convolution cyclic code, Slides of Digital Communication Systems

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Convolutional Codes

Dr. A. W. Umrani

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

State Diagram Representation

• (^) The output depends on the current input and the state of the encoder ( i. e. the contents of the shift register).

Trellis Diagram Representation

• (^) Expansion of state diagram in time.

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.

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.

Example:Example: For the convolutional code example in

the previous lecture, starting from state zero, Decode the following received sequence. Add the weight of the path at each state Compute the two possible paths at each state and select the one with less cumulative Hamming weight

This is called the survival path At the end of the trellis, select the path with the minimum cumulative Hamming weight This is the survival path in this example Decoded Decoded sequence issequence is m=[10 1110]m=[10 1110]

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.

Distance Properties: Illustration

• (^) sequence 2: Hamming weight = 5, dinf = 1
• sequence 3: Hamming weight = 7, dinf = 3.

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: a 0 and a 1.
• (^) 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).

Properties of the Path

Sequence 2: code sequence: .. 00 11 10 11 00 .. state sequence: a 0 b c a 1 Labeled: (D^2 LN)(DL)(D^2 L) = D^5 L^3 N 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: a 0 b d c b c a 1 Labeled: (D^2 LN)(DLN)(DL)(DL)(LN)(D^2 L) = D^7 L^6 N^3 Prop. : w =7, dinf =3, diverges from the allzero path by 6 branches.

Transfer Function

• (^) Input-Output relations: a 0 = 1 b = D^2 LN a 0 + LN c c = DL b + DLN d d = DLN b + DLN d a 1 = D^2 L c
• (^) The transfer function T (D,L,N) = a 1 / a 0

T (D, L, N)

D L

DNL(1 L)

5 3