OFDM With Hybrid Number and Index Modulation
11 OFDM With Hybrid Number and Index Modulation
Ahmad M. Jaradat, Jehad M. Hamamreh, and Huseyin Arslan,
Fellow, IEEE
Abstract —A novel transmission scheme is introduced for ef-ficient data transmission by conveying additional informationbits through jointly changing the index and number of activesubcarriers within each orthogonal frequency division multiplex-ing (OFDM) subblock. The proposed scheme is different fromthe conventional OFDM-subcarrier number modulation (OFDM-SNM) and OFDM-index modulation (OFDM-IM), in which databits are transmitted using either number or index of activesubcarriers. The proposed modulation technique offers superiorspectral and energy efficiency compared to its counterpartsOFDM-SNM and OFDM-IM, especially at low modulation orderssuch as binary phase shift keying (BPSK) that can provide highreliability and low complexity, making it suitable for Internet ofThings (IoT) applications that require better spectral and energyefficiency while enjoying high reliability and low complexity.Bit error rate (BER) performance analysis is provided for theproposed scheme, and Monte Carlo simulations are presented toprove the consistency of simulated BER with the analyzed one.
I. I
NTRODUCTION W IDE scope of use cases along with various demandsand features should be handled by 5G and beyondnetworks [1], [2]. The future networks should have the ca-pabilities of supporting different, contrasting requirements interms of ultra-high reliability, very low latency, very high datarates, improved energy efficiency (EE), and low computationalcomplexity [3], [4]. Thus, it is very critical to define anappropriate modulation for a specific 5G use case [5]–[7].Multicarrier transmission schemes are heavily used in wire-less communications. The fact behind this extensive employ-ment of multicarrier techniques is their attractive features likeeasy equalization, multi-user scheduling, and support for adap-tive modulation and coding techniques, etc [8]. The OFDMtransmission scheme is considered as one of the interestingmulticarrier techniques. The OFDM scheme is employed inbroad range of applications, standards, and communicationsystems [9].However, the conventional OFDM scheme has many limita-tions with respect to a set of performance metrics [10], whichopens the door for multiple options of OFDM-based improvedmodulation schemes with respect to some performance evalua-tion metrics such as reliability, spectral efficiency (SE), EE, etc[7]. Improving SE and EE of modulation options becomes oneof the major issues for the future wireless communications.
A. M. Jaradat is with Department of Electrical and Electron-ics Engineering, Istanbul Medipol University, Istanbul, 34810, Turkey(email: [email protected],[email protected]). J. M. Hamam-reh is with the Department of Electrical and Electronics Engineer-ing, Antalya Bilim University, 07468 Antalya, Turkey (email: [email protected],[email protected]).H¨useyin Arslan is with Department of Electrical and Electronics Engi-neering, Istanbul Medipol University, Istanbul, 34810, Turkey and also withDepartment of Electrical Engineering, University of South Florida, Tampa,FL, 33620, USA (e-mail: [email protected]).
Particularly, achieving significant EE is one important criteriato employ a modulation scheme under the millimeter waveband [11].Different OFDM-based modulation options have been pro-posed in the literature and their classification, comparisonand future directions have been provided in [7]. Amongthe promising OFDM-based modulation options, index andnumber-based OFDM modulation schemes are attractive trans-mission techniques where some subcarriers are convention-ally modulated alongside the additional bits transmitted byexploiting the index or number dimension [7]. The extradegrees of freedom offered by such promising modulationschemes are exploited to enhance the performance of con-ventional OFDM in terms of some performance metrics suchas throughput, reliability, EE, etc. In particular, OFDM-indexmodulation (OFDM-IM) [12] and OFDM-subcarrier numbermodulation (OFDM-SNM) [13] are among the promisingindex, and number-based modulation techniques for OFDM-based waveforms, respectively [7].The OFDM-IM technique exploits specific active subcarrierindices in order to transmit additional information, but thisfixed activation ratio per subblock limits the enhancement ofSE for the conventional OFDM-IM scheme. The design ofOFDM-IM subcarrier activation method (i.e. the codebook)depends on either a look-up table or the combinatorial methodwhich is of high complexity and has not fully utilized thefrequency selectivity for reliability improvement [14]. Severalimproved OFDM-IM schemes have been proposed in [15]–[20] with the main motivation of enhancing the SE of theconventional OFDM-IM scheme.Among these improved OFDM-IM schemes, a general-ized version of OFDM-IM has been developed known asOFDM with generalized index modulation (OFDM-GIM) [21].OFDM-GIM is very much similar in principle to the basicOFDM-IM (where information bits are sent by indices) withthe only difference that instead of using just one fixed ac-tivation ratio throughout the whole OFDM block, differentactivation ratios are used per each subblock within the wholeOFDM block in a deterministic, fixed manner (not randombased on the incoming data). This results in changing thenumber of active subcarriers (but there are no bits sent by thenumber of subcarriers, where the information bits are still sentby indices). Moreover, the selection of the activation ratios inOFDM-GIM must be shared with the receiver so that it canperform the detection of the active subcarrier indices.Some improved OFDM-IM schemes such as dual-modeindex modulation aided OFDM (DM-OFDM) [22] and itsgeneralized version called generalized DM-OFDM (GDM-OFDM) [23] as well as OFDM with multi-mode IM (OFDM-MMIM) [24] overcome the main limitation of OFDM-IM bytransmitting data symbols on all available subcarriers along a r X i v : . [ ee ss . SP ] N ov with IM. Aside from the enhancement in SE provided bythese improved OFDM-IM schemes, it is difficult to mitigateintercarrier interference (ICI) and/or reduce peak-to-averagepower ratio (PAPR) due to activating all OFDM subcarriers.Based on the general shape of the OFDM-IM block, a newnumber-based OFDM transmission scheme called OFDM-SNM is proposed in [13]. This novel modulation schemeimplicitly conveys information by utilizing numbers, insteadof indices, of turned on subcarriers alongside the conventionalsymbols. An enhanced scheme of OFDM-SNM is proposedin [25], which exploits the flexibility enabled by the originalOFDM-SNM scheme [25] by placing the active subcarriersadaptively based on the channel to offer an additional codinggain in the high signal-to-noise ratio (SNR) region. Theadaptation is performed based on the instantaneous channelstate information (CSI) in which the incoming informationbits are dynamically mapped to subcarriers with high channelpower gains.Inspired by these unique features of the OFDM-IM andOFDM-SNM schemes, we introduce a novel alternative trans-mission scheme called OFDM with hybrid number and indexmodulation (OFDM-HNIM) which is based on the combina-tion of both OFDM-SNM and OFDM-IM schemes. Our maincontributions are summarized as follows: • We develop and propose a novel modulation schemecalled OFDM-HNIM in which the available subcarriersare partitioned into subblocks, and information bits areconveyed not only by the modulated subcarriers butimplicitly also by both numbers and indices of activesubcarriers relying on the incoming bits in order toembed extra information besides the M -ary constellationsymbols. Therefore, the SE is significantly improvedcompared to OFDM-SNM and OFDM-IM schemes whilemaintaining low detection complexity. • We derive tight, closed-form expressions for upper boundon the BER of OFDM-HNIM systems assuming ML de-tection. Two different ML detectors have been employedin the proposed OFDM-HNIM scheme: Perfect subcarrieractivation pattern estimation (PSAPE), and imperfectSAP estimation (ISAPE) ML-based detectors. Moreover,log-likelihood ratio (LLR) detector is employed on theproposed OFDM-HNIM scheme. The BER performanceof OFDM-HNIM is further compared with that of theexisting state-of-the-art including OFDM-IM, OFDM-SNM, and conventional OFDM through Monte Carlosimulations.
The merits of the proposed scheme can be stated as follows: • The proposed OFDM-HNIM scheme increases the systemdesign flexibility by creating an extra degree of freedomin the number and index dimensions, which can beexploited for different purposes such as enhancing theoverall SE of the communication system. • Unlike conventional OFDM, DM-OFDM, and OFDM-MMIM where all subcarriers are occupied by non-zeroelements, exploiting the inactive subcarriers featured inthe proposed OFDM-HNIM scheme could be used tolessen interference among subcarriers, and enhance EE by reducing PAPR. Moreover, it is possible to utilizethe inherent features of OFDM-SNM in placing theactive subcarriers in positions where the resultant systemprovides minimum levels of ICI and/or PAPR. • Individual SNM and IM modules could be exploited forthe applications where high throughput or EE are needed,respectively. By utilizing the hybrid mapper of SNMand IM modules, high throughput alongside EE couldbe obtained for some applications.The remainder of this paper is prepared as follows. Theproposed OFDM hybrid scheme with number and indexmodulation scheme is explained in Section II. Performanceevaluation of the OFDM-HNIM scheme in terms of SE,average bit error probability (ABEP), EE, and computationalcomplexity is given in Section III. In Section IV, the simulationresults and comparisons between the proposed hybrid schemeand its competitive schemes are presented. Finally, Section Vpresents concluding remarks.
Notation : Matrices and column vectors are represented bybold, capital and lowercase letters, respectively. (cid:126) , E ( . ) , ( . ) T , ( . ) H , and | . | represent circular convolution operation,expectation, transposition, Hermitian transposition, and ab-solute value, respectively. det( A ) denotes the determinantof A . (cid:0) nk (cid:1) = n ! k !( n − k )! represents the binomial coefficient. A ∼ CN ( µ, σ ) represents the complex Gaussian distributionof A with mean and variance of µ and σ , respectively. Q ( . ) represents the Q-function. ∼ O ( . ) denotes the complexityorder of a technique. H ( A ) represents the entropy of A , and I ( A ) represents the mutual information of A .II. P ROPOSED H YBRID SCHEME
The transmitter block diagram of the proposed OFDM-HNIM system is displayed in Fig. 1. The m incoming bitsare partitioned to G groups using bits splitter, each groupcontains p = p + p + p bits, that are utilized to build anOFDM subblock of L subcarriers length ( L = N F /G ), where N F represents fast Fourier transform (FFT) size. The non-conventional bits ( p and p ) represent the bits conveyed bythe SNM and IM modulator. In each subblock, the SNM andIM mappers can be ordered arbitrary since the hybrid mappercombines both mappings without looking to the order of thesenon-convectional mappers. For convenient representation ofthe basic OFDM-HNIM system model, we represent the SNMbits by p , which are utilized by the SNM mapper to specifynumerically the active OFDM subcarriers for each subblock,and p bits, referred to IM bits, are exploited by the IM mapperto specify the indices of active subcarriers for each OFDMsubblock. Then, the combined SNM and IM mapping in thehybrid mapper determines the subcarrier activation pattern(SAP) using a proper mapping technique.For each subblock g ( g = 1 , , . . . , G ), index and number-dependent variable I ∈ [1 , , · · · , L ] which represents the setof activated subcarriers based on the hybrid mapper accordingto its corresponding p and p bits, and the remaining L − I subcarriers are not active. It should be noted that I is variablenumeric representation for the active subcarriers according to p and p data code. The length of SNM and IM bits is set TABLE IL
OOK - UP TABLE OF THE PROPOSED
OFDM-HNIM
SCHEME WITH p = p = 2 BITS g p p c g I [1 0 0 0] T
12 [0 0] [0 1] [0 1 0 0] T
13 [0 0] [1 0] [0 0 1 0] T
14 [0 0] [1 1] [0 0 0 1] T
15 [0 1] [0 0] [1 1 0 0] T
26 [0 1] [0 1] [1 0 1 0] T
27 [0 1] [1 0] [1 0 0 1] T
28 [0 1] [1 1] [0 1 0 1] T
29 [1 0] [0 0] [1 1 1 0] T
310 [1 0] [0 1] [1 0 1 1] T
311 [1 0] [1 0] [1 1 0 1] T
312 [1 0] [1 1] [0 1 1 1] T
313 [1 1] [0 0] [0 0 0 0] T
014 [1 1] [0 1] [0 0 1 1] T
215 [1 1] [1 0] [0 1 1 0] T
216 [1 1] [1 1] [1 1 1 1] T Secondary
Sub-block
Creator Primary
OFDM
Block
Creator
Conventional
OFDM Modulator 𝑚 bits Bits Splitter
SNM Mapper 𝑝 𝑝 𝑝 IM Mapper 𝑝 M- ary
Modulation 𝑝 𝑝 𝑝 𝑝 Secondary Sub-block
Creator G M- ary Modulation SNM
Mapper
IM Mapper
Hybrid Mapper
Hybrid Mapper
Fig. 1. The proposed OFDM-HNIM transmitter structure. to be fixed in order to simplify the system-level design. TableI presents a bits-to-SAP mapping for small values of L = 4 , I ∈ [1 , , , with p = p = log ( L ) = 2 bits.The SAP foreach subblock g can be written as follows: c g = (cid:2) c c · · · c L (cid:3) T , (1)where c i ∈ , for i = 1 , , · · · , L . There are p = I log ( M ) bits corresponding to specific conventional constellation sym-bols carried over I active subcarriers.It is shown from Table I that a subcarrier in a givenindex of c g patterns is activated with an equal probabilityof 1/2. The presented codebook ( c g ) in Table I looks likea classical codebook generated in binary order; however, theproposed codebook ordering is different from the conventionalbinary ordering. Moreover, the direct binary mapping doesnot provide flexibility, whereas, the proposed scheme can beseen as an adaptive and flexible transmission technique, whosemapping blocks can be employed based on the requirementsof the used application. For example, individual SNM modulecould be used in the applications where higher throughput isneeded at low modulation orders [7]. Also, an IM module could be utilized for high EE applications [7]. To have bothhigh energy and spectral efficiencies in some applications, thecombined SNM and IM modules could be exploited.The OFDM block in OFDM-HNIM system model is built as x F = (cid:2) x F (1) x F (2) ... x F ( N F ) (cid:3) based on the chain of G subblocks. The remaining steps are done as in the classicalOFDM system. By performing inverse fast Fourier transform(IFFT), the output vector is x t . With adding N CP cyclic prefix(CP) samples to the transmitted signal, the resultant wouldbe x CP = (cid:2) x t ( N F − N CP + 1 : N F ) x t (cid:3) . It is assumedthat the transmitted signal passes to a fading channel withimpulse response of h t , affected by additive white Gaussiannoise (AWGN) with noise variance of N o,T in time domain.The received signal over the multi-path channel could berepresented as: y t = x t (cid:126) h t + n t , (2)where n t ∼ CN (0 , N o,T ) represents the AWGN vector. A. OFDM-HNIM receiver
The receiver part of the proposed OFDM-HNIM sys-tem is the reversal of the transmitter part, which in-cludes CP removal, FFT processing, hybrid demapping,and detection. The resultant vector after taking off CPfrom the received signal can be written as y NCP = (cid:2) y NCP (0) y NCP (1) ... y
NCP ( N F − (cid:3) . Next, FFT op-eration is employed and the output signal is y F : y F = x F h F + n F , (3)Subsequently, a frequency domain equalizer of one tapis utilized to properly compensate the multi-path channelfrequency selectivity as y eq = y F / h F , where h F is thefrequency response of the fading channel. Then, a maximum-likelihood (ML) detector is employed to extract the SAP. Inorder to regenerate p and p , similar table is used at thereceiver as the look-up Table I used for the transmitter. It should be noted that there would be similar SAP when g =13 , , , as expected from their SNM mapping, due tosetting a fixed length of p data code. To solve this ambiguityin mapping, extra bits could be transmitted to receiver in orderto differentiate between these exceptional cases. However,this solution is not spectrally efficient since extra non-databits are transmitted which should be avoided. Moreover, theexpected SNM mapping leads to a nonuniform subcarrieractivation which results in unfair protection of transmitted bits,hindering the OFDM-SNM and OFDM-IM relevant schemesfrom attaining their ultimate error performance.The problem of unbalanced activation of subcarriers inthe OFDM-SNM and OFDM-IM schemes is avoided in theproposed look-up Table I by having equiprobable probabilityin each subcarrier without dependency on the transmitterbeforehand information about status of wireless channel [26].Therefore, in order to avoid ambiguity and extra bits signalingat the ML detector as well as satisfying the equiprobable sub-carrier activation (ESA) requirement [26], these exceptionalcases are assigned a unique SAP where their corresponding I ’s are mostly different from what is expected from their SNMmapping.However, the zero-active subcarrier dilemma, where allsubcarriers are switched off and their corresponding datasymbols can not be transmitted, can be seen in Table I at g = 13 corresponding to c g = [0 0 0 0] T . This dilemmacould lead to difficulty in higher layer design, especially thesynchronization process which is the key for OFDM-basedsystems in order to reduce ICI and/or inter symbol interference(ISI) [12], [27]. One efficient solution to such a dilemma couldbe done by exploiting the lexicographic ordering principleand implementing the codebook optimization method as in[28], where the codebooks are optimized in such a way thatthe subcarriers are activated based on their correspondinginstantaneous CSI. The novel codebook design in [28] does notinvolve optimizing the power usage unlike the forward errorcorrection method that adds more computational complexityat the receiver [29].As shown in Table I, the equal probable bit sequenceenables easier detection and designs of higher layer protocolsunlike the dual-mode transmission protocol [27], where themodulation has been employed on a bit sequence with variablelength. Control signaling is not required in [28] unlike themethod in [30] where always-active control subcarrier hasbeen used for control signaling transmission. Therefore, thelexicographic-based codebook design in [28] provides efficientsolution to the observed dilemma as compared to the existingsolutions [27], [29], [30]. In this paper, we focus on thebasic idea and system model of the proposed OFDM-HNIM,whereas employing the novel design in [28] to the proposedscheme could be considered as a future work in order toimprove the system performance. The remaining steps of the OFDM-HNIM receiver are: Thebits conveyed by index and number of subcarriers in eachsubblock are estimated based on the obtained SAP using In this paper, our focus is introducing a new physical layer transmissionscheme. The usage of the proposed modulation scheme at the upper layers isout of this paper scope. hybrid demapper which represents the contrary of the hybridmapper used at the transmitter. Then, constellation symbolsdetection is performed based on the received SAP for eachsubblock. Finally, the detected bits from the hybrid demappingand conventional QAM detection are jointly formed the latestestimated subblock bits. By performing similar operations toall subblocks, the recovered data sequence is acquired for thewhole OFDM block.III. P
ERFORMANCE EVALUATION OF THE PROPOSEDHYBRID SCHEME
Here, evaluation of the hybrid system performance is per-formed based on some metrics such as spectral efficiency, aver-age bit error probability, energy efficiency, and computationalcomplexity.
A. Spectral Efficiency
The achievable rate of the proposed OFDM-HNIM schemecan be calculated based on the mutual information betweenthe channel input and the channel output averaged overthe subcarriers [31]. Since the transmission in the proposedOFDM-HNIM scheme is performed in a subblock level andwe treat the subblocks independently due to having differentsubblocks realizations. Therefore, the mutual information overthe OFDM-HNIM block should be calculated in a subblocklevel as shown in (4) which presents the achievable rate of theproposed OFDM-HNIM scheme: R h = I ( x gF ) L = H ( x gF ) L + H ( x gF | y gF ) L , (4)where I ( x gF ) represents the mutual information of the channelinput of the g -th subblock ( x gF ), H ( x gF ) is the marginal entropyof x gF and H ( x gF | y gF ) is the conditional entropy of x gF and thechannel output of the g -th subblock ( y gF ).The achievable rate formula in (4) can be written in termsof probability density functions (PDFs) of the channel input,channel, and channel output as follows [32]: R h = η h − p L p (cid:88) j =1 E h F g (cid:40) (cid:90) f ( y F g | x F g ( j ) , h F g ) × log (cid:32) f ( x F g ( j ) , y F g | h F g ) f ( y F g | h F g ) (cid:33) d y F g (cid:41) , (5)where f ( y F g | h F g ) = 12 p (cid:80) p j =1 f ( y F g | x F g ( j ) , h F g ) .Furthermore, (5) can be simplified to the following in theanalogy with [31], [32]: R h ≈ η h − p L p (cid:88) j =1 log p (cid:88) w =1 det ( I L + K L U j,w ) , (6)where η h is the SE of the proposed hybrid scheme, and it canbe found in (7), K L = E [ h gF ( h gF ) H ] is the covariance matrixof h gF , I L represents the identity matrix with dimensions L × L , and U j,w = ( x g ( j ) F − x g ( w ) F ) H ( x g ( j ) F − x g ( w ) F )2 N o,F , where x g ( j ) F and x g ( j ) F represent the j -th and w -th realization of thesubblock x gF , respectively. It should be noted that N o,F is thenoise variance in frequency domain. This analytical result willbe verified by simulation as well.The maximum achievable rate (i.e. SE) of the proposedOFDM-HNIM scheme is: η h = (cid:80) Gg =1 (cid:16) log ( L ) + log (cid:0) LI ( g ) (cid:1) + I ( g ) log ( M ) (cid:17) N F + N CP , (7)where I ( g ) represents a numeric variable of SAP in eachsubblock of size L . Since two parts of an incoming non-conventional bits are used for hybrid mapping comparedto only one part of OFDM-SNM and OFDM-IM schemes.Thus, the hybrid scheme improves the SE over the OFDM-IM and OFDM-SNM transmission schemes. For example,Table II shows the maximum achievable rate or SE for theproposed hybrid scheme and its competitive schemes includingOFDM-SNM, OFDM-IM, and conventional OFDM, assuming N F = 64 with N CP = 8 , and BPSK modulation is employed.It should be noted that fixed length of p bits in the hybridmapping is considered for convenient comparison. Also, thenumber of subblocks ( G ) equals to either 16 when L = 4 ( G = N F /L = 64 / ), or ( G = N F /L = 64 / )when L = 8 . For the system settings shown in Table II, theproposed hybrid scheme outperforms its counterparts in termsof SE . Moreover, the SE becomes much more significant inthe hybrid scheme as the subblock length increases. This SEtrend can not be achieved by the conventional OFDM-SNMand OFDM-IM schemes in which SE is degraded as subblocksize rises.Table II shows the SE gain of different OFDM-based mod-ulation schemes under BPSK modulation relative to classicalOFDM system as a baseline . It is clear from Table II thatthe proposed OFDM-HNIM outperformed its counterparts interms of SE at low modulation order ( M = 2 ). A compre-hensive comparison in terms of the average achievable ratebetween the featured OFDM-based modulation options whenthe subblock size of 8 is shown in Fig. 2 where hybrid schemeoutperforms its counterparts including conventional OFDM-SNM and OFDM-IM schemes at different modulation orders.Mathematically, we can deduce the values of M and L that satisfy the condition where the average achievable rate ofthe proposed scheme improves over that of the conventionalOFDM. The average achievable rate of the hybrid schemecan be calculated from (7) with average number of activesubcarriers of I avg = ( L + 1) / over the available OFDMsubcarriers: ¯ η h = N F (cid:16) log ( L ) + log (cid:0) LI avg (cid:1) + I avg log ( M ) (cid:17) L ( N F + N CP ) . (8)By comparing (8) to the average achievable rate of theconventional OFDM which is: The phrase N/A, as shown in Table II, refers to not applicable.
Modulation order A v e r age a c he i v ab l e r a t e ( bp s / H z ) Proposed OFDM-HNIMOFDM-SNMOFDM-IM with I=1OFDM-IM with I=2OFDM-IM with I=3OFDM-IM with I=4
Fig. 2. Average achievable rate of the featured OFDM-based modulationoptions as function of M when the subblock size set to 8. ¯ η OF DM = N F log ( M ) N F + N CP , (9)we observe the trend of M and L for the hybrid schemeand conventional OFDM, then we can set the inequalityresults from considering the average achievable rate of thehybrid scheme greater than that of the conventional OFDM asfollows: log ( L ) + log (cid:18) LI avg (cid:19) + I avg log ( M ) ≥ L log ( M ) . (10)Fig. 3 shows that the average achievable rate of the con-ventional OFDM does not depend on the subblock size sincesubblock-based transmission is not employed by the plainOFDM. However, the hybrid scheme outperforms the plainOFDM at low modulation orders, specifically when M = 2 or M = 4 , for different subblock sizes. However, as M increases,it is less likely that the proposed scheme has a higher averageachievable rate than the conventional OFDM, and this ratedegradation also increases as subblock size increases. Thishappens because of having more probable bits experience deepfading as subblock size increases and it becomes difficult forreceiver to detect these bits in deep-fading condition. B. Average Bit Error Probability
The whole block in the hybrid scheme should be detectedbased on transmission bits conveyed in the number and indexof activated subcarriers and the conventional constellationsymbols [27]. So, the fundamental block error rate (BLER)is computed first and then the average BER calculated basedon the average BLER [28], [33]. Actually, BLER is basicallyexpressed in terms of pairwise error probability (PEP). ThePEP is calculated in two steps, first with the effect of channel( h t ) in which PEP in this case is called conditional PEP, andthe second step is done by getting rid of channel to obtain the TABLE IISE
OF THE FEATURED
OFDM-
BASED MODULATION SCHEMES WITH
BPSKOFDM-based Modulation Scheme
L G × p G × p G × p SE = G × p SE gain over conventional OFDMProposed OFDM-HNIM scheme 4 32 32 32 96 1.3338 24 24 216 264 3.6667OFDM-SNM [13] 4 32 40 N/A 72 18 24 36 N/A 60 0.8333OFDM-IM [12] 4 32 32 N/A 64 0.88898 24 24 N/A 48 0.6667Conventional OFDM 4 N/A N/A N/A 72 18 N/A N/A N/A 72 1
Subblock size A v e r age a c he i v ab l e r a t e ( bp s / H z ) OFDM-HNIM with M=2Plain OFDM with M=2OFDM-HNIM with M=4Plain OFDM with M=4OFDM-HNIM with M=8Plain OFDM with M=8OFDM-HNIM with M=16Plain OFDM with M=16OFDM-HNIM with M=32Plain OFDM with M=32
Fig. 3. Average achievable rate comparison between the hybrid scheme andplain OFDM in terms of subblock size. unconditional PEP. The conditional PEP could be representedby using the Q-function [34]: P ( c g −→ ˆc ˆ g | h t ) = Q ( (cid:115) P t N o,F || h t ( c g (cid:112) I ( g ) − ˆc ˆ g (cid:112) I (ˆ g ) ) || ) , (11)where P t represents the total transmit power, I ( g ) and I (ˆ g ) represent the number of active subcarriers in the g -th and ˆ g -thsubblock, respectively, c g and ˆc ˆ g are transmitted and detectedsequences, respectively. The conditional PEP shown above canbe approximated by using the exponential approximation ofthe Q-function as [35], [36]: Q ( x ) ≈ (cid:88) j =1 ρ j exp( − η j x ) , (12)where ρ = 1 / and ρ = 1 / , η = 1 / and η = 2 / . Thenew formula of conditional PEP becomes [25]: P ( c g −→ ˆc ˆ g | h t ) = (cid:88) j =1 ρ j L (cid:89) l =1 exp( − η j P t N o,F R ( l )∆( l, g, ˆ g )) , (13)where R ( l ) = | h t ( l ) | and ∆( l, g, ˆ g ) = | c g ( l ) (cid:112) I ( g ) − ˆc ˆ g ( l ) (cid:112) I (ˆ g ) | .The unconditional PEP can be found by averaging theconditional PEP over the channel as [37], [38]: P ( c g −→ ˆc ˆ g ) = (cid:88) ˆc ˆ g (cid:54) = c g E h t (cid:8) P ( c g −→ ˆc ˆ g ) | h t ) (cid:9) , (14) With the assistance of the aforementioned PEP and assum-ing equiprobable information bits, the upper bound of ABEPin the hybrid system could be obtained [12], [39]: P b ( E ) = 1 p g n x G (cid:88) g =1 (cid:88) ˆc ˆ g (cid:54) = c g P ( c g −→ ˆc ˆ g ) e ( c g , ˆc ˆ g ) , (15)where p g represents the length of vector that contains infor-mation bits corresponding to OFDM subblock, n x representsnumerically the possible realizations of the transmitted se-quence, and e ( c g , ˆc ˆ g ) denotes errors in information bits due toerroneously choosing ˆc ˆ g rather than c g [12]. This theoreticalBER result will be verified by simulation as well. C. Energy Efficiency
The energy efficiency (EE) implies how efficiently energy isconsumed in a given system. The EE of the proposed OFDM-HNIM scheme is analyzed in terms of the energy savingfactor (
ESF ) achieved by not activating all of the availablesubcarriers in the OFDM block. The evaluation of
ESF forthe featured schemes is discussed here.One main aspect of the overall energy saving is the savingin signal transmission [11]. The EE in terms of
ESF dependson the ratio of saved energy under a given total transmit power( P t ) in a single OFDM symbol. We assume equiprobable sub-carrier activation [26] for convenient comparison between theproposed OFDM-HNIM scheme and its competitive schemes.Moreover, the average number of active subcarriers is consid-ered in our EE analysis. The number of activated subcarriersin an OFDM block impacts both SE and EE, and their ratiosrelation with respect to the traditional OFDM as a referencescheme [40], can be defined as [11] EE r = SE r − ESF , (16)where EE r and SE r represent the EE and SE ratios of thescheme of interest with respect to the conventional OFDMscheme. Maximizing SE and EE are generally two con-flicting objectives as can be clearly seen from (16) [41]. ESF = N a /N v represents the saved energy by activating N a subcarriers out of N v available subcarriers.For example, in the proposed OFDM-HNIM scheme, halfof subcarriers are active in average which is the equivalentto the case of OFDM-IM scheme with half of its availablesubcarriers are active, i.e, subcarrier activation ratio of half( AR = 0 . ). The energy saving resulting from the OFDM-HNIM and OFDM-IM with AR = 0 . is limited to half of s1 s2 s3 s4 s5 OFDM-based modulation scheme SE r a t i o EE r a t i o SErEEr
Fig. 4. The SE and EE ratios of the featured OFDM-based modulation optionswith respect to the plain OFDM. The symbols s1, s2, s3, s4, s5 correspond tothe proposed OFDM-HNIM scheme, OFDM-SNM, OFDM-IM with AR =0 . , OFDM-IM with AR = 0 . , and conventional OFDM, respectively. the OFDM symbol ( ESF = 0 . ). In other words, the activesubcarriers in OFDM-HNIM scheme gain higher power ascompared to the individual subcarriers in classical OFDM un-der the same transmitted power. Hence, the receiver design ofthe proposed OFDM-HNIM scheme would only differentiatebetween the active group with higher power and the inactivegroup regardless of the exact amplitude and phase of eachsubcarrier.On the other hand, the ESF trend in the OFDM-SNM andOFDM-IM depends on the subblock length and AR . It is well-known that the conventional OFDM is not an EE scheme sincemost of the available subcarriers are usually used for datatransmission which results in very low ESF . Fig. 4 showsthe SE r and EE r of the featured OFDM modulation schemeswith respect to classical OFDM. It is clear from Fig. 4 thatthe proposed OFDM-HNIM scheme provides higher SE r and EE r as compared to the OFDM-SNM, OFDM-IM with low AR , and the classical OFDM. D. Complexity Analysis
In the proposed hybrid scheme, the detection for the numberand indices of active subcarriers as well as the conventionalQAM symbols can be performed subblock by subblock with-out introducing any performance loss since the encodingprocesses for all subblocks are independent. Based on ML,a joint decision on the set of active subcarriers as well as theconventional QAM symbols in each g -th subblock as: min s g ,I g || y F − x F h F || , (17)where s g represents the conventional QAM symbol carriedover the active subcarrier in the g -th subblock. The com-putational complexity of this optimal ML detector in termsof complex multiplications is of order ∼ O ( G M n/ ) perbit detected in each OFDM-HNIM block, which becomesimpractical for large values of G , n and M , due to itsexponentially increasing complexity. To simplify the bulky search in this optimal ML detection,two different ML detectors have been implemented; Perfectsubcarrier activation pattern estimation (PSAPE)-based, andimperfect SAP estimation (ISAPE)-based ML detectors. ThePSAPE-based detector ignores the errors due to the incorrectdetection of subcarriers in the received SAP. However, in theISAPE-based detector, the conventional QAM demodulationphase would be unsuccessful to deduce p bits correctlyfor the erroneous SAP detection case. This happens due todemodulation of some unmodulated subcarriers. Therefore, thetransmitted p , p , and p bits are erroneous when the detectedSAP is wrong. On the other hand, if the SAP is correctlydetected, then, p and p is correct but it is not certain that p is correctly estimated. Actually, in case of erroneous conven-tional symbol detection for the received active subcarriers, p bits are affected. Otherwise, a correctly symbol detection foractivated subcarriers in a correctly estimated SAP produces acorrect estimation of p , p , and p bits. The PSAPE-baseddetector is considered in the following simulations due to itslow-complexity compared to that of ISAPE-based detector. Itshould be noted that the performance comparison betweenboth detectors is shown as well. On the other hand, LLRdetector is also employed with the proposed OFDM-HNIMscheme in order to reduce the computational complexity atthe receiver.To compare the computational complexity of the theseproposed ML-based detectors, namely PSAPE and ISAPE-based detectors, with that of the optimal ML detector andLLR detector, we consider the average number of metriccalculations per subcarrier as a performance metric. Table IIIpresents the complexity comparison results for the featuredOFDM-based modulation schemes. As seen from Table III,the computational complexity of the optimal ML detector ishighly susceptible to parameters G , n , and M ; however, thecomplexity of the two proposed ML-based detectors is onlydetermined by G and, apparently, much lower than the optimalML-based detector, and there is much reduction in complexityby employing LLR detector in order of M . Moreover, TableIII shows the comparison between the detection complexityof the proposed hybrid scheme with that of its counterpartsschemes such as OFDM-SNM, OFDM-IM, and plain OFDM.It is clear that the proposed OFDM-HNIM along with LLRdetector is considered as a low-complex scheme with similarorder as that in OFDM-IM and conventional OFDM.IV. S IMULATION R ESULTS
Here, the throughput, BER, and EE performances of thehybrid system are compared to those in OFDM-GIM with I = [1 , , and L = 4 [21], OFDM-IM, OFDM-SNM,and conventional OFDM. It is assumed that FFT size ( N F )is 64, subblock length ( L ) is 4, so the number of subblocks G = N F /L = 16 , and SNM and IM bits are p = p =log ( L ) = 2 bits in each subblock. The modulation usedin the simulation is BPSK. The employed frequency-selectivechannel is Rayleigh distributed with 10 taps and similar SNR(or E b /N o,T ) is considered in order to have fair comparisonbetween the considered schemes, where E b is the bit energy. TABLE IIIC
OMPLEXITY COMPARISON BETWEEN DIFFERENT DETECTORS FOR THE FEATURED
OFDM-
BASED MODULATION SCHEMES
OFDM-based Modulation Scheme Detector type Complexity orderProposed OFDM-HNIM scheme Optimal ML ∼ O ( G M n/ ) PSAPE ∼ O ( G ) ISAPE ∼ O ( G ) LLR ∼ O ( M ) OFDM-SNM ML ∼ O ( L, I, M ) OFDM-IM Near optimal LLR ∼ O ( M ) Conventional OFDM ML ∼ O ( M ) E b /N o,T (dB) T h r oughpu t ( bp s / H z ) Proposed OFDM-HNIMOFDM-GIMOFDM-IMOFDM-SNMClassical OFDM
Fig. 5. Throughput of the proposed scheme and its competitive schemes underBPSK.
It is assumed that the CP length ( N CP ) is longer than theeffective channel impulse response. Moreover, we assume thatthe CSI is not available at the transmitter.The throughput performances of the proposed hybridscheme compared to its competitive schemes under differentmodulation types are shown in Fig. 5 and Fig. 6. Here, thesimulated throughput metric is found by multiplying the SE fora given modulation scheme by simulated BER subtracted fromone. The saturation points of the achievable rates of all theschemes or their SEs fall at very high SNR. These SE valuesdependent on the mapping between incoming bits, SAPs andconventional symbols, as shown in (7).At low order modulation especially BPSK, as shown in Fig.5, the SE of the hybrid scheme is 1.33 bits/s/Hz with SE gainof 0.44 compared to OFDM-IM, and conventional OFDM,0.33 compared to OFDM-SNM, and 0.3 compared to itsOFDM-GIM counterpart. The reason behind the superiority ofthe hybrid scheme over its competitive schemes under BPSKis because of low noise and interference effects in lower ordermodulation as well as additional information sent by indexand number of active subcarriers. This obviously shows thatthe hybrid scheme is an improved spectrally efficient schemecompared to its OFDM-IM and OFDM-SNM counterpartswhere only the indices or numbers of active subcarriers areexploited to convey additional data bits.The throughput performances for the featured OFDM-basedschemes under QPSK is shown in Fig. 6 where SE superiority E b /N o,T (dB) T h r oughpu t ( bp s / H z ) Prposed OFDM-HNIMOFDM-IMOFDM-SNMClassical OFDM
Fig. 6. Throughput of the proposed scheme and its competitive schemes underQPSK. of the OFDM-HNIM scheme is also achieved over its conven-tional OFDM-SNM and OFDM-IM schemes. However, theclassical OFDM outperforms the OFDM-HNIM in terms ofthroughput under QPSK especially at low SNR values dueto sparse distribution of active subcarriers featured in theproposed hybrid scheme, classical OFDM-SNM and OFDM-IM schemes.The BER performance for the featured OFDM-based mod-ulation options when employing a certain conventional mod-ulation type such as BPSK, is shown in Fig. 7. It is assumedthat the same total power is allocated at the transmitter ofeach of the considered schemes, and the original power ofthe inactive subcarriers is evenly reallocated for the activeones. Due to the three different sources of errors occurred byusing the ISAPE-based detector in the hybrid system, its BERperformance would be worse than that of PSAPE detector.As can be seen from Fig. 7 that the LLR detector of theproposed OFDM-HNIM achieves improved BER performancecompared to its competitive schemes. However, employing thetwo proposed ML-based detectors results in less than 1 dBcoding loss in the hybrid scheme compared to its competitiveschemes at low SNR values, and more SNR loss in highSNR region. The reason behind this is the weaker protectionfeatured in the number-modulated and index-modulated bitscompared to the conventional modulation bits at higher valuesof SNR, therefore the number-modulated and index-modulatedbits are more probable to encounter significant error trans- E b /N o,T (dB) -4 -3 -2 -1 B E R Proposed OFDM-HNIM, PSAPEProposed OFDM-HNIM, ISAPEProposed OFDM-HNIM, LLRProposed OFDM-HNIM, Theo.OFDM-GIMOFDM-IMOFDM-SNMClassical OFDM
Fig. 7. BER of the proposed scheme, OFDM-GIM, OFDM-IM, OFDM-SNM,and classical OFDM under frequency-selective Rayleigh channel with BPSK. mission in the high SNR region. Moreover, ratio of number-modulated and index-modulated bits in the proposed hybridscheme is larger than in its competitive schemes, that increasesthe influence of active number and index bits on the BERmore.In Fig. 7, the derived theoretical ABER of the proposedscheme becomes considerably tight with computer simulationsas SNR increases.Fig. 8 presents the complementary cumulative distributionfunctions (CCDFs) for the PAPR of the proposed OFDM-HNIM scheme compared to its counterparts OFDM-basedmodulation schemes. It can be observed from Fig. 8 thatthe PAPR performances of these featured OFDM modulationoptions are high and almost the same [7], [42]. The reasonbehind this high PAPR values featured in the proposed OFDM-HNIM scheme is the sparsity nature of the active subcarriers inthe OFDM-HNIM which is also featured in the conventionalOFDM-SNM and OFDM-IM schemes. These inherent featuresof having some inactive subcarriers in the OFDM-HNIM,OFDM-SNM, and OFDM-IM schemes could be exploitedto reduce the PAPR by designing a proper mapping whileensuring that the PAPR in minimal levels. The PAPR reduc-tion techniques proposed for conventional OFDM [43] maynot be directly applied to non-conventional OFDM schemessuch as OFDM-HNIM scheme. The reason behind this ishaving unique features and characteristic for different OFDMmodulation options [7]. Therefore, we need a proper PAPRreduction method to reduce PAPR of the proposed OFDM-HNIM scheme, which is left as a future work.V. C
ONCLUSION
This paper proposes a new energy and spectrally efficientmulti-carrier transmission scheme named hybrid number andindex modulation that transmits additional information bynumber and index of the active subcarriers beside the con-ventional signal constellation symbols. The OFDM-SNM andOFDM-IM schemes are combined to harvest both of theiradvantages. The proposed hybrid scheme has low-complex
PAPR(dB) -3 -2 -1 CC D F Proposed OFDM-HNIMOFDM-SNMOFDM-IMClassical OFDM
Fig. 8. PAPR performances of the featured OFDM modulation options. transceiver which has outperformed its competitive schemes interms of throughput performance under low order modulationsuch as BPSK, but it loses its dominance as modulationorder increases. Moreover, Monte Carlo simulations have beenconducted and the obtianed results have verified the analysis.Due to having inactive subcarriers within the OFDM symbolfeatured in the proposed hybrid scheme, it could be used tolessen interference among subcarriers, and enhance EE byreducing PAPR. Furthermore, it is possible to design a properpower allocation technique to be integrated with the proposedhybrid scheme to improve its performance further.R
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