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A short and detailed presentation about two types of codes.

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In the tree diagram, a solid line represents input information bit 0 and a dashed line represents input information bit 1.

The corresponding output encoded bits are shown on the branches of the tree.

An input information sequence defines a specific path through the tree diagram from left to right.

- 1. REED SOLOMON CODE AND CONVOLUTION CODE
- 2. REED SOLOMON CODE
- 3. CONTENTS Introduction Properties of RS code RS Encoder RS Decoder Software Implementation Advantages Disadvantages Applications
- 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. 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. The following diagram shows a typical Reed-Solomon codeword: k 2t n Data Parity
- 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. 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. 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. ENCODER
- 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. DECODER
- 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. 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. 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. APPLICATIONS Data Storage Bar Code Satellite Broadcasting Spread-Spectrum System Ultra Wideband(UWB)
- 17. CONVOLUTION CODE
- 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. 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. 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. 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. CONVOLUTIONAL CODE ENCODER + + Shift Register Linear Algebraic Function Generator
- 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. 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. REPRESENTATION OF CONVOLUTION CODES State Diagram Tree Diagram Trellis Diagram
- 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. TREE DIAGRAM 1101 k=1 K=3 n=2
- 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. TRELLIS DIAGRAM
- 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. 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. 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
- 33. THANK YOU E047 E048 E049 E050 E056 E058CREATED

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