On Performance Improvements with Odd-Power (Cross) QAM Mappings in Wireless Networks
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1 San Jose State University From the SelectedWorks of Robert Henry Morelos-Zaragoza April, 2015 On Performance Improvements with Odd-Power (Cross) QAM Mappings in Wireless Networks Quyhn Quach Robert H Morelos-Zaragoza Available at:
2 On Performance Improvements with Odd-Power (Cross) QAM Mappings in Wireless Networks Quyhn Quach and Robert H. Morelos-Zaragoza Abstract Modern wireless networks use QAM modulation with mappings that are derived from square signal constellations in which the number of signal points is an even power of two. In the particular case of IEEE networks, these mappings are 4-QAM, 16-QAM and 64-QAM and are hereto referred as evenpower QAM mappings. This paper considers the performance improvements that are obtained by adding odd-power QAM mappings (obtained from cross QAM constellations) to the set of available mappings. These odd-power mappings are associated with 2 2m-1 -QAM cross constellations that are carved out from larger 2 2m -QAM square constellations. For the specific cases of 8- QAM, 32-QAM and 128-QAM mappings, the improvements in throughput are quantified assuming a Rayleigh fading condition. Furthermore, in order to illustrate the use of bit metrics generated by a demapper, also examined is the performance of bit-interleaved 8-QAM and 32-QAM mappings with binary codes over an AWGN channel. Index Terms Wireless networks; Quadrature-amplitude modulation; Throughput; Demapping; Channel coding. T I. INTRODUCTION HE physical layer of current wireless networks uses bits-tosymbol mappings based on square M-QAM constellations [1-3]. This is to say that the number of signal points is an even power of two: M = 2!ll, 1 ll 5, with the integer m = 2ll equals to the number of bits per symbol. These mappings are referred to as even-power QAM mappings. While it is advantageous to have a square QAM constellation in which each signal point coordinate (in-phase and quadrature matchedfilter outputs) is in turn a signal point in a PAM constellation, recent advances in signal processing platforms and algorithms allow for the extension of the set of available mappings to include odd-power M-QAM mappings with M = 2!ll!!. The purpose of this paper is to explore the advantages of including odd-power QAM mappings in the set of even-power QAM mappings that are already implemented. Two aspects are considered: The first one is the improvement in throughput that results from adding a new set of odd-power QAM mappings. It is shown in particular that, with data transmission in IEEE wireless networks under Rayleigh fading conditions, up to a 9% increase in throughput is possible. Paper submitted to GLOBECOM 2015 on April 1, The authors are with the Department of Electrical Engineering, San José State University, San Jose, CA USA. (Contact robert.morelos-zaragoza@sjsu.edu). Second, as channel coding forms an integral part of modern wireless networks, illustrative examples of demappers and their application in maximum likelihood decoding of a short Reed- Muller code are presented for 8-QAM and 32-QAM mappings. The paper is organized as follows. Section II presents a good approximation on the average bit error probability (ABEP) of QAM mappings under additive white Gaussian noise (AWGN). This expression of the ABEP is used in Section III to determine the increase in throughput that is achieved by the introduction of odd-power QAM mappings. In section IV, for the specific cases of 8-QAM and 32-QAM, the performance is examined of demappers for bit-interleaved odd-power QAM mappings combined with a first-order RM code and a binary LDPC code over an AWGN channel. Finally, section V gives final remarks and directions for future research. II. ODD-POWER M-QAM MAPPINGS The odd-power 2!ll!! -QAM mappings that are considered in this paper are those associated with cross QAM constellations carved out from larger square 2!ll -QAM constellations [4]. Figures 1 to 3 show the 8-QAM, 32-QAM and 128-QAM signal constellations with ll = 2, 3 and 4 respectively studied in this paper. Fig QAM signal constellation. In this section an overview is given of a good approximation on the average bit error probability (ABEP). The expression is used it to determine ranges of values of the average symbol energy-to-noise ratio E! /N! for which a target value of ABEP is not exceeded. This helps to determine the throughput of a wireless network with a given set of available QAM mappings and a probability density function of the received signal energy. This is discussed in section III.
3 TABLE 1 QAM SIZE, ERROR COEFFICIENT AND NORMALIZED MSED M N! D! /E! /3 16 9/4 2/5 32 7/2 2/ /2 2/ /4 2/ /4 2/85 Fig QAM signal constellation. QAM mappings in Figure 4. In Section III these theoretical expressions are used to estimate the increase in throughput that is achievable by the introduction of odd-power QAM mappings. Fig QAM signal constellation. A. Bit error probability of M-QAM mappings Assuming Gray bits-to-signal mappings of 2 m QAM signal constellations, the ABEP P! that is well approximated by the expression P! = N! m Q D! 2N!, (1) where D! is the minimum squared Euclidean distance (MSED) between signal points expressed in terms of the average symbol energy E!, N! /2 is the AWGN power spectral density and N! is the error coefficient (i.e., the average number of nearest neighbors). Figure 4 shows the ABEP curves based on Eq. (1) for BPSK mapping and 2 m QAM mappings with 2 m 8. In the figures, the curves with dashed lines correspond to the BEP of the even-power QAM mappings that are currently used in networks [1], while curves with solid lines to indicate the BEP of odd-power QAM mappings. Table 1 shows the values of constellation size M, error coefficient N!, and normalized MSED D! /E!, for each of the Fig. 4. Average bit error probability of BPSK mapping and 2! -QAM mappings, for 2 m 8. III. THROUGHPUT IMPROVEMENT UNDER RAYLEIGH FADING To illustrate the improvements in throughput that can be obtained with the inclusion of a set of odd-power QAM mappings, a simple Rayleigh fading condition is assumed such that the signal energy-to-noise ratio Γ has an exponential probability density function: p! γ = 1 γ! exp γ γ! u(γ), where γ! is the average symbol energy-to-noise ratio. A value of signal energy-to-noise ratio Γ inside an interval [γ!, γ! ], corresponds to a transmission rate R(γ!, γ! ) in bits per symbol, or bps/hz, using the error performance curves of QAM mappings in Figure 4, such that the ABEP is lower than a target maximum ABEP value.
4 TABLE 2 INTERVALS AND TRANSMISSION RATES FOR A TRAGET P! = 0.01 m γ!! γ!! R(γ!!, γ!! ) is implemented with a binary first-order Reed-Muller code of length N = 8, denoted RM(3.1) code. The idea is simply to illustrate the use of bit metrics with maximum-likelihood softdecision decoding for odd-power mappings. For a given set of available QAM mappings, the average throughput is readily obtained as R = E R = R γ!!, γ!!!! = m!! = m!! p! γ dγ p! γ dγ exp γ!! γ! exp γ!! γ!, Fig. 5. Rate increase ΔR (%) versus average symbol energy-to-noise ratio γ! (db), under Rayleigh fading, for a target ABEP of 0.01 where I denotes the set of available QAM constellations and for simplicity it is assumed that no channel coding is used. In the case of even-power mappings, I! = 1,2,4,6,8, while with the inclusion of odd-power mappings the set is larger and given by I! = 1,2,3,4,5,6,7,8. Table 2 shows the set of values of interval limits γ!, γ! and transmission rate R(γ!, γ! ) for all the QAM mappings in Figure 1 and a target ABEP P! = Let R! and R! denote the average throughput of a wireless network with even-power QAM mappings only and with all QAM mappings. Since the set of available QAM constellations is larger when including cross constellations, I! I!, it is expected that the throughput will be higher. To quantify this increase in information rate, define the rate increase as ΔR = R! R! 1. Figure 5 shows the value of ΔR as a percentage plotted as a function of γ! for a target ABEP of 0.01 with the QAM mappings in Figure 4. The results show that the maximum rate increase is 9% and corresponds to an average signal energy-tonoise ratio of approximately 17 db. IV. DEMAPPING AND CHANNEL CODING In this section, the error performance with bit interleaving and channel coding, in conjunction with odd-power QAM mappings, is illustrated. For simplicity of exposition, attention is focused on 8-QAM and 32-QAM mappings. Channel coding A. Bit-interleaving and Demapping For a given M-QAM mapping with M = 2!, a rectangular interleaver is used heretofore. A total of m binary codewords out of a channel encoder are arranged in an array of m rows and N columns, where N is the code length. The QAM mapper then takes as input every column of the array and maps it to a symbol to be modulated. Therefore, with a binary RM(3,1) code and 8-QAM and 32-QAM mappings respectively, the interleavers are of size 3-by-8 and 5-by-8. The transmitted QAM symbol sequences are of length eight. As pointed out in [5], it is essential that Gray mapping of bits to symbols be applied so that an equivalent wireless link with m parallel and independent binary channels is created. At the receiver end of the link, the eight in-phase and quadrature matched-filter outputs (y!, y! ) are input to a demapper that produces m bit metrics, one metric for every bit used to label the QAM constellation points. A bit metric is given by the logarithm of the ratio of two a-posteriori probabilities, conditioned to bit values 0 and 1 [5], λ B! = log exp! y!!! exp! y!!! where α represents a signal point with B! = 0, β represents a signal point with B! = 1, and y = (y!, y! ) is the pair of matched-filter outputs from the quadrature demodulator.,
5 Figure 6 shows the metrics for the two most-significant label bits B! and B! for the three odd-power mapings examined in this paper. These metrics are the same as QPSK mapping, since these two bits identify the quadrant in which the signal point lies. On the other hand, as opposed to even-power mappings based on square M-QAM signal constellations that be expressed as the Cartesian product of two M-PAM signal constellations, the bit metrics of the m 2 least-significant bits in odd-power QAM mappings depend on both matched filter outputs (y!, y! ). The computation of these metrics is therefore slightly more complex compared to even-power mappings. Figure 7 shows the bit metric for the least significant bit B! for 8-QAM mapping. After the demapping process, the bit metrics are written column-by-column into an m-by-n rectangular array. This process is known as deinterleaving. Each element in a row of this metric array (N metrics) constitutes a BPSK-mapped noisy binary codeword bit that is fed into a binary channel decoder in order to estimate its value. AWGN channel and one with either Rayleigh or Rician fading. The results are shown in Figure 8 with BER representing the bit error rate and E! /N! the average bit energy-to-noise ratio. Fig. 7. Bit metric of the least-significant bit of 8-QAM mapping. Fig. 6. Bit metrics of the two most-significant bits of all the M-QAM mappings considered in this paper. B. Performance with a binary RM(3,1) code In this section, simulation results are presented on the performance of bit-interleaved 8-QAM and 32-QAM mappings using the rectangular interleaver and bit metrics generated by a rectangular demapper described in the pervious section. In the computer simulations, maximum-likelihood decoding was implemented with a Green machine (equivalent to a trellis decoder [6]) for the first-order binary RM(3,1) code of length N = 8. The channel model was assumed is AWGN (i.e., no fading), as the relative difference in error performance with channel coding does not change significantly between an The simulated BER performance curves in Figure 8 show that the bit-interleaved 32-QAM, with a block interleaver of length eight, degrades and eventually reaches an error floor. The reason for this behavior is that, as mentioned in [5], the interleaver length N is not sufficiently long relative to the number of bits per symbol m. Since N = 8 and m = 5 for 32- QAM mapping and the binary RM(3,1) code, the bits metrics become highly correlated and error performance does not improve at higher values of signal energy. The performance of a longer binary projective geometry (273,191) low-density parity-check (LDPC) code [6] with a block interleaver and 32- QAM, shown in Figure 9, does not show a floor because the length of the code is now much larger than the number of bits per symbol. In the simulations, the number of iterations was set to 8 and decoding performed with belief propagation using LLR metrics from the demmaper. V. CONCLUSIONS It has been shown that augmenting the set of available mappings in a wireless network, with the inclusion of oddpower QAM mappings, results in an increased throughput. As an illustration, the improvement in throughput was quantified for data transmission under Rayleigh fading conditions. The results show that augmenting the set of modulations in the IEEE standard [1] by 8-QAM, 32-QAM and 128-QAM mappings can possibly yield an increase of up to 9% in throughput compared to the use of even-power QAM mappings alone, with a relatively low complexity cost incurred in the computation of bit metrics at the demapper. Moreover, the
6 error performance of the combination of bit interlaving and channel coding was examined for the cases of 8-QAM and 32- QAM mappings with a binary RM(3,1) code and 32-QAM with a binary finite-geometry (273,191) LDPC code. Future work includes a theoretical analysis of the rate increase obtained by augmenting the set of available mappings under different signal energy probability density functions due to different fading scenarios. Also contemplated is a more comprehensive study of the combination of odd-power QAM mappings, their associated demappers and binary low-density parity-check (LDPC) codes from the IEEE standard [1]. Fig. 9. Error performance of a binary PG (273,191) LDPC code with a rectangular bit-interleaver and 32-QAM mapping. Fig. 8. Error performance of a binary RM(3,1) code with a rectangular bit-interleaver and 8-QAM and 32-QAM mappings. REFERENCES [1] Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications, IEEE Standard [2] Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications, Amendment 3: Enhancements for Very High Throughput in the 60 GHz Band, IEEE Standard ad [3] Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications, Amendment 4: Enhancements for Very High Throughput for Operation in Bands below 6 GHz, IEEE Standard ac [4] G. J. Foschini, R. D. Gitlin and S. B. Weinstein, Optimization of Two- Dimensional Signal Constellations in the Presence of Gaussian Noise, IEEE Trans. Commun., vol. COM-22, no. 1, pp , Jan [5] G. Caire, G. Taricco, and E. Biglieri, Bit-interleaved coded modulation, IEEE Trans. Info. Theory, vol. 44, pp , May [6] S. Lin and D.J. Costello, Jr., Error Control Coding, 2nd ed., Wiley 2004.
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