![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](https://image.slidesharecdn.com/96ff00c0-d78e-48ac-a0f5-f7c83b410043-160614190256/85/Convolutional-Coding-3-320.jpg)
![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](https://image.slidesharecdn.com/96ff00c0-d78e-48ac-a0f5-f7c83b410043-160614190256/85/Convolutional-Coding-4-320.jpg)
![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:](https://image.slidesharecdn.com/96ff00c0-d78e-48ac-a0f5-f7c83b410043-160614190256/85/Convolutional-Coding-6-320.jpg)
![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](https://image.slidesharecdn.com/96ff00c0-d78e-48ac-a0f5-f7c83b410043-160614190256/85/Convolutional-Coding-7-320.jpg)
Convolutional Coding
- 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