Introduction to Convolutional Codes Convolutional Encoder Structure Convolutional Encoder Representation(Vector, Polynomial, State Diagram and Trellis Representations ) Maximum Likelihood Decoder Viterbi Algorithm MATLAB Simulation Hard and Soft Decisions Bit Error Rate Tradeoff Consumed Time Tradeoff

1. MEH607 Error Control Coding
KOCAELI UNIVERSITY
Graduate School of
Natural and Applied Sciences
Prepared By: Mohammed ABUIBAID
Email: m.a.abuibaid@gmail.com
Submitted to: Dr. SITKI ÖZTÜRK
Electronic and Communication Engineering
Convolutional Error Control Coding
AcademicYear
2015/2016

2. Agenda
1. Introduction to Convolutional Codes
2. Convolutional Encoder Structure
3. Convolutional Encoder Representation
(Vector, Polynomial, State Diagram and Trellis Representations )
4. Maximum Likelihood Decoder
5. Viterbi Algorithm
6. MATLAB Simulation
 Hard and Soft Decisions
 Bit Error Rate Tradeoff
 Consumed Time Tradeoff

3. Error Control Techniques
(ARQ)
Automatic Repeat reQuest
 The receiver sends a feedback to the
transmitter:
 Error is detected (NACK: Not-
Acknowledgement) in the received
packet, then retransmit that data block
 if no errors detected (ACK:
Acknowledgement), don’t resend.
 Uses extra/redundant bits merely for
error detection.
 Full-duplex (two-way) connection
between the Transmitter and the
Receiver.
 Result: Constant reliability, but
varying data rate throughput due to
retransmit.
(FEC)
Forward Error Correction
 The transmitter’s encoder adds
extra/redundant bits to a block of
message data bits to form a Codeword
 The receiver can both detect errors
and automatically correct errors
incurred during transmission, without
retransmission of the data.
 Simplex (one-way) connection
between the Transmitter and the
Receiver.
 Result: Varying reliability, but
constant data rate throughput.
(ARQ+FEC)
Hybrid ARQ
 Full-duplex connection required between
the Transmitter and the Receiver.
 Uses error detection and correction
codes.
In general:
 Wire-line communications (more
reliable) adopts ARQ scheme
 Wireless communications (relatively
less reliable) adopts FEC scheme

4. Classification of FEC Codes

5. FEC Coding Techniques
Block Code
(No Memory)
 It collects k bits in a buffer prior to
processing
 There is no retention within the
encoding system of information
related to the previous samples
points memoryless
 Each output Codeword of an (n, k)
block code depends only on the
current buffer
Convolutional Code
(Memory)
 Why Named Convolutional? Each bit in the output
stream is not only dependent on the current bit,
but also on those processed previously.
 The encoder acts on the serial bit stream as it
enters the transmitter.
 The number of sample points collected prior to
processing is far less than required for a block
code. (delay through the encoder is less)
 Its performance is less sensitive to Signal-to-
Noise Ratio variations than that of block codes.
(preferred in situations of limited power)

6. Convolutional Codes
Convolutional codes offer an approach to error control coding substantially different
from that of block codes.
A Convolutional Encoder:
 encodes the entire data stream, into a single codeword
 does not need to segment the data stream into blocks of fixed size
 is a machine with memory
 is specified by three parameters 𝑛, 𝑘, 𝐾 𝑜𝑟 (𝑘 / 𝑛, 𝐾)
 𝑅 𝑐 = 𝑘 𝑛 is the coding rate, determining the number of data bits per coded bit.
− In practice, usually k=1 is chosen.
 K is the constraint length of the encoder and the encoder has K-1 memory elements.

7. Agenda
1. Introduction to Convolutional Codes
2. Convolutional Encoder Structure
3. Convolutional Encoder Representation
(Vector, Polynomial, State Diagram and Trellis Representations )
4. Maximum Likelihood Decoder
5. Viterbi Algorithm
6. MATLAB Simulation
 Hard and Soft Decisions
 Bit Error Rate Tradeoff
 Consumed Time Tradeoff

8. Convolutional Encoder Structure (rate ½, K=3)
3 Shift-registers where:
− The first one takes the incoming data bit
− The rest, form the memory of the encoder

9. Convolutional Encoder (rate ½, K=3)
1 0 0𝑡1
𝑢1 𝑢2
1 1
0 1 0𝑡4
𝑢1 𝑢2
1 0
0 1 0𝑡2
𝑢1 𝑢2
1 0
0 0 1𝑡5
𝑢1 𝑢2
1 1
1 0 1𝑡3
𝑢1 𝑢2
0 0
0 0 0𝑡6
𝑢1 𝑢2
0 0
Message sequence: m = (101)
Time

10. Effective Code Rate
 Initialize the memory before encoding the first bit (all zero)
 Clear out the memory after encoding the last bit (all zero)
 Hence, a tail of zero-bits is appended to data bits
Effective code rate :
 L is the number of data bits and k=1 is assumed:
𝑅 𝑒𝑓𝑓 =
𝐿
𝑛(𝐿+𝐾−1)
< 𝑅 𝑐 =
1
𝑛
𝑅 𝑒𝑓𝑓 =
3
10
< 𝑅 𝑐 =
1
2

11. Agenda
1. Introduction to Convolutional Codes
2. Convolutional Encoder Structure
3. Convolutional Encoder Representation
(Vector, Polynomial, State Diagram and Trellis Representations )
4. Maximum Likelihood Decoder
5. Viterbi Algorithm
6. MATLAB Simulation
 Hard and Soft Decisions
 Bit Error Rate Tradeoff
 Consumed Time Tradeoff

12. Encoder Vector Representation
 We define n binary vector with K elements (one vector for each modulo-2 adder)
 The i-th element in each vector, is “1” if the i-th stage in the shift register is connected to the
corresponding modulo-2 adder, and “0” otherwise.
 Examples: k=1
1 1 1
1 0 1
𝑔1 = 1 1 1
𝑔2 = [1 0 1]
𝑔1 = 1 0 0
𝑔2 = 1 0 1
𝑔3 = [1 1 1]
Generator matrix
with 2 vectors
Generator matrix
with 3 vectors

13. Encoder Polynomial Representation
 Define n generator polynomials, one for each modulo-2 adder.
 Each polynomial is of degree 𝐾𝑘 − 1 or less and describes the connection of the
shift registers to the corresponding modulo- 2 adder.
 Example: k=1
 The output sequence is found as follows:
1 1 1
1 0 1

14. Encoder Polynomial Representation
Example: m = (101) = 1 + 𝑋2
, 𝑔1 𝑋 =1 + 𝑋 + 𝑋2
𝑔2(𝑋) =1 + 𝑋2
The output sequence is found as follows:
1 1 1
1 0 1

15. Encoder State Diagram Representation
 In a Convolutional encoder, the state is
represented by the content of the
memory. Hence, there are 2 𝐾−1 states.
 A state diagram contains all the states and all
possible transitions between them.
 Only two transitions initiating from a state
 Only two transitions ending up in a state

16. Encoder Trellis Representation
 Trellis diagram is an extension of the state diagram that shows the passage of time.
 Example of a section of trellis for the rate ½ code

17. Encoder Trellis Representation (rate ½ code)
𝑺 𝒐
𝑺 𝟐
𝑺 𝟏
𝑺 𝟑

18. Encoder Trellis Representation (rate ½ code)
𝑺 𝒐
𝑺 𝟐
𝑺 𝟏
𝑺 𝟑

19. Agenda
1. Introduction to Convolutional Codes
2. Convolutional Encoder Structure
3. Convolutional Encoder Representation
(Vector, Polynomial, State Diagram and Trellis Representations )
4. Maximum Likelihood Decoder
5. Viterbi Algorithm
6. MATLAB Simulation
 Hard and Soft Decisions
 Bit Error Rate Tradeoff
 Consumed Time Tradeoff

20. Optimum Decoding (Maximum Likelihood )
 If the input sequence messages are
equally likely, the optimum
decoder which minimizes the
probability of error is the
Maximum Likelihood (ML)
decoder.
 ML decoder, selects a codeword
among all the possible codewords
which maximizes the likelihood
function where is the received
sequence and is one of the
possible codewords
𝟐 𝑳
Codeword
s to search

21. The Viterbi Algorithm
 The Viterbi algorithm performs Maximum Likelihood decoding.
 It finds a path through trellis with the largest metric (maximum correlation or minimum distance).
− It processes the demodulator outputs in an iterative manner.
− At each step in the trellis, it compares the
metric of all paths entering each state, and
keeps only the path with the largest
metric, called the survivor, together with
its metric.
− It proceeds in the trellis by eliminating the
least likely paths.
 It reduces the decoding complexity to 𝐿 2 𝐾−1

22. The Viterbi Algorithm
A. Do the following set up:
 For a data block of L bits, form the trellis.
The trellis has L+K-1 sections or levels and
starts at time 𝑡1 and ends up at time 𝑡 𝐿+𝐾
 Label all the branches in the trellis with their
corresponding branch metric.
 For each state in the trellis at the time 𝑡𝑖
which is denoted by 𝑆 𝑡𝑖 ∈ 1,2, … , 2 𝐾−1 ,
define a parameter Γ 𝑆 𝑡𝑖 , 𝑡𝑖
B. Then, do the following:
1.Set Γ 0, 𝑡1 = 0, and 𝒊 = 𝟐
2.At time 𝑡𝑖, compute the partial path metrics for
all the paths entering each state
3.Set Γ 𝑆 𝑡𝑖 , 𝑡𝑖 equal to the best partial path
metric entering each state at time 𝑡𝑖.
4.Keep the survivor path and delete the dead
paths from the trellis.
5.If 𝑖 < 𝐿 + 𝐾, increase 𝑖 by 1 and return to step 2.
C. Start at state zero at time 𝑡 𝐿+𝐾.
 Follow the surviving branches backwards
through the trellis.
 The path thus defined is unique and correspond
to the ML codeword.

23. Convolutional Trellis Encoder Example (rate ½ code)
Msg. 1 0 1 1 0 0
Rcvd. 11 10 11 00 01 10
CW. 11 11 01 00 01 10
𝑺 𝒐
𝑺 𝟏
𝑺 𝟐
𝑺 𝟑

24. Example: Viterbi Trellis Decoder (𝒊 = 𝟏)
𝑺 𝒐
𝑺 𝟏
𝑺 𝟐
𝑺 𝟑

25. Example: Viterbi Trellis Decoder (𝒊 = 𝟐)
𝑺 𝒐
𝑺 𝟏
𝑺 𝟐
𝑺 𝟑

26. Example: Viterbi Trellis Decoder (𝒊 = 𝟑 , 4)
𝑺 𝒐
𝑺 𝟏
𝑺 𝟐
𝑺 𝟑

27. Example: Viterbi Trellis Decoder (𝒊 = 𝟓 , 𝟔)
𝑺 𝒐
𝑺 𝟏
𝑺 𝟐
𝑺 𝟑

28. Example: Viterbi Trellis Decoder (𝒊 = 𝟕)
𝑺 𝒐
𝑺 𝟏
𝑺 𝟐
𝑺 𝟑

29. Example: Viterbi Trellis Decoder (Trace back)
𝑺 𝒐
𝑺 𝟏
𝑺 𝟐
𝑺 𝟑

30. Error Correcting Code Gain
 It is defined as the reduction of 𝑬 𝒃 𝑵 𝒐 (in dB)
that is needed to obtain the same error rate.
 Example: For a BER of 10−6
𝑬 𝒃 𝑵 𝒐 𝑐 = 11 𝑑𝐵 is needed with coding
𝑬 𝒃 𝑵 𝒐 𝑢 = 13.77 𝑑𝐵 without the coding
The Coding Gain
𝐺 = 13.77 − 11 = 2.77 𝑑𝐵
𝟏𝟎−𝟔
𝟏𝟎−𝟓
𝟏𝟏 𝟏𝟑. 𝟕𝟕

31. Agenda
1. Introduction to Convolutional Codes
2. Convolutional Encoder Structure
3. Convolutional Encoder Representation
(Vector, Polynomial, State Diagram and Trellis Representations )
4. Maximum Likelihood Decoder
5. Viterbi Algorithm
6. MATLAB Simulation
 Hard and Soft Decisions
 Bit Error Rate Tradeoff
 Consumed Time Tradeoff

32. MATLAB Simulation: BER, Time Performance
Convolutional-Viterbi Codec
Generator Vector : [111,101]
Hard and Soft Decisions
Block Codec
Hamming (7,4)
Hard and Soft Decisions
Vs.
Richard Hamming Andrew J. Viterbi

33. Error Performance Trade-offs
Trade-off 1:
Hamming Coding
Hard Decision Vs. Soft Decision
Trade-off 2:
Convolutional Coding
Hard Decision Vs. Soft Decision
Trade-off 3:
Block Coding Vs. Convolutional

34. BER Performance Hard Decision Vs. Soft Decision
Trade-off 2: Viterbi deCodingTrade-off 1: Hamming deCoding

35. Trade-off 3: Hamming and Viterbi BER Performance Trade-offs

36. Time Performance Trade-offs
Trade-off 4:
Block Coding
Hard Decision Vs. Soft Decision
Trade-off 5:
Convolutional Coding
Hard Decision Vs. Soft Decision
Trade-off 6:
Block Coding Vs. Convolutional

37. Time Performance Hard Decision Vs. Soft Decision
Trade-off 5: Viterbi deCodingTrade-off 3: Hamming deCoding

38. Trade-off 6: Hamming and Viterbi Time Performance Trade-offs

39. Mohammed Abuibaid
Live & Breathe Wireless