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# Convolution Codes

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### Convolution Codes

1. 1. Convolution Codes ADVANCED COMMUNICATION SYSTEMS PRATISHTHA SHIRA RAM SEMESTER 8 DEPT. OF ECEPratishtha Shira Ram
2. 2. Codes Codes Source Channel Codes Codes Linear Convolution Block codes codesPratishtha Shira Ram
3. 3. Codes (contd.) 1 Source Codes • The aim of source coding is to take the source data and make it smaller. 2 Channel Codes • The aim of channel coding theory is to find codes which transmit quickly, contain many valid code words and can correct or at least detect many errors.Pratishtha Shira Ram
4. 4. Channel Codes Channel codes are used to add redundancy to the codes. Redundancy adds extra information about the data into the signal. This extra information ensures that the data is received correctly at the information sink. Hence, channel codes increase the BER of the signal. The aim of channel coding The channel code is used to theory is to find codes which protect data sent over it for transmit quickly, contain storage or retrieval even in the many valid code words and presence of noise (errors). can correct or at least detect many errors.Pratishtha Shira Ram
5. 5. Convolution Codes Convolutional codes are characterized by thee parameters: (n, k, m) Where, n= Number of output bits k= Number of input bits m= Number of memory registers Code Rate = k/n =Number of input bits /Number of output bits Constraint length “L”= k(m-1) L represents the no. of bits in the encoder memory that affects the generation of n output bitsPratishtha Shira Ram
6. 6. An example: (2,1,4) Codern= Number of output bits 2k= Number of input bits 1m= Number of memory registers 4 Constraint length L=3 The no. of bits in the shaded registers are called states of code, and are defined by No. of states = 2LPratishtha Shira Ram
7. 7. (2,1,4) Coder The constraint length = 3. The shaded registers hold these bits, while the un-shaded register holds the incoming bit. This means, 3 bits, or 8 different combinations of the bits can be held in the registers. These 8 combinations determine what output will be received as the the 2 output bits v1 and v2.Pratishtha Shira Ram
8. 8. Input sequence: 1,Output: 11 11 10 11 t=0; C Input state =000 t=2;A Input bit=1 Input state =010 Output bit=11 Input bit=0 Output bit=10 t=1;B Input state =100 Input bit=0 D Output bit=11 t=3; Input state =101 Input bit=0 Output bit=11Pratishtha Shira Ram
9. 9. The convolution Providing input of 1 to the coder provides what is called the ”impulse response” of that coder. Here, providing 1 as input (and then „flushing‟ it out from the registers using zeros) gave the output sequence : 11 11 10 11. Similarly, response for input bit 0 would be : 00 00 00 00. The output sequence can be computed simply by convolving the input sequence u with the impulse response g. Or, v=u*g By the principle of linear superposition, a coded sequence can be generated from the impulse response.Pratishtha Shira Ram
10. 10. ExampleTo find the coded sequence for the input 1011, we just have to add the shiftedversions of the individual responses:Input bit Its Impulse Response 1 11 11 10 11 0 00 00 00 00 1 11 11 10 11 1 11 11 10 11 1011 11 11 01 11 01 01 11 This result can be verified by the encoder model too.Pratishtha Shira Ram
11. 11. Output bits and the encoder bits through the(2,1,4) code:Pratishtha Shira Ram
12. 12. Look–up table for the encoderPratishtha Shira Ram
13. 13. Understanding the encoder design There are three graphical ways in which the encoder can be understood better: 1. State diagram 2. Tree diagram 3. Trellis diagram State DiagramPratishtha Shira Ram
14. 14. Tree DiagramPratishtha Shira Ram
15. 15. Trellis DiagramPratishtha Shira Ram
16. 16. Trellis Diagram –Encoded SequencePratishtha Shira Ram
17. 17. Deconvolution of codesThere are 2 basic methods of deconvolution1. Sequential decoding: Fano algorithm - It allows both forward and backward movement through the Trellis diagram flow. -If instead of 11 11 01 11 01 01 11, the code 01 11 01 11 01 01 11 is received, the algorithm will take a start and tally with the outputs it finds on the way. If an output does not tally, it retraces its position back to the previous ambiguous decision.2. Maximum –Likelihood decoding: Viterbi algorithm -The viterbi decoder examines the entire received sequence of a given length. -The decoder then computes a metric for the path and then makes a decision based on it. -The path with the higher metric is kept, and the one with the lower one is discarded. - Generally the Hamming distance is used as the metric here.Pratishtha Shira Ram
18. 18. Sequential decoding: Fano algorithmPratishtha Shira Ram
19. 19. Maximum –Likelihood decoding: Viterbi algorithm [Step 1]Pratishtha Shira Ram
20. 20. Maximum –Likelihood decoding: Viterbi algorithm [Step 2]Pratishtha Shira Ram
21. 21. Maximum –Likelihood decoding: Viterbi algorithm [Step 3]Pratishtha Shira Ram
22. 22. Maximum –Likelihood decoding: Viterbi algorithm [Step 4]Pratishtha Shira Ram
23. 23. Thank you!Pratishtha Shira Ram