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Convolutional Codes

Convolutional Codes. Basic Definitions. k =1, n = 2 , (2,1) Rate-1/2 convolutional code Two-stage register ( M=2 ) Each input bit influences the output for 3 intervals (K=3) K = constraint length of the code = M + 1. Generator Polynomial.

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Convolutional Codes

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  1. Convolutional Codes Convolutional Codes1

  2. Basic Definitions • k =1, n = 2 , (2,1) Rate-1/2 convolutional code • Two-stage register ( M=2 ) • Each input bit influences the output for 3 intervals (K=3) • K = constraint length of the code = M + 1 Convolutional Codes2

  3. Generator Polynomial • A convolutional code may be defined by a set of n generating polynomials for each input bit. • For the circuit under consideration: g1(D) = 1 + D + D2 g2(D) = 1 + D2 • The set {gi(D)} defines the code completely. The length of the shift register is equal to the highest-degree generator polynomial. Convolutional Codes3

  4. State Diagram Representation • The output depends on the current input and the state of the encoder ( i. e. the contents of the shift register). Convolutional Codes4

  5. Trellis Diagram Representation • Expansion of state diagram in time. Convolutional Codes5

  6. Decoding • A message m is encoded into the code sequence c. • Each code sequence represents a path in the trellis diagram. • Minimum Distance Decoding • Upon receiving the received sequence r, search for the path that is closest ( in Hamming distance) to r . Convolutional Codes6

  7. The Viterbi Algorithm • Walk through the trellis and compute the Hamming distance between that branch of r and those in the trellis. • At each level, consider the two paths entering the same node and are identical from this node onwards. From these two paths, the one that is closer to r at this stage will still be so at any time in the future. This path is retained, and the other path is discarded. • Proceeding this way, at each stage one path will be saved for each node. These paths are called the survivors. The decoded sequence (based on MDD) is guaranteed to be one of these survivors. Convolutional Codes7

  8. The Viterbi Algorithm (cont’d) • Each survivor is associated with a metric of the accumulated Hamming distance (the Hamming distance up to this stage). • Carry out this process until the received sequence is considered completely. Choose the survivor with the smallest metric. Convolutional Codes8

  9. 6.3 The Viterbi Algorithm: • The viterbi algorithm is used to decode convolutional codes and any structure or system that can be described by a trellis. • It is a maximum likelihood decoding algorithm that selects the most probable path that maximizes the likelihood function. • The algorithm is based on add-compare-select the best path each time at each state.

  10. Example:For the convolutional code example in the previous lecture, starting from state zero, Decode the following received sequence. At the end of the trellis, select the path with the minimum cumulative Hamming weight This is the survival path in this example Decoded sequence is m=[10 1110] Compute the two possible paths at each state and select the one with less cumulative Hamming weight Add the weight of the path at each state This is called the survival path

  11. Distance Properties of Conv. Codes • Def: The free distance, dfree, is the minimum Hamming distance between any two code sequences. • Criteria for good convolutional codes: • Large free distance, dfree. • Small Hamming distance (i.e. as few differences as possible ) between the input information sequences that produce the minimally separated code sequences. dinf • There is no known constructive way of designing a conv. code of given distance properties. However, a given code can be analyzed to find its distance properties. Convolutional Codes11

  12. Distance Prop. of Conv. Codes (cont’d) • Convolutional codes are linear. Therefore, the Hamming distance between any pair of code sequences corresponds to the Hamming distance between the all-zero code sequence and some nonzero code sequence. Thus for a study of the distance properties it is possible to focus on the Hamming distance between the all-zero code sequence and all nonzero code sequences. • The nonzero sequence of minimum Hamming weight diverges from the all-zero path at some point and remerges with the all-zero path at some later point. Convolutional Codes12

  13. Distance Properties: Illustration • sequence 2: Hamming weight = 5, dinf = 1 • sequence 3: Hamming weight = 7, dinf = 3. Convolutional Codes13

  14. Modified State Diagram • The span of interest to us of a nonzero path starts from the 00 state and ends when the path first returns to the 00 state. Split the 00 state (state a) to two states: a0 and a1. • The branches are labeled with the dummy variables D, L and N, where: The power of D is the Hamming weight (# of 1’s) of the output corresponding to that branch. The power of N is the Hamming weight (# of 1’s) of the information bit(s) corresponding to that branch. The power of L is the length of the branch (always = 1). Convolutional Codes14

  15. Modified State Diagram (cont’d) Convolutional Codes15

  16. Properties of the Path Sequence 2: code sequence: .. 00 11 10 11 00 .. state sequence: a0bca1 Labeled: (D2LN)(DL)(D2L) = D5L3N Prop. : w =5, dinf =1, diverges from the allzero path by 3 branches. Sequence 3: code sequence: .. 00 11 01 01 00 10 11 00 .. state sequence: a0bdcbca1 Labeled: (D2LN)(DLN)(DL)(DL)(LN)(D2L) = D7L6N3 Prop. : w =7, dinf =3, diverges from the allzero path by 6 branches. Convolutional Codes16

  17. Transfer Function • Input-Output relations: a0 = 1 b = D2LN a0 + LNc c = DLb + DLNd d = DLNb + DLNd a1 = D2Lc • The transfer functionT(D,L,N) = a1 /a0 Convolutional Codes17

  18. Transfer Function (cont’d) • Performing long division: T = D5L3N + D6L4N2 + D6L5N2 + D7L5N3 + …. • If interested in the Hamming distance property of the code only, set N = 1 and L = 1 to get the distance transfer function: T (D) = D5 + 2D6 + 4D7 There is one code sequence of weight 5. Therefore dfree=5. There are two code sequences of weight 6, four code sequences of weight 7, …. Convolutional Codes18

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