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Convolutional Codes Representation and Encoding Many known codes can be modified by an extra code symbol or by deleting a symbol * Can create codes of almost any desired rate * Can create codes with slightly improved performance The resulting code can usually be decoded with only a slight modification to the decoder algorithm. Sometimes modification process can be applied multiple times in succession Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE

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Modification to Known Codes 1.Puncturing: delete a parity symbol (n,k) code (n-1,k) code 2.Shortening: delete a message symbol (n,k) code (n-1,k-1) code 3.Expurgating: delete some subset of codewords (n,k) code (n,k-1) code 4.Extending: add an additional parity symbol (n,k) code (n+1,k) code Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE

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Modification to Known Codes… 5.Lengthening: add an additional message symbol (n,k) code (n+1,k+1) code 6.Augmenting: add a subset of additional code words (n,k) code (n,k+1) code Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE

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Interleaving We have assumed so far that bit errors are independent from one bit to the next In mobile radio, fading makes bursts of error likely. Interleaving is used to try to make these errors independent again Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE Depth Of Interleaving 1 6 2 11 3 16 4 21 5 31 7 26 6 5 29 10 30 34 35 Length Order Bits Transmitted Order Bits Received

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Concatenated Codes Two levels of coding Achieves performance of very long code rates while maintaining shorter decoding complexity Overall rate is product of individual code rates Codeword error occurs if both codes fail. Error probability is found by first evaluating the error probability of “inner” decoder and then evaluating the error probability of “outer” decoder. Interleaving is always used with concatenated coding Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE

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Block Diagram of Concatenated Coding Systems Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE Data Bits Outer Encoder Interleave Inner Encoder Modulator Channel De - Modulator Inner Decoder De- Interleave Outer Decoder Data Out

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Practical Application : Coding for CD Each channel is sampled at 44000 samples/second Each sample is quantized with 16 bits Uses a concatenated RS code Both codes constructed over GF(256) (8-bits/symbol) Outer code is a (28,24) shortened RS code Inner code is a (32,28) extended RS code In between coders is a (28,4) cross-interleaver Overall code rate is r = 0.75 Most commercial CD players don’t exploit full power of the error correction coder Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE

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Practical Application: Galileo Deep Space Probe Uses concatenated coding Inner code rate is ½, constraint length 7 convolutinal encoder Outer Code (255,223) RS code over GF(256) – corrects any burst errors from convolutional codes Overall Code Rate is r= 0.437 A block interleaver held 2RS Code words Deep space channel is severely energy limited but not bandwidth limited Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE

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IS-95 CDMA The IS-95 standard employs the rate (64,6) orthogonal (Walsh) code on the reverse link The inner Walsh Code is concatenated with a rate 1/3, constraint length 9 convolutional code Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE Data Transmission in a 3 rd Generation PCS Proposed ETSI standard employs RS Codes concatenated with convolutional codes for data communication Requirements; Ber of the order of 10 -6 Moderate Latency is acceptable CDMA2000 uses turbo codes for data transmission ETSI has optional provisions for Turbo Coding

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Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE A Common Theme from Coding Theory The real issue is the complexity of the decoder. For a binary code, we must match 2 n possible received sequences with code words Only a few practical decoding algorithms have been found: Berlekamp-Massey algorithm for clock codes Viterbi algorithm (and similar technique) for convolutional codes Code designers have focused on finding new codes that work with known algorithms

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Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE Block Versus Convolutional Codes Block codes take k input bits and produce n output bits, where k and n are large there is no data dependency between blocks useful for data communcations Convolutional codes take a small number of input bits and produce a small number of output bits each time period data passes through convolutional codes in a continuous stream useful for low- latency communications

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Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE Convolutional Codes k bits are input, n bits are output Now k & n are very small (usually k=1-3, n=2-6) Input depends not only on current set of k input bits, but also on past input. The number of bits which input depends on is called the "constraint length" K. Frequently, we will see that k=1

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Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE Example of Convolutional Code k=1, n=2, K=3 convolutional code

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Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE Example of Convolutional Code k=2, n=3, K=2 convolutional code

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Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE Representations of Convolutional Codes Encoder Block Diagram (shown above) Generator Representation Trellis Representation State Diagram Representation

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Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE Convolutional Code Generators One generator vector for each of the n output bits: The length of the generator vector for a rate r=k/n code with constraint length K is K The bits in the generator from left to right represent the connections in the encoder circuit. A “1” represents a link from the shift register. A “0” represents no link. Encoder vectors are often given in octal representation

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Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE Example of Convolutional Code k=1, n=2, K=3 convolutional code

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Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE Example of Convolutional Code k=2, n=3, K=2 convolutional code

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Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE State Diagram Representation 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) Arcs connecting states represent allowable transitions Arcs are labeled with output bits transmitted during transition

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Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE Example of State Diagram Representation Of Convolutional Codes k=1, n=2, K=3 convolutional code

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Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE Trellis Representation of Convolutional Code State diagram is “unfolded” a function of time Time indicated by movement towards right 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. Allowable transitions are denoted by connects between states transitions may be labeled with transmitted bits

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Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE Example of Trellis Diagram k=1, n=2, K=3 convolutional code

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Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE Encoding Example Using Trellis Representation k=1, n=2, K=3 convolutional code We begin in state 00: Input Data: 0 1 0 1 1 0 0 Output: 0 0 1 1 0 1 0 0 10 10 1 1

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Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE Distance Structure of a Convolutional Code The Hamming Distance between any two distinct code sequences and is the number of bits in which they differ: The minimum free Hamming distance d free of a convolutional code is the smallest Hamming distance separating any two distinct code sequences:

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Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE Search for good codes We would like convolutional codes with large free distance must avoid “catastrophic codes” Generators for best convolutional codes are generally found via computer search search is constrained to codes with regular structure search is simplified because any permutation of identical generators is equivalent search is simplified because of linearity.

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Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE Best Rate 1/2 Codes

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Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE Best Rate 1/3 Codes

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Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE Best Rate 2/3 Codes

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Error Control Coding, © Brian D. Woerner, reproduced by: Erhan A. INCE Summary of Convolutional Codes Convolutional Codes are useful for real-time applications because they can be continously encoded and decoded We can represent convolutional codes as generators, block diagrams, state diagrams, and trellis diagrams We want to design convolutional codes to maximize free distance while maintaining non-catastrophic performance

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