This presentation was presented by Sachin Motwani (author) for his class presentation for Information Theory & Coding. The topic of discussion was General Convolutional Encoder using block diagram.

1. General
Convolutional
Encoder
(using block diagram)
SACHIN MOTWANI
42276802817|ECE-3
SACHIN MOTWANI | SACHINMOTWANI20@IEEE.ORG | 42276802817| GURU TEGH BAHADUR INSTITUTE OF TECHNOLOGY

2. Contents
Sachin Motwani | sachinmotwani20@ieee.org | 42276802817| Guru Tegh Bahadur Institute of Technology
Slide 3
Convolutional v/s Block Code
Slide 5 & 6
Introduction & Explanation
Slide 7
Parity Equations
Slide 8
Example
Slide 9
Crux
Slide 10
Conclusion

3. Linear Block
Code
Convolutional
Code
k n-k
Information
digits
Parity check
digits
(Redundancy)
k: number of bits in one data element
n: length of codeword
One block of codeword
Sachin Motwani | sachinmotwani20@ieee.org | 42276802817| Guru Tegh Bahadur Institute of Technology

4. Two views of the Convolutional Encoder
Block Diagram View of Convolutional Coding with
Shift Registers
State Machine View of Convolutional Coding
Sachin Motwani | sachinmotwani20@ieee.org | 42276802817| Guru Tegh Bahadur Institute of Technology

5. Introduction
❑ Convolutional Encoder: Any Constant Linear Sequential Circuits
✓Previous inputs are stored in shift registers in the encoder and affect future encodings.
✓Parity bits are computed from message bits
❑ Convolutional Codes: Set of all output sequences resulting from any
set of input sequences beginning at any time.
𝑋 𝑛 𝑋 𝑛−1 𝑋 𝑛−2
Combinational Logic
𝐼𝑛𝑝𝑢𝑡
data
𝐸𝑛𝑐𝑜𝑑𝑒𝑑
𝑜𝑢𝑡𝑝𝑢𝑡
Sachin Motwani | sachinmotwani20@ieee.org | 42276802817| Guru Tegh Bahadur Institute of Technology
✓Bit-by-bit Encoding
✓Simpler Implementation
✓Only parity bit; no
message bit is sent

6. • Combinational logic: modulo-2 operation [a.k.a XOR operation]
• Uses a sliding window to calculate r (number of parity bits per window)^
• Constraint Length (K): Size of the window in bits.
▪ Longer the constraint length, larger the number of parity bits that are influenced by any given
message bit.
• Trade-off:
• In practice, we use r & constraint length as small as possible while providing low enough resulting bit error probability.
• Bit Rate:
𝑁𝑜 𝑜𝑓 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 𝑠𝑙𝑖𝑑𝑒𝑠 𝑜𝑓 𝑤𝑖𝑛𝑑𝑜𝑤 𝑎𝑡 𝑎 𝑡𝑖𝑚𝑒
𝑁𝑜 𝑜𝑓 𝑝𝑎𝑟𝑖𝑡𝑦 𝑏𝑖𝑡𝑠 𝑝𝑒𝑟 𝑤𝑖𝑛𝑑𝑜𝑤
=
1
𝑟
Explanation
Sachin Motwani | sachinmotwani20@ieee.org | 42276802817| Guru Tegh Bahadur Institute of Technology
𝐿𝑎𝑟𝑔𝑒𝑟 𝐶𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡 𝐿𝑒𝑛𝑔𝑡ℎ = 𝑅𝑒𝑑𝑢𝑐𝑒𝑑 𝐵𝑖𝑡 𝐸𝑟𝑟𝑜𝑟
𝐿𝑎𝑟𝑔𝑒𝑟 𝐶𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡 𝐿𝑒𝑛𝑔𝑡ℎ = 𝐷𝑒𝑙𝑎𝑦𝑒𝑑 𝐷𝑒𝑐𝑜𝑑𝑖𝑛𝑔
^ r must always be >1

7. • The equations that govern the way in which parity bits are produced from the sequence of message bits, x.
• In general, the parity equations could be viewed as being produced by composing the message bits, x, & a
generator polynomial, g. The generator polynomials are sets of coefficients of these individual elements
( 𝑥 𝑛 , 𝑥 𝑛 − 1 , x[n-2]…) in each polynomial equation. [In above equations being, (1,1,1) & (1,1,0)].
• We denote 𝑔𝑖, as the K-element generator polynomial for parity bit 𝑝𝑖.
Therefore,
Here, the equation represents convolution of g & x ; hence the term “Convolutional Code”.
Parity Equations
Sachin Motwani | sachinmotwani20@ieee.org | 42276802817| Guru Tegh Bahadur Institute of Technology
# Number of generator
polynomials = number of
generated parity bits, r, in
each sliding window.
𝑝0 𝑛 = 𝑥 𝑛 ⨁𝑥[𝑛 − 1]⨁x[n-2]
𝑝1 𝑛 = 𝑥[𝑛]⨁x[n-1]

8. Convolutional code with two parity bits per message bit (r=2) and constraint
length (shown in the rectangular orange window) K=3.
An Example
0101100101100011…
𝑝0 𝑛 = 𝑥 𝑛 ⨁𝑥[𝑛 − 1]⨁x[n-2]
𝑝1 𝑛 = 𝑥[𝑛]⨁x[n-1]
Sachin Motwani | sachinmotwani20@ieee.org | 42276802817| Guru Tegh Bahadur Institute of Technology

9. Crux of the Encoding Process
Sachin Motwani | sachinmotwani20@ieee.org | 42276802817| Guru Tegh Bahadur Institute of Technology
Encoder looks at K-bits
at a time
Produces r parity bits
according to carefully
chosen functions
Functions operate
over various subsets of
K- bits.
Encoder splits out r
bits
The r bits are sent out
sequentially
Encoder slides the
window by 1

10. Conclusion
Sachin Motwani | sachinmotwani20@ieee.org | 42276802817| Guru Tegh Bahadur Institute of Technology
✓ Difference between Linear Block Code & Convolutional Code
✓ Two types of views of Convolutional Encoder
✓ Discussed Block Diagram view in detail
✓ Parity Equation
✓ Why Convolutional Codes are called so?
✓ Example
Thanks for being a patient audience.