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  1. 1. TURBO CODES Presented by, K Swaraj Gowtham G Srinivasa Rao B Gopi Krishna
  2. 2. Turbo Codes Turbo Code Concepts Log Likelihood Algebra Interleaving & Concatenated Codes Encoding With R S C
  3. 3. IntroductionObjectives  Studying channel coding  Understanding channel capacity  Ways to increase data rate  Provide reliable communication link
  4. 4. Communication System Structural modular approach with various componentsFormatting Source Channel Access Multiplexing ModulationDigitization Coding Coding techniques send receive
  5. 5. CHANNEL CODING Can be categorized Wave form signal design Structuredsequences Better detectible signals Added redundancy Waveform Structured sequence M-ary signaling  Block Antipodal  Convolution Orthogonal  Turbo Trellis coded modulation
  6. 6. Structured Redundency Channel Channel Input word encoder Output word encoder n-bit k-bit codeword Code sequenceRedundancy = (n-k)Code rate = k/n
  7. 7. TURBO CODES A turbo code is a refinement of the concatenated encoding structure plus an iterative algorithm for decoding the associated code sequence. Concatenated coding scheme is a method for achieving large coding gains by combining two or more relatively simple building blocks or component codes.
  8. 8. TURBO CODE CONCEPTSLikelihood Functions: The mathematical foundations of hypothesis testing rests on Baye’s theorem. A Posteriori Probability (APP)of a decision in terms of a continuous-valued random variable x as
  9. 9. Before the experiment, there generallyexists an a priori probability P(d = i). Theexperiment consists of using Equation (1)for computing the APP, P(d = i|x), which canbe thought of as a “refinement” of the priorknowledge about the data, brought about byexamining the received signal x.
  10. 10. The Two-Signal Class Case: H1 > P (d = + 1| x) < P(d=-1|x) H2
  11. 11.  Binary logical elements 1 and 0 are represented electronically by voltages +1 and -1 where ‘d’ represents this voltages. The rightmost function, p(x|d = +1), shows the pdf of the random variable x conditioned on d = +1 being transmitted. The leftmost function, p(x|d = -1), illustrates a similar pdf conditioned on d = -1 being transmitted. A line subtended from Xk an arbitrary value taken from the full range of values of X, intercepts the two likelihood functions, yielding two likelihood values ℓ1 = p(xk|dk = +1) and ℓ2 = p(xk|dk = -1).
  12. 12. General expression for the MAP rule in terms of APPs is
  13. 13. The previous equation is expressed in terms of ratio, yielding the so-called likelihood ratio test, as follows
  14. 14. Log-Likelihood RatioBy logging on both sides to the MAP ruled APPs is To simplify the notation, it is rewritten as
  15. 15. At the decoder it is equal to This equation shows the output LLR of a systematicdecoder Consists of channel measurement , a priorknowledge of the data, and an extrinsic LLR stemmingsolely from the decoder.This soft decoder output L(dˆ ) isa real number that provides a hard decision as well as thereliability of that decision. The sign of L(dˆ ) denotes thehard decision; that is, for positive values of L(dˆ ) decidethat d = +1, and for negative values decide that d = -1.The magnitude of L(dˆ ) denotes the reliability of thatdecision.
  16. 16. Log Likelihood AlgebraFor Statistically independent data d, the sumof two log likelihood ratios are defined as
  17. 17. INTERLEAVING This is the concept that aids much in case of channels with memory. A channel with memory exhibits mutually dependent transmission impairments. A channel with multipath fading is an example for channel with memory.
  18. 18.  Errors caused due to disturbances in these types of channels – Burst Errors. Interleaving only requires a knowledge of span of the memory channel. Interleaving at the Tx’r side and de- interleaving at the Rx’r side causes the burst errors to be corrected.
  19. 19.  The interleaver shuffles the code symbols over a span of several block lengths or constraint lengths. It makes the memory channel look like memoryless one for decoder. Two types of interleavers:- ->Block Interleavers. ->Convolutional Interleavers.
  20. 20. Block Interleaving A block interleaver accepts the coded symbols in blocks from the encoder,permutes the symbols,and then feeds the rearranged ones to the modulator. The minimum end-to-end delay is (2MN-2M+2) symbol times where the encoded sequence is written as M*N array format.
  21. 21.  It needs a memory of 2MN symbol times. The choice of M is dependent on the coding scheme used. The choice of N for t-error- correcting codes must overbound the expected burst length divided by t.
  22. 22. Convolutional Interleaving In this type, the code symbols are sequentially shifted into the bank of N registers; each successive register contains J symbols more storage than the preceding one. In this case, the end-to-end delay is M(N-1) and the memory required is M(N-1)/2.
  23. 23. Concatenated Codes A concatenated code uses two levels on coding : an inner code and an outer code (higher rate).o Popular concatenated codes :- Convolutional codes with Viterbi decoding as the inner code and Reed-Solomon codes as the outer code.
  24. 24.  The purpose is to reduce the overall complexity, yet achieving the required error performance. However, the concatenated system performance is severely degraded by correlated errors among successive smbols.
  25. 25. Encoding with Recursivesystematic codes Turbo codes are generated by parallel concatenation of component convolutional codes Consider an encoder with data rate ½ ,constraint length K ,i/p to encoder dk. The corresponding code word (Uk,Vk) is
  26. 26.  G1 = { g1i } and G2 = { g2i } are the code generators, and dk is represented as a binary digit This encoder can be visualized as a discrete-time finite impulse response (FIR) linear system, giving rise to the familiar nonsystematic convolutional (NSC) code
  27. 27. An example for NSC code G1={111},G2={101}, K=3, bit rate = 1/2.
  28. 28.  At large Eb/N0 values, the error performance of an NSC is better than that of a systematic code infinite impulse response (IIR) convolutional codes [3] has been proposed as building blocks for a turbo code For high code rates RSC codes result in better error performance than the best NSC codes at any value of Eb/N0
  29. 29.  an RSC code, with K = 3, where ak is recursively calculated as g′i is respectively equal to g1i if uk = dk, and to g2i if vk = dk.
  30. 30. An ex for Recursive encoder andits trellis diagram:6(a),6(b)
  31. 31. Trellis diagram
  32. 32.  Example: Recursive Encoders and Their Trellis Diagramsa) Using the RSC encoder in Figure 6(a), verify the section of the trellis structure (diagram) shown in Figure 6(b).b) For the encoder in part a), start with the input data sequence{dk} = 1 1 1 0, and show the step-by- step encoder procedure for finding the output codeword.
  33. 33.  Validation of trellis diagram
  34. 34.  Encoding a bit sequence with RSC encoder
  35. 35. Concatenation of RSC codesGood turbo codes have been constructed from component codes having short lengths(K = 3 to 5).There is no limit to the number of encoders that may be concatenated.we should avoid pairing low-weight codewords from one encoder with low-weight codewords from the other encoder. Many such pairings can be avoided by proper design of the interleaver
  36. 36. Fig :parallel concatenation of RSC codes
  37. 37.  If the component encoders are not recursive, the unit weight input sequence 0 0 … 0 0 1 0 0 … 0 0 will always generate a low-weight codeword at the input of a second encoder for any interleaver design. if the component codes are recursive,a weight-1 input sequence generates an infinite impulse response
  38. 38.  For the case of recursive codes, the weight-1 input sequence does not yield the minimum-weight codeword out of the encoder Turbo code performance is largely influenced by minimum-weight codewords

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