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University of Virginia. Charlottesville, VA 22903. Abstract. Turbo codes have excited the coding community with the promise of performing near channel ca-.

Turbo Trellis Coded Modulation W. J. Blackerty and S. G. Wilson Communications Systems Laboratory Department of Electrical Engineering University of Virginia Charlottesville, VA 22903

Abstract Turbo codes have excited the coding community with the promise of performing near channel capacity by using an iterative decoding technique that relies upon simple constituent codes. However, since the original turbo codes are low rate codes, a signi cant bandwidth penalty can be incurred by systems which utilize turbo codes. Since the constituent codes that make up turbo codes are convolutional codes, a natural extension of the turbo concept, in order to improve bandwidth eciency, is its application to Trellis Coded Modulation (TCM) systems. This paper will discuss the conditions under which encoders with parallel transitions can be used as constituent codes in Turbo TCM systems and present results to compare systems with and without parallel transition constituent encoders.

1 Introduction

Modulation (TCM). In Turbo TCM (T-TCM) systems, the objective is to maximize the minimum Euclidean distance (d2min ) between coded sequences and to maintain small error coecient multipliers as it is with \standard" binary Turbo systems. Ungerboeck's work [2] has shown for encoders with few states the best encoders possess parallel transitions. However, the T-TCM methods presented in [3] and [4] either exclude encoders with parallel transitions as constituent codes or do not discuss conditions in which these encoders can be used. This paper will 

present various bandwidth ecient turbo code schemes,



discuss conditions such that encoders with parallel can be used,



describe necessary modi cations to the decoder of [3] so that constituent codes with parallel transitions may be used, and

Turbo coding, introduced in [1], is a coding tech-  present simulation results to investigate the e ectiveness of using encoders with parallel nique that uses an iterative decoding algorithm transitions. to give performance near channel capacity. Since Turbo codes use convolutional codes as their constituent codes, a natural extension of the Turbo 2 Turbo TCM Encoders concept which improves bandwidth eciency is its application to systems using Trellis Coded In traditional TCM systems, the encoder at Sponsor: National Science Foundation tempts to produce a coded modulation sequence y Corresponding Author: [email protected] of message symbols, each with m bits, so that

the Euclidean distance between any pair of valid sequences is large. Rather then using a signal encoder with large constraint length, the turbo coding technique suggests the use of multiple, e.g. two, simple encoders operating on permuted versions of the same message of length N . Although the free Euclidean distance of one encoder may be small, by properly designing the permutation of the data sequence, those sequences that produce modest distance in one encoder will produce a larger distance in the other encoder.1 In the context of convolutional codes, this is equivalent to short error events in one encoder trellis producing long error events in the other trellis. The fundamental issues in the design of the T-TCM encoder are therefore 



The \merging" of the separate encoder outputs into a single QAM/PSK coded sequence, and the design of the interleaver.

2.1 Proposed Encoder Schemes

Thus far, three di erent encoder structures have been suggested. In [5], the authors simply map the output of a binary turbo encoder onto an M-ary modulator. Decoding is done by forming LLRs of each bit in the signal constellation and then allowing the turbo decoders to use these LLRs as data. Although this system utilizes a bandwidth ecient modulation scheme, the encoder and modulator are not designed cooperatively as in TCM systems. In [3], a Turbo-TCM system was presented where a single M-ary symbol is transmitted at each trellis interval, e.g. 2 information bits into 8-PSK. The high level bits of the signal set partition are de ned by the message symbol. The low level bit(s) of the modulator sequence are produced by a punctured version of the turbo

encoder parity output. A generalized version of this encoder is depicted in gure 1. It should be Encoder

M-ary

I -1

M-ary

I

Encoder

Figure 1: T-TCM Encoder of [3] noted that the interleaver is constrained to interleave symbols. A third bandwidth ecient modulation scheme has been suggested in [4]. The encoder of this system, an example of which is shown in gure 2, works in a slightly di erent manner then the encoder of [3]. Here each encoder forms a I-Channel 4-AM Encoder 1 I1

I2

Q-Channel Encoder 2

4-AM

Figure 2: T-TCM Encoder of [4] modulation symbol from the parity stream produced by the encoder and a subset of the systematic information. These symbols are then sent, as depicted in gure 2, using in-phase and quadrature modulation to form a composite symbol or by serializing the symbols for transmission, e.g. sequentially sending two 8-PSK symbols.

2.2 Parallel Transition Encoder Conditions

As discussed above, for constituent encoders with small memory order, the best (in terms of free distance) encoders will possess parallel transitions. For this reason, it may be desirable to use encoders with parallel transitions as the constituent encoders in the T-TCM schemes dis1 In turbo systems, it is important to not only have cussed in [3] and [4]. However, in doing this, large free distance by also small error multipliers for the one must ensure that the free distance of the enlow weight sequences. tire system is not limited by a codeword that was

holds, then it is possible to exchange lines in such a way each input line in uences the state of at least one encoder. Given this, it is possible to use constituent encoders that possess onestep error events, but guarantee that at least one encoder has a multiple-step error event for each codeword. The encoder of [3] has an interleaver which is constrained to permute only symbols. Due to this constraint on the interleaver, one-step error events in one constituent encoder will become one step error events in all constituent encoders. However, it is possible to modify the encoder of [3], so that this does not occur. Figure 3 presents modi ed version of the encoder of [3] that will nf  1 (1) awork with parallel transitions. The only di erProof: If nf  1 then, with proper interleaver selection, each bit will in uence the state of at least one encoder. If this occurs, then at least one encoder will be perturbed from the all zeros state which means that one-step transitions do not occur in all n encoders. If nf < 1 then at least one bit, no matter what interleaver is selected, does not in uence the state of any en- Figure 3: Turbo TCM Encoder that allows parcoder. Therefore, by selecting this bit to be a allel transitions 1 and all other bits to be a 0, each encoder will remain in the all zero state and the only event that occurs in each encoder is a single-step error event. ence between the systems of gure 1 and gure 3 is the addition of the \mapper" after the interleaver. The purpose to this device is to map 2.3 Encoder Adaptation the original, interleaved, input symbols onto a With the above condition in mind, it is possible new set of symbols within the same alphabet. to now discuss conditions under which the previ- This mapping may be as simple as exchanging ously discussed T-TCM encoders can be imple- bit lines or as general as swapping symbols but mented with encoders that possess parallel tran- the mapping must be reversible and done such sitions. that symbols that would cause a one-step error The interleaver that is used in the encoder of event in the rst encoder will cause a longer er[4] forces bits to maintain \membership" in the ror event in the second encoder. By adding the input line in which they were presented to the en- mapping, the system is no longer limited by the coder. Due to this constraint on the interleaver, parallel transition case. With the exception of lines may only be \exchanged" in the process of the additional stage of mapping, the operation ensuring that each bit is encoded by at least one of this encoder is identical to the operation of of the encoders. If the condition of equation (1) the encoder presented in [3].

produced by a single one-step error event in each trellis. Otherwise, the encoder has not bene ted from the interleaving and presence of multiple encoders. It is possible to design the interleaver for a TTCM system such that at least one trellis has a multiple-step error event for every input message if the following condition holds. Let f be the fraction of the bits that enter a constituent encoder and in uence the state of the encoder. If there are n encoders present in a system, it is possible to design the interleaver such that at least one trellis has a multiple-step error event for every input message if and only if

M-ARY

Encoder

I

M-ARY

M

Encoder

D

3 Decoder Modi cations It is interesting to discuss the modi cations to the T-TCM decoder of [3] that are necessary to allow for the use of constituent encoders that have parallel transitions. Figure 4 depicts the decoder presented in [3]. The `*' character indicates the position of the switch when the parity of the current symbol was not produced by the encoder that is matched to the decoder in question. noisy channel symbols metric "0"

*

metric s (1-m) log 2 *

MAP

I first decoding all others

-

+

metric I "0"

*

-1

I

MAP Hard Decision

m-1

2 =

-

+

-1

now assign this con dence level to the symbol j since, if i was in the original data stream, j was encoded by the second encoder. Along the same lines as this modi cation, since the hard decision is eventually made based on the output of the second decoder, it is necessary to not only deinterleave the symbols but also \unmap" them. The second modi cation that must be made is for the the block labeled `metric s.' This box computes the likelihood that each possible input symbol was sent by computing the likelihood between the channel symbol formed by the input in question and each possible parity value and then averaging these likelihoods. This operation is only performed for punctured (relative to the rst encoder) symbol positions. Since these symbols are generated by the second decoder, which is working on mapped version of the input data stream, the likelihoods must be assigned to the appropriate positions in the vector to accurately re ect the probability that each symbol was sent. For example, let us continue to assume that the input symbol i is mapped to the input symbol j . To appropriately evaluate the `metric s' value for the symbol i, likelihoods must be computed as if symbol j formed the MSBs of the M-ary symbol.

I

output

Figure 4: Non Parallel T-TCM decoder There are two modi cations which need to be made so that this decoder will properly decode a Turbo-TCM system using parallel transitions. The more obvious modi cation stems from the fact that we have added the mapping operation to the encoder. The presence of the mapping means that we must swap values within each APP vector after we have interleaved. For example, suppose that input symbol i is mapped to input symbol j before it is passed to the second encoder. After the rst decoder has been run, we have a level of con dence that the input symbol at a particular time was i. However, for the second decoder to operate correctly, we must

4 Simulation Results The following plots compare the performance of T-TCM systems of the type presented in [3] with and without parallel transitions. In all cases the systems in question used 16-QAM modulation, have identical interleavers and the following 8state TCM constituent encoders are used: Encoder Type h0(D) h1(D) h2(D) h3(D) Parallel 118 28 48 08 Non-Parallel 118 28 48 108 The encoder with parallel transitions is taken from [6] and the encoder without parallel transitions is taken from [3].

4.1 Example 1

−1

10

The rst simulation is done for a system using a 291 3-bit symbol packet and the following mapping function

−3

10

Pb

Original Symbol 0 1 2 3 4 5 6 7 Mapped Symbol 0 1 4 5 2 3 6 7

−2

10

−4

10

Parallel Transitions No Parallel Transitions First Iteration

−5

10

which is equivalent to exchanging the rst two bit lines that enter encoder 2. −1

Second Iteration Seventh Iteration

−6

10

5

5.5

6

6.5 Eb/No (dB)

7

7.5

8

Figure 6: 597 Symbol Packet Simulation Results

10

−2

10

as the number of iterations increase, the performance of the two 597 symbol packet systems becomes comparable. It also interesting to compare the two examples presented here. As expected, the system with the larger packet size has the better performance of the two systems.

−3

Pb

10

−4

10

Parallel Transitions No Parallel Transitions First Iteration

−5

10

Second Iteration Seventh Iteration

−6

10

5

5.5

6

6.5 Eb/No (dB)

7

7.5

8

5 Conclusions

Figure 5: 291 Symbol Packet Simulation Results This paper has reviewed various bandwidth ecient turbo coding schemes and presented condiFigure 5 presents simulation results that com- tions under which encoders having one-step erpare these two systems. Based on these results, it ror events can be used as constituent encoders. should be noted that the system that uses paral- Modi cations to the encoder and decoder of [3] lel transitions shows improvement over the non- have also been presented along with simulation parallel transition system during the rst two it- results that demonstrate that the use of encoders erations of the decoder. However, as the number with one-step error events can improve T-TCM of iterations is increased, the performance of the system performance. two systems becomes comparable for this frame size.

4.2 Example 2

References

[1] C. Berrou, A. Glavieux, and P. ThitimaFigure 6 shows the results for the same sysjshima, \Near Shannon limit error-correcting tems as in example 1 when a larger packet (597 coding and decoding: Turbo Codes," in ICC, symbols) is used. Again, the simulation results pp. 1064{1070, 1993. show that the system with parallel transitions outperforms the system without parallel transi- [2] G. Ungerboeck, \Trellis-coded modulation tions during the rst several iterations. However, with redundant signal sets, part II: State of

[3]

[4]

[5] [6]

the art," IEEE Communications Magazine, vol. 25, pp. 12{21, Feb. 1987. P. Robertson and T. Worz, \A novel coded modulation scheme employing turbo codes," in URSI & ITG Convference `Kleinheubacher Tagung', (Kleinheubach, Germany), Oct. 1995. S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, \Bandwidth ecient parallel concatentated coding schemes," Electronics Letters, vol. 31, Nov. 23 1995. S. L. Go et al., \Turbo-codes and high spectral eciency modulation," in ICC, pp. 645{ 649, 1994. S. G. Wilson, Digital Modulation and Coding. Englewood Cli s, New Jersey: Prentice-Hall, 1995.

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