Convolution Coding

a

b c

d

Input sequence all ones will give encoded output as

Compare with all zero output sequence – Hamming distance is just 4

Catastrophic codes• Self loop with Input 1 but

output all zeros.• k – input bits not all

zeroes, but output all zeroes.

a

bc

d

a

b c

d

Decoding

• Symbol by symbol estimation - Max Aposteriori Probability (MAP) based Estimation. This is a Suboptimal procedure. The Probability of errors have been derived for various demodulation schemes based on distances in N- dimensional Euclidean space (Euclidean distances) for M - symbols.

• Maximum likelihood Sequence based Estimation procedure (MLSE). This is an optimal procedure. Received sequence consists of ‘mn’ bits. The ‘m’ branches, n code bits output per branch has 2^mn possible sequences. Need to select from a subset of 2^mk possible sequences for the Rate (k/n) code. Received sequence compared with this subset of sequence in the Trellis based on one of the two metrics

• Hamming distance (hard decision decoding)• Euclidean distance (Soft decision decoding)

Received sequence ‘a’ parsed over ‘m’ branches

‘b’ is a path thru the Trellis

g1 = [1 0 0] , g2 = [1 0 1] , g3 = [ 1 1 1]

Encoder, STD, TFD, TF, Dfree? Is it catastrophic?

g1 = [1 0 0] , g2 = [1 0 1] , g3 = [ 1 1 1]

Encoder, STD, TFD, TF, Dfree? Is it catastotrophic?

a

bc

d

Received sequence { 110110110010101100 }Trellis Path { 111 110 101 101 010 011 000 } and Info seq is { 11110 00 }

g1 = [1 0 1] , g2 = [1 1 1] , g3 = [ 1 1 1]

Encoder, STD, TFD, TF, Dfree? Is it catastrophic?

Path through the Trellis for 011011

Received CW sequence { 101001011110111} find the user sequence

Modulo 2 Addition of terms in product

Section 10.5 Haykin’s book, Example 10.5, pages 655 - 658

Time domain approach

Z (D)- Transform domain approach