Serially concatenated turbo codes have been proposed to satisfy requirements for low bit- and word-error rates and for low (in comparison with related previous codes) complexity of coding and decoding algorithms and thus low complexity of coding and decoding circuitry. These codes are applicable to such high-level modulations as octonary phase-shift keying (8PSK) and 16-state quadrature amplitude modulation (16QAM); the signal product obtained by applying one of these codes to one of these modulations is denoted, generally, as "serially concatenated trellis-coded modulation" ("SCTCM"). These codes could be particularly beneficial for communication systems that must be designed and operated subject to limitations on bandwidth and power.

Some background information is prerequisite to a meaningful summary of this development. Trellis-coded modulation (TCM) is now a well- established technique in digital communications. A turbo code combines binary component codes (which typically include trellis codes) with interleaving. A turbo code of the type that has been studied prior to this development is composed of parallel concatenated convolutional codes (PCCCs) implemented by two or more constituent systematic encoders joined through one or more interleavers. The input information bits feed the first encoder and, after having been scrambled by the interleaver, enter the second encoder. A code word of a parallel concatenated code consists of the input bits to the first encoder followed by the parity check bits of both encoders. The suboptimal iterative decoding structure for such a code is modular, and consists of a set of concatenated decoding modules — one for each constituent code — connected through an interleaver identical to the one in the encoder side. Each decoder performs weighted soft decoding of the input sequence. PCCCs yield very large coding gains at the cost of a reduction in the data rate and/or an increase in bandwidth.As its full name suggests, SCTCM merges serially concatenated convolutional codes (SCCCs) with TCM. SCTCM is believed to offer the potential to achieve low bit-error rates (≤10^{–9}), in part because the error floors of SCCCs are lower than those of PCCCs.

It is important to note that the proposed serial concatenated coding scheme differs from "classical" concatenated coding schemes. In a classical scheme, the role of the interleaver between two encoders is merely to break up bursts of errors produced by the inner decoder, and no attempt is made to consider the combination of the two encoders and the interleaver as a single entity. In SCTCM, on the other hand, one seeks to optimize the whole serial structure.

No attempt at such optimization was made in the past, in part because optimizing an overall code with large deterministic interleavers is prohibitively complex. However, by introducing the concept of a uniform interleaver, it is possible to draw some criteria to optimize the component codes for the construction of powerful serially concatenated codes with large block size. Another reason optimization of overall codes was not attempted is that optimum decoding of complex codes is practically impossible; only the use of suboptimum iterative decoding techniques makes it possible to decode such complex codes. The decoder in an SCTCM system would utilize an adapted version of iterative decoding used in PCCC schemes.

The upper part of the figure is a block diagram of an encoder in an SCTCM system that yields a bit-rate-to-bandwidth ratio of *b* (bits/second)/Hertz when ideal Nyquist pulse shaping is used. The outer encoder implements a rate-[2*b*/(2*b*+1)] binary convolutional code (or a short block code) with maximum free Hamming distance (or minimum distance). The interleaver (Π) permutes the output of the outer encoder. The interleaved data enter the inner encoder, which implements a rate-[(2*b*+1)/(2*b*+2)] recursive convolutional code. The 2*b*+2 output bits are then mapped to two symbols, each belonging to a 2^{b}^{+1}-point two- dimensional constellation. This results in four-dimensional modulation. In this way, 2*b* information bits are used for every two modulation symbol intervals; in other words, there are *b* information bits per modulation symbol. The inner code and the mapping are jointly optimized on the basis of maximizing the effective free Euclidean distance of the inner TCM.

Unfortunately, the decoder associated with such an encoder would be unacceptably complex and thus unsuitable for high-speed operation. This is because the number of transitions per state for the inner TCM is 2^{2b+1} and so the number of edges in the trellis section of the decoder would have to equal to 2^{2b+1} × the number of states.

The lower part of the figure is a block diagram of an SCTCM encoder for an *M*-point two-dimensional constellation that would enable the use of a decoder of lower complexity. This encoder yields a bit-rate-to-bandwidth ratio of *bm*/(*b*+1) (bits/second)/Hertz [where *m* º log_{2}*M* and *M* is the number of points in a two-dimensional signal constellation] when ideal Nyquist pulse shaping is used. The outer encoder implements a rate-[*b*/(*b*+1)] binary convolutional code (or a short block code) with maximum free Hamming distance (or minimum distance). The interleaver (Π) permutes the output of the outer encoder. The interleaved data enter the inner encoder, which implements a rate-(*m*/*m*) [rate-1] recursive convolutional code. The *m* output bits are then mapped to one symbol that belongs to a 2* ^{m}*-level modulation. Because the inner code does not have redundancy, it is useless by itself; however, the combination of the inner and outer codes with the interleaver results in very powerful code. For MQAM where

*M*=

*N*

^{2}, further reduction in complexity is possible. This can be done by assigning the

*m*= log

_{2}

*N*output bits of the inner encoder alternately to the inphase and quadrature components of N

^{2}QAM modulation. In this case, the bit-rate-to-bandwidth ratio will be

2*bm*/(*b+*1).

The advantage of this generic design can be made more apparent by citing an example of *b* = 3 for 16QAM, for which *m* = 2. In this example, the number of transitions per state of the inner TCM is only 4, which is only 1/32 of the corresponding number for the previous case.

*This work was done by Dariush Divsalar, Sam Dolinar, and Fabrizio Pollara of Caltech for *NASA's Jet Propulsion Laboratory*. For further information, access the Technical Support Package (TSP) *

**free on-line at www.techbriefs.com/tsp**

*under the Information Sciences category.*

In accordance with Public Law 96-517, the contractor has elected to retain title to this invention. Inquiries concerning rights for its commercial use should be addressed to:

Intellectual Property group

JPL

Mail Stop 202-233

4800 Oak Grove Drive

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(818) 354-2240

*Refer to NPO-20878, volume and number of this *NASA Tech Briefs *issue, and the page number.*

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###### Serial-Turbo-Trellis-Coded Modulation With Rate-1 Inner Code

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