TURBO CODES

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TURBO CODES

• Michelle Stoll

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A Milestone in ECCs

• Based on convolutional codes
• multiple encoders used serially to create a
  codeword
• defined as triple (n, k, m)
• n encoded bits generated for every k data bits
  recd, where m represents the number of memory
  registers used

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Enhancements

• Added features include
• concatenated recursive systematic encoders
• pseudo-random interleavers
• soft input/soft output (SISO) iterative decoding

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Accolades

• TCs nearly achieve Shannons channel capacity
  limit first to get within 0.7 dB
• Do not require high transmission power to deliver
  low bit error rate
• Considered most powerful class of ECCs to-date

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Sidebar Shannon Limit

• Defines the fundamental transmission capacity of
  a communication channel
• Claude Shannon from Bell Labs proved
  mathematically that totally random sets of
  codewords could achieve channel capacity,
  theoretically permitting error-free transmission

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Shannon Limit, cont

• Use of random sets of codewords not a practical
  solution
• channel capacity can only be attained when k data
  bits mapped to n code symbols approach infinity
• Cost of a code, in terms of computation required
  to decode it, increases closer to the Shannon
  limit
• Coding paradox find good codewords the deliver
  BERs close to the Shannon limit, but not overly
  complex
• ECCs addressing both have been elusive for years
• until advent of TCs, best codes were outside 2 dB
  of Shannons Limit
• All codes are good, except the ones we can think
  of.
• Folk theorem

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Performance Bounds
The performance floor is in the vicinity of a BER
of 10-5
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Turbo Code History

• Claude Berrou, Alain Glavieux, and Punja
  Thitimajshima presented their paper Near Shannon
  Limit Error-Correcting Coding and Decoding
  Turbo Codes in 1993
• Their results were received with great skepticism
  
• in fact, the paper was initially rejected
• independent researchers later verified their
  simulated BER performance

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Anatomy Encoder

• Two encoders, parallel concatenation of codes
• can use the same clock, decreasing delay
• d blocks of n bits length sent to each encoder
• encoder 1 receives bits as-is, encodes the parity
  bits y1, and concatenates them with original data
  bits
• encoder 2 receives pre-shuffled bit string from
  interleaver, encodes the parity bits y2
• multiplexer receives a string of size 3n of
  parity bits and original data bits from encoder
  1, and parity bits from encoder 2

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Turbo Encoder Schematic
Example original data 01101 Encoder 1
creates parity bits 10110 and appends original
01101 Encoder 2 receives pre-shuffled bit string
and create parity bits 11100 Multiplexer
receives 011011011011100
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Non-Uniform Interleaver

• Irregular permutation map used to produce a
  pseudo-random interleaver no block interleaving
• Nonuniform interleaving assures a maximum
  scattering of data, introducing quasi-random
  behavior in the code
• recall Shannons observation
• Operates between modular encoders to permute all
  poor input sequences (low-weight CWs) into good
  input sequences producing large-weight output CWs

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Anatomy Decoder

• Decoder is most complex aspect of turbo codes
• But imposes the greatest latency in the process
  as it is serial, iterative
• Two constituent decoders are trying to solve the
  same problem from different perspectives
• Decoders make soft decisions about data
  integrity, passing the extrinsic bit reliability
  information back and forth
• Hence the name turbo in reference to a turbo
  engine

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Decoder, cont

• Inspects analog signal level of the received bits
• then turns the signal into integers which lend
  confidence to what the value should actually be
• Next, examines parity bits and assigns bit
  reliabilities for each bit
• Bit reliabilities are expressed as log likelihood
  ratios that vary between a positive and negative
  bound
• in practice, this bound is quite large, between
  -127 and 127
• the closer the LLR is to one side, the greater
  the confidence assigned one way or the other.

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Decoder, contLog Likelihood Ratio (LLR)

• The probability that a data bit d 1, Pr d
  1, is expressed as
• What is passed from one decoder to the other are
  bit reliabilities
• its computations with respect to the estimation
  of d, without taking its own input into account
• The input related to d is thus a single shared
  piece of information

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Decoder, cont

• Decoder modules dec1 and dec2 receive input
• dec1 passes its bit reliability estimate to
  interleaver, dec2
• if dec1 successful, it wouldve passed few or no
  errors to dec2
• Decoder module dec2 processes its input as well
  as the bit reliability from dec1
• refines the confidence estimate, then passes to
  de-interleaver
• This completes the first iteration.
• If no further refinements needed (i.e. acceptable
  confidence) the data is decoded and passed to
  upper layer
• Otherwise, the output is passed back to dec1 for
  another iteration

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Turbo Decoder Schematic
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Decoding Drawbacks

• To achieve near-optimum results, a relatively
  large number of decoding iterations are required
  (on the order of 10 to 20)
• This increases computational complexity and
  output delay
• one way to mitigate delay is to use a stop rule
• Select some pre-determined number of interations
  to perform
• if convergence is detected before the number is
  reached, stop

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Puncturing

• Another way to address latency is through code
  puncturing
• puncturing will change the code rate, k/n,
  without changing any of its attributes
• instead of transmitting certain redundant values
  in the codeword these values are simply not
  transmitted
• i.e. a Rate 1/2 code can be increased to a Rate
  2/3 code by dropping every other output bit from
  the parity stream

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Complexity

• Because the decoder is comprised of two
  constituent decoders, it is twice as complex as a
  conventional decoder when performing a single
  iteration
• two iterations require twice the computation,
  rendering it four times as complex as a
  conventional decoder

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Latency

• Latency on the decoding side is the biggest
  drawback of Turbo Codes
• Decoding performance is influenced by three broad
  factors interleaver size, number of iterations,
  and the choice of decoding algorithm
• these can be manipulated, with consequences

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Ongoing Research

• Turbo coding is responsible for a renaissance in
  coding research
• Turbo codes, turbo code hybrids being applied to
  numerous problems
• Multipath propagation
• Low-density parity check (LDPC)
• Software implementation!
• turbo decoding at 300 kbits/second using 10
  iterations per frame. With a stopping rule in
  place, the speed can be doubled or tripled

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Turbo Codes in Practice

• Turbo codes have made steady inroads into a
  variety of practical applications
• deep space
• mobile radio
• digital video
• long-haul terrestrial wireless
• satellite communications
• Not practical for real-time, voice

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More Information

• Excellent high-level overview
• Guizzo, Erico. Closing in on the Perfect
  Code, IEEE Spectrum, March 2004.
• Very informative four-part series on various
  aspects of TCs
• Gumas, Charles Constantine. Turbo Codes ev up
  error-correcting performance. (part I in the
  series) EE Times Network at http//archive.chipcen
  ter.com/dsp/DSP000419F1.html
• The paper that started it all
• Berrou, Glavieux, and Thitimajshima. Near
  Shannon Limit Error-Correcting Coding and
  Decoding Turbo Codes Ecole Superieure des
  Telecommunications de Bretagne, France. 1993.
• Complete bibliography soon available on my CS522
  page