Performance of Nonuniform M-ary QAM Constellation on Nonlinear Channels

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1 Performance of Nonuniform M-ary QAM Constellation on Nonlinear Channels Nghia H. Ngo, S. Adrian Barbulescu and Steven S. Pietrobon Abstract This paper investigates the effects of the distribution of a high bandwidth efficiency M-ary QAM signal constellation on the error performance for nonlinear channels. A simple nonlinear channel model where the phase and amplitude of the transmitted samples are distorted by a nonlinear function with additive white Gaussian noise (AWGN) in the downlink was simulated. Simulation results for binary turbo coded modulation (TCM) show a 3 db improvement when the distribution of the signal constellation is. Index Terms Nonequiprobable constellation, high order modulation, nonlinear channel, spectral efficiency II. THE SYSTEM MODEL The block diagram in Fig. illustrates a simple model for a nonlinear channel using an M-ary QAM constellation. The nonlinear system model assumes the transmitted signal is narrow band relative to the satellite transponder bandwidth. Therefore, the memory of the satellite input and output MUX filters is negligible. Detailed descriptions for each block are given in the next sections. Transmitter I. INTRODUCTION Digital communications over satellites is constrained by two important factors; power and bandwidth. The increasing demand for this type of communication has led to where the utilisation of these scarce resources needs to be optimised. To serve this purpose, high order QAM modulation is an attractive method to achieve both bandwidth and power efficiency. µ For the Gaussian channel, constellation shaping techniques µ refer to the selection of the signal shape where the average energy is reduced and the constellation shape is Gaussian like in distribution. Thus, the system gain is obtained by the sum of both coding gain and shaping gain. To achieve the shaping gain, there are a number of proposed solutions. Forney in [] suggested a method of constellation shaping called trellis shaping where a gain of about db can be achieved with a simple 4-state shaping code. Another approach proposed by Kschischang and Pasupathy [2] uses variable rate codes. In [3], the signal constellation is divided into sub-regions with different sizes. The idea of using an uniform space constellation with non-uniform distribution of signal points was explored by Calderbank and Ozarou [4]. In [5], the authors proposed a solution where the distance among signal points are varied while the probability of transmission of each signal point is the same. A recent publication [6] follows the method presented in [2] and obtained a gain of almost.6 db. Motivated by this technique, we extend this work to nonlinear channels and make a comparison with uniform signal set coding. We observe a significant gain of almost 3 db. The structure of this paper is as follows. Section II describes the system model for a nonlinear satellite and the technique used in our investigation. Some simulation results are present in Section III and Section IV concludes the paper. The authors are with the Institute for Telecommunications Research, University of South Australia, Mawson Lakes SA 595, Australia. Steven S. Pietrobon is also with Small World Communications, 6 First Ave., Payneham South SA 57, Australia. Nghia.Ngo@postgrads.unisa.edu.au, Adrian.Barbulescu@unisa.edu.au, steven@sworld.com.au. Source Ù Turbo Encoder Receiver Signal Mapper Sink Turbo Soft Decoder Demapper Ù Fig.. General System Model. A. The Transmitter The transmitter in Fig. employs two identical recursive systematic convolutional codes (RSC) in a parallel-concatenated scheme [7], i.e., PCCC. The component RSC code is rate /2. The systematic bit and the two parity bit sequences are transmitted and this makes the system code rate equal to /3. For different code rates, we use a puncture block after the demultiplexer and apply it to the parity output bits. An interleaver interleaves the data between the two RSCs with a block length of bits using an Ë-random scheme with Ë Ô µ. The input binary sequence Ù is encoded by this PCCCC encoder. The output consists of the systematic bit ¼ and the parity bits and from the first and second RSC, respectively. These bits are demultiplexed times, punctured if required and grouped into bit sequences as shown in Fig. 2. The interleavers,, ¼ Ñ µ, connect the turbo encoder and the M-ary signal mapper. They are also of Ë- random type and each of them is of size Ö Ñµ where is the block length, Ö is the code rate and Ñ is the number of bits per symbol. Interleaved bits µ are mapped onto an M-ary signal in two different ways. In the case of equiprobable signalling, we Nonlinear Channel

2 Å Ù Turbo Fig. 2. Encoder ¼ Demultiplexer The Transmitter. ¼ ¼ ¼ ¼µ ¼ µ M ary ¼ ¼ Ü ¼ Signal ¼ ¼ µ Mapper ¼ ¼ Ñ Ñ µ Ñ µ Ñ Puncture TABLE I MAPPING RULE FOR -D ËÒÐ ÓÒ ¼ 5 3 x 9 x x 7 x x 5 x x 3 x x x x x x - x x x -3 x x x -5 x x -7 x x -9 x x - x -3-5 group Ñ encoded bits and map them on one of Ñ signal points using Gray mapping. When signalling is used, a group containing more than Ñ encoded bits is mapped to one of Ñ signal points with some bits being punctured according to the mapping rule present in Table I [6]. Table I shows the mapping for each dimension of a 256-QAM signal set. An x means the bit in that position is punctured. In this case, a group of 2 bits is used to produce a 256 QAM symbol (6 bits per dimension). A rate /3 turbo code with is used to achieve a spectral efficiency of 4 bit/s/hz. In the case of equiprobable signalling, every symbol is represented by 8 bits (4 bits per dimension). A rate /2 turbo code is used to achieve the same spectral efficiency as in the case. In both schemes, the information bits are always mapped onto the most protected positions of the signal constellation. The role of the mapping is to use some particular low energy ¼µ µ µ Ñ µ Ñ µ Ù signal points more often than the others. This would make the probability of each point in the constellation be nonuniform. In fact, with this mapping rule, the distribution of the signal points follows the Maxwell-Boltzmann (MB) distribution. According to [6], this distribution provides a good approximation for the optimal mutual information under a power constraint and a finite constellation. B. The Nonlinear Channel A typical nonlinear channel consists of transmitter and receiver filters. These filters would make the system memory extend over several symbols. A simple model for nonlinear channels as used in this paper only considers the distortion effect of the nonlinear device and ignores the other effects of these filters. We also assume perfect coherent detection and that the channel is ISI free. These assumptions make the channel memoryless. A block diagram for this channel is given in Fig. 3. Fig. 3. Nonlinear Device The Nonlinear Channel. Å Å Ý Ò ¼ µ The block in Fig. 3 indicates the normalisation process. The M-ary QAM symbols go through a nonlinear device where the phase and amplitude conversion are characterised by two standard functions given by Saleh [8]. These functions are Öµ Öµ «Ö Ö () «Ö Ö (2) where Ö is the amplitude, Öµ is the AM/AM conversion and Öµ is the AM/PM conversion. Parameters for these equations are «,, «and. The amplifier input backoff (IBO) in db is defined as the diference between the amplifier input saturation power and the input signal power. Value of the IBO indicates effect of the input signal on the nonlinear region where the amplifier operates and it also measures amount of the distortion occurs. For each value of IBO in db, the amplitude Ö in () and (2) is scaled by ¼ ÁǼ. The level of the signal constellation s distortion is shown in Fig. 4 The effects of distribution at different values of IBO can be seen in Figures 5, 6, and 7. It is clear that for high values of IBO, the system operates near the linear region and the distorted signal points are close to their original points. When IBO is decreased, the system operates in the nonlinear region where the high energy points are squeezed close together. This would reduce the Euclidian distance among the outer signal points. Consequently, it should be harder for the receiver to

3 ¼ ¼ 2.5 IBO db IBO 6dB IBO 3dB square 256 QAM mapping ¼ ¼ QAM Constellation on nonlinear channel at IBO db square constelation equiprobable Ñ.5 ¼µ Ñ.5 ¼µ µ µ µ µ Ñ µ Ñ µ.5 µ Ò ¼ µ Ý.5 Å.5 Ñ µ Ò ¼ µ Ý.5 Å Fig. 4. Equiprobable constellation at different input backoff values. Fig. 5. Nonlinear constellation at IBO of db equiprobable vesus. decode what was transmitted. When a distribution is applied, the average energy is reduced since the high energy points are used less frequently. As a result, a nonuniform constellation seems to suffer from less distortion than the uniform one, even at low IBO. In Fig. 7, all the outer points are squeezed together for both cases. The distance among the inner points for the case is larger and this results in better overall performance. The noise added to the transmitted signal after being converted by the nonlinear device is white Gaussian noise with zero mean and variance. The probability distribution for this channel is Ý Ý µ Ô ÜÔ µ where is a distorted constellation point. C. The Receiver The receiver in this paper uses the model of iterative demapping and decoding which was studied widely in [9 ]. The block diagram is shown in Fig. 8. The study of iterative demaping and decoding in [9] showed that the use of extrinsic information produced by the decoder can further improve performance of the system with relatively small added complexity. The output from the soft-demapper is the log-likelihood ratio (LLR) for each encoded bits and is denoted by ݵ µ ÐÒ ¼Ýµ Using Bayes rule, we can derive the probability values as (3) (4) ݵ ݵ ݵ ݵ ݵ Ý µ µ µ ݵ where is one of Ñ signal points and µ ¼. Thus, the LLRs become µ ÐÒ We have ÐÒ Ý µ µ ¼ Ý µ ¼µ ÜÔ Ý µ µ ¼ ÜÔ Ý µ ¼µ µ Ñ ¼ (5) (6) µ (7) where µ is the extrinsic information derived from the turbo decoder, ¼, ¼ Ñ is the binary represent position of and Ñ is the number of non-punctured bits of. After de-interleaving, depuncturing and multiplexing, these LLRs values are input to the turbo decoder. In this case, the output extrinsic information of the encoded bits produced by the turbo decoder are interleaved by identical interleavers as

4 ¼.5 ¼ Ñ.5 ¼µ 256 QAM Constellation on nonlinear channel at IBO 6dB ¼ µ square constelation equiprobable ¼ Ù ¼ ¼ ¼ ¼ Ñ ¼µ QAM Constellation on nonlinear channel at IBO 3dB square constelation equiprobable µ µ Ñ µ µ.5 Ò ¼ µ Ý Å Ñ µ Ü Ñ µ Fig. 6. Nonlinear constellation at IBO of 6 db equiprobable vs.. Fig. 7. Nonlinear constellation at IBO of 3 db equiprobable vesus Ò ¼ µ from the transmitter and fed back to the soft-demapper in or- ¼µ Ý µ µ ¼ Ñ µ Ü ¼ Ñ µ.5 Ñ Ü Ò ¼ µ ¼µ ݵ der to calculate the updated values for the PCCC decoder in the Å next iteration. Initially, the extrinsic information is assumed to be constant, i.e., µ =.5. The original method in [6] used one dimensional PAM. When a 2-dimensional M-ary QAM signal set is considered, we use the same mapping principle for each I and Q dimension. The updated probabilities are the product of two -dimension signal points. The LLRs output from the soft-demapper after being deinterleaved and multiplexed are fed into a normal binary turbo decoder using the log-map algorithm. Å µ Ý Soft Demapper µ µ Ñ Ñ µ Depuncture Demultiplexer Turbo Decoder Ù III. SIMULATION RESULTS Parameters for the code used in this simulation are taken from [2], which is the optimal encoder for various turbo-code rates and memory size. The encoder memory is four with feed forward and feedback polynomials of [37, 23] in octal notation. The block length is 892 bits and iterations of iterative demapping and decoding are considered. Simulations were carried out for both linear and nonlinear channels. Comparisons are made based on the same spectral efficiency. The code rates for equiprobable and signal sets are different. Fig. 9 shows the performance of turbo coding with rate /3 using a 256 QAM signal set and rate /2 for the same QAM constellation using an equiprobable signal set. The spectral efficiency for both cases is 4 bit/s/hz. Simulation results for the linear AWGN channel reveal that we can achieve about db better performance at a BER of ¼ when signalling is used. The same coding parameters are used for the nonlinear channel, except for the constellation. In this case, we distorted the Fig. 8. The Receiver. Puncture & Interleave Extrinsic Inform. square 256 QAM Gray constellation by the two functions presented in the previous section at different values IBO of db, 6 db and 3 db. Results are shown in Fig. 9. We observed a significant improvement between and equiprobable signalling schemes. At 3 db IBO, a gain of 3 db is observed at a BER of ¼. At db IBO we can achieve a gain of.5 db. A drawback of this technique is the flooring effect of the BER curve, most likely due to the squeezing of the outer points. Further investigations are underway to lower the error floor. IV. CONCLUSION In this paper, a non-uniform distribution of signal constellation points over a nonlinear channel was investigated. The

5 Ñ µ µ Ò ¼ µ Ý Å Ý ¼µ µ BER Performance of 256 QAM rate /2 on linear and nonlinear channel at different IBO, blocksize 892 bits [] S. Le Goff, A. Glavieux and C. Beurrou, Turbo-codes and high spectral efficiency, ICC 94, New Orleans, USA, pp , May 994. [2] S. Benedetto, R. Garello, and G. Montorsi, A search for good convolutional codes to be used in the construction of turbo codes, IEEE Trans. Commun., vol. 46, pp. -5, Sep µ Ñ µ Ù Ñ db IBO equiprob. 6dB IBO equiprob. 3dB IBO equiprob. linear equiprob. linear nonequiprob. db IBO nonequiprob. 6dB IBO nonequiprob. 3dB IBO nonequiprob EbNo in db Fig. 9. BER performance of 256 QAM on linear and nonlinear channel, equiprobable vesus. signal points in the nonlinear channel are greatly distorted when the system operates at low IBO. By applying a non-uniform distribution where the more distorted high energy points are used less frequently, the average energy can be reduced and therefore a significant gain of almost 3 db can be achieved. A distinct advantage of the non-uniform scheme is shown. ACKNOWLEDGEMENTS The authors would like to thank Craig Burnet for his insigntful comments. REFERENCES [] G. D. Forney, Jr., Trellis shaping, IEEE Trans. Inform. Theory, vol. 38, pp. 28-3, Mar [2] F. Kschischang and S. Pasupathy, Optimal nonuniform signalling for Gaussian channel, IEEE Trans. Inform. Theory, vol. 39, pp , May 993. [3] J. Livingston, Shaping using variable-size regions, IEEE Trans. Inform. Theory, vol. 38, pp , July 992. [4] A. Calderbank and L. Orazou, Nonequiprobable signaling on Gaussian channels, IEEE Trans. Inform. Theory, vol. 36, pp , July 99. [5] C. Fragauli, R. D Wesel, D. Sommer and G. P. Fettweis, Turbo codes with non-uniform constellation, IEEE Int. Conf. Comm., Helsinki, Finland, vol., pp. 7-73, June 2. [6] D. Raphaeli and A. Gurevitz, Constellation shaping for pragmatic turbocoded modulation with high spectral efficiency, IEEE Trans. Comm., vol 52, pp , Mar. 24. [7] C. Berrou, A. Glavieux, and P. Thitimajshima, Near Shannon limit error correcting coding and decoding, IEEE Int. Conf. Commun, Geneva, Switzerland, pp. 64-7, May 993. [8] A. M. Saleh, Frequency independent and frequency dependent nonlinear models of TWT amplifiers, IEEE Trans. Commun., vol. COM-29, pp , Nov. 98. [9] S. ten Brink, J. Speidei, and R. H. Yan, Iterative demapping and decoding for multilevel modulation, GLOBECOM 98, Sydney, Australia, pp , Nov [] S. Benedetto, G. Montorosi, D. Divsalar and F. Pollara, Serial concatenation of interleaved codes: Performance analysis, design and iterative decoding, TDA Progress Report 42-26, pp. -26, Aug. 996.