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# Reed Soloman and convolution codes

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A short and detailed presentation about two types of codes.

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### Reed Soloman and convolution codes

1. 1. REED SOLOMON CODE AND CONVOLUTION CODE
2. 2. REED SOLOMON CODE
3. 3. CONTENTS  Introduction  Properties of RS code  RS Encoder  RS Decoder  Software Implementation  Advantages  Disadvantages  Applications
4. 4. INTRODUCTION  Reed–Solomon codes are an important group of error-correcting codes introduced by Irving S. Reed and Gustave Solomon in 1960.  RS codes operate on the information by dividing the message stream into blocks of data, adding redundancy per block depending only on the current inputs.  It is capable to correct both burst errors (where a series of bits in the codeword are received in error) and erasures.
5. 5. PROPERTIES OF RS CODE  RS codes are generally represented as RS (n, k), with s-bit symbols.  Block Length: n No. of Original Message symbols: k Number of Parity Digits: n – k  A Reed-Solomon decoder can correct up to t symbols that contain errors in a codeword, where 2t = n-k.  The relationship between the symbol size, m, and the size of the codeword n, is given by n=2s-1
6. 6.  The following diagram shows a typical Reed-Solomon codeword: k 2t n Data Parity
7. 7. Example:-  RS(255,223) with 8-bit symbols.  Each codeword contains 255 code word bytes, of which 223 bytes are data and 32 bytes are parity. For this code: n = 255, k = 223, s = 8 2t = 32, t = 16  The decoder can correct any 16 symbol errors in the code word: i.e. errors upto 16 bytes anywhere in the codeword can be automatically corrected.
8. 8.  Given a symbol size s, the maximum codeword length (n) for a Reed-Solomon code is n = 2s – 1  For example, the maximum length of a code with 8-bit symbols (s=8) is 255 bytes.  Reed-Solomon codes may be shortened by (conceptually) making a number of data symbols zero at the encoder, not transmitting them, and then re-inserting them at the decoder.
9. 9.  Example: The (255,223) code described above can be shortened to (200,168). The encoder takes a block of 168 data bytes, (conceptually) adds 55 zero bytes, creates a (255,223) codeword and transmits only the 168 data bytes and 32 parity bytes.
10. 10. ENCODER
11. 11.  Message Polynomial- c(x) = m(x). xn-k  RS generator Polynomial- g(x) = g0 + g1. x+ g2 x2+ …. + g2t-1. x2t-1 + x2t
12. 12. DECODER
13. 13. SOFTWARE IMPLEMENTATION  Until recently, software implementations in real-time required too much computational power for all but the simplest of Reed-Solomon codes (i.e. codes with small values of t).  The following Table gives some example benchmark figures on a 166 MHz Pentium PC: Code Data rate RS(255,251) 12 Mb/s RS(255,239) 2.7 Mb/s RS(255,223) 1.1 Mb/s
14. 14. ADVANTAGES  Reed-Solomon codes are most widely used to correcting burst errors.  Coding gain is very high.  The Coding rate is very high for Reed Solomon code so it is suitable for many applications including storage and transmission.
15. 15. DISADVANTAGES  Unlike BCH codes , RS Codes does not perform considerably well in BPSK modulation schemes.  Bit Error Ratio(BER) for Reed-Solomon Codes is not as good as BCH codes.
16. 16. APPLICATIONS  Data Storage  Bar Code  Satellite Broadcasting  Spread-Spectrum System  Ultra Wideband(UWB)
17. 17. CONVOLUTION CODE
18. 18. ERROR CORRECTION CODE  There are four important error correction codes that find applications in digital transmission. They are : 18 Block Parity Hamming Code Interleaved Code Convolutional Code
19. 19. INTRODUCTION  Convolutional codes are introduced in 1955 by Elias.  Convolution coding is a popular error-correcting coding method used in digital communications. A message is convoluted, and then transmitted into a noisy channel.  This convolution operation encodes some redundant information into the transmitted signal. 19
20. 20. CONVOLUTIONAL ENCODER  Convolutional encoding of data is accomplished using a shift register and associated combinatorial logic that performs modulo-two addition.  A shift register is merely a chain of flip-flops.
21. 21. PARAMETERS OF CONVOLUTION ENCODER  Convolutional codes are commonly specified by three parameters:  n = number of output bits  k = number of input bits  K = number of shift registers  Code Rate: The quantity k/n is called as code rate. It is a measure of the efficiency of the code.  Constraint Length: The quantity L(or K) is called the constraint length of the code. It represents the number of bits in the encoder memory that affect the generation of the n output bits.
22. 22. CONVOLUTIONAL CODE ENCODER + + Shift Register Linear Algebraic Function Generator
23. 23. CONVOLUTION CODE ENCODER + + Constraint Length (K) = 3Code Rate = k/nNo of input bits (k)= 1state = K-1state = 2k=1 K=3 n=2 No of linear Algebraic Function Generator(n) = 2
24. 24. CONVOLUTION CODE ENCODER + 0 0 0 + 11001 k=1 K=3 n=2 Input 1 0 0 1 1 State 10 01 00 10 11 Outpu 11 10 11 11 01 1 1 111100110111 1
25. 25. REPRESENTATION OF CONVOLUTION CODES  State Diagram  Tree Diagram  Trellis Diagram
26. 26. STATE DIAGRAM  Contents of shift registers make up "state" of code:  Most recent input is most significant bit of state.  Oldest input is least significant bit of state.  (this convention is sometimes reverse)
27. 27. TREE DIAGRAM 1101 k=1 K=3 n=2
28. 28. TRELLIS DIAGRAM REPRESENTATION  The trellis diagram is basically a redrawing of the state diagram. It shows all possible state transitions at each time step. Then we connect each state to the next state.  There are only two choices possible at each state. These are determined by the arrival of either a 0 or a 1 bit.  The arrows show the input bit.  The arrows going upwards represent a 0 bit and going downwards represent a 1 bit.
29. 29. TRELLIS DIAGRAM
30. 30. DIFFERENCE BETWEEN BLOCK CODE AND CONVOLUTION CODE  The difference between block codes and convolution codes is the encoding principle.  In the block codes, the information bits are followed by the parity bits while in convolution codes the information bits are spread along the sequence.  The block codes can be applied only for the block of data whereas convolution coding can be applied to a continuous data stream as well as to blocks of data.
31. 31. ADVANTAGES  Convolution coding is a popular error-correcting coding method used in digital communications.  The convolution operation encodes some redundant information into the transmitted signal.  It is simple and has good performance with low implementation cost.
32. 32. FACTORS AND PROPERTIES  The performance of a convolutional code depends on the coding rate and the constraint length.  Longer constraint length K More powerful code More coding gain  Smaller coding rate R=k/n More powerful code due to extra redundancy Less bandwidth efficiency
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