Some Reflections on the Design of Bandwidth Efficient Turbo Codes Daniel J. Costello Jr., Adrish Banerjee, Thomas E. Fuja, and Peter C. Massey Abstract— In this paper we review several approaches to bandwidth efficient turbo coding that have appeared in the recent literature. In addition, some new designs using bit interleaved coded modulation are introduced, including asymmetric designs and low-complexity multiple turbo code designs.
I. Introduction In the area of error control coding, parallel concatenated codes, or turbo codes [1], have been the most exciting development of the past decade. However, conventional turbo codes are low rate codes. This implies that they require considerable bandwidth expansion, which may not be acceptable in some applications. In this paper, we review several recently published schemes for combining turbo coding with coded modulation to develop power efficient error control techniques without sacrificing bandwidth. (Early work on bandwidth efficient turbo coding was previously reviewed in [2].) There are various approaches that can be used to extend turbo coding to higher spectral efficiencies. This paper focuses on bit interleaved coded modulation [3]. This technique involves the concatenation of an encoder for a binary code with a memoryless modulator over a 2m -ary signal set, connected by a bit interleaver, along with a one-to-one mapping function between the interleaved coded bits and the signal set (see Figure 1). In this paper we present some new bandwidth efficient code designs based on bit interleaved coded modulation using asymmetric turbo codes and multiple turbo codes. These codes not only have excellent performance in the waterfall region, but some of them also have lower error floors and lesser decoding complexity than conventional bandwidth efficient turbo code designs. The paper is organized as follows. In Section II, various approaches to bandwidth efficient turbo coding are reviewed. Section III describes new asymmetric designs for bitinterleaved coded modulation. In Section IV, new low state complexity bandwidth efficient code designs based on multiple turbo codes are presented. Finally, Section V includes some concluding remarks. The authors are with the Coding Research Group, University of Notre Dame, Notre Dame, IN 46556. E-mail:
[email protected]
systematic bits d
k
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Fig. 1. Encoder Structure: Turbo coded modulation
II. Different Approaches to Bandwidth Efficient Coding Using Turbo Codes Turbo coding can be combined with coded modulation to develop power efficient coding techniques without sacrificing bandwidth efficiency. Some well known techniques are briefly described below. A. Turbo Coded Modulation Bit interleaved coded modulation using turbo codes, also known as turbo coded modulation, was originally proposed by Le Goff, Glavieux, and Berrou [4]. This approach uses a binary turbo encoder that is linked to a signal mapper after its output bits are suitably punctured and multiplexed to achieve the desired number of information bits per transmitted symbol. Gray mapping is used between the coded bits and the modulation symbols ([4], [5], [6], [7], [8]) (see Figure 1). A coherent additive white Gaussian noise (AWGN) channel model is assumed. At the receiver, the received noisy symbols are demapped and likelihood ratios associated with each received bit are calculated and used as soft inputs to a binary turbo decoder (see Figure 2). The turbo coded modulation scheme is simple and can be applied to a wide range of signal constellations and code rates. Caire et al. show that there is practically no loss in terms of channel capacity as long as Gray mapping is used in conjunction with bit interleaving [3]. B. Turbo Trellis Coded Modulation This technique, originally proposed by Robertson and W¨ orz [9], consists of the parallel concatenation of two trellis coded modulation (TCM) schemes [10] using rate k/(k+1) convolutional codes, in the same fashion as binary turbo codes (see also [11]). The interleaving works on groups of bits instead of individual bits. The second encoder’s output symbols are de-interleaved to insure that the ordering of the
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Fig. 5. Encoder Structure: Multilevel coding Fig. 3. Encoder Structure: Turbo trellis coded modulation
information bits, which partly define each transmitted symbol, correspond to that of the first encoder. Finally, symbols from a 2k+1 -ary signal constellation are transmitted alternately from the first and second encoders. Thus each information bit pair is transmitted in exactly one transmitted symbol with the parity bit alternately chosen from the first and second encoders (see Figure 3). C. Parallel Concatenated Trellis Coded Modulation In this scheme, two parallel concatenated rate k/(k+1) convolutional codes use all their information bits to produce the parity bit which is sent to their respective symbol mappers, but the information bits are punctured in such a fashion that one half of them are used in the first trellis code and the other half in the second trellis code [12]. This is done to limit the number of points in the signal constellation to 2(k/2)+1 . Also, the information bits transmitted by the two trellis encoders are interleaved by different bit interleavers (see Figure 4). In [13], a variation of this scheme is presented where a symbol interleaver is used instead of bit interleavers, resulting in improved BER performance in the waterfall region. D. Multilevel Coding Wachsmann and Huber use binary turbo codes as constituent codes in a multi-level coding arrangement [14] (see Figure 5). A 2m -ary signal constellation is binary partitioned in several stages, with the selection of each subset governed by the out-
puts of m independent encoders. They use the fact that the sum of the capacities of m equivalent binary channels associated with each of the levels is equal to the capacity of the channel corresponding to the 2m -ary signal constellation. These capacities may be used to determine the rates of the various constituent encoders. Mittelholzer, Lin, and Massey [15] propose a multilevel coding scheme that relies on 4-ary partitioning of a 2m -ary QAM signal constellation. Isaki and Imai also design a multilevel coding scheme that uses turbo codes at the individual levels. Further, they use iterative decoding based on the “turbo principle” to decode the multilevel code [16].
E. Self-Concatenated Trellis Coded Modulation Self concatenated TCM was introduced by Benedetto et al. [17] and independently by Loeliger [18] for the special case of a single interleaver and binary modulation. A rate bq/(bq + 1) recursive systematic convolutional encoder is used, where b + 1 outputs are mapped to the signal constellation. Each of the b bit streams are interleaved by (q − 1) interleavers. The interleaved bits are not transmitted; however, they affect the state transitions and the outputs of the encoder. The advantage here is that only one constituent code is required. Figure 6 shows an example of self-concatenated TCM with b = 3, q = 2, and 16-QAM signal mapping (see also [19]).
Π2 Π3
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Fig. 6. Example of self-concatenated TCM −5
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F. Other Bandwidth Efficient Designs Based on Iterative Decoding Several other bandwidth efficient code designs based on iterative decoding have appeared recently in the literature. Benedetto et al. extend the idea of serial concatenation to bandwidth efficient modulation ([20], [7]). Ping et al. design low complexity parallel concatenated two-state TCM schemes using concatenated tree codes [21]. ten Brink et al. propose an iterative de-mapping and decoding scheme which employs a simple bit interleaved convolutional code along with anti-Gray mapping [22]. Some bandwidth efficient coded modulation schemes using low density parity check codes (LDPCC’s) have also been proposed. Narayanan and Li design bit interleaved coded modulation and multilevel coding schemes using randomly constructed LDPCC’s [23]. Sridhara and Fuja use algebraic LDPCC’s in a similar setting [24]. Hou et al. design capacity approaching multilevel coding schemes using irregular LDPCC’s [25]; they jointly optimize the code rates and the degree distribution pairs of the constituent LDPCC’s at each level of the multilevel scheme. G. Some Reflections on the Above Schemes Most of the proposed bandwidth efficient turbo coding schemes perform within 1 dB of channel capacity for large interleavers and their performance is comparable to one another for the same interleaver lengths. Bit interleaved turbo coded modulation is especially interesting due to its simplicity for practical implementation. Since it uses a binary turbo code, it gives great flexibility in implementing a wide range of bandwidth efficiencies by merely changing the puncturing pattern and modulation scheme without changing the decoder module. This has advantages in modern mobile communication systems where one needs to adapt the transmission rate according to the channel conditions and user demands. Moreover, the bit interleaver between the coded bits and the mapper (in the encoder structure) provides the diversity necessary to achieve good performance on fading channels. In the following two sections we present some new designs for bit interleaved turbo coded modulation that have improved performance
−6
10
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3
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3.4 Eb/N0(dB)
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Fig. 7. BER performance comparison of turbo coded modulation schemes employing asymmetric turbo codes
compared to earlier schemes. III. Turbo Coded Modulation Employing Asymmetric Turbo Codes The designs of [4] were based on identical (symmetric) constituent encoders for turbo codes. Symmetric encoder designs for turbo codes usually have good performance in either the waterfall region or the error floor region of the BER curve, but good performance for all SNR’s is difficult to achieve with symmetric codes. Asymmetric designs, on the other hand, consist of different constituent encoders in a turbo configuration [26]. They exploit the properties of weak and strong codes in a mutually beneficial fashion [27]. The notion of weak and strong codes is based on how the individual codes perform as conventional convolutional codes in a turbo configuration. Weak constituent codes produce better extrinsic estimates for the information bits when the a-priori inputs are less reliable. This is consistent with the observation that weak codes (or even no coding) outperform strong codes, at low SNR. In the turbo cofiguration, this starts the iterative decoder towards convergence. Strong constituent codes, on the other hand, provide good extrinsic estimates for the information bits when the a-priori inputs are more reliable; this helps an iterative decoder to converge to the maximum likelihood solution for high SNR’s. Figure 7 shows the BER performance of one such turbo code (P4-P16), an asymmetric configuration 1+D 2 of a primitive 4-state [1 1+D+D 2 ] code (P4) with a 2
4
+D primitive 16-state [1 1+D+D 1+D 3 +D 4 ] code (P16). To achieve an overall rate R=1/2, parity bits are alternately punctured from both encoders. The coded bits are Gray mapped to a 16-QAM signal constellation to achieve an overall spectral efficiency of 2
4
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Fig. 9. Puncturing patterns for bit interleaved turbo coded modulation
1 0
Fig. 8. Gray mapping for a 16-QAM constellation
bits/symbol (see Figure 8). The BER performance 1+D 2 is compared with primitive 4-state [1 1+D+D 2 ] (P4), 1+D 2 +D 3 1+D+D 3 ] (P8), and primitive 1+D+D 2 +D 4 1+D 3 +D 4 ] (P16) codes, all used in
primitive 8-state [1
16-state [1 a symmetric configuration for a block size N=2048 symbols with 30 decoding iterations. In Figure 7, it can be seen that the waterfall region performance for the asymmetric P4-P16 turbo code is about 0.10.2 dB better than for any of the corresponding symmetric designs. This improved performance in the waterfall region of the asymmetric turbo code can be predicted from the mutual information based EXIT charts, which can be used to visualize the growth of extrinsic information with increasingly more reliable a-priori L-values ([28]). We also see from Figure 7 that the asymmetric code design achieves good error floor performance. A. Big Numerator-Little Denominator Codes Another asymmetric turbo code design uses Big Numerator-Little Denominator (BN-LD) constituent codes ([27], [29], [30]). A rate R=1/2 BNLD encoder [1 n(D) d(D) ] is one for which the degree of the numerator polynomial n(D) is greater than the degree of the denominator polynomial d(D). A low degree denominator generates short periodic cycles, which typically produce better extrinsic estimates than codes with larger cycle lengths when the a-priori inputs are less reliable, resulting in better performance in the waterfall region. However, the finishing strength of the iterative decoding process is influenced by the weight of this periodic cycle, which can be increased by using a high degree numerator. The big numerator is used to append a feedforward portion to the recursive code
which helps the decoder converge to the maximum likelihood solution. An example of such a code is 2 3 +D 4 +D 5 ] Big Numeratorthe 32-state [1 1+D +D 1+D Accumulator (BNA32). The BNA32 code in a symmetric configuration performs poorly [31], but in an asymmetric configuration with the 16-state P16 code, it achieves 0.1-0.2 dB performance improvement in the waterfall region (see Figure 7). It’s performance in the error floor region is not as good as that of the P4-P16 asymmetric design, however. B. Doubly Asymmetric Codes A doubly asymmetric turbo code is not only asymmetric with respect to the encoders for the constituent codes, but it is also asymmetric with respect to their individual rates [27]. The idea behind having an asymmetric rate is to allow a strong code to operate at a lower rate in an asymmetric configuration, so that it generates better initial extrinsic estimates for low a-priori inputs; this results in more puncturing of the weaker code. For example, to design a rate R=1/2 binary turbo code to be used in conjection with 16-QAM modulation, one can use two rate 2/3 convolutional codes as constituent codes, but if we allow the stronger code to have a lower rate, we can achieve better waterfall performance. One such code is a doubly asymmetric combination of a rate 1/3 primitive 64-state 2 +D 4 +D 5 +D 6 1+D 2 +D 3 +D 4 +D 6 [1 1+D+D 1+D 2 +D 3 +D 5 +D 6 1+D 2 +D 3 +D 5 +D 6 ] P64 code with the rate 1/2 BNA32 code. To achieve an overall rate of 1/2, the strong code, P64, is punctured to rate 3/5, while the weak code, BNA32, is punctured to rate 1. The puncturing pattern is shown in Figure 9(a). The resulting binary code is Gray mapped to a 16-QAM signal constellation. The BER performance on an AWGN channel after 30 iterations is shown in Figure 7. This scheme provides around 0.1 dB further improvement in the waterfall region compared to the other asymmetric designs, and it also achieves good error floor performance.
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Rate 2/4 Gray mapped to 16−QAM N=2048 symbols, Iterations=30
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Fig. 10. Encoder for bit interleaved coded modulation using multiple turbo codes −5
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IV. Turbo Coded Modulation Employing Multiple Turbo Codes Multiple turbo codes are a class of parallel concatenated codes with three or more constituent encoders separated by multiple interleavers [32] (see Figure 10). A multiple turbo code based approach to turbo coded modulation allows us to design codes with low state complexity and very good performance in the waterfall and error floor regions. In the next two subsections, we discuss two such classes of codes, namely, partially systematic multiple turbo codes and nonsystematic multiple turbo codes. A. Partially Systematic Multiple Turbo Codes Recently, P. Massey and D. Costello ([33], [34]) have shown that a bank of parallel low complexity encoders connected by several interleavers (i.e., a multiple turbo code) can perform as well as a more complex conventional (single interleaver) turbo code of the same rate for binary modulation. For multiple turbo codes, there are several decoding strategies possible depending upon how the extrinsic L-values from different decoders are combined. We use what is known as an extended serial mode, since it results in the best performance for a given number of iterations [35]. In this decoding strategy, the decoders work in a round robin fashion, with each constituent APP decoder accepting a combination of the most recent extrinsic estimates from the other decoders. One new design employing multiple turbo 2 codes uses a 4-state [1 1+D+D 1+D ] Big NumeratorAccumulator (BNA) ([27], [29]) constituent encoder in a symmetric parallel concatenation configuration with two random interleavers [33]. The overall rate of the unpunctured code is R=1/4. To increase the rate to R=1/2, half the information bits as well as half the parity bits are punctured. Since only some of the information bits are actually transmitted, this is referred to as a partially systematic code. The puncturing pattern is shown is Figure 9(b). Figure 11 shows the BER performance of this BNA multiple turbo code (BNA3parallel). The spectral efficiency is 2 bits/symbol.
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Fig. 11. BER performance of turbo coded modulation schemes employing multiple turbo codes
The BER performance is compared with primitive 1+D 2 4-state [1 1+D+D 2 ] (P4-2parallel), primitive 8-state [1
1+D 2 +D 3 1+D+D 3 ] (P8-2parallel), and primitive 16-state 1+D+D 2 +D 4 1+D 3 +D 4 ] (P16-2parallel) codes used in a con-
[1 ventional (single interleaver) symmetric configuration for the same block size and number of iterations. In Figure 11, it can be seen that the BER performance for the BNA-3parallel turbo code is more than 0.1 dB better than the conventional symmetric designs in the waterfall region and it also has good error floor performance. It is important to note that this improved performance is achieved with less overall decoding complexity. B. Nonsystematic Multiple Turbo Codes The multiple turbo code configuration can be expanded to include more than two interleavers [36]. To design rate R=1/2 codes, one needs to puncture more bits, possibly including all the information bits. If no information bits are transmitted, a nonsystematic multiple turbo code results. Alternately, a nonsystematic recursive constituent encoder can be used in place of the usual systematic recursive encoder. P. Massey and D. Costello [36] observed that not all nonsystematic recursive constituent encoders work well with iterative decoding. However, encoders that have the recursive quicklook-in (RQLI) property exhibit good convergence behavior. This property requires the existence of a feedforward (nonrecursive) encoder inverse with only two non-zero connections (or taps). This is an extension of the QLI property for nonsystematic feedforward (nonrecursive) encoders originally introduced by J. Massey and D. Costello [37]. A special case is the class of self-RQLI encoders. A rate 1 recursive encoder, G(D) = n(D) d(D) , is referred to as self-RQLI if a feedforward inverse having only two
4
taps exists. This condition is satisfied by a rate 1 Dα , for any recursive encoder of the form G(D) = d(D) non-negative integer α and any denominator polynomial of weight 2. Clearly, the accumulator (ACC) 1 [38] is self-RQLI. rate 1 encoder G(D) = 1+D One such nonsystematic multiple turbo code uses an asymmetric combination of four 2-state constituent encoders with three interleavers [34]. It employs a parallel concatenation of three (self-RQLI) 1 ] and a feedforward accumulators (ACC) [1 1+D (FF) [1 1 + D] encoder. The overall rate of the unpunctured code is R=1/5. The puncturing pattern used for this code is shown in Figure 9(c). The coded bits are Gray mapped to a 16-QAM signal constellation to achieve an overall spectral efficiency of 2 bits/symbol. The self-RQLI property of the constituent encoders is not utilized explicitly by the decoder, but is implicitly used in the decoding process to construct soft channel estimates of the (not transmitted) information bits. The ACC encoder helps to achieve good initial extrinsic estimates, while the FF encoder aids in faster convergence. Figure 11 compares the performance of this code (ACC-3,FF-4parallel) with conventional (single interleaver) symmetric P4-2parallel, P8-2parallel, and P16-2parallel turbo coded modulation schemes. The new scheme results in about 0.1 dB improvement in the waterfall region over the conventional symmetric turbo code designs; moreover, it has a much lower state complexity. The new scheme also has good error floor performance. As a final note, the nonsystematic 4-parallel multiple turbo scheme achieves performance similar to the partially systematic 3-parallel multiple turbo scheme presented in the previous section, but with even lower complexity. (It uses only two state codes.)
Acknowledgments This research was supported in part by NSF grant CCR00-75514, NSF grant CCR-99-96222, NASA grant NAG5-10503, and MIT Lincoln Laboratory grant CX-24535. References [1]
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V. Conclusions For high bandwidth efficiency, bit interleaved turbo coded modulation schemes using asymmetric and multiple turbo codes allow us to design schemes with low state complexity and excellent waterfall and good error floor performance. In particular, we have demonstrated that good performance can be achieved using only two-state constituent codes. These new bit interleaved turbo coded modulation schemes can also be used in combination with automatic repeat request (ARQ) protocols to improve the reliability of two-way systems. In the ARQ scenario, a multiple turbo code based turbo coded modulation scheme has the added advantage that the punctured parity bits can be used for retransmission, which allows the receiver to create more powerful lower rate codes for decoding [39].
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