Q-ary Multi-Mode OFDM with Index Modulation
aa r X i v : . [ ee ss . SP ] M a r Q -ary Multi-Mode OFDM with Index Modulation Ferhat Yarkin
Student Member, IEEE and Justin P. Coon
Senior Member, IEEE
Abstract —In this paper, we propose a novel orthogonal fre-quency division multiplexing with index modulation (OFDM-IM) scheme, which we call Q -ary multi-mode OFDM-IM ( Q -MM-OFDM-IM). In the proposed scheme, Q disjoint M -aryconstellations are used repeatedly on each subcarrier, and amaximum-distance separable code is applied to the indices ofthese constellations to achieve the highest number of indexsymbols. A low-complexity subcarrier-wise detection is shownpossible for the proposed scheme. Spectral efficiency (SE) andthe error rate performance of the proposed scheme are furtheranalyzed. It is shown that the proposed scheme exhibits a veryflexible structure that is capable of encompassing conventionalOFDM as a special case. It is also shown that the proposed schemeis capable of considerably outperforming the other OFDM-IMschemes and conventional OFDM in terms of error and SEperformance while preserving a low-complexity structure. Index Terms —Orthogonal frequency division multiplexing(OFDM), index modulation (IM), maximum-distance separable(MDS) code.
I. I
NTRODUCTION
The studies, so far, show that index modulation (IM) tech-niques exhibit important advantages compared to conventionalmodulation techniques. Specifically, when IM is applied toorthogonal frequency division multiplexing (OFDM) tech-nique, a better error performance and improved data ratecompared to conventional OFDM are shown to be possible[1]–[6]. Moreover, an application of a recent IM scheme,called set partition modulation (SPM), to OFDM brings abouta marginal enhancement in error rate at high signal-to-noiseratio (SNR) values and a substantial improvement in data ratewhen compared to other OFDM-IM benchmarks [7].It is well-known that, for a fading channel, the error per-formance of a codebook is limited by the minimum Euclideandistance between codeword pairs that have the minimumHamming distance [8]. Moreover, such minimum Euclideanand Hamming distances, respectively, determine the codingand diversity gains of error probability curves related to thecodebook. Hence, the superior error performance provided bythe OFDM-IM schemes is based on the fact that the informa-tion bits are mapped not only to signal space as in conventionalOFDM but also to the index domain . Such a mapping enablesthe OFDM-IM schemes to have a higher minimum Euclideandistance in the signal space than conventional OFDM. In thisregard, conventional OFDM-IM [1], [2] encodes data into F. Yarkin and J. P. Coon are with the Department of Engineering Sci-ence, University of Oxford, Parks Road, Oxford, OX1 3PJ, U.K. E-mail: { ferhat.yarkin and justin.coon } @eng.ox.ac.uk Note that the minimum Hamming distance between conventional modu-lation symbols drawn from the signal space is limited to one; however, suchdistance is two for index symbols drawn from the index domain. Since theoverall OFDM-IM symbols are formed by both conventional modulation andindex symbols, the minimum Hamming distance between these symbols islimited to one. the combinations of active subcarriers, thus the total poweris distributed across all subcarriers. Moreover, dual-modeOFDM-IM (DM-OFDM-IM) [3], [4], multi-mode OFDM-IM(MM-OFDM-IM) [5], [6] and OFDM-SPM [7] use disjointconstellations on all subcarriers. The presence of the indexcodewords makes these schemes capable of achieving the samespectral efficiency (SE) as that of conventional OFDM byemploying lower order modulation. On the other hand, thepotential of index domain is exploited more by the studies thatincrease the number of index symbols with combinatorial toolssuch as permutation and set partitioning [5]–[7]. However, itis not obvious whether the existing OFDM-IM schemes canachieve the full potential of the index domain by producingthe largest number of possible index symbols.Against this background, we propose a novel MM-OFDM-IM scheme named Q -ary MM-OFDM-IM ( Q -MM-OFDM-IM) which is capable of exploiting the full potential ofthe index domain by employing Q disjoint constellationsrepeatedly on each subcarrier. In this scheme, unlike theMM-OFDM-IM scheme in [5] that uses the permutations ofdisjoint constellations to form the index symbols, we employa completely different approach that uses a maximum-distanceseparable (MDS) code on the disjoint constellations to achievethe highest number of index symbols. Then, for the proposedscheme, we present a sub-optimal maximum likelihood (ML)detector. We also investigate the SE and bit error rate (BER) of Q -MM-OFDM-IM in this paper and obtain an upper-bound onthe BER. Our analytical, as well as numerical findings, showthat the proposed scheme can achieve a substantially betterperformance than OFDM-SPM, MM-OFDM-IM, OFDM-IMand conventional OFDM in terms of SE and BER whileexhibiting a very simple structure and high flexibility.II. S YSTEM M ODEL
In this section, we present the system model of the Q -MM-OFDM-IM scheme. A. Transmitter m input bits enter the transmitter, and these bits are dividedinto B = m/f blocks, each having f input bits. Similarly,the total number of subcarriers N T is also divided into B = N T /N blocks, each having N subcarriers.Since each bit and each subcarrier block has the samemapping operation, we focus on a single block, the b th block(where b ∈ (cid:8) , , . . . , B (cid:9) ), in what follows. In the b th block,the f information bits are further divided into two parts, oneof them having f bits and the other one having f bits with f + f = f . Letting Q be a positive integer, the first f bits are used to determine the one of Q disjoint M -ary constellations M q , q ∈ (cid:8) , , . . . , Q − (cid:9) , or in other words modes, whichwill be used on each subcarrier. Unlike the other OFDM-IMschemes, which employ different modes on their subcarriers,the proposed scheme is capable of using any mode on anysubcarrier, repeatedly.
Hence, we have Q N mode patterns intotal on N subcarriers.Unlike the index symbols of the OFDM-IM schemes, theminimum Hamming distance between Q N patterns is limitedto one. In this context, it is important to attain the highestnumber of the index symbols that achieve the same minimumHamming distance as those of the OFDM-IM schemes. Sucha number is bounded by the Singleton bound [9]. With B Q ( d, N ) and d denoting the maximum number of possiblecodewords in a Q -ary block code of length N and minimumHamming distance d between such codewords, the Singletonbound states that B Q ( d, N ) ≤ Q N − d +1 . (1)To have the same minimum Hamming distance betweenthe index symbols as the index symbols of the OFDM-IM schemes, one needs to pick d = 2 . In that case, theSingleton bound becomes B Q (2 , N ) ≤ Q N − . This boundcan be achieved by a simple maximum-distance separable(MDS) code [9]. Such a code forms the first N − elements, I τ , τ ∈ (cid:8) , . . . , N − (cid:9) , by using the integers, , , . . . , Q − as symbols, i.e., I τ ∈ (cid:8) , , . . . , Q − (cid:9) , and the last symbol, I N is chosen from the same integers by letting the code bethose N -tuples summing to zero under modulo- Q arithmetic,i.e., ( I + I + . . . + I N ) mod Q = 0 . Hence, there are Q N − such N -tuples. Then, we map these N -tuples to the one ofthe index sets in the b th block, I b := (cid:8) I , I , . . . , I N (cid:9) , andthe n th element of the index set is used to determine theindex of the M -ary constellation on the n th subcarrier where n ∈ (cid:8) , , . . . , N (cid:9) . f bits are further mapped to one of theseindex sets, and therefore f = ⌊ log Q N − ⌋ . Note that theseoperations result in Q N − index symbols whose minimumHamming distance is two. Moreover, the mapping of f bitsto the index symbols can be implemented by using a look-uptable.Let us consider an example of how we form the indexsymbols and implement the look-up table between these code-words and the information bits for the proposed scheme when Q = N = 3 . We use the integers, , , , to form MDS codesas shown in the leftmost column of Table I. The correspondingtriple codes are given in the second column (from the left) ofthe table. By mapping them to the index symbol vectors, weconstruct the final index sets of the Q -MM-OFDM-IM schemeas shown in the table. Moreover, f bits are used to determinethe specific index set. For two disjoint constellations M q and M ˆ q where q, ˆ q ∈ (cid:8) , , . . . , Q − (cid:9) and q = ˆ q , M q ∩ M ˆ q = ∅ . Note also that we choose the size ofeach constellation as M , i.e., |M q | = M, ∀ q ∈ (cid:8) , , . . . , Q − (cid:9) , forconvenience. Note that each index pattern of the Q -MM-OFDM-IM scheme can beregarded as a Q -ary block code of length N since the index bits are mapped to N -tuples whose elements are chosen among Q disjoint constellations. Hence,the Singleton bound is valid for the number of index symbols of the Q -MM-OFDM-IM scheme. Table II NDEX S YMBOL G ENERATION AND L OOK -U P T ABLE E XAMPLE FOR THE
Q-MM-OFDM-GSPM
SCHEME WHEN Q = N = 3 . ( N − -tuple ( I , I ) N -tupleMDS Code Q -MM-OFDM-IMIndex Set f bits (0 ,
0) (0 , , (cid:8) , , (cid:9) [0 0 0](0 ,
1) (0 , , (cid:8) , , (cid:9) [0 0 1](0 ,
2) (0 , , (cid:8) , , (cid:9) [0 1 0](1 ,
0) (1 , , (cid:8) , , (cid:9) [0 1 1](1 ,
1) (1 , , (cid:8) , , (cid:9) [1 0 0](1 ,
2) (1 , , (cid:8) , , (cid:9) [1 0 1](2 ,
0) (2 , , (cid:8) , , (cid:9) [1 1 0](2 ,
1) (2 , , (cid:8) , , (cid:9) [1 1 1](2 ,
2) (2 , , (cid:8) , , (cid:9) unused Once the index set, I b , is determined by f bits, theremaining f bits are used to modulate symbols on the N subcarriers by using the disjoint M -ary constellations regard-ing the determined index set. Hence, the Q -MM-OFDM-IMsymbol vector corresponding to the b th block can be written as s b = [ s b , s b . . . , s bN ] where s bn ∈ M I n , I n ∈ (cid:8) , , . . . , Q − (cid:9) is the n th element of the set I b . Since |M q | = M, ∀ q ∈ (cid:8) , , . . . , Q − (cid:9) , f = N log M . After obtaining sym-bol vectors for all blocks, an OFDM block creator formsthe overall symbol vector s := [ s (1) , s (2) , . . . , s ( N T )] T =[ s , . . . , s b , . . . , s B ] T ∈ C N T × . After this point, exactly thesame operations as conventional OFDM are applied . B. Receiver
At the receiver, the received signal is down-converted, andthe cyclic prefix is then removed from each received basebandsymbol vector before processing with an FFT. After employinga N T -point FFT operation, the frequency-domain receivedsignal vector can be written as y := [ y (1) , y (2) , . . . , y ( N T )] T = p E S Sh + n (2)where E S is the energy of the transmitted symbol vectorand S = diag ( s ) . Moreover, h and n are N T × channeland noise vectors, respectively. Elements of these vectorsfollow the complex-valued Gaussian distributions CN (0 , and CN (0 , N ) , respectively, where N is the noise variance.Since the encoding procedure for each block is independentof others, decoding can be performed independently at thereceiver. Hence, using maximum likelihood (ML) detection,the detected symbol vector for the b th block can be written as (ˆ I b , ˆ s b ) = arg min I b , s b || y b − p E S S b h b || (3)where y b = [ y (( b − N + 1) , . . . , y ( bN )] T , S b = diag ( s b ) and h b = [ h (( b − N + 1) , . . . , h ( bN )] T . By following the useful design guidelines in [5], we obtain the disjointPSK constellations M q by rotating each constellation with the angle of qπ/ ( MQ ) , q = 0 , . . . , N − , to maximize the distance between constella-tion points. To obtain disjoint QAM constellations, likewise [5], we employthe well-known set partitioning technique in [10]. We assume that the elements of s are interleaved sufficiently and themaximum spacing is achieved for the subcarriers. It is also assumed that eachmodulated symbol carried by a subcarrier has unit energy, i.e., E[ | s ( t ) | ] = 1 , t = 1 , . . . , N T . Optimum ML detection complexity of the proposed schemeis of order O ( Q N − M N ) since we have Q N − and M N index and M -ary modulation symbols, respectively, on N subcarriers. Hence, such a detection mechanism is impracticalwhen N and Q are high. To overcome the high complexityof the optimum ML detector, we design a low-complexitysuboptimal ML detector which operates on N − subcarriersindependently while the disjoint constellation on the remainingsubcarrier is decided according to the disjoint constellations onthese N − subcarriers based on the MDS code. The proposedlow-complexity ML (LC-ML) detection is based on the factthat each disjoint constellation can be used on each subcarrierrepeatedly, and the MDS code forms the index symbols ina way that the sum of elements of corresponding N -tuplesis zero under modulo- Q arithmetic. The proposed LC-MLdetector for a Q -MM-OFDM-IM block can be explained asfollows:1) Sort the channel gains of N subcarriers in descendingorder. In other words, | h ( λ ) | ≥ . . . ≥ | h ( λ N ) | where λ n ∈ (cid:8) , . . . , N (cid:9) .2) Determine the constellation, M I λ , λ ∈ (cid:8) λ , . . . , λ N − (cid:9) and the modulation symbol, s λ ∈ M I λ , on each subcar-rier by substituting all possible constellation symbols,which are drawn from the union of all disjoint constel-lations, M = M ∪ . . . ∪ M Q − , based on an MLdetector except for the subcarrier that has the smallestchannel gain, i.e., λ N th subcarrier. The detected disjointconstellation index and the modulation symbol for the λ th subcarrier of an Q -MM-OFDM-IM block can bewritten as (cid:0) ˆ I λ , ˆ s λ (cid:1) = arg min I λ , s λ | y ( λ ) − h ( λ ) s λ | . (4)3) Estimate the constellation index, ˆ I λ N , on the λ N thaccording to the equation ( ˆ I λ + ˆ I λ + . . . + ˆ I λ N ) mod Q = 0 . (5)4) Determine the modulation symbol, ˆ s λ N , on the λ N thsubcarrier by substituting M symbols belonging to theconstellation M ˆ I λN .The proposed LC-ML detector compares QM ( N − squared Euclidean distances for N − subcarriers, whichhave the highest N − channel gains, and M squaredEuclidean distances for the subcarrier that has the smallestchannel gain. The total number of squared Euclidean distancecomparisons for a Q -MM-OFDM-IM subcarrier is given by QM − QM/N + M/N . Hence, the computational complexityof the proposed detector is of order O ( QM ) . It can be shownthat the proposed ML detector exhibits a lower complex-ity than the subcarrier-wise log-likelihood detector of MM-OFDM-IM [5] when Q = N and the constellation size ofMM-OFDM-IM is chosen in a way that the overall SEs ofboth schemes are equal. Here, we omit the block superscript for convenience since the decodingcan be performed independently for each block.
III. P
ERFORMANCE A NALYSIS
In this section, we analyze the SE and BER of the proposedscheme.
A. Spectral Efficiency
As defined above, the proposed scheme transmits f = ⌊ log Q N − ⌋ and f = N log M bits by the index symbolsand the M -ary constellation symbols on N subcarriers, re-spectively. Hence, by ignoring cyclic prefix length, the SE ofthe proposed scheme can be given by η = f + f N = ⌊ log Q N − ⌋ + N log MN (6)This SE scales as η ∼ log ( QM ) . Remark.
The proposed scheme exhibits outstanding flexi-bility since Q , N and M can be adjusted independentlyto achieve a desired SE for the proposed scheme. On theother hand, it is easy to notice that the proposed scheme isequivalent to conventional OFDM when Q = 1 . Moreover,when Q = N , we arrive at N N − index codewords for theproposed scheme and the SE scales as η ∼ log ( N M ) , whichis the highest SE among the OFDM-IM schemes having N disjoint constellations and subcarriers. Proposition 1. As N → ∞ , the index codewords of Q -MM-OFDM-IM are capable of achieving the same SE as the overallSEs of the MM-OFDM-IM and OFDM - Ordered Full SPM(OFDM-OFSPM) schemes, which employ N disjoint M -aryconstellations, when Q = N M/e and Q = N M/ ( e ln 2) ,respectively. Proof:
The proof can easily be obtained by looking atthe SEs of MM-OFDM-IM and OFDM-OFSPM, and the SEprovided by only the index symbols of the Q -MM-OFDM-IMscheme. They can be given, respectively, as η ∼ log N − log e + log M [5], η ∼ log N − log ( e/ ln 2) + log M [7]and η ∼ log Q when N → ∞ .Based on the fact in Proposition 1, the proposed schemecan achieve the same SE as that of the MM-OFDM-IMscheme without employing conventional modulation whileutilizing fewer constellation points. Such a result makes theproposed scheme not only spectrally more efficient but alsomore reliable in terms of error rate than MM-OFDM-IM sincethe index symbols are capable of achieving higher minimumHamming distance than the conventional modulation symbols. B. Bit-Error Rate
An upper-bound on the average BER is given by the well-known union bound P b ≤ f f f X i =1 2 f X j =1 P ( S i → S j ) D ( S i → S j ) (7)where P ( S i → S j ) stands for the pairwise error probability(PEP) regarding the erroneous detection of S i as S j where i = j , i, j ∈ (cid:8) , . . . , Q N − M N (cid:9) , S i = diag ( s i ) and S j = diag ( s j ) and D ( S i → S j ) is the number of bits in error for the corresponding pairwise error event. One can use the samePEP expression as in [1] and substitute the Q -MM-OFDM-IMcodewords to obtain the upper bound on the average BER.As explained in the previous section, the minimum Ham-ming distance between index symbols of the Q -MM-OFDM-IM is equal to two. However, the minimum Hamming distancebetween the conventional modulation symbols is limited toone. Hence, as in other OFDM-IM schemes, the average BERexpression of the proposed scheme will be dominated by theminimum Euclidean distance between the modulation symbolsat high SNR values, and therefore, the diversity order of theBER curves is limited to one. However, the proposed schemeis capable of producing a large number of index symbols sincethe number of these symbols is a power of Q . Hence, one canuse only the index symbols of the proposed scheme to attain acodebook. In this case, the diversity order of the BER curvesbecomes equal to two.IV. N UMERICAL R ESULTS
In this section, we compare uncoded and coded BERperformance of the proposed scheme with that of the OFDM-IM benchmarks and conventional OFDM. For coded schemes,we use a rate-1/3 turbo code that is specified by Third-Generation Partnership Project (3GPP) [11]. We also showthe effectiveness of the proposed LC-ML detector comparedto the optimum ML detector.In figures, “ Q -MM-OFDM-IM ( Q, N, M ) ” and “ Q -MM-OFDM-IM ( Q, N, M ) , QAM” signify the proposed schemesemploying Q disjoint M -ary PSK and QAM constellations,respectively, on N subcarriers, whereas “OFDM-OFSPM ( N, M ) ” is the variant of OFDM-SPM [7] employing all setpartitions and having N subcarriers as well as N disjoint M -PSK constellations in each OFDM block. “OFDM-IM ( N, K a , M ) ” stands for the conventional OFDM-IM schemein which K a out of N subcarriers are activated to send M -PSK modulated symbols in each block. Finally, “MM-OFDM-IM ( N, M ) ” represents a multi-mode scheme having N subcarriers along with N disjoint M -PSK constellationsin each block. Note also that “ Q -MM-OFDM-IM ( Q, N, ”and “ Q -MM-OFDM-IM ( Q, N, , QAM” employ Q disjointconstellations, each of which involves only one constellationpoint. This means that such schemes include only indexsymbols and the points in a Q -ary (PSK for the former andQAM for the latter) constellation form these symbols.Fig. 1 compares the uncoded and coded BER performanceof the proposed schemes , Q -MM-OFDM-IM (4 , , and Q -MM-OFDM-IM (8 , , , with OFDM-OFSPM (4 , , MM-OFDM-IM (4 , , OFDM-IM (4 , , and OFDM (QPSK).SEs for uncoded Q -MM-OFDM-IM (4 , , ; Q -MM-OFDM-IM (8 , , and OFDM-OFSPM (4 , ; MM-OFDM-IM (4 , , OFDM-IM (4 , , and OFDM (QPSK) are 2.5 bps,2.25 bps and 2 bps, respectively. In the uncoded case, the Q -MM-OFDM-IM (8 , , scheme is capable of achieving thesame SE as OFDM-OFSPM (4 , by employing only index In Fig. 1, BER curves of the uncoded schemes are obtained by employingthe optimum ML detector, whereas the coded schemes perform hard-decisiondecoding based on the optimum ML detector.
SNR (dB) -5 -4 -3 -2 -1 BE R UncodedCoded
Figure 1. Uncoded and coded BER comparison of Q -MM-OFDM-IM (4 , , and Q -MM-OFDM-IM (8 , , with OFDM-OFSPM (4 , , MM-OFDM-IM (4 , , OFDM-IM (4 , , and OFDM (QPSK). SNR (dB) -6 -5 -4 -3 -2 -1 BE R Figure 2. BER comparison of the proposed LC-ML detector with optimumML detector for Q -MM-OFDM-IM (8 , , , Q -MM-OFDM-IM (4 , , and Q -MM-OFDM-IM (8 , , . symbols, and it considerably outperforms the other schemesby introducing an additional diversity gain. Moreover, Q -MM-OFDM-IM (4 , , outperforms OFDM-OFSPM (4 , andMM-OFDM-IM (4 , slightly, and OFDM-IM (4 , , andOFDM (QPSK) considerably at high SNR values although itachieves the highest SE. In the coded case, the BER curvesare shifted to the left. Hence, the proposed schemes areoutperformed by the OFDM-IM and OFDM schemes for alarger BER range compared to the uncoded case. However,a similar behavior to the uncoded case is observed as theproposed schemes start to outperform conventional OFDM atrelatively high SNR. Moreover, depending on the bit-to-indexmapping and the constellation indices, the coded MM-OFDM-IM scheme outperforms the coded proposed schemes.Fig. 2 demonstrates the uncoded BER performance of theproposed LC-ML detector compared to the optimum MLdetector based on (3) for the Q -MM-OFDM scheme when Q ∈ (cid:8) , (cid:9) , N = 4 , and M ∈ (cid:8) , (cid:9) . In this figure, “ Q -MM-OFDM-IM ( Q, N, M ) , LC-ML” and “ Q -MM-OFDM-IM ( Q, N, M ) , ML” stand for the Q -MM-OFDM-IM schemes SNR (dB) -5 -4 -3 -2 -1 BE R UncodedCoded
Figure 3. Uncoded and coded BER comparison of Q -MM-OFDM-IM ( Q, N, M ) and Q -MM-OFDM-IM ( Q, N, M ) , QAM with OFDM-OFSPM (4 , , QAM, OFDM-OFSPM (4 , , MM-OFDM-IM (4 , , OFDM-IM (4 , , and OFDM (8-PSK) for N = 4 , Q ∈ (cid:8) , (cid:9) , M ∈ (cid:8) , (cid:9) . applying the proposed LC-ML and optimum ML detectors,respectively. As seen from the figure, the performance ofthe proposed LC-ML detector is very close to that of theoptimum ML detector, especially at low and high SNR values.Moreover, the performance loss in terms of SNR can begiven approximately as 1.4 dB, 1 dB and 1.4 dB for Q -MM-OFDM-IM (8 , , , Q -MM-OFDM-IM (4 , , and Q -MM-OFDM-IM (8 , , , respectively, at a BER value of − .The figure also shows the theoretical upper-bound results,“ Q -MM-OFDM-IM ( Q, N, M ) , Upper-bound”, for the proposedscheme. As observed from the figure, upper-bound curves areconsistent with computer simulations, especially at high SNR.In Fig. 3, we compare the uncoded and coded BER perfor-mance of the proposed schemes , Q -MM-OFDM-IM (8 , , ,QAM, Q -MM-OFDM-IM (8 , , and Q -MM-OFDM-IM (16 , , , QAM, with OFDM-OFSPM (4 , , QAM, OFDM-OFSPM (4 , , MM-OFDM-IM (4 , , OFDM-IM (4 , , and OFDM (8-PSK) schemes. Here, in the uncoded case,except for Q -MM-OFDM-IM (16 , , , the Q -MM-OFDM-IM and OFDM-OFSPM schemes exhibit the same SE of 3.25bps whereas the remaining schemes, except for OFDM-IM,have the same SE of 3 bps. Also, the SE for OFDM-IM (4 , , is 2.75 bps. As seen from the figure, Q -MM-OFDM-IM (8 , , , QAM, Q -MM-OFDM-IM (8 , , achieve anoutstanding BER performance at high SNR and outperformthe OFDM-OFSPM (4 , , QAM, OFDM-OFSPM (4 , , andMM-OFDM-IM (4 , schemes as well as the OFDM-IM (4 , , , and OFDM (8-PSK) schemes by providing almost5 dB and 10 dB SNR gains, respectively, at a BER valueof − . These results arise from the fact that the Q -MM-OFDM-IM schemes achieve a similar SE as those of the otherschemes by employing a lower order modulation. Such an In Fig. 3, the BER curves regarding the uncoded and coded Q -MM-OFDM-IM schemes are obtained by employing the proposed LC-MLdetector and performing hard-decision decoding based on the proposed LC-ML detector, respectively. The remaining uncoded and coded schemes employthe optimum ML detector and perform hard-decision decoding based on theoptimum ML detector at the receiver, respectively. order reduction results in higher minimum Euclidean distancefor the modulation symbols as well as better error performanceat high SNR values. Moreover, Q -MM-OFDM-IM (16 , , achieves the best BER performance with an SNR gain ofmore than 10 dB at a BER value of − compared tothe OFDM-OFSPM (4 , , QAM, OFDM-OFSPM (4 , , andMM-OFDM-IM (4 , schemes. Such a scheme achieves anadditional diversity order due to the index symbols, whereasthe diversity orders of the other schemes are limited by theminimum Hamming distance between the conventional modu-lation symbols. In the coded case, OFDM-IM (4,3,8) achievesthe best performance. However, at a relatively high SNR, Q -MM-OFDM-IM (16 , , achieves a satisfactory performancewith higher SE compared to the benchmarks.V. C ONCLUSION
In this paper, we proposed a novel IM scheme that wecall Q -MM-OFDM-IM. We showed that the proposed schemeencompasses conventional OFDM as a special case, and it iscapable of providing the highest number of index symbolsamong other OFDM-IM schemes. We investigated the SE andBER performance of the proposed scheme and demonstratedthat the proposed scheme can provide substantial performanceimprovement compared to MM-OFDM-IM, OFDM-OFSPM,OFDM-IM, and OFDM.As future work, the proposed scheme could be generalizedby utilizing in-phase and quadrature dimensions of the modu-lation symbols as in [5] and employing disjoint constellationswith variable size as in [6]. ACKNOWLEDGEMENTS
The authors wish to acknowledge the support of the Bris-tol Innovation & Research Laboratory of Toshiba ResearchEurope Ltd. They also would like to thank Dr. J. Vonk forhis valuable contributions to the discussions on the number ofindex symbols. R
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