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- 1. DIGITAL COMMUNICATIONS Block 3 Channel Coding Francisco J. Escribano, 2014-15
- 2. 2 Fundamentals of error control ● Error control: – error detection (ARQ schemes) – correction (FEC schemes) ● Channel model: – discrete inputs, – discrete (hard, rn ) or continuous (soft, λn ) outputs, – memoryless. Channel encoder Channel Channel error corrector bn bn rn cn λn Channel error detector RTx
- 3. 3 Fundamentals of error control ● Enabling detection/correction: – Adding redundancy to the information: for every k bits, transmit n, n>k. ● Shannon's theorem (1948): 1) If , for ε>0, there is n, R=k/n const., so that Pb <ε. 2) If Pb is acceptable, rates R<R(Pb )=C/(1-H2 (Pb )) are achievable. 3) For any Pb , rates greater than R(Pb ) are not achievable. ● Problem: Shannon's theorem is not constructive. R<C=max pX I ( X ;Y )
- 4. 4 Fundamentals of error control ● Added redundancy is structured redundancy. ● This relies on sound algebraic & geometrical basis. ● Our initial approach: – Algebra over the Galois Field of order 2, GF(2)={0,1}. – GF(2) is a proper field, GF(2)m is a vector field of dim. m. – Dot product · :logical AND. Sum +(-) : logical XOR. – Scalar product: b, d ∈ GF(2)m b·dT =b1 ·d1 +...+bm ·dm – Product by scalars: a ∈ GF(2), b ∈ GF(2)m a·b=(a·b1 ..a·bm ) – It is also possible to define a matrix algebra over GF(2).
- 5. 5 Fundamentals of error control ● Given a vector b ∈ GF(2)m , its binary weight is w(b)=number of 1's in b. ● It is possible to define a distance over vector field GF(2)m , called Hamming distance: dH (b,d)=w(b+d); b, d ∈ GF(2)m ● Hamming distance is a proper distance and accounts for the number of differing positions between vectors. ● Geometrical view: (0110) (1011) (1110) (1010)
- 6. 6 Fundamentals of error control ● A given encoder produces n bit outputs for each k bit inputs: – R=k/n<1 is the rate of the code. ● The information rate decreases by R by the use of a code. R'b =R·Rb < Rb (bit/s) ● If used jointly with a modulation with spectral efficiency η=Rb /B (bit/s/Hz), the efficiency decreases by R η'=R·η < η (bit/s/Hz) ● In terms of Pb , the achievable Eb /N0 region in AWGN is lower bounded by: Eb N 0 (dB)⩾10⋅log10 ( 1 η' ⋅(2η' −1))
- 7. 7 Fundamentals of error control Source: http://www.comtechefdata.com/technologies/fec/ldpc
- 8. 8 Fundamentals of error control ● How a channel code can improve Pb (BER in statistical terms). ● Cost: loss in resources (spectral efficiency, power).
- 9. 9 Binary Linear block codes
- 10. 10 Binary Linear block codes ● An (n,k) linear block code (LBC) is a subspace C(n,k) < GF(2)n with dim(C(n,k))=k. ● C(n,k) contains 2k vectors c=(c1 …cn ). ● R=k/n is the rate of the LBC. ● n-k is the redundancy of the LBC – we would only need vectors with k components to specify the same amount of information.
- 11. 11 Binary Linear block codes ● Recall vector theory: – A basis for C(n,k) has k vectors over GF(2)n – C(n,k) is orthogonal to an n-k dimensional subspace over GF(2)n (its null subspace). ● c ∈ C(n,k) can be both specified as: – c=b1 ·g1 +...+bk ·gk , where {gj }j=1,...,k is the basis, and (b1 ...bk ) are its coordinates over it. – c such that the scalar products c·hi T are null, when {hi }i=1,...,n-k is a basis of the null subspace.
- 12. 12 Binary Linear block codes ● Arranging in matrix form, an LBC C(n,k) can be specified by – G={gij }i=1,...,k, j=1,...,n , c=b·G, b ∈ GF(2)k . – H={hij }i=1,...,n-k, j=1,...,n , c·HT =0. ● G is a k×n generator matrix of the LBC C(n,k). ● H is a (n-k)×n parity-check matrix of the LBC C(n,k). – In other approach, it can be shown that the rows in H stand for linearly independent parity-check equations. – The row rank of H for an LBC should be n-k. ● Note that gj ∈ C(n,k), and so G·HT =0.
- 13. 13 Binary Linear block codes ● The encoder is given by G – Note that a number of different G generate the same LBC ● For any input information block with length k, it yields a codeword with length n. ● An encoder is systematic if b is contained in c=(b1 ...bk | ck+1 ...cn ), so that ck+1 ...cn are the n-k parity bits. – Systematicity is a property of the encoder, not of the LBC C(n,k) itself. – GS =[Ik | P] is a systematic generator matrix. LBC encoder G b=(b1 ...bk ) c=(c1 ...cn )=b·G
- 14. 14 Binary Linear block codes ● How to obtain G from H or H from G. – G rows are k vectors linearly independent over GF(2)n – H rows are n-k vectors linearly independent over GF(2)n – They are related through G·HT =0 (a) ● (a) does not yield a sufficient set of equations, given H or G. – A number of vector sets comply with it (basis sets are not unique). ● Given G, put it in systematic form by combining rows (the code will be the same, but the encoding does change). – If GS =[Ik | P], then HS =[PT | In-k ] complies with (a). ● Conversely, given H, put it in systematic form by combining rows. – If HS =[In-k | P], then GS =[PT | Ik ] complies with (a). ● Parity check submatrix P can be on the left or on the right side (but on opposite sides of H and G simultaneously for a given LBC).
- 15. 15 Binary Linear block codes ● Note that, by taking 2k vectors out of 2n , we are getting apart the binary words. ● Minimum Hamming distance between input words is ● Recall that we have added n-k redundancy bits, so that dmin (GF(2)k )=min bi≠bj {dH (bi ,bj)|bi ,bj∈GF(2)k }=1 d min (C(n ,k ))=min ci ≠c j {dH (ci ,c j)|ci ,c j∈C(n ,k )}>1 dmin (C(n ,k ))⩽n−k+1 (Singleton bound)
- 16. 16 Binary Linear block codes ● The channel model corresponds to a BSC (binary symmetric channel) ● p=P(ci ≠ri ) is the bit error probability of the modulation in AWGN. Modulator Channel BSC(p) AWGN channel r=(r1 ...rn )c=(c1 ...cn ) Hard demodulator 0 1 0 1 p 1-p 1-p p
- 17. 17 Binary Linear block codes ● The received word is r=c+e, where P(ei =1)=p. – e is the error vector introduced by the noisy channel – w(e) is the number of errors in r wrt original word c – P(w(e)=t)=pt ·(1-p)n-t , because the channel is memoryless. ● At the receiver side, we can compute the syndrome s=(s1 ...sn-k ) as s=r·HT =(c+e)·HT =c·HT +e·HT =e·HT . ● r ∈ C(n,k) ⇔ s=0. Channel decoder H s=(s1 ...sn-k )=r·HT r=(r1 ...rn )
- 18. 18 Binary Linear block codes Two possibilities at the receiver side: ● a) Error detection (ARQ schemes): – If s≠0, there are errors, so ask for retransmission. ● b) Error correction (FEC schemes): – Decode an estimated ĉ ∈ C(n,k), so that dH (ĉ,r) is the minimum over all codewords in C(n,k) (closest neighbor decoding). – ĉ is the most probable word under the assumption that p is small (otherwise, the decoding fails). (0110) (1011) (1110) (1010) ĉ1 e1 e2 r2 r1 c ĉ2 OK
- 19. 19 Binary Linear block codes ● Detection and correction capabilities (worst case) of an LBC with dmin (C(n,k)). – a) It can detect error events e with binary weight up to w(e)|max,det =d=dmin (C(n,k))-1 – b) It can correct error events e with binary weight up to w(e)|max,corr =t=⎣(dmin (C(n,k))-1)/2⎦ ● It is possible to implement a joint strategy: – A dmin (C(n,k))=4 code can simultaneously correct all error patterns with w(e)=1, and detect all error patterns with w(e)=2.
- 20. 20 Binary Linear block codes ● The minimum distance dmin (C(n,k)) is a property of the set of codewords in C(n,k), independent from the encoding (G). ● As the code is linear, dH (ci ,cj )=dH (ci +cj ,cj +cj )=dH (ci +cj ,0). – ci , cj , ci +cj , 0 ∈ C(n,k) ● dmin (C(n,k))=min{w(c) | c ∈ C(n,k), c≠0} – i.e., corresponds to the minimum word weight over all codewords different from null. ● dmin (C(n,k)) can be calculated from H: – It is the minimum number of different columns of H adding to 0. – It is the column rank of H + 1.
- 21. 21 Binary Linear block codes ● Detection limits: probability of undetected errors? – Note that an LBC contains 2k codewords, and the received word corresponds to any of the 2n possibilities in GF(2)n . – An LBC detects up to 2n -2k error patterns. ● An undetected error occurs if r=c+e with e≠0 ∈ C(n,k) – In this case, r·HT =0. – Ai is the number of codewords in C(n,k) with weight i: it is called the weight spectrum of the LBC. Pu (E)= ∑ i=dmin n Ai⋅p i ⋅(1− p) n−i
- 22. 22 Binary Linear block codes ● On correction, an LBC considers syndrome s=r·HT . – Assume correction capabilities up to w(e)=t, and EC to be the set of correctable error patterns. – A syndrome table associates a unique si over the 2n-k possibilities to a unique error pattern ei ∈ EC with w(ei )≤t. – If si =r·HT , decode ĉ=r+ei . – Given an encoder G, estimate information vector b such that b·G=ĉ. ● If the number of correctable errors #(EC )<2n-k , there are 2n-k - #(EC ) syndromes usable in detection, but not in correction. – At most, an LBC can correct 2n-k error patterns. ^ ^
- 23. 23 Binary Linear block codes ● A w(e)≤t error correcting LBC has a probability of correcting erroneously bounded by – This is an upper bound, since not all the codewords are separated by the minimum distance of the code. ● Calculating the resulting P'b of an LBC is not an easy task, and it depends heavily on how the encoding is made through G. ● LBC codes are mainly used in detection tasks (ARQ). P(E)⩽ ∑ i=t+1 n (n i )⋅p i ⋅(1− p) n−i
- 24. 24 Binary Linear block codes ● Observe that both coding & decoding can be performed with low complexity hardware (combinational logic: gates). ● Examples of LBC – Repetition codes – Single parity check codes – Hamming codes – Cyclic redundancy codes – Reed-Muller codes – Golay codes – Product codes – Interleaved codes ● Some of them will be examined in the lab.
- 25. 25 Binary Linear block codes ● An example of performance: RHam =4/7, RGol =1/2.
- 26. 26 Multiple-error correction block codes
- 27. 27 Multiple-error-correction block codes ● There are a number of more powerful block codes, based on higher order Galois fields. – They use symbols over GF(r), with r>2. – Now the operations are defined as mod r. – An (n,k) linear block code with symbols from GF(r) is again a k- dimensional subspace of the vector space GF(r)n . ● They are used mainly for correction, and have applications in channels or systems where error bursts are frequent, i.e. – Storage systems (erasure channel). – Communication channels with deep fades. ● They are frequently used in concatenation with other codes that fail in short bursts when correcting (i.e. convolutional codes). ● We are going to study two important instances of such broad class of codes: Reed-Solomon, and BCH codes.
- 28. 28 Multiple-error-correction block codes ● An (n,k,t)q Reed-Solomon (R-S) code is defined as a mapping from GF(q)k ↔GF(q)n , with the following parameters: – Block length n=q-1 symbols, for input block length of k<n symbols. – Alphabet size q=pr , where p is a prime number. ● Minimum distance of the code dmin =n-k+1 (it achieves the Singleton bound for linear block codes). – It can correct up to t symbol errors, dmin =2t+1. ● This is a whole class of codes. – For any q=pr , n=q-1 and k=q-1-2t, there exists a Reed- Solomon code meeting all these criteria.
- 29. 29 Multiple-error-correction block codes ● The R-S code is built with the help of polynomial algebra. ● The message is mapped to a polynomial with given coefficients ● To get the corresponding codeword, the encoding function works evaluating the polynomial at n distinct given points Note that all the operations are performed over GF(q). s=(s1...sk ) ∈ GF (q)k p(a)=∑ i=1 k zi⋅a i−1 ; a, zi ∈ GF (q) c=(p(a1)...p(an)) ∈ GF(q) n
- 30. 30 Multiple-error-correction block codes ● We can rewrite the encoding in a more familiar form where A is the transpose of a Vandermonde matrix with structure c=z⋅A , z=(z1 ...zk ) A= ( 1 1 ⋯ 1 a1 a2 ⋯ an a1 2 a2 2 ⋯ an 2 ⋮ ⋮ ⋱ ⋮ a1 (k−1) a2 (k −1) ⋯ an (k −1) )
- 31. 31 Multiple-error-correction block codes ● The number of polynomials in GF(q) of degree less than k is clearly qk , exactly the possible number of messages. – This guarantees that each information word can be mapped to a unique codeword by choosing a convenient mapping s↔z. ● It is only required that a1 ,..,an are distinct points in GF(q), (these are the points where the polynomial p(a) is evaluated to build the codeword) – The points can be chosen to meet certain properties. – Either the polynomial can be chosen in a given way.
- 32. 32 Multiple-error-correction block codes ● One way to build the framework for R-S codes consists in choosing a1 ,..,an as n distinct points in GF(q) and build p(a) by forcing the condition ● The polynomial is characterized as the only polynomial of degree less than k that meets the above mentioned condition, and can be found by using known algebraic methods (Lagrange interpolation). ● The codeword is given as This is an instance of R-S systematic encoding. p(ai)=si ∀ i=1,...k cs=(s1...sk , p(ak+1)...p(an))
- 33. 33 Multiple-error-correction block codes ● In other possible construction, the polynomial is given by the mapping z=s ● And the points in GF(q) are chosen to meet certain convenient properties. – Let α be a primitive root of GF(q). This means that, for any b ∊ GF(q), except 0, there exists an integer u / αu =b (mod q). – ai =αi , i=1,...,q-1 (this spans GF(q), except 0). p(a)=∑ i=1 k si⋅ai−1 , a ∈ GF (q)
- 34. 34 Multiple-error-correction block codes ● Now we can rewrite where this time A is the transpose of a Vandermonde matrix with structure c=s⋅A , s=(s1...sk ) A= ( 1 1 ⋯ 1 α α2 ⋯ αn (α) 2 (α 2 ) 2 ⋯ (α n ) 2 ⋮ ⋮ ⋱ ⋮ (α)(k −1) (α2 )(k−1) ⋯ (αn )(k−1) )
- 35. 35 Multiple-error-correction block codes ● In this last case, it can be demonstrated that the parity- check matrix of the resulting R-S code is – It can correct t or fewer random symbol errors over a span of n=q-1 symbols. H= ( 1 α α 2 ⋯ α (q−2) 1 α2 (α2 )2 ⋯ (α2 )(q−2) 1 α3 (α3 )2 ⋯ (α3 )(q−2) ⋮ ⋮ ⋮ ⋱ ⋮ 1 α 2t (α 2t ) 2 ⋯ (α 2t ) (q−2) )
- 36. 36 Multiple-error-correction block codes ● The weight distribution (spectrum) of R-S codes has closed form ● We can derive bounds for the probability of undetected errors for a symmetric DMC with q-ary input/output alphabets, and probability of correct reception 1-ε. Ai= (q−1 i )q −2t {(q−1) i +∑ j=0 2t (−1) i+ j (i j)(q 2t −q i )} 2t+1⩽i⩽q−1 Pu (E)<q−2t before any correction step Pu (E,λ)<q −2t ∑ h=0 λ (q−1 h )(q−1) h if used first to correct λ errors 0⩽ϵ⩽ q q−1 0<λ<t
- 37. 37 Multiple-error-correction block codes ● Other kind of linear block code defined over higher-order Galois fields is the class of BCH codes. – Named after Raj Bose and D. K. Ray-Chaudhuri. ● An (n,k,t)q BCH code is again defined over GF(q), where q=pr , and p is prime. ● We have m, n, q, d=2t+1, l, so that – 2≤d≤n. l will be considered later. – gcd(n,q)=1 (“gcd” → greatest common divisor) – m is the multiplicative order of q modulo n; m is thus the smallest integer meeting qm =1 (mod n). – t is the number of errors that may be corrected.
- 38. 38 Multiple-error-correction block codes ● Let α be a primitive n-th root of 1 in GF(qm ). This means αn =1 (mod qm ). ● Let mi (x) the minimal polynomial for GF(q) of αi , ∀ i. – This is the monic polynomial of least degree having αi as a root. – Monic → the coefficient of the highest power of x is 1. ● Then, a BCH code is defined by a so-called generator polynomial where “lcm” stands for least common multiple. Its degree is at most (d-1)m=2mt. g(x)=lcm(ml(x)...ml+d−2(x))
- 39. 39 Multiple-error-correction block codes ● The encoding with a generator polynomial is done by building a polynomial containing the information symbols ● Then ● We may do systematic encoding as s (x)=∑ i=1 k si⋅x i−1 s=(s1...sk ) ∈ GF (q)k → c(x)=∑ i=1 n ci⋅x i−1 =s(x)⋅g (x), ci ∈ GF(q) → c=(c1 ...cn ) cs (x)=x n−k ⋅s(x)⏟ systematic symbols +x n−k ⋅s (x) mod g(x)⏟ redundancy symbols
- 40. 40 Multiple-error-correction block codes ● The case with l=1, and n=qm -1 is called a primitive BCH code. – The number of parity check symbols is n-k≤(d-1)m=2mt. – The minimum distance is dmin ≥d=2t+1. ● If m=1, then we have a Reed-Solomon code (of the “primitive root” kind)! – R-S codes can be seen as a subclass of BCH codes. – In this case, it can be verified that the R-S code may be defined by means of a generator polynomial, in the form g (x)=(x−α)(x−α 2 )...(x−α 2t )= =g1+g2 x+g3 x 2 +...+g2t−2 x 2t−1 +x 2t
- 41. 41 Multiple-error-correction block codes ● The parity check matrix for a primitive BCH code over GF(qm ) is – This code can correct t or fewer random symbol errors, d=2t+1, over a span of n=qm -1 symbol positions. H= ( 1 α α 2 ⋯ α (n−1) 1 α 2 (α 2 ) 2 ⋯ (α 2 ) (n−1) 1 α3 (α3 )2 ⋯ (α3 )(n−1) ⋮ ⋮ ⋮ ⋱ ⋮ 1 α (d−1) (α (d−1) ) 2 ⋯ (α (d−1) ) (n−1) )
- 42. 42 Multiple-error-correction block codes ● Why these codes can correct error bursts in the channel? BCH/R-S encoder BSC(p) BCH/R-S decoder Map bits to q-ary symbols Map q-ary symbols to bits Map q-ary symbols to bits Map bits to q-ary symbols s=(s1 ...sk ) r=(r1 ...rn ) b=(b1 ...bk·log2(q) ) b'=(b'1 ...b'k·log2(q) ) s'=(s'1 ...s'k ) c=(c1 ...cn ) cb rb =cb +eb p=Pb
- 43. 43 Multiple-error-correction block codes ● Why these codes can correct error bursts in the channel? BCH/R-S encoder BSC(p) BCH/R-S decoder Map bits to q-ary symbols Map q-ary symbols to bits Map q-ary symbols to bits Map bits to q-ary symbols s=(s1 ...sk ) r=(r1 ...rn ) b=(b1 ...bk·log2(q) ) b'=(b'1 ...b'k·log2(q) ) s'=(s'1 ...s'k ) c=(c1 ...cn ) Other channel encoder/decoder, modulator/demodulator, medium access technique cb rb =cb +eb p=Pb
- 44. 44 Multiple-error-correction block codes ● Why these codes can correct error bursts in the channel? BCH/R-S encoder BSC(p) BCH/R-S decoder Map bits to q-ary symbols Map q-ary symbols to bits Map q-ary symbols to bits Map bits to q-ary symbols eb =(...1111...) s=(s1 ...sk ) r=(r1 ...rn ) b=(b1 ...bk·log2(q) ) b'=(b'1 ...b'k·log2(q) ) s'=(s'1 ...s'k ) c=(c1 ...cn ) Other channel encoder/decoder, modulator/demodulator, medium access technique cb rb =cb +eb p=Pb
- 45. 45 Multiple-error-correction block codes ● Why these codes can correct error bursts in the channel? BCH/R-S encoder BSC(p) BCH/R-S decoder Map bits to q-ary symbols Map q-ary symbols to bits Map q-ary symbols to bits Map bits to q-ary symbols eb =(...1111...) s=(s1 ...sk ) r=(r1 ...rn ) b=(b1 ...bk·log2(q) ) b'=(b'1 ...b'k·log2(q) ) s'=(s'1 ...s'k ) c=(c1 ...cn ) Other channel encoder/decoder, modulator/demodulator, medium access technique If bit error burst falls within a single symbol ri in GF(q), or at most spans over t symbols in GF(q) within word r, it can be corrected! cb rb =cb +eb p=Pb
- 46. 46 Multiple-error-correction block codes ● Note that algebra is now far more involved and more complex than with binary LBC. – This is part of the price to pay to get better data integrity protection. – Other logical price to pay is the reduction in data rate, R. k and n are measured in symbols, but, as the mapping and demapping is performed from GF(2) to GF(q), and viceversa, the end-to-end effective data rate is again ● But there is still something to do to get the best from R-S and BCH codes: decoding is substantially more complex! Rb ' =Rb⋅ k n
- 47. 47 Multiple-error-correction block codes ● We are going to see a simple instance of decoding for these families of nonbinary LBC. ● We address the general BCH case, as R-S can be seen as an instance of the former. ● Correction can be performed by identifying the pairs ● For this, we can resort to the syndrome r (x)=c(x)+e(x) is the received codeword e(x)=ej1 x j1 +...+ejν x jν 0⩽ j1⩽...⩽jν⩽n−1 (x ji ,eji ) ∀ i=1,...,ν (S1... S2t), where Si=r (α i )=e(α i ) ∈ GF(q m )
- 48. 48 Multiple-error-correction block codes Sl=δ1β1 l +...δνβν l , i=l ,..2t δi=eji , βi=αji ● We can build a set of equations ● Based on this, a BCH or an R-S may be decoded on 4 steps: 1. Compute the syndrome vector. 2. Determine the so-called error-location polynomial. 3. Determine the so-called error-value evaluator. 4. Evaluate error-location numbers ( ) and error values ( ), and perform correction. ● The error-location polynomial is defined as ji eji σ (x)=(1−β1 x)...(1−βν x)=∑ l=0 ν σl x l , where σ0=1
- 49. 49 Multiple-error-correction block codes ● We can find the error-location polynomial with the help of the Berlekamp's algorithm, using the syndrome vector. – It works iteratively, in 2t steps. – Details can be found in the references. ● Once determined, its roots can be found by substituting the elements of GF(qm ) cyclically in σ(x). – If , is an error-location number. – The errors are thus located at such positions. ● On the other hand, the error-value evaluator is defined as σ (αi )=0 α −i =α q m −i−1 q m −i−1 Z0(x)=∑ l=1 ν δlβl ∏ i=1, i≠l ν (1−βi x)
- 50. 50 Multiple-error-correction block codes ● It can be shown that the error-value evaluator can be calculated as a function of known quantities ● After some algebra, the error values are determined as – Where the denominator is the derivative of σ(x). ● With the error values and the error locations, the error vector is estimated and correction may be performed Z0(x)=S1+(S2+σ1 S1)x+(S3+σ1 S2+σ2 S1)x2 + +...+(Sν+σ1 Sν−1+...+σν−1 S1)x ν δk = −Z0 (βk −1 ) σ ' (βk −1 ) ^e (x)=∑ i=1 ν δi x ji → ^c (x)=r(x)−^e(x)
- 51. 51 Multiple-error-correction block codes ● BCH and R-S codes may be very powerful, but the amount of algebra required is very high. – The supporting theory is very complex, and designing and analyzing these codes require mastering algebra and geometry over finite-size fields. – All the operations are to be understood in GF(q) or GF(qm ), when corresponding. ● Nonbinary LBC of the kind described are usually employed in sophisticated FEC strategies for specific channels. – Binary LBC are more usual in ARQ strategies.
- 52. 52 Multiple-error-correction block codes ● Example BPSK+AWGN: RHam =246/255, RGol =1/2, RR-S =155/255.
- 53. 53 Convolutional codes
- 54. 54 Convolutional codes ● A binary convolutional code (CC) is another kind of linear channel code class. ● The encoding can be described in terms of a finite state machine (FSM). – A CC can eventually produce sequences of infinite length. – A CC encoder has memory. General structure: MEMORY: ml bits for l-th input Forward logic (coded bits) Backward logic (feedback) k input streams n output streams not mandatory not mandatory Systematic output
- 55. 55 Convolutional codes ● The memory is organized as a shift register. – Number of positions for input l: memory ml . – ml =νl is the constraint length of the l-th input/register. – The register effects step by step delays on the input: recall discrete LTI systems theory. ● A CC encoder produces sequences, not just blocks of data. – Sequence-based properties vs. block-based properties. 1 2 3 4 ml l-th input stream input at instant i to backward logic to forward logic d i (l) d i−1 (l) d i−2 (l) d i−3 (l) d i−4 (l) d i−ml (l)
- 56. 56 Convolutional codes ● Both forward and backward logic is boolean logic. – Very easy: each operation adds up (XOR) a number of memory positions, from each of the k inputs. ● , is 1 when the p-th register position for the l- th input is added to get the j-th output. ci ( j) =∑ l=1 k ∑ q=i i−ml gl , q−i ( j) ⋅d q (l) j-th output at instant i inputs from all the k registers at instant i gl , p ( j) , p=0,... , ml Same structure for backward logic
- 57. 57 Convolutional codes ● Parameters of a CC so far: – k input streams – n output streams – k shift registers with length ml each, l=1,...,k – νl =ml is the constraint length of the l-th register – m=maxl {νl } is the memory order of the code – ν=ν1 +...+νk is the overall constraint length of the code ● A CC is denoted as (n,k,ν). – As usual, its rate is R=k/n, where k and n take normally small values.
- 58. 58 Convolutional codes ● The backward / forward logic may be specified in the form of generator sequences. – Theses sequences are the impulse responses of each output j wrt each input l. ● Observe that: – connects the l-th input directly to the j-th output – just delays the l-th input to the j-th output q time steps. gl ( j) =(gl , 0 ( j) ,... , gl ,ml ( j) ) gl ( j) =(1,0,... ,0) gl ( j) =(0,... ,1(qth ),... ,0)
- 59. 59 Convolutional codes ● Given the presence of the shift register, the generator sequences are better denoted as generator polynomials ● We can thus write, for example gl ( j) =(gl ,0 ( j) ,... , gl , ml ( j) )≡ gl ( j) (D)=∑ q=0 ml gl, q ( j) ⋅D q gl ( j) =(1,0,... ,0) ≡ gl ( j) (D)=1 gl ( j) =(0,... ,1(qth ),... ,0) ≡ gl ( j) (D)=Dq gl ( j) =(1,1,0,... ,0) ≡ gl ( j) (D)=1+D
- 60. 60 Convolutional codes ● As all operations involved are linear, a binary CC is linear and the sequences produced constitute CC codewords. ● A feedforward CC (without backward logic - feedback) can be denoted in matrix from as G(D)= ( g1 (1) (D) g1 (2) (D) ⋯ g1 (n) (D) g2 (1) (D) g2 (2) (D) ⋯ g2 (n) (D) ⋮ ⋮ ⋱ ⋮ gk (1) (D) gk (2) (D) ⋯ gk (n) (D) )
- 61. 61 Convolutional codes ● If each input has a feedback logic given as the code is denoted as gl (0) (D)=∑q=0 ml gl , q (0) ⋅Dq G(D)= ( g1 (1) (D) g1 (0) (D) g1 (2) (D) g1 (0) (D) ⋯ g0 (n) (D) g1 (0) (D) g2 (1) (D) g2 (0) (D) g2 (2) (D) g2 (0) (D) ⋯ g2 (n) (D) g2 (0) (D) ⋮ ⋮ ⋱ ⋮ gk (1) (D) gk (0) (D) gk (2) (D) gk (0) (D) ⋯ gk (n) (D) gk (0) (D) )
- 62. 62 Convolutional codes ● We can generalize the concept of parity-check matrix H(D). – An (n,k,ν) CC is fully specified by G(D) or H(D). ● Based on the matrix description, there are a good deal linear tools for design, analysis and evaluation of a given CC. ● A regular CC can be described as a (canonical) all-feedforward CC and through an equivalent feedback (recursive) CC. – Note that a recursive CC is related to an IIR filter. ● Even though k and n could be very small, a CC has a very rich algebraic structure. – This is closely related to the constraint length of the CC. – Each output bit is related to the present and past inputs via powerful algebraic methods.
- 63. 63 Convolutional codes ● Given G(D), a CC can be classified as: – Systematic and feedforward (NSC). – Systematic and recursive (RSC). – Non-systematic and feedforward. – Non-systematic and recursive. ● RSC is a popular class of CC, because it provides an infinite output for a finite-weight input (IIR behavior). ● Each NSC can be converted straightforwardly to a RSC with similar error correcting properties. ● CC encoders are easy to implement with standard hardware: shift registers + combinational logic.
- 64. 64 Convolutional codes ● As with the case of nonbinary LBC, we may use the polynomial representation to perform coding & decoding. – But now we have encoding with memory, spanning over theoretically infinite length sequences → not practical. c(D)=b(D )⋅G(D) b(D)=(b(1) (D)... b(k) (D)); b(i) (D )=∑ j=0 bj (i) Dj b j (i) is the i−th input bit stream c(D)=(c(1) (D)...c(n) (D)); c(l) (D)=∑ h=0 ch (l) Dh ch (l) is the l−th output bit stream
- 65. 65 Convolutional codes ● We do not need to look into the algebraic details of G(D) and H(D) to study: – Coding – Decoding – Error correcting capabilities ● A CC encoder is a FSM! The ν memory positions store a content (among 2ν possible ones) at instant i-1 Coder is said to be at state s(i-1) The ν memory positions store a new content at instant i Coder is said to be at state s(i) k input bits determine the shifting of the registers And we get n related output bits
- 66. 66 Convolutional codes ● The finite-state behavior of the CC can be captured by the concept of trellis. – For any starting state, we have 2k possible edges leading to a corresponding set of ending states. ss =s(i-1) s=1,...,2ν se =s(i) e=1,...,2ν input bi =(bi,1 ...bi,k ) output ci =(ci,1 ...ci,n )
- 67. 67 Convolutional codes ● The trellis illustrates the encoding process in 2 axis: – X-axis: time / Y-axis: states ● Example for a (2,1,3) CC: – For a finite-size input data sequence, a CC can be forced to finish at a known state (often 0) by adding terminating (dummy) bits. – Note that one section (e.g. i-1 → i) fully specifies the CC. output 00 output 01s2 s1 s3 s4 s6 s5 s7 s8 s2 s3 s4 s6 s5 s7 s8 s1 i-1 i i+1 input 0 input 1
- 68. 68 Convolutional codes ● The trellis illustrates the encoding process in 2 axis: – X-axis: time / Y-axis: states ● Example for a (2,1,3) CC: – For a finite-size input data sequence, a CC can be forced to finish at a known state (often 0) by adding terminating (dummy) bits. – Note that one section (e.g. i-1 → i) fully specifies the CC. output 00 output 01s2 s1 s3 s4 s6 s5 s7 s8 s2 s3 s4 s6 s5 s7 s8 s1 i-1 i i+1 input 0 input 1 Memory:Memory: same input,same input, different outputsdifferent outputs
- 69. 69 Convolutional codes ● The trellis description allows us – To build the encoder – To build the decoder – To get the properties of the code ● The encoder: H(D)↔G(D) Registers Combinational logick n CLK ss =s(i-1) s=1,...,2ν se =s(i) e=1,...,2νinput bi =(bi,1 ...bi,k ) output ci =(ci,1 ...ci,n )
- 70. 70 Convolutional codes ● The decoder is far more complicated – Long sequences – Memory: dependence with past states ● In fact, CC were already well known before there existed a practical good method to decode them: the Viterbi algorithm. – It is a Maximum Likelihood Sequence Estimation (MLSE) algorithm with many applications. ● Problem: for a length N>>n sequence at the receiver side – There are 2ν ·2N·k/n paths through the trellis to match with the received data. – Even if the coder starting state is known (often 0), there are still 2N·k/n paths to walk through in a brute force approach.
- 71. 71 Convolutional codes ● Viterbi algorithm setup. s2 s3 s4 s6 s5 s7 s8 s1 i-1 i input bi → output ci (s(i-1) ,bi ) start s(i-1) → end s(i) (s(i-1) ,bi ) received data ri Key facts: ● The encoding corresponds to a Markov chain model: P(s(i) )=P(s(i) |s(i-1) )·P(s(i-1) ). ● Total likelihood P(r|b) can be factorized as a product of probabilities. ● Given , P(ri |s(i) ,s(i-1) ) depends only on the channel kind (AWGN, BSC...). ● Transition from s(i-1) to s(i) (linked in the trellis) depends on the probability of bi : P(s(i) |s(i-1) )=2-k if the source is iid. ● P(s(i) |s(i-1) )=0 if they are not linked in the trellis (finite state machine: deterministic). s (i−1) → bi s (i)
- 72. 72 Convolutional codes ● The total likelihood can be recursively calculated as: ● In the BSC(p), the observation (branch) metric would be related to: ● Maximum likelihood (ML) criterion: P(ri|s (i) ,s (i−1) )=P(ri|ci)→w (ri+ci)=dH (ri ,ci) P(r∣b)=∏i=1 N /n P(ri∣s (i) ,s (i−1) )⋅P(s (i) ∣s (i−1) )⋅P(s (i−1) ) ̂b=arg{max b [P (r∣b)]}
- 73. 73 Convolutional codes ● We know that the brute force approach to ML criterion is at least O(2N·k/n ). ● The Viterbi algorithm works recursively from 1 to N/n on the basis that – Many paths can be pruned out (transition probability=0). – During forward recursion, we only keep the paths with highest probability: the path probability goes easily to 0 from the moment a term metric ⨯ transition probability is very small. – When recursion reaches i=N/n, the surviving path guarantees the ML criterion (optimal for ML sequence estimation!). ● The Viterbi algorithm complexity goes down to O(N·22ν ).
- 74. 74 Convolutional codes ● The algorithm recursive rule is ● {Vj (i) } stores the most probable state sequence wrt observation r V j (0) =P(s (0) =sj); V j (i) =P(ri∣s (i) =sj ,s (i−1) =smax )⋅ max s(i−1) =smax {P(s (i) =sj∣s (i−1) =sl)⋅V l (i−1) } s2 s1 s3 s4 s6 s5 s7 s8 s2 s3 s4 s6 s5 s7 s8 s1 i-1 i i+1 MAX MAX
- 75. 75 Convolutional codes ● The algorithm recursive rule is ● {Vj (i) } stores the most probable state sequence wrt observation r V j (0) =P(s (0) =sj); V j (i) =P(ri∣s (i) =sj ,s (i−1) =smax )⋅ max s(i−1) =smax {P(s (i) =sj∣s (i−1) =sl)⋅V l (i−1) } Probability of the most probable state sequence corresponding to the i-1 previous observations s2 s1 s3 s4 s6 s5 s7 s8 s2 s3 s4 s6 s5 s7 s8 s1 i-1 i i+1 MAX MAX
- 76. 76 Convolutional codes ● The algorithm recursive rule is ● {Vj (i) } stores the most probable state sequence wrt observation r V j (0) =P(s (0) =sj); V j (i) =P(ri∣s (i) =sj ,s (i−1) =smax )⋅ max s(i−1) =smax {P(s (i) =sj∣s (i−1) =sl)⋅V l (i−1) } Probability of the most probable state sequence corresponding to the i-1 previous observations s2 s1 s3 s4 s6 s5 s7 s8 s2 s3 s4 s6 s5 s7 s8 s1 i-1 i i+1 MAX MAX Note that we may better work with logs: products ↔ additions Criterion remains the same
- 77. 77 Convolutional codes ● Note that we have considered the algorithm when the demodulator yields hard outputs – ri is a vector of n estimated bits (BSC(p) equivalent channel). ● In AWGN, we can do better to decode a CC – We can provide soft (probabilistic) estimations for the observation metric. – For an iid source, we can easily get an observation transition metric based on the probability of each bi,l =0,1, l=1,...,k, associated to a possible transition. – There is a gain of around 2 dB in Eb /N0 . – LBC decoders can also accept soft inputs (non syndrome-based decoders). – We will examine an example of soft decoding of CC in the lab.
- 78. 78 Convolutional codes ● We are now familiar with the encoder and the decoder – Encoder: FSM (registers, combinational logic). – Decoder: Viterbi algorithm (for practical reasons, suboptimal adaptations are usually employed). ● But what about performance? ● First... – CC are mainly intended for FEC, not for ARQ schemes. – In a long sequence (=CC codeword), the probability of having at least one error is very high... – And... are we going to retransmit the whole sequence?
- 79. 79 Convolutional codes ● Given that we truncate the sequence to N bits and CC is linear – We may analyze the system as an equivalent (N,N·k/n) LBC. – But... equivalent matrices G and H would not be practical. ● Remember FSM: we can locate error loops in the trellis. i i+1 i+2 i+3 b b+e
- 80. 80 Convolutional codes ● The same error loop may occur irrespective of s(i-1) and b. b b+e i i+1 i+2 i+3 b b+e
- 81. 81 Convolutional codes ● Examining the minimal length loops and taking into account this uniform error property we can get dmin of a CC. – For a CC forced to end at 0 state for a finite input data sequence, dmin is called dfree . ● We can draw a lot of information by building an encoder state diagram: error loops, codeword weight spectrum... Diagram of a (2,1,3) CC, from Lin & Costello (2004).
- 82. 82 Convolutional codes ● With a fairly amount of algebra, related to FSM, modified encoder state diagrams and so on, it is possible to get an upper bound for optimal MLSE decoding. ● For the BSC(p), the bound can be calculated as Pb⩽∑ d Bd⋅erfc(√d R Eb N0 ) Bd is the total number of nonzero information bits associated with CC codewords of weight d, divided by the number of information bits k per unit time... A lot of algebra behind... BPSK in AWGN Pb⩽∑ d Bd⋅(2√ p⋅(1−p)) d
- 83. 83 Convolutional codes ● There are easier, suboptimal ways to decode a CC, and performance will vary accordingly. ● A CC may be punctured to match other rates lower than R=k/n, but the resulting equivalent CC is clearly weaker. – Performance-rate trade-off. – Puncturing is a very usual tool that provides flexibility to the usage of CC's in practice. Native convolutional encoderk n Puncturing algorithm (prune bits) n'<n
- 84. 84 Convolutional codes ● Examples with BPSK+AWGN using ML bounds.
- 85. 85 Turbo codes
- 86. 86 Turbo codes ● Canonically, turbo codes (TC) are parallel concatenated convolutional codes (PCCC). ● Coding concatenation has been known and employed for decades, but TC added a joint efficient decoding. – Example of concatenated coding with independent decoding is the use of ARQ + FEC hybrid strategies (CRC + CC). CC1 CC2 ? k input streams n=n1 +n2 output streams Rate R=k/(n1 +n2 ) b c=c1∪c2
- 87. 87 Turbo codes ● Canonically, turbo codes (TC) are parallel concatenated convolutional codes (PCCC). ● Coding concatenation has been known and employed for decades, but TC added a joint efficient decoding. – Example of concatenated coding with independent decoding is the use of ARQ + FEC hybrid strategies (CRC + CC). CC1 CC2 ? k input streams n=n1 +n2 output streams We will see this is a key element... Rate R=k/(n1 +n2 ) b c=c1∪c2
- 88. 88 Turbo codes ● We have seen that standard CC decoding with Viterbi algorithm relied on MLSE criterion. – This is optimal when binary data at CC input is iid. ● For CC, we also have decoders that provide probabilistic (soft) outputs. – They convert a priori soft values + channel output soft estimations into updated a posteriori soft values. – They are optimal from the maximum a posteriori (MAP) criterion point of view. – They are called soft input-soft output (SISO) decoders.
- 89. 89 Turbo codes ● What's in a SISO? SISO (for a CC) 0 1 0 1 P(bi=b)= 1 2 P(bi=b∣r ) r ● Note that the SISO works on a bit by bit basis, but produces a sequence of APP's. Probability density function of bi
- 90. 90 Turbo codes ● What's in a SISO? SISO (for a CC) 0 1 0 1 P(bi=b)= 1 2 P(bi=b∣r ) r Soft demodulated values from channel ● Note that the SISO works on a bit by bit basis, but produces a sequence of APP's. Probability density function of bi
- 91. 91 Turbo codes ● What's in a SISO? SISO (for a CC) 0 1 0 1 P(bi=b)= 1 2 P(bi=b∣r ) r Soft demodulated values from channel A priori probabilities (APR) ● Note that the SISO works on a bit by bit basis, but produces a sequence of APP's. Probability density function of bi
- 92. 92 Turbo codes ● What's in a SISO? SISO (for a CC) 0 1 0 1 P(bi=b)= 1 2 P(bi=b∣r ) r Soft demodulated values from channel A priori probabilities (APR) A posteriori probabilities (APP) updated with channel information ● Note that the SISO works on a bit by bit basis, but produces a sequence of APP's. Probability density function of bi
- 93. 93 Turbo codes ● The algorithm inside the SISO is some suboptimal version of the MAP BCJR algorithm. – BCJR computes the APP values through a forward-backward dynamics → it works over finite length data blocks, not over (potentially) infinite length sequences (like pure CCs). – BCJR works on a trellis: recall transition metrics, transition probabilities and so on. – Assume the block length is N: trellis starts at , ends at . αi ( j)=P(s (i) =sj ,r1,⋯,ri) βi ( j)=P(ri+1 ,⋯,rN∣s (i) =s j) γi ( j , k )=P(ri , s(i) =sj∣s(i−1) =sk ) s(0) s(N )
- 94. 94 Turbo codes ● The algorithm inside the SISO is some suboptimal version of the MAP BCJR algorithm. – BCJR computes the APP values through a forward-backward dynamics → it works over finite length data blocks, not over (potentially) infinite length sequences (like pure CCs). – BCJR works on a trellis: recall transition metrics, transition probabilities and so on. – Assume the block length is N: trellis starts at , ends at . αi ( j)=P(s (i) =sj ,r1,⋯,ri) βi ( j)=P(ri+1 ,⋯,rN∣s (i) =s j) γi ( j , k )=P(ri , s(i) =sj∣s(i−1) =sk ) FORWARD term s(0) s(N )
- 95. 95 Turbo codes ● The algorithm inside the SISO is some suboptimal version of the MAP BCJR algorithm. – BCJR computes the APP values through a forward-backward dynamics → it works over finite length data blocks, not over (potentially) infinite length sequences (like pure CCs). – BCJR works on a trellis: recall transition metrics, transition probabilities and so on. – Assume the block length is N: trellis starts at , ends at . αi ( j)=P(s (i) =sj ,r1,⋯,ri) βi ( j)=P(ri+1 ,⋯,rN∣s (i) =s j) γi ( j , k )=P(ri , s(i) =sj∣s(i−1) =sk ) FORWARD term BACKWARD term s(0) s(N )
- 96. 96 Turbo codes ● The algorithm inside the SISO is some suboptimal version of the MAP BCJR algorithm. – BCJR computes the APP values through a forward-backward dynamics → it works over finite length data blocks, not over (potentially) infinite length sequences (like pure CCs). – BCJR works on a trellis: recall transition metrics, transition probabilities and so on. – Assume the block length is N: trellis starts at , ends at . αi ( j)=P(s (i) =sj ,r1,⋯,ri) βi ( j)=P(ri+1 ,⋯,rN∣s (i) =s j) γi ( j , k )=P(ri , s(i) =sj∣s(i−1) =sk ) FORWARD term BACKWARD term TRANSITION s(0) s(N )
- 97. 97 Turbo codes ● The algorithm inside the SISO is some suboptimal version of the MAP BCJR algorithm. – BCJR computes the APP values through a forward-backward dynamics → it works over finite length data blocks, not over (potentially) infinite length sequences (like pure CCs). – BCJR works on a trellis: recall transition metrics, transition probabilities and so on. – Assume the block length is N: trellis starts at , ends at . αi ( j)=P(s (i) =sj ,r1,⋯,ri) βi ( j)=P(ri+1 ,⋯,rN∣s (i) =s j) γi ( j , k )=P(ri , s(i) =sj∣s(i−1) =sk ) FORWARD term BACKWARD term TRANSITION s(0) s(N ) Remember, n components for an (n,k,ν) CC
- 98. 98 Turbo codes ● BCJR algorithm in action: – Forward step i=1,...,N: – Backward step i=N-1,...,0: – Compute the joint probability sequence i=1,...,N: α0 ( j)=P(s (0) =s j); αi ( j)=∑ k=1 2 ν αi−1 (k )⋅γi (k , j) βN ( j)=P(s (N ) =s j); βi ( j)=∑ k=1 2 ν βi+1 (k )⋅γi+1 ( j ,k ) P(s (i−1) =sj ,s (i) =sk ,r)=βi (k )⋅γi ( j ,k )⋅αi−1 ( j)
- 99. 99 Turbo codes ● Finally, the APP's can be calculated as: ● Decision criterion based on these APP's: P(bi=b∣r )= 1 p(r ) ⋅ ∑ s (i−1) → bi =b s (i) P(s (i−1) =s j ,s (i) =sk ,r) log (P(bi=1∣r) P(bi=0∣r ))=log ( ∑ s(i−1) → bi =1 s(i) P(s (i−1) =sj ,s (i) =sk ,r) ∑ s(i−1) → bi =0 s(i) P(s (i−1) =sj ,s (i) =sk ,r))> ̂bi=1 0 <̂bi=0 0
- 100. 100 Turbo codes ● Finally, the APP's can be calculated as: ● Decision criterion based on these APP's: P(bi=b∣r )= 1 p(r ) ⋅ ∑ s (i−1) → bi =b s (i) P(s (i−1) =s j ,s (i) =sk ,r) log (P(bi=1∣r) P(bi=0∣r ))=log ( ∑ s(i−1) → bi =1 s(i) P(s (i−1) =sj ,s (i) =sk ,r) ∑ s(i−1) → bi =0 s(i) P(s (i−1) =sj ,s (i) =sk ,r))> ̂bi=1 0 <̂bi=0 0 Its modulus is the reliability of the decision
- 101. 101 Turbo codes ● How do we get γi (j,l)? ● This probability takes into account – The restrictions of the trellis (CC). – The estimations from the channel. γi ( j , l)=P(ri ,s (i) =s j∣s (i−1) =sl)= = p(ri∣s(i) =s j ,s(i−1) =sl)⋅P(s(i) =sj∣s(i−1) =sl)
- 102. 102 Turbo codes ● How do we get γi (j,l)? ● This probability takes into account – The restrictions of the trellis (CC). – The estimations from the channel. γi ( j , l)=P(ri ,s (i) =s j∣s (i−1) =sl)= = p(ri∣s(i) =s j ,s(i−1) =sl)⋅P(s(i) =sj∣s(i−1) =sl) =0 if transition is not possible =1/2k if transition is possible (binary trellis, k inputs)
- 103. 103 Turbo codes ● How do we get γi (j,l)? ● This probability takes into account – The restrictions of the trellis (CC). – The estimations from the channel. γi ( j , l)=P(ri ,s (i) =s j∣s (i−1) =sl)= = p(ri∣s(i) =s j ,s(i−1) =sl)⋅P(s(i) =sj∣s(i−1) =sl) =0 if transition is not possible =1/2k if transition is possible (binary trellis, k inputs)in AWGN for unmodulated ci,m 1 (2πσ 2 ) n/2 ⋅e −∑ m =1 n (ri ,m −ci ,m ) 2 2σ 2
- 104. 104 Turbo codes ● Idea: what about feeding APP values as APR values for other decoder whose coder had the same inputs? SISO (for CC2 ) 0 1 0 1 P(bi=b∣r1 ) P(bi=b∣r2) r2 From CC1 SISO
- 105. 105 Turbo codes ● Idea: what about feeding APP values as APR values for other decoder whose coder had the same inputs? SISO (for CC2 ) 0 1 0 1 P(bi=b∣r1 ) P(bi=b∣r2) r2 From CC1 SISO This will happen under some conditions
- 106. 106 Turbo codes ● APP's from first SISO used as APR's for second SISO increase updated APP's reliability iff – APR's are uncorrelated wrt channel estimations for second decoder. – This is achieved by permuting input data for each encoder. CC1 CC2 k input streams n=n1 +n2 output streams Rate R=k/(n1 +n2 ) b c=c1∪c2 Π d
- 107. 107 Turbo codes ● APP's from first SISO used as APR's for second SISO increase updated APP's reliability iff – APR's are uncorrelated wrt channel estimations for second decoder. – This is achieved by permuting input data for each encoder. CC1 CC2 k input streams n=n1 +n2 output streams INTERLEAVER (permutor) Rate R=k/(n1 +n2 ) b c=c1∪c2 Π d
- 108. 108 Turbo codes ● The interleaver preserves the data (b), but changes its position within the second stream (d). – Note that this compels the TC to work with blocks of N=size(Π) bits. – The decoder has to know the specific interleaver used at the encoder. b1 b2 b3 b4 bN d π (2) d π (N ) dπ (3) dπ (1) d π (4) d π (i)=bi
- 109. 109 Turbo codes ● The mentioned process is applied iteratively (l=1,...). – Iterative decoder → this may be a drawback, since it adds latency (delay). – Note the feedback connection: it is the same principle as in the turbo engines (that's why they are called “turbo”!). SISO 1 SISO 2 Π −1 Π r2 r1 APP1 (l) APR2 (l) APP2 (l) APR1 (l+1) from channel
- 110. 110 Turbo codes ● The mentioned process is applied iteratively (l=1,...). – Iterative decoder → this may be a drawback, since it adds latency (delay). – Note the feedback connection: it is the same principle as in the turbo engines (that's why they are called “turbo”!). SISO 1 SISO 2 Π −1 Π r2 r1 APP1 (l) APR2 (l) APP2 (l) APR1 (l+1) from channel
- 111. 111 Turbo codes ● The mentioned process is applied iteratively (l=1,...). – Iterative decoder → this may be a drawback, since it adds latency (delay). – Note the feedback connection: it is the same principle as in the turbo engines (that's why they are called “turbo”!). SISO 1 SISO 2 Π −1 Π r2 r1 APP1 (l) APR2 (l) APP2 (l) APR1 (l+1) from channel
- 112. 112 Turbo codes ● The mentioned process is applied iteratively (l=1,...). – Iterative decoder → this may be a drawback, since it adds latency (delay). – Note the feedback connection: it is the same principle as in the turbo engines (that's why they are called “turbo”!). SISO 1 SISO 2 Π −1 Π r2 r1 APP1 (l) APR2 (l) APP2 (l) APR1 (l+1) from channel
- 113. 113 Turbo codes ● The mentioned process is applied iteratively (l=1,...). – Iterative decoder → this may be a drawback, since it adds latency (delay). – Note the feedback connection: it is the same principle as in the turbo engines (that's why they are called “turbo”!). SISO 1 SISO 2 Π −1 Π r2 r1 APP1 (l) APR2 (l) APP2 (l) APR1 (l+1) from channel
- 114. 114 Turbo codes ● The mentioned process is applied iteratively (l=1,...). – Iterative decoder → this may be a drawback, since it adds latency (delay). – Note the feedback connection: it is the same principle as in the turbo engines (that's why they are called “turbo”!). SISO 1 SISO 2 Π −1 Π r2 r1 APP1 (l) APR2 (l) APP2 (l) APR1 (l+1) from channel Initial APR1 (l=0) is taken with P(bi =b)=1/2
- 115. 115 Turbo codes ● When the interleaver is adequately chosen and the CC's employed are RSC, the typical BER behavior is – Note the two distinct zones: waterfall region / error floor.
- 116. 116 Turbo codes ● The location of the waterfall region can be analyzed by the so- called density evolution method – Based on the exchange of mutual information between SISO's. ● The error floor can be lower bounded by the minimum Hamming distance of the TC – Contrary to CC's, TC relies on reducing multiplicities rather than just trying to increase minimum distance. Pbfloor > wmin⋅M min N ⋅erfc(√d min R Eb N 0 )
- 117. 117 Turbo codes ● The location of the waterfall region can be analyzed by the so- called density evolution method – Based on the exchange of mutual information between SISO's. ● The error floor can be lower bounded by the minimum Hamming distance of the TC – Contrary to CC's, TC relies on reducing multiplicities rather than just trying to increase minimum distance. Pbfloor > wmin⋅M min N ⋅erfc(√d min R Eb N 0 ) Hamming weight of the error with minimum distance
- 118. 118 Turbo codes ● The location of the waterfall region can be analyzed by the so- called density evolution method – Based on the exchange of mutual information between SISO's. ● The error floor can be lower bounded by the minimum Hamming distance of the TC – Contrary to CC's, TC relies on reducing multiplicities rather than just trying to increase minimum distance. Pbfloor > wmin⋅M min N ⋅erfc(√d min R Eb N 0 ) Hamming weight of the error with minimum distance Error multiplicity (low value!!)
- 119. 119 Turbo codes ● The location of the waterfall region can be analyzed by the so- called density evolution method – Based on the exchange of mutual information between SISO's. ● The error floor can be lower bounded by the minimum Hamming distance of the TC – Contrary to CC's, TC relies on reducing multiplicities rather than just trying to increase minimum distance. Pbfloor > wmin⋅M min N ⋅erfc(√d min R Eb N 0 ) Hamming weight of the error with minimum distance Error multiplicity (low value!!) Interleaver gain (only if recursive CC's!!)
- 120. 120 Turbo codes ● Examples of 3G TC. Note that TC's are intended for FEC...
- 121. 121 Low Density Parity Check codes
- 122. 122 Low Density Parity Check codes ● LDPC codes are just another kind of channel codes derived from less complex ones. – While TC's were initially an extension of CC systems, LDPC codes are an extension of the concept of binary LBC, but they are not exactly our known LBC. ● Formally, an LDPC code is an LBC whose parity check matrix is large and sparse. – Almost all matrix elements are 0!!!!!!!!!! – Very often, the LDPC parity check matrices are randomly generated, subject to some constraints on sparsity... – Recall that LBC relied on extreme powerful algebra related to carefully and well chosen matrix structures.
- 123. 123 Low Density Parity Check codes ● Formally, a (ρ,γ)-regular LDPC code is defined as the null space of a parity check matrix J⨯n H that meets these constraints: a) Each row contains ρ 1's. b) Each column contains γ 1's. c) λ, the number of 1's in common between any two columns, is 0 or 1. d) ρ and γ are small compared with n and J. ● These properties give name to this class of codes: their matrices have a low density of 1's. ● The density r of H is defined as r=ρ/n=γ/J.
- 124. 124 Low Density Parity Check codes H= [ 11110 000 00 00 000 00 00 0 0 000 111100 00 000 00 00 0 0 000 00 001111 000 00 00 0 0 000 00 00 000 011110 00 0 0 000 00 00 000 00 00 01111 10 0010 00 100 0100 00 00 0 01 000 100 010 000 00 100 0 0 0100 010 00 000 100 010 0 0 0010 00 00 0100 010 001 0 0 000 00 010 0010 00 100 01 10 000 100 00 0100 00 010 0 01 000 010 001 000 010 00 0 0 0100 00 100 0010 00 001 0 0 0010 00 010 000 100 100 0 0 000 100 001 000 010 00 01 ] ● Example of a (4,3)-regular LPDC parity check matrix
- 125. 125 Low Density Parity Check codes H= [ 11110 000 00 00 000 00 00 0 0 000 111100 00 000 00 00 0 0 000 00 001111 000 00 00 0 0 000 00 00 000 011110 00 0 0 000 00 00 000 00 00 01111 10 0010 00 100 0100 00 00 0 01 000 100 010 000 00 100 0 0 0100 010 00 000 100 010 0 0 0010 00 00 0100 010 001 0 0 000 00 010 0010 00 100 01 10 000 100 00 0100 00 010 0 01 000 010 001 000 010 00 0 0 0100 00 100 0010 00 001 0 0 0010 00 010 000 100 100 0 0 000 100 001 000 010 00 01 ] ● Example of a (4,3)-regular LPDC parity check matrix 15⨯20
- 126. 126 Low Density Parity Check codes H= [ 11110 000 00 00 000 00 00 0 0 000 111100 00 000 00 00 0 0 000 00 001111 000 00 00 0 0 000 00 00 000 011110 00 0 0 000 00 00 000 00 00 01111 10 0010 00 100 0100 00 00 0 01 000 100 010 000 00 100 0 0 0100 010 00 000 100 010 0 0 0010 00 00 0100 010 001 0 0 000 00 010 0010 00 100 01 10 000 100 00 0100 00 010 0 01 000 010 001 000 010 00 0 0 0100 00 100 0010 00 001 0 0 0010 00 010 000 100 100 0 0 000 100 001 000 010 00 01 ] ● Example of a (4,3)-regular LPDC parity check matrix This H defines a (20,7) LBC!!!15⨯20
- 127. 127 Low Density Parity Check codes H= [ 11110 000 00 00 000 00 00 0 0 000 111100 00 000 00 00 0 0 000 00 001111 000 00 00 0 0 000 00 00 000 011110 00 0 0 000 00 00 000 00 00 01111 10 0010 00 100 0100 00 00 0 01 000 100 010 000 00 100 0 0 0100 010 00 000 100 010 0 0 0010 00 00 0100 010 001 0 0 000 00 010 0010 00 100 01 10 000 100 00 0100 00 010 0 01 000 010 001 000 010 00 0 0 0100 00 100 0010 00 001 0 0 0010 00 010 000 100 100 0 0 000 100 001 000 010 00 01 ] ● Example of a (4,3)-regular LPDC parity check matrix This H defines a (20,7) LBC!!! r=4/20=3/15=0.2 15⨯20
- 128. 128 Low Density Parity Check codes H= [ 11110 000 00 00 000 00 00 0 0 000 111100 00 000 00 00 0 0 000 00 001111 000 00 00 0 0 000 00 00 000 011110 00 0 0 000 00 00 000 00 00 01111 10 0010 00 100 0100 00 00 0 01 000 100 010 000 00 100 0 0 0100 010 00 000 100 010 0 0 0010 00 00 0100 010 001 0 0 000 00 010 0010 00 100 01 10 000 100 00 0100 00 010 0 01 000 010 001 000 010 00 0 0 0100 00 100 0010 00 001 0 0 0010 00 010 000 100 100 0 0 000 100 001 000 010 00 01 ] ● Example of a (4,3)-regular LPDC parity check matrix This H defines a (20,7) LBC!!! r=4/20=3/15=0.2 Sparse! 15⨯20
- 129. 129 Low Density Parity Check codes H= [ 11110 000 00 00 000 00 00 0 0 000 111100 00 000 00 00 0 0 000 00 001111 000 00 00 0 0 000 00 00 000 011110 00 0 0 000 00 00 000 00 00 01111 10 0010 00 100 0100 00 00 0 01 000 100 010 000 00 100 0 0 0100 010 00 000 100 010 0 0 0010 00 00 0100 010 001 0 0 000 00 010 0010 00 100 01 10 000 100 00 0100 00 010 0 01 000 010 001 000 010 00 0 0 0100 00 100 0010 00 001 0 0 0010 00 010 000 100 100 0 0 000 100 001 000 010 00 01 ] ● Example of a (4,3)-regular LPDC parity check matrix This H defines a (20,7) LBC!!! r=4/20=3/15=0.2 Sparse! λ=0,1 15⨯20
- 130. 130 Low Density Parity Check codes ● Note that the J rows of H are not necessarily linearly independent over GF(2). – To determine the dimension k of the code, it is mandatory to find the row rank of H = n-k < J. – That's the reason why in the previous example H defined a (20,7) LBC instead of a (20,5) LBC as could be expected! ● The construction of large H for LDPC with high rates and good properties is a complex subject. – Some methods relay on smaller Hi used as building blocks, plus random permutations or combinatorial manipulations; resulting matrices with bad properties are discarded. – Other methods relay on finite geometries and lot of algebra.
- 131. 131 Low Density Parity Check codes ● LDPC codes yield performances equal or even better than TC's, but without the problem of their relatively high error floor. – Both LDPC codes and TC's are capacity approaching codes. ● As in the case of TC, their interest is in part related to the fact that – The encoding can be easily done (even when H or G are large, the low density of 1's reduces complexity of the encoder). – At the decoder side, there are powerful algorithms that can take full advantage of the properties of the LDPC code.
- 132. 132 Low Density Parity Check codes ● There are several algorithms to decode LDPC codes. – Hard decoding. – Soft decoding. – Mixed approaches. ● We are going to examine two important instances thereof: – Majority-logic (MLG) decoding; hard decoding, the simplest one (lowest complexity). – Sum-product algorithm (SPA); soft decoding, best error performance (but high complexity!). ● Key concepts: Tanner graphs & belief propagation.
- 133. 133 Low Density Parity Check codes ● MLG decoding: hard decoding; r=c+e → received word. – The simplest instance of MLG decoding is the decoding of a repetition code by the rule “choose 0 if 0's are dominant, 1 if otherwise”. ● Given a (ρ,γ)-regular LDPC code, for every bit position i=1,...,n, there is a set of γ rows that have a 1 in position i, and do not have any other common 1 position among them... Ai={h1 (i) ,⋯,hγ (i) }
- 134. 134 Low Density Parity Check codes ● We can form the set of syndrome equations ● Si gives a set of γ checksums orthogonal on ei . ● ei is decoded as 1 if the majority of the checksums give 1; 0 in the opposite case. ● Repeating this for all i, we estimate ê, and ĉ=r+ê. – Correct decoding of ei is guaranteed if there are less than γ/2 errors in e. Si={si=r⋅hj (i)T =e⋅hj (i)T , h j (i) ∈ Ai , i=1,⋯, γ}
- 135. 135 Low Density Parity Check codes ● Tanner graphs. Example for a (7,3) LBC. ● It is a bipartite graph with interesting properties for decoding. – A variable node is connected to a check node iff the corresponding code bit is checked by the corresponding parity sum equation. + + + ++++ c1 c2 c3 c4 c5 c6 c7 s1 s2 s3 s4 s5 s6 s7
- 136. 136 Low Density Parity Check codes ● Tanner graphs. Example for a (7,3) LBC. ● It is a bipartite graph with interesting properties for decoding. – A variable node is connected to a check node iff the corresponding code bit is checked by the corresponding parity sum equation. + + + ++++ c1 c2 c3 c4 c5 c6 c7 s1 s2 s3 s4 s5 s6 s7 Variable nodes or code-bit vertices
- 137. 137 Low Density Parity Check codes ● Tanner graphs. Example for a (7,3) LBC. ● It is a bipartite graph with interesting properties for decoding. – A variable node is connected to a check node iff the corresponding code bit is checked by the corresponding parity sum equation. + + + ++++ c1 c2 c3 c4 c5 c6 c7 s1 s2 s3 s4 s5 s6 s7 Variable nodes or code-bit vertices Check nodes or check-sum vertices
- 138. 138 Low Density Parity Check codes ● Tanner graphs. Example for a (7,3) LBC. ● It is a bipartite graph with interesting properties for decoding. – A variable node is connected to a check node iff the corresponding code bit is checked by the corresponding parity sum equation. + + + ++++ c1 c2 c3 c4 c5 c6 c7 s1 s2 s3 s4 s5 s6 s7 Variable nodes or code-bit vertices Check nodes or check-sum vertices The absence of short loops is necessary for iterative decoding
- 139. 139 Low Density Parity Check codes ● Based on the Tanner graph of an LDPC code, it is possible to make iterative soft decoding (SPA). ● SPA is performed by belief propagation (which is an instance of a message passing algorithm). + + + ++++ c1 c2 c3 c4 c5 c6 c7 s1 s2 s3 s4 s5 s6 s7
- 140. 140 Low Density Parity Check codes ● Based on the Tanner graph of an LDPC code, it is possible to make iterative soft decoding (SPA). ● SPA is performed by belief propagation (which is an instance of a message passing algorithm). + + + ++++ c1 c2 c3 c4 c5 c6 c7 s1 s2 s3 s4 s5 s6 s7 “Messages” (soft values) are passed to and from related variable and check nodes
- 141. 141 Low Density Parity Check codes ● Based on the Tanner graph of an LDPC code, it is possible to make iterative soft decoding (SPA). ● SPA is performed by belief propagation (which is an instance of a message passing algorithm). + + + ++++ c1 c2 c3 c4 c5 c6 c7 s1 s2 s3 s4 s5 s6 s7 “Messages” (soft values) are passed to and from related variable and check nodes This process, applied iteratively and under some rules, yields P(ci∣λ )
- 142. 142 Low Density Parity Check codes ● Based on the Tanner graph of an LDPC code, it is possible to make iterative soft decoding (SPA). ● SPA is performed by belief propagation (which is an instance of a message passing algorithm). + + + ++++ c1 c2 c3 c4 c5 c6 c7 s1 s2 s3 s4 s5 s6 s7 “Messages” (soft values) are passed to and from related variable and check nodes This process, applied iteratively and under some rules, yields P(ci∣λ ) λ soft values
- 143. 143 Low Density Parity Check codes ● Based on the Tanner graph of an LDPC code, it is possible to make iterative soft decoding (SPA). ● SPA is performed by belief propagation (which is an instance of a message passing algorithm). + + + ++++ c1 c2 c3 c4 c5 c6 c7 s1 s2 s3 s4 s5 s6 s7 “Messages” (soft values) are passed to and from related variable and check nodes This process, applied iteratively and under some rules, yields P(ci∣λ ) λ soft values
- 144. 144 Low Density Parity Check codes ● Based on the Tanner graph of an LDPC code, it is possible to make iterative soft decoding (SPA). ● SPA is performed by belief propagation (which is an instance of a message passing algorithm). + + + ++++ c1 c2 c3 c4 c5 c6 c7 s1 s2 s3 s4 s5 s6 s7 “Messages” (soft values) are passed to and from related variable and check nodes This process, applied iteratively and under some rules, yields P(ci∣λ ) N(c5 ): check nodes neighbors of variable node c5 λ soft values
- 145. 145 Low Density Parity Check codes ● Based on the Tanner graph of an LDPC code, it is possible to make iterative soft decoding (SPA). ● SPA is performed by belief propagation (which is an instance of a message passing algorithm). + + + ++++ c1 c2 c3 c4 c5 c6 c7 s1 s2 s3 s4 s5 s6 s7 “Messages” (soft values) are passed to and from related variable and check nodes This process, applied iteratively and under some rules, yields P(ci∣λ ) N(c5 ): check nodes neighbors of variable node c5 λ soft values
- 146. 146 Low Density Parity Check codes ● Based on the Tanner graph of an LDPC code, it is possible to make iterative soft decoding (SPA). ● SPA is performed by belief propagation (which is an instance of a message passing algorithm). + + + ++++ c1 c2 c3 c4 c5 c6 c7 s1 s2 s3 s4 s5 s6 s7 “Messages” (soft values) are passed to and from related variable and check nodes This process, applied iteratively and under some rules, yields P(ci∣λ ) N(c5 ): check nodes neighbors of variable node c5 N(s7 ) λ soft values
- 147. 147 Low Density Parity Check codes ● If we get P(ci | λ), we have an estimation of the codeword sent ĉ. ● The decoding aims at calculating this through the marginalization ● Brute-force approach for LDPC is impractical, hence the iterative solution through SPA. Messages interchanged at step l: P(ci∣λ)= ∑ c' :c'i=ci P(c '∣λ) μci→s j (l) (ci=c)=αi , j (l) ⋅P(ci=c∣λi)⋅ ∏ sk ∈N (ci ) sk ≠s j μsk →ci (l−1) (ci=c) μsj →ci (l) (ci=c)= ∑ c∖ck ∈N (s j) ck≠ci P(s j=0∣ci=c ,c)⋅ ∏ c' k∈N (sj) μck →sj (l) (c 'k=ck )
- 148. 148 Low Density Parity Check codes ● If we get P(ci | λ), we have an estimation of the codeword sent ĉ. ● The decoding aims at calculating this through the marginalization ● Brute-force approach for LDPC is impractical, hence the iterative solution through SPA. Messages interchanged at step l: P(ci∣λ)= ∑ c' :c'i=ci P(c '∣λ) μci→s j (l) (ci=c)=αi , j (l) ⋅P(ci=c∣λi)⋅ ∏ sk ∈N (ci ) sk ≠s j μsk →ci (l−1) (ci=c) μsj →ci (l) (ci=c)= ∑ c∖ck ∈N (s j) ck≠ci P(s j=0∣ci=c ,c)⋅ ∏ c' k∈N (sj) μck →sj (l) (c 'k=ck ) From variable node to check node
- 149. 149 Low Density Parity Check codes ● If we get P(ci | λ), we have an estimation of the codeword sent ĉ. ● The decoding aims at calculating this through the marginalization ● Brute-force approach for LDPC is impractical, hence the iterative solution through SPA. Messages interchanged at step l: P(ci∣λ)= ∑ c' :c'i=ci P(c '∣λ) μci→s j (l) (ci=c)=αi , j (l) ⋅P(ci=c∣λi)⋅ ∏ sk ∈N (ci ) sk ≠s j μsk →ci (l−1) (ci=c) μsj →ci (l) (ci=c)= ∑ c∖ck ∈N (s j) ck≠ci P(s j=0∣ci=c ,c)⋅ ∏ c' k∈N (sj) μck →sj (l) (c 'k=ck ) From variable node to check node From check node to variable node
- 150. 150 Low Density Parity Check codes ● Note that: – is a normalization constant. – plugs into the SPA the values from the channel → it is the APR info. ● : APP value. ● Based on the final values of P(ci | λ), a candidate ĉ is chosen and ĉ·HT is tested. If 0, the information word is decoded. αi , j (l) P(ci=c∣λi ) P (l) (ci=c∣λ)=βi (l) ⋅P(ci=c∣λi)⋅ ∏ sj∈N (ci) μsj→ ci (l) (ci=c)
- 151. 151 Low Density Parity Check codes ● Note that: – is a normalization constant. – plugs into the SPA the values from the channel → it is the APR info. ● : APP value. ● Based on the final values of P(ci | λ), a candidate ĉ is chosen and ĉ·HT is tested. If 0, the information word is decoded. αi , j (l) P(ci=c∣λi ) P (l) (ci=c∣λ)=βi (l) ⋅P(ci=c∣λi)⋅ ∏ sj∈N (ci) μsj→ ci (l) (ci=c) Normalization
- 152. 152 Low Density Parity Check codes ● LDPC BER performance examples (DVBS2 standard).
- 153. 153 Low Density Parity Check codes ● LDPC BER performance examples (DVBS2 standard). Short n=16200
- 154. 154 Low Density Parity Check codes ● LDPC BER performance examples (DVBS2 standard). Short n=16200 Long n=64800
- 155. 155 Coded modulations
- 156. 156 Coded modulations ● We have considered up to this point channel coding and decoding isolated from the modulation process. – Codewords feed any kind of modulator. – Symbols go through a channel (medium). – The info recovered from received modulated symbols is fed to the suitable channel decoder ● As hard decisions. ● As soft values (probabilistic estimations). – The abstractions of BSC(p) (hard demodulation) or soft values from AWGN ( ⋉ exp[-|ri -sj |2 /(2σ2 )] ) -and the like for other cases- are enough for such an approach. ● Note that there are other important channel kinds not considered so far.
- 157. 157 Coded modulations ● Coded modulations are systems where channel coding and modulation are treated as a whole. – Joint coding/modulation. – Joint decoding/demodulation. ● This offers potential advantages (recall the improvements made when the demodulator outputs more elaborated information -soft values vs. hard decisions). – We combine gains in BER with spectral efficiency! ● As a drawback, the systems become more complex. – More difficult to design and analyze.
- 158. 158 Coded modulations ● TCM (trellis coded modulation). – Normally, it combines a CC encoder and the modulation symbol mapper. output mk output m j s2 s1 s3 s4 s6 s5 s7 s8 s2 s3 s4 s6 s5 s7 s8 s1 i-1 i i+1
- 159. 159 Coded modulations ● If the modulation symbol mapper is well matched to the CC trellis, and the decoder is accordingly designed to take advantage of it, – TCM provides high spectral efficiency. – TCM can be robust in AWGN channels, and against fading and multipath effects. ● In the 80's, TCM become the standard for telephone line data modems. – No other system could provide better performance over the twisted pair cable before the introduction of DMT and ADSL. ● However, the flexibility of providing separated channel coding and modulation subsystems is still preferred nowadays. – Under the concept of Adaptive Coding & Modulation (ACM).
- 160. 160 Coded modulations ● Other possibility of coded modulation, evolved from TCM and from the concatenated coding & iterative decoding framework is Bit-Interleaved Coded Modulation (BICM). – What about if we use an interleaver between the channel coder (normally a CC) and the modulation symbol mapper? – A soft demodulator can also accept APR values and update as APP's its soft outputs in an iterative process! CC Π Soft demapper Channel corrupted outputs APR values (interleaved from CC SISO) APP values (to interleaver and CC SISO)
- 161. 161 Coded modulations ● As TCM, BICM has special good behavior (even better!) – In channels where spectral efficiency is required. – In dispersive channels (multipath, fading). – Iterative decoding yields a steep waterfall region. – Being a serial concatenated system, the error floor is very low (contrary to the parallel concatenated systems). ● BICM has already found applications in standards such as DVB-T2. ● The drawback is the higher latency and complexity of the decoding.
- 162. 162 Coded modulations ● Examples of BICM.
- 163. 163 References ● S. Lin, D. Costello, ERROR CONTROL CODING, Prentice Hall, 2004. ● S. B. Wicker, ERROR CONTROL SYSTEMS FOR DIGITAL COMMUNICATION AND STORAGE, Prentice Hall, 1995.

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