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Turbo Codes Vidya T Ramachandran April 22, 2008

Turbo Codes Vidya T Ramachandran April 22, 2008. Abstract.

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Turbo Codes Vidya T Ramachandran April 22, 2008

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  1. Turbo Codes Vidya T Ramachandran April 22, 2008

  2. Abstract • In 1993, group of French researchers, Berrou, Glavieux and Thitimajshima presented a new class of error correction codes, termed as “Turbo Codes”. These codes were shown to achieve a performance in terms of Bit Error Rate (BER) within 0.7 dB of the Shannon capacity limit. Turbo-codes promise the attainment of the ‘Holy Grail’ of communication theory. They have a very wide range of applications mainly in wireless communications, ranging from the third generation mobile systems to deep-space exploration.

  3. Outline • Introduction • Channel Capacity • Why turbo codes perform so well • Review of Convolutional codes • RSC Encoding • Turbo Code Architecture • Encoder • Interleaver • Decoder • Example: UMTS Turbo Encoder-Interleaver-Decoder • Performance • Practical Issues • Applications • Conclusion

  4. Introduction • Powerful class of error correcting codes • Iterative decoding • Discovered by Berrou, Glavieux and Thitimajshima in 1993 at ICC, Genève, Switzerland • Advantages • Come closest to approaching the Shannon capacity limit on maximum achievable data transfer rate over a noisy channel • For a certain BER, power can be decreased • Drawbacks • High decoder complexity • Relatively high latency due to interleaving and iterations when decoding • Applications in areas where power saving is required or low SNR is available Claude Berrou Alain Glavieux

  5. Coding Timeline From: http://userver.ftw.at/%7Ejossy/turbo/2004/lecture01.pdf Clearly, the coding world can be divided into 2 time zones: BTC & ATC – Before and After Turbo Codes…

  6. Channel Capacity – Shannon’s Limit Claude Shannon, “A mathematical theory of communication,” Bell Systems Technical Journal, 1948 Every channel has associated with it a capacity “C” – Measured in bits per channel use (modulated symbol) The channel capacity is an upper bound on information rate “r” – There exists a code of rate r < C that achieves reliable communications. – Showed that the Bit Error probability approaches zero as block length n of the code goes to infinity by selecting a rate r < C code at random. Shannon playing with mechanical mouse Shannon in his office at Bell Labs

  7. Why Turbo codes perform so well • Linear Code - code for which the modulo-2 sum of two valid code words (XOR-ing each bit position) is also a valid code word. • Good linear code • has mostly high-weight code words except, all- zeros code word. • Desired as they are distinct, so easier for decoder to distinguish. • Use turbo encoder with Interleaver to reduce low-weight code words • One of the two encoders will occasionally produce a low-weight output, the probability that both encoders simultaneously produce a low-weight output is extremely small. • Random codes achieve the best performance • Shannon showed that as n→∞, random codes achieve channel capacity. • However, not feasible because code must contain enough structure so that decoding can be realized with actual hardware. • With turbo codes: • The non-uniform Interleaver adds apparent randomness to the code. • Yet, they contain enough structure so that decoding is feasible.

  8. Review: Convolutional codes Input data shifted into and along shift register k bits at once k binary inputs n binary outputs K-1 delay elements (linear shift registers) Coefficients are either 1 or 0 Operation - Addition: XOR rate r = k / n K = Constraint length – number of bits that each output depends on. Non-systematic codes encoder’s input bits do not appear at its output Correct by using systematic RSC codes Rate r = ½ Convolutional encoder with constraint length K = 3 From: M.C. Valenti, “Turbo Codes and Iterative Processing,” in Proc. IEEE New Zealand Wireless Communications Symposium, (Auckland New Zealand), Nov. 1998

  9. RSC - Recursive Systematic Convolutional Encoding An RSC encoder is constructed from a standard convolutional encoder by feeding back one of the outputs. An RSC code is systematic. – The input bits appear directly in the output. An RSC encoder is an Infinite Impulse Response(IIR) Filter. –An arbitrary input will cause a “good” (high weight) output with high probability – Some inputs will cause “bad” (low weight) outputs. From: M.C. Valenti, “Turbo Codes and Iterative Processing,” in Proc. IEEE New Zealand Wireless Communications Symposium, (Auckland New Zealand), Nov. 1998.

  10. Turbo Encoder • From: A study of turbo codes for UMTS third generation cellular standard by Teodor Iliev, University of Rousse, ACM International Conference Proceedings, 2007, Volume 285. Two constituent parallel RSC encoders to interleaved versions of the same information u to be transmitted. A non-uniform Interleaver scrambles the ordering of bits at the input of the second encoder. Uses a pseudo-random interleaving pattern. Increase the code rate via a puncturing technique which enables to select the coded bits following a particular pattern.

  11. Parallel Concatenated Encoding • It consists of two conventional feedback shift-register-based convolutional encoders whose inputs are separated by an Interleaver. • The key to this technique lies in the recursive nature of the encoder and the impact of the Interleaver on the data bits. • The two constituent encoders are coding the same information sequence u but in a different order. • For each input binary information symbol ui , we keep the systematic output of xsi = ui of the first RSC encoder, and the parity outputs, x1pi and x2piof both RSC encoders • All these symbols are then multiplexed in order to form the following turbo-coded sequence: {...,ui ,x1pi ,x2pi,ui+1 ,x1pi +1 ,x2pi+1 ,ui+2 ,x1pi +2 ,x2pi+1,...} • Code rate is thus R=1/3. • Increase the code rate via a puncturing technique to say, ½ as follows: {...,ui , x1pi , ui+1 , x1pi +1 , ui+2 , x1pi +2 ,...} • Uses tail bits (sequence of 3 zeros) to return encoders back to the all-zeros state

  12. UMTS Turbo Encoder From: M.C. Valenti and J. Sun, “Turbo Codes,” Chapter 12 of Handbook of RF and Wireless Technologies, (editor: F. Dowla), Newnes, 2004, Pages 375-78.

  13. Output Stream Format • Input three feedback bits generated immediately after encoding k-bit code word • Output stream format: • X1 Z1 Z’1X2 Z2 Z’2 …XL ZL Z’L; XL+1 ZL+1 XL+2 ZL+2 XL+3 ZL+3;X’L+1Z’L+1 X’L+2Z’L+2 X’L+3Z’L+3 • L data bits and 3 tail bits for 3 tail bits for • their associated upper encoder lower encoder • 2L parity bits & their 3 parity bits & their 3 parity bits • (total of 3L bits) • Total number of coded bits = 3L + 12 • Code rate: r = L / (3L + 12)  1/3

  14. UMTS Interleaver: • Device that rearranges the order of the data bits in a prescribed, but irregular, manner. • Although the same set of data bits is present at the output of the Interleaver, the order of these bits has been changed, much like a shuffled deck of cards • Without the Interleaver, the two constituent encoders would receive the data in the exact same order and thus—assuming identical constituent encoders—their outputs would be the same. • Thus, the output of the second encoder will almost surely be different than the output of the first encoder • Quite different than the rectangular interleaves that are commonly used in wireless systems to help break up deep fades • Rectangular channel Interleaver tries to space the data out according to a regular pattern, a turbo code Interleaver tries to randomize the ordering of the data in an irregular manner.

  15. UMTS Interleaver: Inserting Data into Matrix Data is fed row-wise into a R by C matrix. – R = 5, 10, or 20. – 8 ≤C ≤256 – If L < RC then matrix is padded with dummy characters. Slide from : Iterative Solutions Coded Modulation Library (ISCML) - Theory of Operation ppt, Oct. 3, 2005, Matthew Valenti at http://www.iterativesolutions.com/idownload.htm

  16. UMTS Interleaver: Reading Data From Matrix Intra-Row Permutations Data is permuted within each row. Data is read from matrix column-wise. X’1 = X40 X’38 = X24 X’2 = X26 X’39 = X16 X’3 = X18 …X’40 = X8 Slide from : Iterative Solutions Coded Modulation Library (ISCML) – Theory of Operation ppt, Oct. 3, 2005, Matthew Valenti at http://www.iterativesolutions.com/idownload.htm

  17. UMTS Interleaver: Inter-Row Permutations Rows are permuted. – If R = 5 or 10, the matrix is reflected about the middle row. – For R=20 the rule is more complicated and depends on L. Slide from : Iterative Solutions Coded Modulation Library (ISCML) – Theory of Operation ppt, Oct. 3, 2005, Matthew Valenti at http://www.iterativesolutions.com/idownload.htm

  18. Turbo Decoder • Divide-and-conquer approach From: M.C. Valenti and J. Sun, “Turbo Codes,” Chapter 12 of Handbook of RF and Wireless Technologies, (editor: F. Dowla), Newnes, 2004, Pages 375-78.

  19. Iterative Decoding • Ui - modulating code bit – 0 or 1 – hard value • Yi- corresponding received signal – any value – soft value • Log-Likelihood Ratio (LLR) is used as input to decoder • R ( Ui ) = ln P(Yi | Ui = 1) • P(Yi | Ui = 0) • For BPSK over AWGN channel with noise variance of σ2, • LLR = R (Ui ) = 2 Yi / σ2 • For each data bit Xi , decoder computes the following LLR: •  ( Xi ) = ln P(Xi = 1| Y1 . . . Yn) • P(Xi = 0| Y1 . . . Yn) • Soft input soft output (SISO) processors compute 2 LLR estimates. •  ( Xi ) = 1( Xi ) + 2( Xi ) • Final LLR estimate = LLR from upper encoder + LLR from lower encoder

  20. SISO Processor • Iterative decoding • first SISO processor passes its LLR output to the input of the second SISO processor and vice versa • Improve performance by sharing LLR estimates between the 2 SISO processors. • feedback operation reminiscent of the feedback between exhaust and intake compressor in a turbo engine. • Use extrinsic information w( Xi ) to prevent positive feedback • subtract the systematic input of each SISO from its output prior to sharing information with the other decoder. • SISO processor uses a trellis diagram to represent all possible sequences of encoder states • sweeps through the labeled trellis in a prescribed manner to obtain LLR estimates of each data bit using: • Soft output Viterbi algorithm (SOVA) • maximum a posteriori (MAP) algorithm • SISO processor uses logarithmic version of the MAP algorithm called log-MAP

  21. Log-MAP Algorithm: • Log-MAP is similar to the Viterbi algorithm. • Multiplications become additions. • Additions become special “max*” operator (Jacobi logarithm) • max∗(x, y) = ln (ex + ey) • = max (x, y) + ln (1 + e−|y−x|) • = max (x, y) + fc (|y − x|) • Implementation: • Sweep through the trellis in forward direction using modified Viterbi algorithm. • Sweep through the trellis in backward direction using modified Viterbi algorithm. • Determine LLR for each trellis section. • Determine output extrinsic info for each trellis section. • Flavors of Log-MAP algorithm are max-log-MAP algorithm, constant-log-MAP and linear-log-MAP

  22. Characteristics of Turbo Codes • Turbo codes have extraordinary performance at low SNR. • Very close to the Shannon limit. • Due to a low multiplicity of low weight code words. • However, turbo codes have a BER “floor”. • This is due to their low minimum distance. • Performance improves for larger block sizes. • Larger block sizes mean more latency (delay). • However, larger block sizes are not more complex to decode. • The BER floor is lower for larger frame/Interleaver sizes • The complexity of a constraint length KTC turbo code is the same as a K = KCC convolutional code, where: • KCC ≈2 + KTC + log2 * (number decoder iterations)

  23. Performance as a function of number of iterations Simulation Setup: R = ½ ; K= 5 N= 256 X 256 Interleaver matrix G1 = {1 1 1 1 1} ; G2 = {1 0 0 0 1} Expected: Shannon’s Limit Binary Modulation R = ½ @ Eb/No = 0 dB BER (Pe) = 0 (or 10-5) Obtained: For SNR > 0, BER decreases as a function of decoding step p For p = 18, @ Eb/No = 0.7 dB BER (Pe)  10-5  Performances are at 0.7 dB from Shannon’s Limit ! From: BERROU, C., GLAVIEUX, A., and THITIMAJSHIMA, P.: “Near Shannon limit error-correcting coding: turbo codes”, Proc. IEEE International Conference Communication, Geneva, Switzerland, 1993, pp.1064–1070

  24. Performance as a function of the Interleaver size R = 1/2 ; K=5 ; 18 decoder iterations As frame size increases, performance improves. However, as Interleaver size increases, decoder latency also increases. High latency means low BER! Tradeoff between performance and latency Exploited in wireless communication systems. From: M.C. Valenti, “Turbo Codes and Iterative Processing,” in Proc. IEEE New Zealand Wireless Communications Symposium, (Auckland New Zealand), Nov. 1998.

  25. Performance comparison of two rate ½ codes Convolutional code K = 15 ; free distance* (dfree) = 18 Turbo code K = 5 ; L = 65,536 ; free distance = 6 “Error Floor” effect at low BER for turbo codes. Increasing dfree improves bit error performance. Turbo codes have a comparatively lower dfree due to small number of free distance code words. Hence, it maybe better to use convolutional codes at high SNR values. From: M.C. Valenti, “Turbo Codes and Iterative Processing,” in Proc. IEEE New Zealand Wireless Communications Symposium, (Auckland New Zealand), Nov. 1998. *dfree - minimum hamming weight of all non-zero code words

  26. Practical Implementation Issues • Error floor • BER curve begins to flatten at higher SNR • due to the presence of a few low-weight code words that become significant only at high SNRs. • hinders the ability of a turbo code to achieve extremely small BERs. • Solution: serially concatenated convolutional codes (SCCC) • excellent performance at high SNR – error floor pushed way down to BER  10-10, but worse at low SNR • Latency • Increased delay due to large Interleaver sizes in encoder/decoder • Encoder/Decoder Delay = ( information bit period ) X ( latency ) • Example : For 8kbps (speech transmissions), N = 65,536 bits • delay = 65536 / 8 = 8192 ms  8s • unacceptable delay in voice services.

  27. Issues • Complexity • If max-log-MAP algorithm is used, then each half-iteration would require that the Viterbi algorithm be executed twice. • Example: If 8 full-iterations are executed, then the Viterbi algorithm will be invoked 32 times. • contrast to the decoding of a conventional convolutional code, which only requires the Viterbi algorithm to be executed once. • Solution: • Turbo decoding progresses until a fixed number of iterations have completed. • However, the decoder will often converge early. • Can stop once it has converged (i.e. BER = 0). • Stopping early can greatly increase the throughput. • For a simulation, it is reasonable to automatically halt once the BER goes to zero. • Simply halt the decoder iterations once the entire frame has been completely corrected.

  28. Issues • Memory Limitations • Storing the entire beta trellis can require a significant amount of memory: • 8L states • For L=5114 there will have to be 40,912 stored values. • The values only need to be single-precision floats. • An alternative is to use a “sliding window” approach. • The metrics for only a portion of the code trellis are saved in memory • Rather than running the MAP algorithm over the entire trellis, it is only run over each window • Channel estimation • For an AWGN channel, the SNR must be known. • For a fading channel with random amplitude fluctuations, the per-bit gain of the channel must also be known. • complicated by the fact that turbo codes typically operate at very low SNR.

  29. Applications • Development of other new codes • Serial Concatenated Block Codes • LDPC - Low-Density Parity Check Codes, or Gallagher codes • RCA - Repeat-Accumulate Code • Deep-space exploration • At the cost of lower bandwidth efficiencies & low BER, delay not important • Used in the Mars Exploration: Pathfinder mission,1997 • FEC Coding in UMTS: 3G third generation mobile radio standard • for application both to speech services, where latency must be minimized (BER = 4 X 10-2) and to data services that must provide very low BER (10-5). • Turbo iterative decoding principle • allows separate decoding and synchronization in receivers without increasing complexity.

  30. Conclusions • Practical means of attaining the Shannon capacity bounds for a communication channel • Parallel concatenated encoding • Iterative decoding • Interleaver design, heart of turbo coding. • Significant latency introduced. • Larger Interleaver size, means a longer decoding delay, gives a lower bit error rate. • Best suited at low BER in the range of 10-4 to 10-6 • Applications ranging from mobile phones to satellite systems. • Not the ultimate error correction codes, but groundbreaking, giving rise to new codes.

  31. References • [1] BERROU, C., GLAVIEUX, A., and THITIMAJSHIMA, P.: “Near Shannon limit error-correcting coding: turbo codes”, Proc. IEEE International Conference Communication, Geneva, Switzerland, 1993, pp.1064–1070 • [2] M.C. Valenti and J. Sun, “Turbo Codes,” Chapter 12 of Handbook of RF and Wireless Technologies, (editor: F. Dowla), Newnes, 2004, Pages 375-78. • [3] Burr, A., Dept. of Electron., York Univ., “Turbo-codes: the ultimate error control codes?” IEEE ELECTRONICS & COMMUNICATION ENGINEERING JOURNAL, Aug 2001, Volume: 13, Issue 4, Pages: 155-165 • [4] M.C. Valenti, “Turbo Codes and Iterative Processing,” in Proc. IEEE New Zealand Wireless Communications Symposium, (Auckland New Zealand), Nov. 1998. • http://www.csee.wvu.edu/~mvalenti/turbo.html • [5] Iterative Solutions Coded Modulation Library (ISCML ) - open source toolbox for simulation of modern communication systems - Theory of Operation power presentation, Oct. 3, 2005, Matthew Valenti. http://www.iterativesolutions.com/idownload.htm • [6] Bernard Sklar, “A primer on turbo code concepts,” IEEE Communications Magazine, vol. 35, no. 12, pp. 94-102, Dec. 1997.

  32. References • [7] Lecture at TU Vienna: Theory and Design of Turbo and Related Codes, Jossy Sayir and Gottfried Lechner, Senior Researchers at Vienna Research Center for Telecommunications (FTW) • [8] S. A. Barbulescu and S. S. Pietrobon, "Turbo codes: A tutorial on a new class of powerful error correcting coding schemes, Part 2: Decoder design and performance," J. Elec. and Electron. Eng., Australia, vol. 19, pp. 143-152, Sep. 1999 • [9] Application and standardization of turbo codes in third-generation high-speed wireless data services by Lee, L.-N.; Hammons, A.R., Jr.; Feng-Wen Sun; Eroz, M.;Vehicular Technology, IEEE Transactions on Volume 49,  Issue 6,  Nov. 2000 Page(s):2198 - 2207 • [10] Turbo code performance and design trade-offs by Achiba, R.; Mortazavi, M.; Fizell, W.;MILCOM 2000. 21st Century Military Communications Conference ProceedingsVolume 1, 22-25 Oct. 2000 Page(s):174 - 180 vol.1 • [11] Class presentation slides on “LDPC Codes and Trellis Coded Modulation” by Dr.Sam Shanmughan in EECS 865 (Wireless Communication), Fall ‘07.

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