1. CONVOLUTIONAL CODING
Ashish Kumar Meshram: mt1402102002
Devendra Magraiya: mt1402102004
Mohit Singh: mt1402102008
M.Tech. Communication & Signal Processing
Discipline of Electrical EngineeringIIT – Indore | EE646 | Information & Coding Theory

2. 01IIT – Indore | EE646 | Information & Coding Theory
Contents
References5
1 Introduction
Convolutional Encoder2
3 Viterbi Decoder
4 Implementation Issues

3. Introduction
02IIT – Indore | EE646 | Information & Coding Theory
A convolutional code is specified by three parameters: (𝑛, 𝑘, 𝑣)
the codeword length𝑛 →
𝑘 →
𝑣 →
the message length
the constraint length
The generators for this code are more conveniently
given in octal form
𝐺 =
1 0 1
1 1 1
← 𝑔1
← 𝑔2
↔ 𝐺 = [ 5, 7]
Generator Matrix
𝑛 = 2, 𝑘 = 1, 𝑣 = 3
𝐷 𝐷
⊕
⊕
𝑚 𝑘
𝑚 𝑘−1 𝑚 𝑘−2
𝑐 𝑘
(1)
= 𝑚 𝑘 ⊕ 𝑚 𝑘−2
𝑐 𝑘
(2)
= 𝑚 𝑘 ⊕ 𝑚 𝑘−1 ⊕ 𝑚 𝑘−2
Code Rate, 𝑅 𝑐 =
𝑘
𝑛
= 1/2

4. Convolution Encoder
03IIT – Indore | EE646 | Information & Coding Theory
𝐷 𝐷
⊕
⊕
𝑚 𝑘
𝑚 𝑘−1 𝑚 𝑘−2
𝑐 𝑘
(1)
= 𝑚 𝑘 ⊕ 𝑚 𝑘−2
𝑐 𝑘
(2)
= 𝑚 𝑘 ⊕ 𝑚 𝑘−1 ⊕ 𝑚 𝑘−2
𝑐(1)
= [1 1 1 1 1 0 0 0 1]
𝑐(2)
= [1 0 0 1 1 1 0 1 1 ]
𝑚 = [1 1 0 0 1 0 1]
𝐼𝑛𝑝𝑢𝑡 𝐵𝑖𝑡𝑠:
𝑐 = [11 10 10 11 11 01 00 01 11]
𝐶𝑜𝑑𝑒 𝑊𝑜𝑟𝑑𝑠:
𝐼𝑛𝑡𝑒𝑟𝑙𝑒𝑎𝑣𝑒𝑑 𝐵𝑖𝑡𝑠

5. State Diagram & Trellis
04IIT – Indore | EE646 | Information & Coding Theory

6. Viterbi Decoder
05IIT – Indore | EE646 | Information & Coding Theory
𝑆𝑜𝑢𝑟𝑐𝑒 𝐸𝑛𝑐𝑜𝑑𝑒𝑟
𝐶ℎ𝑎𝑛𝑛𝑒𝑙
𝐷𝑒𝑐𝑜𝑑𝑒𝑟 𝐷𝑒𝑚𝑜𝑑𝑢𝑙𝑎𝑡𝑖𝑜𝑛𝐷𝑒𝑠𝑡𝑖𝑛𝑎𝑡𝑖𝑜𝑛
𝑀𝑜𝑑𝑢𝑙𝑎𝑡𝑖𝑜𝑛
𝑟 = [11 11 01 00 10 11]
𝑅𝑒𝑐𝑒𝑖𝑣𝑒𝑑 𝐵𝑖𝑡𝑠:
𝑚 = [1 1 0 1 0 0]
𝐼𝑛𝑝𝑢𝑡 𝐵𝑖𝑡𝑠:
𝑚 = [1 1 0 1 0 0]
𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑 𝐵𝑖𝑡𝑠:
𝑐 = [11 01 01 00 10 11]
𝐶𝑜𝑑𝑒 𝑊𝑜𝑟𝑑𝑠:

7. References
06IIT – Indore | EE646 | Information & Coding Theory
[1]. Daniel J Costello, Error Control Coding, Shu Lin, 2e
[2]. Tood K Moon, Error Correction Coding Mathematical Methods and Algorithms

8. THANKS Ashish Meshram, Devendra Magraiya, Mohit Singh