Convolutional Codes Representation and Encoding

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Convolutional CodesRepresentation 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

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Modification to Known Codes

• Puncturing delete a parity symbol
• (n,k) code ? (n-1,k) code
• Shortening delete a message symbol
• (n,k) code ? (n-1,k-1) code
• Expurgating delete some subset of codewords
• (n,k) code ? (n,k-1) code
• Extending add an additional parity symbol
• (n,k) code ? (n1,k) code

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Modification to Known Codes

• 5. Lengthening add an additional message symbol
  
• (n,k) code ? (n1,k1) code
• 6. Augmenting add a subset of additional code
  words
• (n,k) code ? (n,k1) code

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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

Depth Of Interleaving
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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

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Block Diagram of Concatenated Coding Systems
Data Bits
Outer Encoder
Inner Encoder
Modulator
Interleave
Channel
Data Out
Inner Decoder
Outer Decoder
De-Modulator
De- Interleave
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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 dont exploit full
  power of the error correction coder

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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

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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

Data Transmission in a 3rd 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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• A Common Theme from Coding Theory
• The real issue is the complexity of the decoder.
• For a binary code, we must match 2n 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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• 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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• Convolutional Codes
• k bits are input, n bits are output
• Now k n are very small (usually k1-3, n2-6)
• 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 k1

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Example of Convolutional Code k1, n2, K3
convolutional code
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Example of Convolutional Code k2, n3, K2
convolutional code
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• Representations of Convolutional Codes
• Encoder Block Diagram (shown above)
• Generator Representation
• Trellis Representation
• State Diagram Representation

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• Convolutional Code Generators
• One generator vector for each of the n output
  bits
• The length of the generator vector for a rate
  rk/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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Example of Convolutional Code k1, n2, K3
convolutional code
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Example of Convolutional Code k2, n3, K2
convolutional code
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• 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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Example of State Diagram Representation Of
Convolutional Codes k1, n2, K3
convolutional code
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• 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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Example of Trellis Diagram k1, n2, K3
convolutional code
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Encoding Example Using Trellis Representation k1,
n2, K3 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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• 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 dfree of a
  convolutional code is the smallest Hamming
  distance separating any two distinct code
  sequences

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• 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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Best Rate 1/2 Codes
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Best Rate 1/3 Codes
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Best Rate 2/3 Codes
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• 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