Convolutional Coding In telecommunication, a convolutional code is a type of error- correcting code in which m-bit information symbol to be encoded is.

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Presentation transcript:

Convolutional Coding In telecommunication, a convolutional code is a type of error- correcting code in which m-bit information symbol to be encoded is transformed into n-bit symbol. Convolutional codes are used extensively in numerous applications in order to achieve reliable data transfer, including digital video, radio, mobile communication, and satellite communication. These codes are often implemented in concatenation with a hard- decision code, particularly Reed Solomon. Author Phani Swathi Chitta Mentor Prof. Saravanan Vijayakumaran Course Name: Error Correcting Codes Level : UG

Learning Objectives After interacting with this Learning Object, the learner will be able to: Explain the convolutional encoding and Viterbi (decoding) Algorithms

Definitions of the components/Keywords: A convolutional encoder is a finite state machine. An encoder with n registers will have 2 n states. Convolutional codes have memory that uses previous bits to encode or decode following bits The most commonly known graphical representation of a code is the trellis representation. A code trellis diagram is simply an edge labeled directed graph in which every path represents a code sequence. This representation has resulted in a wide range of applications of convolutional codes for error control in digital communications. Viterbi algorithm is used for decoding a bit stream that has been encoded using forward error correction based on a convolutional code. Viterbi decoding compares the hamming distance between the branch code and the received code. Path producing larger hamming distance is eliminated. In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different.

Master Layout /00 1/11 0/01 0/11 1/10 1/01 0/10 1/00 Encoder State diagram Trellis Diagram Place a dropdown box for encoder and decoder Part 1 – Encoder Part 2 – Decoder

Step 1: I nstruction for the animator T ext to be displayed in the working area (DT) The encoder figure in the master layout is shown first. Along with that show the state diagram without arrows(show only the ovals) After first sentence in DT, red 0 must appear. Then the circles should blink and blue 0s must appear. The initial state of encoder is 00 which represents the contents of the shift register in the encoder. Input 0is given to the encoder and the corresponding outputs of modulo 2 adders are Input data:

Step 2: I nstruction for the animator T ext to be displayed in the working area (DT) Then right shift the bits so that red 0 appears in first box. Then the second figure should appear. The text in DT should be displayed with the second figure. This is the state diagram. 0/00 represents input/output Input data: /00 0 0

Step 3: I nstruction for the animator T ext to be displayed in the working area (DT) After first sentence in DT, red 1 must appear. Then the circles should blink and blue 1s must appear. Input 1is given to the encoder and the corresponding outputs of modulo 2 adders are Input data:

Step 4: I nstruction for the animator T ext to be displayed in the working area (DT) Then right shift the bits so that red 1 appears in first box. Then the second figure should appear. The text in DT should be displayed with the second figure. This is the state diagram /00 1/ Input data:

Step 5: I nstruction for the animator T ext to be displayed in the working area (DT) After first sentence in DT, red 0 must appear. Then the circles should blink and blue 01 must appear. Input 0is given to the encoder and the corresponding outputs of modulo 2 adders are Input data:

Step 6: I nstruction for the animator T ext to be displayed in the working area (DT) Then right shift the bits so that red 0 appears in first box. Then the second figure should appear. The text in DT should be displayed with the second figure. This is the state diagram /00 1/11 0/ Input data:

Step 7: I nstruction for the animator T ext to be displayed in the working area (DT) After first sentence in DT, red 0 must appear. Then the circles should blink and blue 1s must appear. Input 0is given to the encoder and the corresponding outputs of modulo 2 adders are Input data:

Step 8: I nstruction for the animator T ext to be displayed in the working area (DT) Then right shift the bits so that red 0 appears in first box. Then the second figure should appear. The text in DT should be displayed with the second figure. This is the state diagram /00 1/11 0/01 0/ Input data:

Step 9: I nstruction for the animator T ext to be displayed in the working area (DT) After first sentence in DT, red 1 must appear. Then the circles should blink and blue 1s must appear. Input 1is given to the encoder and the corresponding outputs of modulo 2 adders are Input data:

Step 10: I nstruction for the animator T ext to be displayed in the working area (DT) Then right shift the bits so that red 0 appears in first box. Then the second figure should appear. The text in DT should be displayed with the second figure. This is the state diagram /00 1/11 0/01 0/ Input data:

Step 11: I nstruction for the animator T ext to be displayed in the working area (DT) After first sentence in DT, red 1 must appear. Then the circles should blink and blue 10 must appear. Input 1is given to the encoder and the corresponding outputs of modulo 2 adders are Input data:

Step 12: I nstruction for the animator T ext to be displayed in the working area (DT) Then right shift the bits so that red 1 appears in first box. Then the second figure should appear. The text in DT should be displayed with the second figure. This is the state diagram /00 1/11 0/01 0/11 1/ Input data:

Step 13: I nstruction for the animator T ext to be displayed in the working area (DT) After first sentence in DT, red 1 must appear. Then the circles should blink and blue 01 must appear. Input 1is given to the encoder and the corresponding outputs of modulo 2 adders are Input data:

Step 14: I nstruction for the animator T ext to be displayed in the working area (DT) Then right shift the bits so that red 1 appears in first box. Then the second figure should appear. The text in DT should be displayed with the second figure. This is the state diagram /00 1/11 0/01 0/11 1/10 1/ Input data:

Step 15: I nstruction for the animator T ext to be displayed in the working area (DT) After first sentence in DT, red 0 must appear. Then the circles should blink and blue 10 must appear. Input 0is given to the encoder and the corresponding outputs of modulo 2 adders are Input data:

Step 16: I nstruction for the animator T ext to be displayed in the working area (DT) Then right shift the bits so that red 0 appears in first box. Then the second figure should appear. The text in DT should be displayed with the second figure. This is the state diagram /00 1/11 0/01 0/11 1/10 1/01 0/ Input data:

Step 17: I nstruction for the animator T ext to be displayed in the working area (DT) After first sentence in DT, red 1 must appear. Then the circles should blink and blue 0s must appear. Input 1is given to the encoder and the corresponding outputs of modulo 2 adders are Input data:

Step 18: I nstruction for the animator T ext to be displayed in the working area (DT) Then right shift the bits so that red 1 appears in first box. Then the second figure should appear. The text in DT should be displayed with the second figure. This is the state diagram /00 1/11 0/01 0/11 1/10 1/01 0/10 1/ Input data: 1 0 0

Step 19: Trellis Diagram I nstruction for the animator T ext to be displayed in the working area (DT) The figure in step 19 must be shown (red lines should also be shown as black lines). Then the red lines must be shown. This is the trellis diagram with all the possible transitions 0/00 1/11 0/00 1/11 0/01 1/10 0/11 1/00 0/10 1/01 t=0 t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=8 t=9 0/11 1/11 1/10 1/01 0/10 1/00

Step 20: I nstruction for the animator T ext to be displayed in the working area (DT) The figure in step 20 is shown. The text in DT is displayed. The trellis diagram for the given input The output sequence is t=0t=1t=2t=3t=4t=5t=6t=7t=8 t=9 0/00 1/11 0/11 1/11 1/10 0/10 1/00 1/01 0/01 Input data:

Master Layout Part 1 – Encoder Part 2 – Decoder Trellis Diagram The output sequence is 11, 10, 10, 11, 11, 01, 00, 01

Step 1: I nstruction for the animator T ext to be displayed in the working area (DT) Consider an example When the data sequence is applied to the encoder, the coded output bit sequence is The coded output sequence passes through a channel, producing the received sequence r= [ ]. The two underlined bits are flipped by noise in the channel.

Step 2: I nstruction for the animator T ext to be displayed in the working area (DT) Except the dotted lines everything should appear After the text in DT is displayed, the dotted lines must appear The received sequence is r 0 = 11. Computing the metric to each state at time t=1 by finding the hamming distance between r 0 and the possible transmitted sequence along the branches of the first stage of the trellis. Since state 0 is the initial state, there are only two paths, with path metrices 2 and 0. r 0 = t=0 t=1 2 0

Step 3: I nstruction for the animator T ext to be displayed in the working area (DT) Except the new dotted lines everything should appear After the text in DT is displayed, the new dotted lines must appear The received sequence is r 1 = 10. Each path at time t=1 is extended, adding the path metric to each branch metric. r 1 = t=0 t=1 t=

Step 4: I nstruction for the animator T ext to be displayed in the working area (DT) Except the new dotted lines everything should appear After the text in DT is displayed, the new dotted lines must appear The received sequence is r 2 = 00. Each path at time t=2 is extended, adding the path metric to each branch metric. There are multiple paths to each node at time t=3 r 2 = t=0 t=1 t=2 t=

Step 5: I nstruction for the animator T ext to be displayed in the working area (DT) After the text in DT is displayed, some dotted lines must disappear and the figure in step 5 is shown. Select the path to each node with best metric and eliminate the other paths. r 2 = t=0 t=1 t=2 t=

Step 6: I nstruction for the animator T ext to be displayed in the working area (DT) Except the new dotted lines everything should appear After the text in DT is displayed, the new dotted lines must appear The received sequence is r 3 = 10. Each path at time t=3 is extended, adding the path metric to each branch metric. Here, the best path to each state is selected. In selecting the best paths, some of the paths to some states at earlier times have no successors, these paths can be deleted now. r 3 = t=0 t=1 t=2 t=3 t=

Step 7: I nstruction for the animator T ext to be displayed in the working area (DT) After the text in DT is displayed, some dotted lines must disappear and the figure in step 7 is shown. This is the resulting figure after selecting the best paths. r 3 = t=0 t=1 t=2 t=3 t=

Step 8: I nstruction for the animator T ext to be displayed in the working area (DT) Except the new dotted lines everything should appear After the text in DT is displayed, the new dotted lines must appear The received sequence is r 4 = 11. Each path at time t=4 is extended, adding the path metric to each branch metric. In this case, there are multiple paths into states 2 and 3 with same path metrics but only one of the paths must be selected. So the choice can be made arbitrarily. r 4 = t=0 t=1 t=2 t=3 t=4 t=

Step 9: I nstruction for the animator T ext to be displayed in the working area (DT) After the text in DT is displayed, some dotted lines must disappear and the figure in step 9 is shown. This is the resulting figure after selecting the best paths. r 4 = t=0 t=1 t=2 t=3 t=4 t=

Step 10: I nstruction for the animator T ext to be displayed in the working area (DT) Except the new dotted lines everything should appear After the text in DT is displayed, the new dotted lines must appear The received sequence is r 5 = 01. Each path at time t=5 is extended, adding the path metric to each branch metric. r 5 = t=0 t=1 t=2 t=3 t=4 t=5 t=

Step 11: I nstruction for the animator T ext to be displayed in the working area (DT) After the text in DT is displayed, some dotted lines must disappear and the figure in step 11 is shown. This is the resulting figure after selecting the best paths. r 5 = t=0 t=1 t=2 t=3 t=4 t=5 t=

Step 12: I nstruction for the animator T ext to be displayed in the working area (DT) Except the new dotted lines everything should appear After the text in DT is displayed, the new dotted lines must appear The received sequence is r 6 = 00. Each path at time t=6 is extended, adding the path metric to each branch metric. r 6 = t=0 t=1 t=2 t=3 t=4 t=5 t=6 t=

Step 13: I nstruction for the animator T ext to be displayed in the working area (DT) After the text in DT is displayed, some dotted lines must disappear and the figure in step 13 is shown. This is the resulting figure after selecting the best paths. r 6 = t=0 t=1 t=2 t=3 t=4 t=5 t=6 t=

Step 14: I nstruction for the animator T ext to be displayed in the working area (DT) Except the new dotted lines everything should appear After the text in DT is displayed, the new dotted lines must appear The received sequence is r 7 = 01. Each path at time t=7 is extended, adding the path metric to each branch metric. r 7 = t=0 t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=

Step 15: I nstruction for the animator T ext to be displayed in the working area (DT) After the text in DT is displayed, some dotted lines must disappear and the figure in step 15 is shown. This is the resulting figure after selecting the best paths. r 7 = t=0 t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=

Step 16: I nstruction for the animator T ext to be displayed in the working area (DT) After the text in DT is displayed, the solid line must be shown. Selection of the final path with the best metric is done. The input/output pairs are indicated on each branch. The recovered input bit sequence is same as the original bit sequence. Thus, out of the sequence of 16bits, two bit errors have been corrected. r 7 = t=0 t=1 t=2 t=3 t=4 t=5 t=6 t=7 t= /11 3 0/10 0/11 1/11 0/01 1/00 0/01 1/10

Introduction Credits 42 Definitions Test your understanding (questionnaire) ‏ Lets Sum up (summary) ‏ Want to know more… (Further Reading) ‏ Try it yourself Interactivity: Analogy Slide 1 Slide 3 Slide 44,45 Slide 47 Slide 46 Electrical Engineering + + Place an input box to enter the input data Place a dropdown box with options 0 and Encoder State Diagram Encoder:

Introduction Credits 43 Definitions Test your understanding (questionnaire) ‏ Lets Sum up (summary) ‏ Want to know more… (Further Reading) ‏ Try it yourself Interactivity: Analogy Slide 1 Slide 3 Slide 44,45 Slide 47 Slide 46 Electrical Engineering Place an input box to enter the 16-bit output data sequence Place a dropdown box with options 00, 10, 01, 11 Decoder: Trellis Diagram

Questionnaire 1. An encoder with n registers will have ________ states. Answers: a) 2 n b) 2 n +1 ‏ 2. For the given encoder, what will be the output sequence? Input data: Answers: a) b) c) d) ‏

Questionnaire 3.The received sequence is Among the four choices given below, which is the sequence nearest to the received sequence in terms of Hamming distance? Answers: a) b) c) d) The answers are given in red

Links for further reading Reference websites: Books: Error correction coding – Todd K. Moon, John wiley & sons,INC Research papers:

Summary A convolutional encoder is a finite state machine. An encoder with n binary cells will have 2 n states. The most commonly known graphical representation of a code is the trellis representation. A code trellis diagram is simply an edge labeled directed graph in which every path represents a code sequence. This representation has resulted in a wide range of applications of convolutional codes for error control in digital communications. Viterbi algorithmfor decoding a bitstream that has been encoded using forward error correctionbased on a convolutional code.