This repository has codes pertaining to the above question written in MATLAB

In this we have demonstrated the convolutional encoding and decoding process and finally calculated the percentage of error detection and correction

In data communication, a convolutional code is a type of error-correcting code that generates parity symbols via the sliding application of a boolean polynomial function to a data stream. The sliding application represents the 'convolution' of the encoder over the data, which gives rise to the term 'convolutional coding.' The sliding nature of the convolutional codes facilitates trellis decoding using a time-invariant trellis. Time invariant trellis decoding allows convolutional codes to be maximum-likelihood soft-decision decoded with reasonable complexity.

MATLAB must be installed to run the program.
To install MATLAB follow the official site instructions https://in.mathworks.com/downloads/

1. Clone the repository

       git clone https://github.com/shushantkumar/Convolutional-Encoding-Decoding.git
   

2. Move into the directory

       cd Convolutional-Encoding-Decoding/
   

3. To run the the program execute file ex.m passing parameter as 4

       run ex(4)        //for 4bit dataword generalization
   

1. ex.m -This is the main function which takes number of bits in dataword as input and returns the percentage of errors detected and corrected
2. encode.m - This function encodes input bit with the help of generating functions
3. decoding.m - This funtion calculates number of cases where error can be corrected
4. detection.m - This funtion calculates number of cases where error can be detected
5. tabl.m - This function calculates and returns the state table of the convolution code for every state with transition on 0 or 1.
6. onebit.m - Function to introduce 1 bit error in codeword
7. twobit.m - Function to introduce 2 bit error in codeword
8. threebit.m - Function to introduce 3 bit error in codeword
9. fourbit.m - Function to introduce 4 bit error in codeword
10. fivebit.m - Function to introduce 5 bit error in codeword

The function ex.m takes an integer p as input which denotes the number of bits in dataword. Then all the possible binary numbers possible of p input bits number is generated. All the generated datawords of p bits is then encoded. Then in a loop for all the codewords all the 1,2,3,4 and 5 bits error are introduced and then sent to function decoding.m and detection.m for finding number of corrections and detections respectively. Then total percentage is calculated and average is found for each bit errors. Finally graph is plotted for percentage detection and correction. The most important function is tabl.m that stores all the generated codeword from each state if the input bit is 0 and 1.
Detailed Explanation given as comments in respective functions

For p = 4 as number of bits in datawords following were the result:-
Time taken (avg.) = 2minutes

Percentage detection separately for all 4bit possible datawords

```
    Error bits     1bit  2bit  3bit  4bit  5bit
    0000           100   100   100   100   100
    0001           100   100   100   100   100
    0010           100   100   100   100   100
    0011           100   100   100   100   100
    0100           100   100   100   100   100
    0101           100   100   100   100   100
    0110           100   100   100   100   100
    0111           100   100   100   100   100
    1000           100   100   100   100   100
    1001           100   100   100   100   100
    1010           100   100   100   100   100
    1011           100   100   100   100   100
    1100           100   100   100   100   100
    1101           100   100   100   100   100
    1110           100   100   100   100   100
    1111           100   100   100   100   100
```

Percentage correction separately for all 4bit possible datawords

```

    Error bits     1bit      2bit      3bit      4bit      5bit
    0000           100.0000  97.8947   93.6842   87.4923   79.5666
    0001           90.0000   78.9474   67.3684   55.7895   44.6981
    0010           90.0000   78.9474   67.3684   55.7895   44.6981
    0011           80.0000   62.1053   46.6667   33.8287   23.5552
    0100           90.0000   78.9474   67.3684   55.7895   44.6981
    0101           80.0000   62.1053   46.6667   33.8287   23.5552
    0110           80.0000   62.1053   46.6667   33.8287   23.5552
    0111           70.0000   47.3684   30.8772   19.2982   11.4938
    1000           90.0000   78.9474   67.3684   55.7895   44.6981
    1001           80.0000   62.1053   46.6667   33.8287   23.5552
    1010           80.0000   62.1053   46.6667   33.8287   23.5552
    1011           70.0000   47.3684   30.8772   19.2982   11.4938
    1100           80.0000   62.1053   46.6667   33.8287   23.5552
    1101           70.0000   47.3684   30.8772   19.2982   11.4938
    1110           70.0000   47.3684   30.8772   19.2982   11.4938
    1111           60.0000   34.7368   19.2982   10.2167    5.1084
   
```

Average Percentage detection

```
  Errors 1bit  2bit  3bit  4bit  5bit
         100   100   100   100   100
```

Average Percentage correction

```
  Errors 1bit      2bit      3bit      4bit      5bit
         80.0000   63.1579   49.1228   37.5645   28.1734
```

Bargraph of percentage average Error Detection and Correction for all possible 4bit datawords

Histogram Representation of Error Correction of codeword generated from 4bit dataword using sequential decoding