1. Turbo Codes

2. 2 A Need for Better Codes Designing a channel code is always a tradeoff between energy efficiency and bandwidth efficiency. Lower rate Codes  correct more errors  the communication system can operate with a lower transmit power, transmit over longer distances, tolerate more interference, use smaller antennas and transmit at a higher data rate.  have a large overhead and are hence more heavy on bandwidth consumption  decoding complexity grows exponentially with code length, and long (low-rate) codes set high computational requirements to conventional decoders. Encoding is easy but decoding is hard

3. 3 Claude Shannon’s Limit

4. 4 Motivation If the transmission rate, the bandwidth and the noise power are fixed, we get a lower bound on the amount of energy that must be expended to convey one bit of information. Hence, Shannon capacity sets a limit to the energy efficiency of a code. Although Shannon developed his theory in the 1940s, several decades later the code designs were unable to come close to the theoretical bound. Even in the beginning of the 1990s, the gap between this theoretical bound and practical implementations was still at best about 3dB. “practical codes required about twice as much energy as the theoretical predicted minimum”  new codes were sought that would allow for easier decoding  using a code with mostly high-weight code words  combining simple codes in a parallel fashion, so that each part the code can be decoded separately with less complex decoders and each decoder can gain from information exchange with others.

5. 5 Turbo Codes Berrou & Glavieux 1993 International Conf. on Commun. (ICC) Rate ½ performance within 0.5 dB of Shannon capacity. Patent held by France Telecom. Features: Parallel code concatenation Can also use a serial concatenation Nonuniform interleaving Recursive systematic encoding Usually RSC convolutional codes are used. Can use block codes. Iterative decoding algorithm. Optimal approaches: BCJR/MAP, SISO, log-MAP Suboptimal approaches: max-log-MAP, SOVA

6. 6 Concatenated Coding

7. 7 Error Propagation If a decoding error occurs in a codeword,  results in a number of subsequent data errors  the next decoder may not be able to correct the errors  the performance might be improved if these errors were distributed between a number of separate codewords  Can be achieved using an interleaver/de-interleaver.

8. 8 Interleaver/de-interleaver

9. 9 If the rows of the interleaver are at least as long as the outer codewords, and the columns at least as long as the inner data blocks, each data bit of an inner codeword falls into a different outer codeword. Hence, if the outer code is able to correct at least one error, it can always cope with single decoding errors in the inner code.

10. 10 Example So, how to overcome this ?

11. 11 Iterative decoding  If the output of the outer decoder were reapplied to the inner decoder it would detect that some errors remained, since the columns would not be codewords of the inner code  Iterative decoder: to reapply the decoded word not just to the inner code, but also to the outer, and repeat as many times as necessary.  However, it is clear that this would be in danger of simply generating further errors. One further ingredient is required for the iterative decoder.

12. 12 Soft-In, Soft-Out (SISO) decoding  The performance of a decoder is significantly enhanced if, in addition to the ‘hard decision’ made by the demodulator on the current symbol, some additional ‘soft information’ on the reliability of that decision is passed to the decoder.  For example, if the received signal is close to a decision threshold (say between 0 and 1) in the demodulator, then that decision has low reliability, and the decoder should be able to change it when searching for the most probable codeword. Making use of this information in a conventional decoder, called soft decision decoding, leads to a performance improvement of around 2dB in most cases.