-lx- yl) is small whenx x andy y (x, y) is = ln log2.(eit+iseyclear ) = max(x, y) + log x1 and + e y x or when x and large that for this term for which max se, we can use the approximation I We can nowmax*{x, simplify y} the computations max{x, y} in the log-domain: (8.8-32) Â can use the approximation ar conditions we u u log Michael Lentmaier May 7, 2014 e log a e log g e log b = max (log a + log g + log b ) max*{x, y, z} max{x, y, z} EDI042 Error Control Coding: Chapter Proakis-Salehi Chapter / / 25 (8.8-33)

22 Turbo Codes May 7, 2014 Proakis-Salehi Chapter 8 21 / 25

23 Turbo Codes Turbo Codes are parallel concatenated convolutional codes. Input information bits Output - convolutional Pseudorandom block interleaver length=n convolutional - Linear r-- encoder I bits puncturmg - Linear - encoder 2 Optional Parity check PI Output Output Panty check bitsp2 May 7, 2014 Proakis-Salehi Chapter 8 22 / 25

24 decoder is shown in Figure A typical plot of the performance of the iterative decoding algorithm for turbo codes is given in Figure It is clearly seen that the first few iterations noticeably improve the performance. It is seen from these plots that three regions are distinguishable. For the low-snr region where the error probability changes very slowly as a function of Eb/No and the number of iterations, for moderate SNRs the error probability drops neral we Ican useof a iterative SISO decoding module foriterations each Pb decreases consistently. This rapidly with increasing Eb/ No and over many Idea decoding: ponent code:region is called the waterfall region or the turbo cliff region. Finally, for moderately 1. individually decode component codes large Eb/ No values, the code an error floor which is typically achieved with a L2. (u) L (v) exchange the symbol estimates between them a a Le (u) Le (v) nput Iterative Soft-Output Decoder Decoding for Turbo Codes SISO Decoder Lch (v) PP decoder based on the BCJR algorithm is typically used y' ome approximation) see Chapter 3 output of an APP decoder in the log-domain is given by FIGURE Block L(vidiagram ) = Laof (vai )turbo + Ldecoder. ch (vi ) + Le (vi ) {z } {z } intrinsic -based extrinsic L(u;) I The intrinsic decoder in the SPC example is antoapp part corresponds the adecoder priori and channel reliabilities of the symbol vi itself nt: I The extrinsic part corresponds to an estimate of vi extrinsic values in the basedare onexchanged other symbols vj,iterations j 6= i May 7, 2014 EDI042 Error Control Coding: Chapter Proakis-Salehi Chapter / / 25

25 Iterative Decoding for Turbo Codes Uncoded BPSK P b I=6 I=3 I=2 I=1 I=18 May 7, 2014 Proakis-Salehi Chapter 8 24 / 25

26 EXIT Chart study of Iterative Decoding Memory 4, (0 1, G) = (023, 037) -+- First decoder, 0 8 db --- Second decoder, 0.8 db -lr- First decoder, 0.1 db -e- Second decoder, 0.1 db output le2 of second decoder= input /at of first encoder May 7, 2014 Proakis-Salehi Chapter 8 25 / 25