Beyond Powers of Two: Hexagonal Modulation and Non-Binary Coding for Wireless Communication Systems
aa r X i v : . [ ee ss . SP ] D ec Beyond Powers of Two: Hexagonal Modulation andNon-Binary Coding for Wireless CommunicationSystems
Zhe Yang , Lin Cai , Aaron Gulliver , Liang He and Jianping Pan School of Computer Science and Engineering, Northwestern Polytechnical University, Xi’an, China Dept. of Electrical and Computer Engineering, University of Victoria, Canada University of Michigan, Ann Arbor, MI, USA Dept. of Computer Science, University of Victoria, Canada
Abstract —Adaptive modulation and coding (AMC) is widelyemployed in modern wireless communication systems to improvethe transmission efficiency by adjusting the transmission rateaccording to the channel conditions. Thus, AMC can providevery efficient use of channel resources especially over fadingchannels. Quadrature Amplitude Modulation (QAM) is an ef-ficient and widely employed digital modulation technique. Ittypically employs a rectangular signal constellation. Thereforethe decision regions of the constellation are square partitionsof the two-dimensional signal space. However, it is well knownthat hexagons rather than squares provide the most compactregular tiling in two dimensions. A compact tiling means adense packing of the constellation points and thus more energyefficient data transmission. Hexagonal modulation can be difficultto implement because it does not fit well with the usual power-of-two symbol sizes employed with binary data. To overcomethis problem, non-binary coding is combined with hexagonalmodulation in this paper to provide a system which is compatiblewith binary data. The feasibility and efficiency are evaluatedusing a software-defined radio (SDR) based prototype. Extensivesimulation results are presented which show that this approachcan provide improved energy efficiency and spectrum utilizationin wireless communication systems.
I. I
NTRODUCTION
Wireless communications have become an essential compo-nent of modern information systems. According to the CiscoVisual Networking Index and numerous studies based onmarket status and trends, mobile data traffic has increased18-fold from 2011 to 2016 [1]. This ever-growing demandfor mobile data creates significant demands on the scarcewireless spectrum and limited power of mobile devices. Thus,improving the spectral and energy efficiency of wireless com-munication systems is an important challenge for the researchcommunity and industry.Wireless channels typically suffer from path-loss and theeffects of multipath fading and shadowing, resulting in widevariations in received signal quality. As a consequence, manymodern digital communication systems employ adaptive mod-ulation and coding (AMC) to adjust the transmission rateaccording to the time-varying channel conditions. AMC can
Corresponding Author: Prof. Lin Cai, Dept. of Electrical and ComputerEngineering, University of Victoria, Victoria, BC V8W 2Y2, Canada, Email:[email protected]. efficiently utilize the available bandwidth while meeting thebit-error-rate (BER) requirements. Quadrature amplitude mod-ulation (QAM) is widely employed to transmit more than onebit per modulation symbol. The most common QAM signalconstellations are QPSK and 16-, 64- and 256-QAM, whichcarry 2, 4, 6 and 8 bits per symbol, respectively [12].QAM demodulation converts the received signal, whichmay be affected by fading, noise and interference, to bits.The signal space is partitioned into decision regions for thispurpose, and demodulation is achieved by determining theregion that contains the received signal. With conventionalQAM, the signal space is partitioned into rectangular decisionregions; however, it is well known that a two-dimensionalregular tiling with hexagons is the most efficient packing interms of compactness [7]. Therefore, if hexagonal decisionregions are employed to partition the signal space, referred toas hexagonal quadrature amplitude modulation (H-QAM), thespectrum and/or energy efficiency can be improved. H-QAMmaximizes the minimum distance between signals in the con-stellation and thus minimizes the symbol error probability fora given average signal energy as well as the peak-to-averagepower ratio, which is important for OFDM systems [2], [8],[17], [24]. In [27], a hexagonal lattice was employed inthe time-frequency domain to enhance system performance.However, there has been little interest in H-QAM because ofthe inherent difficulty in using H-QAM with binary data. Thenumber of constellation points may not be a power-of-two,while existing information systems are based on binary data.A number of approaches have been proposed to overcomethis problem [17], [24]. One solution is to convert the binarydata stream to non-binary symbols, e.g. using a binary-inputand ternary-output (BITO) code at the transmitter and reversethe procedure at the receiver [8], [24]. Another approach isto leave some symbols unused [17]. Both of these techniquescannot fully utilize the gains possible with H-QAM, especiallywhen the size of the signal constellation is small. Thus, currentH-QAM solutions alone cannot provide sufficient performanceimprovement for wireless communication systems, although ithas been adopted in optical systems [26].To efficiently explore the potential of H-QAM for wire-less communication systems, we propose to go beyond the conventional binary bit-mapping and coding. Thus, in thispaper we employ H-QAM with ternary digits (trits). Ternaryarchitectures have previously been considered in computingand storage systems due to their higher radix economy and thethree usable states for certain electromagnetic materials [15],[16]. Although ternary communication and computing systemshave not yet reached commercial viability, their future use hasbeen predicted by Knuth [13].The main contributions of this paper are as follows.1) New H-QAM modulation schemes are proposed basedon hexagonal tiling and the corresponding BER perfor-mance is evaluated. These new schemes contain 3, 6, 8,and 12 constellation points to represent 1 trit, 1 bit plus1 trit, 3 bits, and 2 bits plus 1 trit, respectively.2) Ternary convolutional coding is used to protect the tritsdirectly. For H-QAM with hybrid bit and trit informa-tion, we consider a combination of binary and ternarycoding. The BER performance is evaluated for differentmodulation schemes, including conventional rectangularQAM, with code rates 1/2 and 3/4 to conform to theIEEE 802.11 standard [19]. These results show thatthe the new modulation and coding schemes not onlyprovide finer granularity adjustment for AMC, but alsocan replace some of the existing schemes by achieving ahigher throughput with a lower BER for the given SNRregion.3) A prototype H-QAM wireless communication system ispresented which employs non-binary information map-ping using GNU Radio and USRP2, a commonly usedsoftware-defined radio (SDR) platform [4]. To the bestof our knowledge, this is the first H-QAM based pro-totype communication system which demonstrates thefeasibility and efficacy of hexagonal signal constellationswith non-binary coding. Further, extensive simulationresults are presented to demonstrate the efficiency ofthe proposed scheme, which can provide considerableperformance gains compared to existing binary systems.In summary, this paper demonstrates the efficiency of H-QAM with non-binary symbol mapping and error controlcoding in wireless communication systems. This deviates fromconventional approaches that employ rectangular constella-tions with binary coding.The remainder of this paper is organized as follows. InSection II, we discuss the background and related work. Sec-tion III presents the non-binary communication system, includ-ing the hexagonal symbol constellation structure, information-to-symbol mapping, non-binary error correction coding andinterleaving, and the packetization interface with the upperlayers. The system performance is investigated in Section IVthrough extensive trace-driven simulation, and the prototypesystem and measurement results are described in Section V.Finally, some conclusions are given in Section VI along withsuggestions for future work.II. B
ACKGROUND AND R ELATED W ORK
A. Hexagonal Signal Constellations
Modulation is the process of converting a data streamto waveforms suitable for transmission through a commu- nication channel by varying one or more of the waveformproperties, e.g. amplitude, phase, or frequency. For bandwidthlimited bandpass modulation, quadrature amplitude modula-tion (QAM) is commonly employed. Typical QAM constel-lations can be considered as rectangular partitions of thetwo-dimension signal space. It is well known that regularhexagons provide the densest two-dimension packing, andthis has motivated research into the potential of H-QAM [8],[9], [11], [17], [18], [24], [28]. Existing work on hexagonalmodulation usually considers one of the following approaches.The first considers binary data so that each modulation symbolrepresents an integer number of bits. However, this schemerequires that the number of constellation points be a power-of-two. Since the number of H-QAM constellation points isnot a power-of-two [8], some of the constellation points arenot used, which is a waste of available resources.The second approach uses all points in the hexagonalconstellation for transmission to maximize the per-symbolthroughput. It has been shown in [24] that hexagonal-18 QAMmodulation (H18-QAM) requires less energy per-bit than 16-QAM. As
18 = 2 × × , one H18-QAM symbol can bedecomposed into one bit and two trits. To accommodate abinary data stream, binary symbols can be mapped to ternarysymbols using binary-input ternary-output (BITO) convolu-tional or turbo codes to also provide error correction [24].However, this conversion is not flexible and suffers from poorperformance since it fails to offer adequate protection for theternary digits. According to the simulation results, the codederror performance of H18-QAM is . dB lower than thatof the coded 16-QAM, so in this case hexagonal modulationprovides no improvement over rectangular modulation.From the existing literature, it can concluded that usingbinary data with non-binary modulation may be suboptimal assome constellation points are not used, and employing BITOcodes for error correction leads to inflexibility in multiplexingbits and trits. An alternative approach is to combine H-QAM with non-binary, in particular ternary, coding, whichis advocated in this paper. The effectiveness of H-QAM inmulti-media transmission using H-QAM to transmit layeredvideo has previously been demonstrated [28]. B. Ternary Computing and Communications
Radix economy is used to measure the cost of storing ortransmitting numbers in a given base [10]. It is defined asthe number of digits needed to represent a number N inbase b multiplied by the radix b . A base with a lower radixeconomy has a higher efficiency. It has been proven that aradix of three provides the lowest radix economy among allinteger bases. This implies that a ternary base outperformsthe widely used binary base in terms of both storage andcommunications. Ternary based data recording/storing systemshave been investigated [15], [16], and the the cost of storingnumbers can be minimized if a ternary base is used [10].Computers using balanced ternary logic were implemented inthe late 1950s and were shown to be more efficient comparedwith the binary based computers. Knuth has predicted thatthe elegance and efficiency of ternary logic will result in itsemergence in the future [13]. T h r o u g hpu t Received SNR (dB)
Channel capacity (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
Fig. 1: The achievable throughput with different modulationschemes.As discussed in Section II-A, the use of ternary codes withhexagonal constellations can fully realize the potential of H-QAM and improve the performance of wireless communica-tion systems. Thus, we have conducted an extensive literaturesurvey to find the best known ternary convolutional codes, andthese codes are punctured to obtain different code rates. Thedetails will be given in Section III-C.
C. Modulation vs. Coding Gain
The motivation of AMC is to combine different modulationschemes and code rates to fully utilize the capacity of thetime-varying channel. The code rate can be adjusted to correctdifferent numbers of bits in error according to the BERafter demodulation (uncoded BER). If this BER is below agiven threshold, coding can be employed to reduce it to anegligible level. Otherwise, it can be very difficult to reducethe BER to an acceptable level even with a powerful codingscheme, and the BER with and without coding may actuallybe similar. To overcome this issue, for a given SNR at thereceiver, an appropriate modulation scheme can be used toensure that the uncoded BER after demodulation is belowa desired threshold (often − for wireless systems), andthen error control coding can be applied to further reducethe BER to an acceptable region (e.g., below − ). As errorcontrol coding may not be effective unless the uncoded BERis sufficiently low, a better modulation scheme can improvethe overall system performance. For wireless communicationsystems employing AMC, several combinations of modulationand coding schemes are usually adopted to fit different channelconditions. In this paper, we also consider different code ratesfor H-QAM. Introducing H-QAM modulation with coding canoutperform and thus replace some of the existing rectangularQAM based AMC schemes.In addition, if the SNR gap between two AMC schemes islarge, e.g., QPSK with a rate 3/4 code and 16-QAM with a rate1/2 code as in 802.11 standard [6], this gap can be filled withan H-QAM based scheme. As illustrated in Figure 1, sincethe SNR at the receiver is continuous, there is space (shadedareas) for new modulation schemes (using new or existingcoding schemes) to improve the system performance, whichmotivates the work reported in this paper. III. S YSTEM D ESIGN
As mentioned in the previous section, the number of con-stellation points in the most compact hexagonal constellationsis not always an integer power-of-two. Therefore, to fullyutilize these constellations for modulation, both bits and tritsshould be transmitted. This requires a reinvestigation of themodulation constellation geometry, the mapping of bits andtrits to constellation points, the error control coding, and themultiplexing of bits and trits.
A. Constellation Geometry
A signal can be represented in the signal space domainusing an in-phase and quadrature-phase (I/Q) constellationdiagram. For a constellation with N points, the informationcarried in each symbol equals log N bits. The distancebetween a constellation point to the origin, d , is proportionalto the square root of the transmitted symbol energy. In theabsence of noise, the received signal constellation has thesame shape as the transmitted constellation, except that, at thereceiver, the distance from a constellation point to the origin isproportional to the square root of the received symbol energy.In the following, constellation refers to the constellation at thereceiver unless otherwise stated.In an additive white Gaussian noise (AWGN) channel,a received symbol follows a two-dimensional Gaussian dis-tribution centered at the corresponding constellation point.A Voronoi diagram can be used to determine the decisionboundary of each symbol. The probability that a symbol isdemodulated in error is equal to the probability that the re-ceived symbol lies outside the decision region of the intendedsymbol.Given the fact that the Gaussian distribution decays ex-ponentially and the BER after demodulation should be suf-ficiently low (e.g., below − ), the symbol error probabilityfor H-QAM can be approximated as SER = 2 Q s r N , (1)where Q ( · ) is the Q-function, r is the shortest distance fromthe constellation point to its decision boundary, and N isthe noise spectral density. As r is equal to half of theminimum Euclidean distance in the signal space between twoconstellation points, the BER is determined by the minimumEuclidean distance between constellation points.A good modulation constellation should convey more in-formation under the same average power (symbol energy)and BER constraints. Using (1), this can be converted to thefollowing geometry problem. In a circle of radius √ E + r , packas many non-overlapping circles with radius r as possible,where E is the maximum received symbol energy and r isdetermined by the BER constraint.For a sufficiently large radius, √ E , the optimal packing isa hexagonal tiling. Comparing this tiling with the rectangulartiling widely used in existing QAM schemes, a hexagonaltiling can cover the region more efficiently, which leads toapproximately a . dB gain [5]. ! !" ! ! $" Fig. 2: TPSK, H6-QAM, and H8-QAM constellations. !"
Fig. 3: Constellation and mapping for H12-QAM.
1) The Proposed H-QAM Geometry:
In this paper, wepropose TPSK, H6-QAM, H8-QAM, and H12-QAM schemesas design examples to demonstrate the benefits of hexagonalmodulation combined with ternary coding. Conditioned onmaximizing the minimum Euclidean distance, the average symbol energy should be minimized. This is equivalent tominimizing the average of the squares of the distances from theconstellation points to the origin under the condition that theminimum distance of each constellation point to its decisionboundary is no smaller than r , i.e. min P i d i M , (2)where d i is the distance of the i -th constellation point to theorigin and M is the number of constellation points. Thus, thepoints should be as close to the origin as possible.For the TPSK constellation shown in Figure 2 (a), eachsymbol represents one trit, or log ≈ . bits of informa-tion. It is straightforward to arrange the constellation pointsas the end points of an equilateral triangle, and set the originto the center of the triangle. The minimum Euclidean distancefrom each point to its decision boundary is then r = p E s / ,where E s equals the average symbol energy. For H6-QAM,each symbol carries one trit plus one bit of information,and the constellation arrangement is shown in Figure 2 (b).The minimum Euclidean distance from a constellation pointto its decision boundary is r = p E s / , which is muchlarger than that of 6-PSK ( p E s / ), and thus the symbolerror performance of H6-QAM is better. Thus, H6-QAM cantransmit the same amount of information as 6-PSK as theyboth contain constellation points, but with a lower symbolerror rate.For H8-QAM, each symbol carries three bits of information,and the constellation arrangement is shown in Figure 2 (c),where the origin is located at the midpoint of an edge of a hexagon. The minimum Euclidean distance from a con-stellation point to its decision boundary is r = p E s / ,which is larger than that of rectangular 8-QAM ( p E s / ).For H12-QAM, each symbol carries one trit plus two bitsof information, and the proposed constellation arrangement isshown in Figure 3 (a), where the origin is the joint vertex ofthe center hexagons. The minimum Euclidean distance from aconstellation point to its decision boundary is r = p E s / .Note that TPSK, H6-QAM and H12-QAM are rotationallysymmetric by ◦ , which provides additional benefits as willbe shown in Section V. B. Constellation Mapping
Given the geometry of the constellation points, the next stepis to map the bits and trits to the constellation point suchthat the BER is minimized. Compared with the conventionalrectangular modulation, there are more neighboring points(with the smallest distance to a constellation point) usinghexagonal modulation, so careful mapping is required to limitthe number of bit and/or trit errors due to a symbol error. It isnot straightforward to obtain a Gray type mapping (with onlyone bit/trit difference between neighboring points), becausethe number of neighboring constellation points with hexagonaltiling often exceeds the number of digits (bits and/or trits)represented by each symbol.The following design principle is used here to obtain goodmappings. If starting from a bit, the constellation points aredivided into two clusters. Similarly, the constellation pointsare divided into three clusters if starting from a trit. Then, ‘0’and ‘1’ (for a bit) or ‘0’, ‘1’ and ‘2’ (for a trit) are arbitrarilyassigned to each of the clusters. For the remaining bits or trits,binary or ternary numbers are first assigned to the points inone cluster, and then in turn to the points in the other clusters.The same number is assigned to neighboring points in differentclusters as much as possible. For example, with the H12-QAMconstellation in Figure 3 (b), starting with a trit, the pointsare divided into three clusters, and are assigned ‘0’, ‘1’ and‘2’, respectively. Within the first cluster, a Gray-type mappingis used to assign ‘00’, ‘01’, ‘11’ and ‘10’ to the four points. Forthe second cluster, ‘100’ and ‘101’ are assigned to the pointsneighboring ‘000’ and ‘001’. Similarly, for the third cluster,‘201’ and ‘211’ are assigned to the points neighboring ‘101’and ‘111’. Although this approach to constellation mappingdoes not in general produce a Gray mapping, the resultspresented in Section IV show that it still leads to a significantperformance improvement. C. Ternary Error Control Coding and Interleaving
Given a noisy channel, a received constellation point maydiffer from the transmitted one, which will result in bit/triterrors. Interleaving and error control coding can be used tomitigate these errors.
1) Non-binary Convolutional Codes:
Convolutional codinghas been widely used in wireless systems such as 802.11because of the relatively simple implementation and good per-formance improvements [6]. A binary convolutional encodercan be represented by the parameters ( n, k, m ) where k and n (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) input u output v2output v1 Fig. 4: A ternary convolutional encoder [25].are the numbers of input and output bits, respectively, and m is the encoder memory size. The corresponding code rate is k/n . Three code rates, / , / and / , are employed in theIEEE 802.11 standard. At the receiver, the received coded bitstream is decoded to recover the original message bit stream.The Viterbi algorithm provides maximum-likelihood decodingis widely used in practice because of the low implementationcomplexity and satisfactory performance.A ternary ( n, k, m ) convolutional encoder maps k input tritsto n output trits [25]. A ternary convolutional encoder can besimply implemented using shift registers and modulo-3 adders,as shown in Figure 4. In the figure, the shaded squares arememory elements, the circled numbers are the coefficients thatthe trits are multiplied by, and ⊕ represents a modulo-3 adder.As there is one input trit stream u and two output trit streamsv1 and v2, the code rate is / . Similar to binary convolutionalcodes, puncturing can be employed to obtain different coderates to adjust the protection level according to the modulationscheme, channel condition and required BER. Without loss ofgenerality and to be consistent with the 802.11 standard, arate 3/4 punctured ternary convolutional code is considered.The puncturing pattern is the same as that used for the binaryconvolutional code, [1 1 1 0 0 1] , where means the codeddigit at that position is punctured.The operations in a ternary Viterbi decoder are in the Galoisfield of three elements, GF (3) . The complexity associatedwith a convolutional code is primarily in the decoder. Aternary decoder using the Viterbi algorithm requires O (3 m L T ) memory space and O (3 m +1) L C ) computation time, where L T is the trace back length of the decoder and L C is the codeblock length, respectively. The computational complexity of aternary convolutional decoder is comparable to that of a binarydecoder with a similar number of states.
2) Interleaving:
Many error correcting codes such as con-volutional codes cannot tolerate burst errors, so it is desirableto separate these errors using an interleaver. An interleavercan also help mitigate errors when Gray mapping is notemployed. Bits or trits are interleaved within a packet as theperformance results show that this intra-packet interleavingprovides a performance gain for fading channel and does notintroduce significant delay.
Remark:
Several error control coding designs exist in theliterature that have been shown to provide near-capacity per- formance, e.g. low-density parity check (LDPC) codes [22],accumulate-repeat-accumulate (ARA) codes [21], and ratelesscodes [3]. Further research work is required to extend thesedesigns to ternary codes to further improve the performanceof H-QAM AMC schemes.
D. System Architecture
Using the IEEE 802.11a standard as an example, Figure 5shows the H-QAM system architecture. The shaded blocks in-dicate new modules or existing modules that required updated.The messages are divided into two queues, one of which isconverted to a trit stream and the other is kept as a bit stream.The bits/trits conversion module converts the bit sequence to atrit sequence (details of this conversion will be discussed later),and this is the input to the ternary convolutional code (TCC)encoder. The cross-layer controller uses the channel stateinformation (CSI) to determine the modulation and codingscheme to be used, and this is included in the Physical LayerConvergence Procedure (PLCP) header [6].The PLCP header cotains the physical layer control infor-mation. The rate field of the PLCP header is used to inform thereceiver of the modulation and coding scheme to be employed.There are different adaptive modulation and coding (AMC)schemes in the current IEEE 802.11a standard, which isrepresented by a -bit rate field. There is one reserved bit inthe header which can be combined with the -bit rate field torepresent up to different modulation and coding schemes.This is sufficient to include the new AMC schemes based onH-QAM and ternary convolutional coding proposed in thispaper, as some of new AMC schemes replace existing AMCschemes based on rectangular QAM and binary convolutionalcoding. The proposed non-binary H-QAM communicationsystem is compatible with conventional systems and does notrequire additional communication overhead as will be shownlater. IV. P ERFORMANCE E VALUATION
A. Uncoded BER Performance
A key performance indicator is the BER w.r.t the receivedSNR over an AWGN channel. We consider a received SNRfrom to dB and examine the uncoded BER for differentmodulation schemes. The received signal to noise ratio is E s /N where E s is the received energy per symbol and N is the noise power spectral density. Monte Carlo simulationwith Matlab was used to obtain the BER results shown inFigure 6. Each H-QAM constellation point has more neighborsthan with rectangular QAM, so the resulting non-Gray codemapping will degrade the BER performance of H-QAM, butH-QAM is still an efficient modulation design in terms ofthe symbol error rate. For example, H8-QAM outperformsrectangular 8-QAM by . dB at a BER of − , and this gapincreases for smaller BERs. This demonstrates the advantageof using hexagonal constellations.Using − as the BER benchmark, the addition of TPSK,H6-QAM, H8-QAM and H12-QAM provides more choicesto adapt the modulation according to the received SNR. Forinstance, TPSK can be used to replace BPSK in the SNR ... ... MAC (bits) Mod RFFrontend(trits) RFFrontend De−inter−leaverDeMod
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CSI PLCP Header Trits/Bits
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Fig. 5: The new system architecture based on IEEE 802.11a. ✵ ✷ ✹ ✻ ✽ ✶✵ ✶✷ ✶✹ ✶✻ ✶✽✶✵✲(cid:0)✶✵✲✁✶✵✲✂✶✵✲✄✶✵☎ ❊s✴✆✝❇✞✟ ✠✡☛☞◗✡☛☞✽✌◗✍✎✶✻✌◗✍✎❍✏✡☛☞❍✻✌◗✍✎❍✽✌◗✍✎❍✶✷✌◗✍✎
Fig. 6: BER without error control coding.range [6 . , dB to achieve a throughput gain. Similarly,H6-QAM, H8-QAM, and H12-QAM can replace QPSK inthe SNR range of [9 . , . dB to achieve a to throughput gain. Similarly, H-QAM constellations such asH27-QAM and H54-QAM should provide throughput gainswhen the received SNR is higher. B. Coded BER Performance
For the coded BER performance, an AWGN channel isconsidered with the rate / ternary convolutional code shownin Figure 4. This code has = 81 states, so for a faircomparison a rate / binary convolutional code is used with = 64 states. This binary code is employed in the IEEE802.11 standard [6]. To evaluate the performance of H-QAMbased AMC, we also consider punctured convolutional coding ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶(cid:0) ✶✶ ✶✁✶(cid:0)✲✂✶(cid:0)✲✄✶(cid:0)✲☎✶(cid:0)✲✆✶(cid:0)✲✝✶(cid:0)✲✞✶(cid:0)✲✟ ❊s✴✠✵❈✡☛☞☛✌✍✎✏☞✑✒✡✑✓✔✕✖☞ ❇✗✘✙ ✚✛✜✢ ✶✴✁ ✣✣❇✗✘✙ ✚✛✜✢ ✸✴✹ ✣✣❚✗✘✙ ✚✛✜✢ ✶✴✁ ❚✣✣❚✗✘✙ ✚✛✜✢ ✸✴✹ ❚✣✣◗✗✘✙ ✚✛✜✢ ✶✴✁ ❚✣✣◗✗✘✙ ✚✛✜✢ ✸✴✹ ❚✣✣ Fig. 7: Coded BER with BPSK, TPSK, and QPSK modulation.with the puncture pattern discussed in Sec. III-C1, whichprovides a code rate of / .The BERs for BPSK, TPSK and QPSK with different coderates are shown in Figure 7, and the BERs for QPSK, H6-QAM, H8-QAM, H12-QAM and 16-QAM with different coderates are given in Figure 8. Comparing Figures 6 and 7, whenthe SNR is below dB, the BER for BPSK with or withoutcoding is similar ( − or above). In addition, when the SNRis below dB, the BER for TPSK with or without coding issimilar ( − or above). This indicates that error correctioncoding is effective only when the uncoded BER is sufficientlylow.Figures 7 and 8 show that for a given code rate and BER,the required SNR increases with respect to the number ofconstellation points. This is because the denser the constel-lation, the smaller the minimum Euclidean distance. However, ✹ ✻ ✽ ✶(cid:0) ✶✁ ✶✹ ✶✻ ✶✽✶(cid:0)✲✂✶(cid:0)✲✄✶(cid:0)✲☎✶(cid:0)✲✆✶(cid:0)✲✝✶(cid:0)✲✞✶(cid:0)✲✟ ❊s✴✠✵❈✡☛☞☛✌✍✎✏☞✑✒✡✑✓✔✕✖☞ ◗✗✘✙ ✚✛✜✢ ✣✤✥ ✦✦◗✗✘✙ ✚✛✜✢ ✧✤★ ✦✦❍✩◗✪✫ ✚✛✜✢ ✣✤✥ ✦✦❍✩◗✪✫ ✚✛✜✢ ✧✤★ ✦✦❍✬◗✪✫ ✚✛✜✢ ✣✤✥ ✦✦❍✬◗✪✫ ✚✛✜✢ ✧✤★ ✦✦❍✣✥◗✪✫ ✚✛✜✢ ✣✤✥ ✦✦❍✣✥◗✪✫ ✚✛✜✢ ✧✤★ ✦✦✣✩◗✪✫ ✚✛✜✢ ✣✤✥ ✦✦✣✩◗✪✫ ✚✛✜✢ ✧✤★ ✦✦ Fig. 8: Coded BER with QPSK, H6-QM, H8-QAM, H12-QAM, and 16-QAM modulation.TABLE I: Comparison of Modulation and Coding Schemes
Modulation Code rate Throughput (b/sym) SNR (dB)
BPSK* 1/2 0.5 > > > > > > > > > > > > > > some combinations of H-QAM and coding outperform therectangular QAM combinations in terms of both throughput(bits per symbol) and BER. For instance, the BER performanceof TPSK with a rate / code is better than that of QPSKwith a rate / code, with more than . dB improvementat a BER of − . In addition, TPSK with a rate / codehas a throughput of . b/sym (bits per symbol), whichis higher than that of QPSK with a rate / code, b/sym.Thus, H-QAM provides a . throughput gain at a lowerSNR, so TPSK with rate / coding can replace QPSK withrate / coding in AMC. Similarly, H12-QAM with rate / coding can replace 16-QAM with rate / coding. Using − as the threshold for coded BER, the required SNR and thecorresponding throughput are given in Table I.Considering AMC, TPSK and H12-QAM with rate / coding can be used to replace QPSK and 16-QAM with rate / coding, respectively, and H6-QAM and H8-QAM withrate / coding can be used in the SNR gap between QPSKwith rate / coding and 16-QAM with rate / coding. Thisprovides a finer grain set of choices for AMC. These newAMC schemes can be indicated in the PLCP header for properdemodulation and decoding at the receiver. We next study ✽ ✶✶ ✶(cid:0) ✶✁✵✵✂✄✶✶✂✄✷✷✂✄ ❆☎✆✝✞✟✆ ✠✡☛❚☞✌✍✎✏☞✑✎✒✓✔✕✒✖✗✖✘✙✚ ❈✛✜✢✣✜✤✥✛✜✦✧ ★✩❈ ✪✥✤✫ ✬✣✭✮✣✯✤ ❈✰✱❍✲✳★✩ ★✴✸✹✣✜✤✣✺ ★✩❈ ✪✥✤✫ ✬✣✭✮✣✯✤ ❈✰✱❈✛✜✢✣✜✤✥✛✜✦✧ ★✩❈ ✪✥✤✫ ✱✹✻✣✭✮✣✯✤ ❈✰✱❍✲✳★✩ ✦✴✸✹✣✜✤✣✺ ★✩❈ ✪✥✤✫ ✱✹✻✣✭✮✣✯✤ ❈✰✱ Fig. 9: Single link throughput comparison.these new modulation and coding schemes in terms of systemthroughput and efficiency.
C. Single Link Throughput
The new modulation and coding combinations are nowconsidered in an AMC system. A Rician fading channelwith Rician factor K = 6 dB is considered for a singlecommunications link. The conventional AMC set containsBPSK, QPSK and 16-QAM with rate / and / codingaccording to the IEEE 802.11 standard [6]. The augmentedAMC set includes the existing QAM and new H-QAM basedtransmission schemes marked by * in Table I. For a faircomparison, we use the same symbol rate and energy for allthe modulation and coding schemes. In the simulations, datapackets of size kB are transmitted, and , packets aretransmitted to obtain the average performance. Separate intra-packet interleaving is used for the bits and trits. The transmitterand receiver structures are as given in Figure 5.A practical issue with AMC is the imperfect estimation ofchannel conditions. If the received SNR is underestimated,the sender may select a modulation and coding scheme witha lower throughput (number of bits per received symbol).Conversely, if the received SNR is overestimated, it may resultin a higher BER than the required threshold, which is evenmore undesirable. The impact of channel estimation errorson the system performance is thus of critical importance. Toexamine this impact, channel estimation errors are modeled asa Gaussian random variable with zero mean and unit variance.To reduce the probability that the received SNR is overes-timated (which may severely degrade system performance),the transmitter uses the estimated SNR minus its standarddeviation to select the modulation and coding scheme.Figure 9 compares the link throughput using AMC with andwithout the proposed H-QAM schemes for an average receivedSNR of , , and dB. The average throughput wasobtained for Monte Carlo iterations to average the effects of fading. With perfect channel estimation, the proposed non-binary communication system outperforms the conventionalsystem by . , . , . and . when the averageSNR is , , and dB, respectively. The performance ofboth conventional QAM and H-QAM degrades with imperfectchannel estimation, but the proposed system still achievesthroughput gains of . , . , . and for anaverage SNR of , , and dB, respectively. Theseresults show that the proposed H-QAM AMC can providebetter BER performance and also higher throughput. Further,it has a finer granularity than conventional QAM AMC w.r.t.the SNR. D. Network Throughput
The system performance is now evaluated in aninfrastructure-based network where an access point (AP) iscentrally located to serve all users in the network, e.g. a WiFinetwork, and the mobile users are randomly distributed. Thewireless channels suffer from independent Rician block fading.The path-loss exponent is α = 3 , and all transmitted symbolshave the same average energy. We consider the downlinkperformance where the AP transmits packets (with size kB)to all mobile users in a round-robin manner. It is assumed thatthe AP has all channel information which is used to select theAMC scheme for each packet according to the estimated SNR.The average SNR is set to dB when users are at the boundaryof the network.The system performance was evaluated using Monte Carlosimulation with different node densities. For each density,1000 simulations were run using random topologies, and theaverage network throughput was determined in terms of bitsper symbol. Figure 10 presents the results for and users,which correspond to sparse and dense networks, respectively.These results show that the proposed non-binary H-QAMschemes can increase the average network throughput by morethan . and . for the and user cases, respec-tively, with perfect or imperfect channel state information. Inaddition to the throughput gain, for a given average transmittedsymbol energy the per bit energy is also reduced by . and . , respectively.V. P ROTOTYPE S YSTEM AND M EASUREMENTS
A prototype for the non-binary H-QAM communicationsystem was developed using the software-defined radio (SDR)platform USRP2 [4] and GNU Radio. One USRP2 was con-nected to the laptop host (DELL E5400) as the transmitterand another to the PC host (DELL OPTIPLEX 755) as thereceiver. The USRP2-based OFDM implementation [20] wasaugmented with the proposed H-QAM. The carrier frequency,number of subcarriers and FFT length are . GHz, and , respectively.The three new hexagonal modulation designs, TPSK, H6-QAM and H8-QAM, were implemented using the bits/tritsconversion presented previously. This conversion will intro-duce minimal overhead and thus there is a small performanceloss [24]. Using a long bit sequence will reduce this loss but ✺ ✸(cid:0)✶✁✶✶✁✂✶✁✸✶✁✄✶✁✺✶✁☎✶✁✆✶✁✝✶✁✞✂ ◆✟✠✡☛☞ ✌✍ ✎✏☛☞✏✑✒✓✔✕✖✗✘✙✖✕✚✛✙✜✚✓✢✣✤✓✥✦✥✧★✩ ❈✪✫✬✭✫✮✯✪✫✰✱ ✲✳❈ ✴✯✮✵ ✷✭✹✻✭✼✮ ❈✽✾❍✿❀✲✳ ✲❁❂❃✭✫✮✭❄ ✲✳❈ ✴✯✮✵ ✷✭✹✻✭✼✮ ❈✽✾❈✪✫✬✭✫✮✯✪✫✰✱ ✲✳❈ ✴✯✮✵ ✾❃❅✭✹✻✭✼✮ ❈✽✾❍✿❀✲✳ ✲❁❂❃✭✫✮✭❄ ✲✳❈ ✴✯✮✵ ✾❃❅✭✹✻✭✼✮ ❈✽✾ Fig. 10: The network throughput with and users.TABLE II: The experimental BER results Modulation Uncoded BER Coded BER
BPSK . × − < − TPSK . × − < − QPSK . × − . × − H6-QAM . × − . × − H8-QAM . × − . × − . × − . × − increase the delay and complexity. The conversion efficiencyis defined as η = l b l t log (3) , (3)where l b is the length of the input bit sequence and l t is thelength of the output trit sequence. Converting bits to tritsprovides an efficiency of
11 log
27 log = 99 . [14]. As the blockof bits is small, this can easily be implemented using a lookuptable with entries. After bits/trits conversion, the data ismapped to modulation symbols.One thousand test frames were transmitted where eachframe contains blocks of data, and each block contains bytes of data with a -byte CRC. The transmitter converts allor some of the received bits in each block to trits dependingon the modulation employed. The receiver demodulates thereceived symbols to bits and/or trits according to the mod-ulation scheme used as indicated in the PLCP header. Thenthe trits are converted back to bits and the CRC is checkedto determine whether there are any transmission errors inthe block. If a block fails the CRC check, the number oferrors is obtained by comparing it with the original block. TheBER after demodulation can then be calculated using the totalnumber of errors. To obtain the coded BER, the received bitsor trits are further processed using binary and ternary Viterbidecoders.The BER results are shown in Table II. As expected, theperformance of TPSK is better than that of QPSK and H6-QAM, and H8-QAM performs better than 8PSK. These results demonstrate the feasibility and simplicity of deploying theproposed H-QAM. Note that the uncoded BER performance ofTPSK is slightly better than that of BPSK. This is because thenumber of bits transmitted with a TPSK symbol is significantlyhigher [23]. Further, the ◦ phase difference between TPSKsymbols is much smaller than that of BPSK, which is apractical benefit of TPSK modulation. Despite the limitationsof the USRP2 hardware and the effects of the fading channel,the measured results given for an indoor environment confirmthe feasibility of employing H-QAM.VI. C ONCLUSION
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