convolutional error control coding
TRANSCRIPT
MEH607 Error Control Coding
KOCAELI UNIVERSITYGraduate School of
Natural and Applied Sciences
Prepared By: Mohammed ABUIBAIDEmail: [email protected] to: Dr. SITKI ÖZTÜRK
Electronic and Communication Engineering
Convolutional Error Control Coding
Aca
dem
ic Y
ear
20
15
/20
16
Agenda1. Introduction to Convolutional Codes2. Convolutional Encoder Structure3. Convolutional Encoder Representation
(Vector, Polynomial, State Diagram and Trellis Representations )
4. Maximum Likelihood Decoder5. Viterbi Algorithm6. MATLAB Simulation
Hard and Soft Decisions Bit Error Rate Tradeoff Consumed Time Tradeoff
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
FEC Coding TechniquesBlock 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)
Convolutional CodesConvolutional 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 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.
Agenda1. Introduction to Convolutional Codes2. Convolutional Encoder Structure3. Convolutional Encoder Representation
(Vector, Polynomial, State Diagram and Trellis Representations )
4. Maximum Likelihood Decoder5. Viterbi Algorithm6. MATLAB Simulation
Hard and Soft Decisions Bit Error Rate Tradeoff Consumed Time Tradeoff
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
Convolutional Encoder ( rate ½, K=3 )
1 0 0𝑡1
0 1 0𝑡 4
0 1 0𝑡 2
0 0 1𝑡5
1 0 1𝑡 3
0 0 0𝑡 6
Message sequence: m = (101)
Time
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:
Agenda1. Introduction to Convolutional Codes2. Convolutional Encoder Structure3. Convolutional Encoder Representation
(Vector, Polynomial, State Diagram and Trellis Representations )
4. Maximum Likelihood Decoder5. Viterbi Algorithm6. MATLAB Simulation
Hard and Soft Decisions Bit Error Rate Tradeoff Consumed Time Tradeoff
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 1Generator matrix with 2 vectors
Generator matrix with 3 vectors
Encoder Polynomial Representation Define n generator polynomials, one for each modulo-2 adder. Each polynomial is of degree 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
Encoder Polynomial Representation
Example: m = (101) =
The output sequence is found as follows:
1 1 1
1 0 1
Encoder State Diagram Representation In a Convolutional encoder, the state is
represented by the content of thememory. Hence, there are 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
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
Agenda1. Introduction to Convolutional Codes2. Convolutional Encoder Structure3. Convolutional Encoder Representation
(Vector, Polynomial, State Diagram and Trellis Representations )
4. Maximum Likelihood Decoder5. Viterbi Algorithm6. MATLAB Simulation
Hard and Soft Decisions Bit Error Rate Tradeoff Consumed Time Tradeoff
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
Codewords to search
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
The Viterbi AlgorithmA. 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 and ends up at time
Label all the branches in the trellis with their corresponding branch metric.
For each state in the trellis at the time which is denoted by , define a parameter
B. Then, do the following:1. Set , and
2. At time , compute the partial path metrics for all the paths entering each state
3. Set equal to the best partial path metric entering each state at time .
4. Keep the survivor path and delete the dead paths from the trellis.
5. If , increase by 1 and return to step 2.
C. Start at state zero at time . Follow the surviving branches backwards through the
trellis.
The path thus defined is unique and correspond to the ML codeword.
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
Error Correcting Code Gain
It is defined as the reduction of (in dB) that is needed to obtain the same error rate.
Example: For a BER of is needed with coding without the coding The Coding Gain
𝟏𝟎−𝟔
𝟏𝟎−𝟓
𝟏𝟏 𝟏𝟑 .𝟕𝟕
Agenda1. Introduction to Convolutional Codes2. Convolutional Encoder Structure3. Convolutional Encoder Representation
(Vector, Polynomial, State Diagram and Trellis Representations )
4. Maximum Likelihood Decoder5. Viterbi Algorithm6. MATLAB Simulation
Hard and Soft Decisions Bit Error Rate Tradeoff Consumed Time Tradeoff
MATLAB Simulation: BER, Time Performance
Convolutional-Viterbi CodecGenerator Vector : [111,101]
Hard and Soft Decisions
Block CodecHamming (7,4)
Hard and Soft Decisions
Vs.
Richard Hamming Andrew J. Viterbi
Error Performance Trade-offsTrade-off 1:
Hamming Coding Hard Decision Vs. Soft Decision
Trade-off 2:Convolutional Coding
Hard Decision Vs. Soft DecisionTrade-off 3:
Block Coding Vs. Convolutional
BER Performance Hard Decision Vs. Soft Decision
Eb/No, dB0 2 4 6 8 10
Bit
Err
or R
ate
10-5
10-4
10-3
10-2
10-1
Viterbi Decoding
theory - uncodedHard ViterbiSoft Viterbi
Eb/No, dB0 2 4 6 8 10
Bit
Err
or R
ate
10-5
10-4
10-3
10-2
10-1
BER of Hamming (7,4) code
theory - uncodedHard (7,4) HammingSoft (7,4) Hamming
Trade-off 2: Viterbi deCoding Trade-off 1: Hamming deCoding
Trade-off 3: Hamming and Viterbi BER Performance Trade-offs
Eb/No, dB0 1 2 3 4 5 6 7 8 9 10
Bit
Err
or R
ate
10-5
10-4
10-3
10-2
10-1
Theory - UncodedHard (7,4) HammingSoft (7,4) HammingHard Viterbi (111,101)Soft Viterbi(111,101)
Time Performance Trade-offsTrade-off 4:
Block Coding Hard Decision Vs. Soft Decision
Trade-off 5:Convolutional Coding
Hard Decision Vs. Soft DecisionTrade-off 6:
Block Coding Vs. Convolutional
Time Performance Hard Decision Vs. Soft Decision
Trade-off 5: Viterbi deCoding Trade-off 3: Hamming deCoding
Eb/No, dB0 2 4 6 8 10
Con
sum
ed T
ime
0
0.2
0.4
0.6
0.8
1Hamming Decoding
Hard (7,4) HammingSoft (7,4) Hamming
Eb/No, dB0 2 4 6 8 10
Con
sum
ed T
ime
0
10
20
30
40
50
60
70
80Viterbi Decoding
Hard ViterbiSoft Viterbi
Trade-off 6: Hamming and Viterbi Time Performance Trade-offs
Eb/No, dB0 1 2 3 4 5 6 7 8 9 10
Con
sum
ed T
ime
0
10
20
30
40
50
60
70
80Hard (7,4) HammingSoft (7,4) HammingHard ViterbiSoft Viterbi