Houjun Liu

SU-ENGR76 MAY282024

Convolutional Code

The output sequences of Convolutional Code behaves like a five bit difference in Hamming Distance.


  • brute force decoding: precompute, for a sequence length of \(k\), compute \(2^{k}\) sequneces and what they should correspond to in our target code — of course, this is not computationally feasible
  • Virtirbi Algorithm: time complexity — \(4(k+2)\), \(8(k+2)\) 4 possible blocks of 2, and 8 comparisons for Hamming Distance: in general \(k_0\)

in general

  • \(k_0\) of source symbols entering the decoder
  • \(n_0\) of symbols produced by decoder at each step
  • constrain length \(m_0\), how many bits are considered

for the Convolutional Code setup we discussed, we have \(k_0=1\), \(n_0=2\), and \(m_0 = 3\) (one bit produces 2 bits, and we consider 3 bits per step.)

Comparison of Codes

repetition code1/3233
Hamming Code4/71673
Convolutional Code1/2≈ 5

Bit Error Rate

KEY GOAL: if I hand you an error rate of the uncoded transmission, what would be the bit error rate of a given resulting error-correction code?