Orthogonal Frequency Division Multiplexing with Index Modulation Ertuğrul Başar, Member, IEEE, Ümit Aygölü, Member, IEEE,Erdalpanayırcı,Fellow, IEEE,And H

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5536 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 22, NOVEMBER 15, 2013 Orthogonal Frequency Division Multiplexing With Index Modulation Ertuğrul Başar, Member, IEEE, Ümit Aygölü, Member, IEEE,ErdalPanayırcı,Fellow, IEEE,and H. Vincent Poor, Fellow, IEEE

Abstract—In this paper, a novel orthogonal frequency division the most popular multicarrier transmission technique in wire- multiplexing (OFDM) scheme, called OFDM with index modula- less communications and has become an integral part of IEEE tion (OFDM-IM), is proposed for operation over frequency-selec- 802.16 standards, namely Mobile Worldwide Interoperability tive and rapidly time-varying fading channels. In this scheme, the information is conveyed not only by -ary signal constellations Microwave Systems for Next-Generation Wireless Communi- as in classical OFDM, but also by the indices of the subcarriers, cation Systems (WiMAX) and the Long Term Evolution (LTE) which are activated according to the incoming bit stream. Different project. low complexity transceiver structures based on maximum likeli- In frequency selective fading channels with mobile terminals hood detection or log-likelihood ratio calculation are proposed and reaching high vehicular speeds, the subchannel orthogonality a theoretical error performance analysis is provided for the new scheme operating under ideal channel conditions. Then, the pro- is lost due to rapid variation of the wireless channel during the posed scheme is adapted to realistic channel conditions such as im- transmission of the OFDM block, and this leads to inter-channel perfect channel state information and very high mobility cases by interference (ICI) which affects the system implementation and modifying the receiver structure. The approximate pairwise error performance considerably. Consequently, the design of OFDM probability of OFDM-IM is derived under channel estimation er- systems that work effectively under high mobility conditions, rors. For the mobility case, several interference unaware/aware detection methods are proposed for the new scheme. It is shown via is a challenging problem since mobility support is one of the computer simulations that the proposed scheme achieves signifi- key features of next generation broadband wireless communica- cantly better error performance than classical OFDM due to the tion systems. Recently, the channel estimation and equalization information bits carried by the indices of OFDM subcarriers under problems have been comprehensively studied in the literature both ideal and realistic channel conditions. for high mobility [1], [2]. Index Terms—Frequency selective channels, maximum like- Multiple-input multiple-output (MIMO) transmission tech- lihood (ML) detection, mobility, orthogonal frequency division niques have been also implemented in many practical applica- multiplexing (OFDM), spatial modulation. tions, due to their benefits over single antenna systems. More recently, a novel concept known as spatial modulation (SM), which uses the spatial domaintoconveyinformationinaddition I. INTRODUCTION to the classical signal constellations, has emerged as a promising ULTICARRIER transmission has become a key tech- MIMO transmission technique [3]–[5]. The SM technique has M nology for wideband digital communications in recent been proposed as an alternative to existing MIMO transmission years and has been included in many wireless standards to sat- strategies such as Vertical Bell Laboratories Layered Space- isfy the increasing demand for high rate communication sys- Time (V-BLAST) and space-time coding which are widely used tems operating on frequency selective fading channels. Orthog- in today’s wireless standards. The fundamental principle of SM onal frequency division multiplexing (OFDM), which can ef- is an extension of two dimensional signal constellations (such fectively combat the intersymbol symbol interference caused as -ary phase shift keying ( -PSK) and -ary quadrature by the frequency selectivity of the wireless channel, has been amplitude modulation ( -QAM), where is the constellation size) to a new third dimension, which is the spatial (antenna) dimension. Therefore, in the SM scheme, the informa- Manuscript received January 29, 2013; revised May 12, 2013 and July 10, 2013; accepted August 14, 2013. Date of publication August 28, 2013; date tion is conveyed both by the amplitude/phase modulation tech- of current version October 10, 2013. The associate editor coordinating the re- niques and by the selection of antenna indices. The SM principle view of this paper and approving it for publication was Prof. Huaiyu Dai. This has attracted considerable recent attention from researchers and research is was supported in part by the U.S. Air Force Office of ScientificRe- search under MURI Grant FA 9550-09-1-0643. This paper was presented in several different SM-like transmission methods have been pro- part at the IEEE Global Communications Conference, Anaheim, CA, USA, De- posed and their performance analyses are given under perfect cember 2012, and in part at the First International Black Sea Conference on and imperfect channel state information (CSI) in recent works Communications and Networking, Batumi, Georgia, July 2013. E. Başar and Ü. Aygölü are with Faculty of Electrical and Electronics [6]–[12]. Engineering, Istanbul Technical University, Istanbul 34381, Turkey (e-mail: The application of the SM principle to the subcarriers of an [email protected]; [email protected]). OFDM system has been proposed in [13]. However, in this E. Panayırcı is with Department of Electrical and Electronics Engineering, Kadir Has University, Istanbul 34381, Turkey (e-mail: [email protected]). scheme, the number of active OFDM subcarriers varies for each H. V. Poor is with the Department of Electrical Engineering, Princeton Uni- OFDM block, and furthermore, a kind of perfect feedforward versity, Princeton, NJ, 08544 USA (e-mail: [email protected]). is assumed from the transmitter to the receiver via the excess Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. subcarriers to explicitly signal the mapping method for the sub- Digital Object Identifier 10.1109/TSP.2013.2279771 carrier index selecting bits. Therefore, this scheme appears to

1053-587X © 2013 IEEE BAŞAR et al.: ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING WITH INDEX MODULATION 5537 be quite optimistic in terms of practical implementation. An channel. This fact is also validated by computer simula- enhanced subcarier index modulation OFDM (ESIM-OFDM) tions under ideal and realistic channel conditions. scheme has been proposed in [14] which can operate without • Unlike the ESIM-OFDM scheme, in which the number of requiring feedforward signaling from the transmitter to the re- active subcarriers is fixed, the OFDM-IM scheme provides ceiver. However, this scheme requires higher order modulations an interesting trade-off between complexity, spectral effi- to reach the same spectral efficiency as that of classical OFDM. ciency and performance by the change of the number of In this paper, taking a different approach from those in active subcarriers. Furthermore, in some cases, the spec- [13] and [14], we propose a novel transmission scheme called tral efficiency of the OFDM-IM scheme can exceed that of OFDM with index modulation (OFDM-IM) for frequency classical OFDM without increasing the size of the signal selective fading channels. In this scheme, information is con- constellation by properly choosing the number of active veyed not only by -ary signal constellations as in classical subcarriers. OFDM, but also by the indices of the subcarriers, which are The rest of the paper can be summarized as follows. In activated according to the incoming information bits. Unlike Section II, the system model of OFDM-IM is presented. In the scheme of [13], feedforward signaling from transmitter to Section III, we propose different implementation approaches the receiver is not required in our scheme in order to success- for OFDM-IM. The theoretical error performance of OFDM-IM fully detect the transmitted information bits. Opposite to the is investigated in Section IV. In Section V, we present new scheme of [14], a general method, by which the number of detection methods for the OFDM-IM scheme operating under active subcarriers can be adjusted, and the incoming bits can be realistic channel conditions. Computer simulation results are systematically mapped to these active subcarriers, is presented giveninSectionVI.Finally,Section VII concludes the paper.1 in the OFDM-IM scheme. Different mapping and detection techniques are proposed for the new scheme. First, a simple II. SYSTEM MODEL OF OFDM-IM look-up table is implemented to map the incoming information Let us first consider an OFDM-IM scheme operating over a bits to the subcarrier indices and a maximum likelihood (ML) frequency-selective Rayleighfadingchannel.Atotalof infor- detector is employed at the receiver. Then, in order to cope with mation bits enter the OFDM-IM transmitter for the transmission the increasing encoder/decoder complexity with the increasing of each OFDM block. These bits are then split into groups number of information bits transmitted in the spatial domain each containing bits, i.e., . Each group of -bits is of the OFDM block, a simple yet effective technique based on mappedtoanOFDM subblock of length ,where combinatorial number theory is used to map the information and is the number of OFDM subcarriers, i.e., the size of bits to the antenna indices, and a log-likelihood ratio (LLR) the fast Fourier transform (FFT). Unlike classical OFDM, this detector is employed at the receiver to determine the most mapping operation is not only performed by means of the mod- likely active subcarriers as well as corresponding constellation ulated symbols, but also by the indices of the subcarriers. In- symbols. A theoretical error performance analysis based on spired by the SM concept, additional information bits are trans- pairwise error probability (PEP) calculation is provided for the mitted by a subset of the OFDM subcarrier indices. For each new scheme operating under ideal channel conditions. subblock, only out of available indices are employed for In the second part of the paper, the proposed scheme is inves- this purpose and they are determined by a selection procedure tigated under realistic channel conditions. First, an upper bound from a predefined set of active indices, based on the first on the PEP of the proposed scheme is derived under channel bits of the incoming -bit sequence. This selection procedure estimation errors in which a mismatched ML detector is used is implemented by using two different mapping techniques in for data detection. Second, the proposed scheme is substantially the proposed scheme. First, a simple look-up table, which pro- modified to operate under channel conditions in which the mo- vides active indices for corresponding bits, is considered for bile terminals can reach high mobility. Considering a special mapping operation. However, for larger numbers of informa- structure of the channel matrix for the high mobility case, three tion bits transmitted in the index domain of the OFDM block, novel ML detection based detectors, which can be classified as the use of a look-up table becomes infeasible; therefore, a simple interference unaware or aware, are proposed for the OFDM-IM and effective technique based on combinatorial number theory scheme. In addition to these detectors, a minimum mean square is used to map the information bits to the subcarrier indices. Fur- error (MMSE) detector, which operates in conjunction with an ther details can be found in Section III. We set the symbols cor- LLR detector, is proposed. The new scheme detects the higher 1Notation: Bold, lowercase and capital letters are used for column vectors number of transmitted information bits successfully in the spa- and matrices, respectively. and denote transposition and Hermitian tial domain. transposition, respectively. and denote the determinant and The main advantages of OFDM-IM over classical OFDM and rank of , respectively. is the th eigenvalue of ,where is the largest eigenvalue. is a submatrix of with di- ESIM-OFDM can be summarized as follows mensions ,where is composed of the rows and • The proposed scheme benefits from the frequency selec- columns of with indices and , respectively. tivity of the channel by exploiting subcarrier indices as a and are the identity and zero matrices with dimensions and , respectively. stands for the Frobenius norm. The prob- source of information. Therefore, the error performance of ability of an event is denoted by and stands for expectation. The the OFDM-IM scheme is significantly better than that of probability density function (p.d.f.) of a random vector is denoted by . classical OFDM due to the higher diversity orders attained represents the distribution of a circularly symmetric complex Gaussian r.v. with variance . denotes the tail probability of the stan- for the bits transmitted in the spatial domain of the OFDM dard Gaussian distribution. denotes the binomial coefficient and is block mainly provided by the frequency selectivity of the the floor function. denotes the complex signal constellation of size . 5538 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 22, NOVEMBER 15, 2013

Fig. 1. Block diagram of the OFDM-IM transmitter. responding to the inactive subcarriers to zero, and therefore, we have unit average power. The OFDM block creator creates all do not transmit data with them. The remaining of the subblocks by taking into account and for all first bits of this sequence are mapped onto the -ary signal constel- anditthenformsthe main OFDM block lation to determine the data symbols that modulate the subcarriers having active indices; therefore, we have .In (5) other words, in the OFDM-IM scheme, the information is con- where , , by concatenating these veyed by both of the -ary constellation symbols and the in- subblocks. Unlike the classical OFDM, in our scheme con- dices of the subcarriers that are modulated by these constellation tains some zero terms whose positions carry information. symbols. Due to the fact that we do not use all of the available After this point, the same procedures as those of classical subcarriers, we compensate for the loss in the total number of OFDM are applied. The OFDM block is processed by the in- transmitted bits by transmitting additional bits in the index do- verse FFT (IFFT) algorithm: main of the OFDM block. The block diagram of the OFDM-IM transmitter is given in (6) Fig. 1. For each subblock ,theincoming bits are transferred to the index selector, which chooses active indices out of where is the time domain OFDM block, is the discrete available indices, where the selected indices are given by Fourier transform (DFT) matrix with and the term is used for the normalization (1) (at the receiver, the FFT demodulator employs a normalization where for and . factor of ). At the output of the IFFT, a cyclic prefix(CP) Therefore, for the total number of information bits carried by of length samples is the positions of the active indices in the OFDM block, we have appended to the beginning of the OFDM block. After parallel to serial (P/S) and digital-to-analog conversion, the signal is sent (2) through a frequency-selective Rayleigh fading channel which can be represented by the channel impulse response (CIR) co- In other words, has possible realizations. On the efficients other hand, the total number of information bits carried by the -ary signal constellation symbols is given by (7)

(3) where , are circularly symmetric complex Gaussian random variables with the distribution. As- since the total number of active subcarriers is in our suming that the channel remains constant during the transmis- scheme. Consequently, a total of bits are trans- sion of an OFDM block and the CP length is larger than , mitted by a single block of the OFDM-IM scheme. The vector the equivalent frequency domain input-output relationship of of the modulated symbols at the output of the -ary mapper the OFDM scheme is given by (modulator), which carries bits, is given by (8) (4) where , and are the received signals, the where , , . We assume that channel fading coefficients and the noise samples in the fre- , i.e., the signal constellation is normalized to quency domain, whose vector presentations are given as , BAŞAR et al.: ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING WITH INDEX MODULATION 5539 and , respectively. The distributions of and which is the same as that of the classical OFDM detector. In are and , respectively, where is the order to prevent numerical overflow, the Jacobian logarithm noise variance in the frequency domain, which is related by the [15] can be used in (12). As an example, for and bi- noise variance in the time domain by nary-phase shift keying (BPSK) modulation, (12) simplifies to

(9) (13)

We define the signal-to-noise ratio (SNR) as where and where is the average transmitted energy per . bit. The spectral efficiency of the OFDM-IM scheme is given For higher order modulations, to prevent numerical by [bits/s/Hz]. overflow we use the identity The receiver’s task is to detect the indices of the active subcar- , riers and the corresponding information symbols by processing where , . Unlike classical OFDM, a simple ML . After calculation of the LLR values, for each decision on is not sufficient based on only in our subblock, the receiver decides on active indices out of them scheme due to the index information carried by the OFDM-IM having maximum LLR values. This detector is classified as subblocks. In the following, we investigate two different types near-ML since the receiver does not know the possible values of detection algorithms for the OFDM-IM scheme: of . Although this is a desired feature for higher values of 1) ML Detector: The ML detector considers all possible sub- and , the detector may decide on a catastrophic set of active block realizations by searching for all possible subcarrier index indices which is not included in since for combinations and the signal constellation points in order to make ,and index combinations are unused at the a joint decision on the active indices and the constellation sym- transmitter. bols for each subblock by minimizing the following metric: After detection of the active indices by one of the detectors presented above, the information is passed to the “index (10) demapper”, at the receiver which performs the opposite action of the “index selector” block giveninFig.1,toprovideanes- timate of the index-selecting bits. Demodulation of the con- where and for are the received sig- stellation symbols is straightforward once the active indices are nals and the corresponding fading coefficients for the subblock determined. ,i.e., , , respectively. It can be easily shown that the total computational III. IMPLEMENTATION OF THE OFDM-IM SCHEME complexity of the ML detector in (10), in terms of complex mul- In this subsection, we focus on the index selector and index tiplications, is per subblock since and have demapper blocks and provide different implementations of and different realizations, respectively. Therefore, this ML them. As stated in Section II, the index selector block maps the detector becomes impractical for larger values of and due to incoming bits to a combination of active indices out of its exponentially growing decoding complexity. possible candidates, and the task of the index demapper is to 2) Log-Likelihood Ratio (LLR) Detector: The LLR detector provide an estimate of these bits by processing the detected of the OFDM-IM scheme provides the logarithm of the ratio of active indices provided by either the ML or LLR OFDM-IM a posteriori probabilities of the frequency domain symbols by detector. considering the fact that their values can be either non-zero or It is worth mentioning that the OFDM-IM scheme can be im- zero. This ratio, which is given below, gives information on the plemented without using a bit splitter at the beginning, i.e., by active status of the corresponding index for : using a single group which results in . However, in this case, can take very large values which make the (11) implementation of the overall system difficult. Therefore, in- steadofdealingwithasingleOFDMblockwithhigherdimen- where . In other words, a larger value means it sions, we split this block into smaller subblocks to ease the index is more probable that index is selected by the index selector selection and detection processes at the transmitter and receiver at the transmitter, i.e., it is active. Using Bayes’ formula and sides, respectively. The following mappers are proposed for the considering that and new scheme: , (11) can be expressed as 1) Look-Up Table Method: In this mapping method, a look-up table of size is created to use at both transmitter and receiver sides. At the transmitter, the look-up table provides the corresponding indices for the incoming bits for each subblock, and it performs the opposite operation at the receiver. (12) A look-up table example is presented in Table I for , ,and ,where .Since ,two The computational complexity of the LLR detector in (12), in combinations out of six are discarded. Although a very efficient terms of complex multiplications, is per subcarrier, and simple method for smaller values, this mapping method 5540 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 22, NOVEMBER 15, 2013

TABLE I As an example, for , , , the following ALOOK-UP TABLE EXAMPLE FOR , AND sequences can be calculated:

. .

. is not feasible for higher values of and due to the size of . the table. We employ this method with the ML detector since the receiver has to know the set of possible indices for ML decoding, i.e., it requires a look-up table. On the other hand, a look-up table cannot be used with the LLR detector presented The algorithm, which finds the lexicographically ordered se- in Section II since the receiver cannot decide on active indices quences for all , can be explained as follows: start by choosing if the detected indices do not exist in the table. the maximal that satisfies , and then choose the We give the following remark regarding the implementation maximal that satisfies of the OFDM-IM scheme with a reduced-complexity ML de- and so on [17]. In our scheme, for each subblock, we first con- coding. vert the bits entering the index selector to a decimal number Remark: The exponentially growing decoding complexity of , and then feed this decimal number to the combinatorial al- the actual ML decoder can be reduced by using a special LLR gorithm to select the active indices as . At the receiver detector that operates in conjunction with a look-up table. Let us side, after determining active indices, we can easily get back to denote the set of possible active indices by for the decimal number using (15). This number is then applied which ,where for . to a -bit decimal-to-binary converter. We employ this method As an example, for the look-up table given in Table I, we have with the LLR detector for higher values to avoid look-up ta- , , , .After bles. However, it can give a catastrophic result at the exit of the the calculation of all LLR values using (12), for each subblock decimal-to-binary converter if ; nevertheless, we use this , the receiver can calculate the following LLR sums for all detector for higher bit-rates. possible set of active indices using the corresponding look-up table as IV. PERFORMANCE ANALYSISOFTHEOFDM-IM SCHEME In this section, we analytically evaluate the average bit error (14) probability (ABEP) of the OFDM-IM scheme using the ML decoder with a look-up table. The channel coefficients in the frequency domain are related for . Considering Table I, for the first subblock to the coefficients in the time domain by we have , , ,and . After calculation (16) of LLR sums for each subblock, the receiver makes a decision on the set of active indices by choosing the set with the where is the zero-padded version of the vector with maximum LLR sum, i.e., and obtains the cor- length ,i.e., responding set of indices, and finally detects the corresponding (17) -ary constellation symbols. As we will show in the sequel, our simulation results indicate that this reduced-complexity ML de- It can easily be shown that , ,followsthe coder exhibits the same BER performance as that of the actual distribution , since taking the Fourier transform of a ML detector presented in Section II with higher decoding com- Gaussian vector gives another Gaussian vector. However, the plexity. On the other hand, for the cases where a look-up table is elements of are no longer uncorrelated. The correlation ma- not feasible, the actual LLR decoder of the OFDM-IM scheme trix of is given as can be implemented by the following method. 2) Combinatorial Method: The combinational number system provides a one-to-one mapping between natural num- (18) bers and -combinations, for all and [16], [17], i.e., where it maps a natural number to a strictly decreasing sequence ,where .Inotherwords, (19) for fixed and ,all can be presented by asequence of length , which takes elements from the set is an all-zero matrix except for its first diagonal elements according to the following equation: whichareallequalto . It should be noted that becomes a diagonal matrix if , which is very unlikely for a practical (15) OFDM scheme. Nevertheless, since is a Hermitian Toeplitz BAŞAR et al.: ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING WITH INDEX MODULATION 5541 matrix, the PEP events within different subblocks are identical, TABLE II and it is sufficient to investigate the PEP events within a single AND VALUES FOR WITH VARYING AND VALUES subblock to determine the overall system performance. Without loss of generality, we can choose the first subblock, and intro- duce the following matrix notation for the input-output relationship in the frequency domain:

(20) where , is an all-zero matrix except for its main diagonal elements denoted by , and .Letus deﬁne .Infact,thisisan submatrix cen- where (25) and (26) are related via (24), and (27) is obtained tered along the main diagonal of the matrix .Thusitisvalid from the identity for all subblocks. If is transmitted and it is erroneously de- (29) tected as , the receiver can make decision errors on both active indices and constellation symbols. The well-known conditional where the dimensions of and are and , pairwise error probability (CPEP) expression for the model in respectively. We have the following remarks: (20) is given as [18] Remark 1 (Diversity Order of the System): Let us deﬁne for ,2.Since (21) (30) where and . We can approximate quite well using [19] where , for high SNR values , we can rewrite (28) as (22)

Thus, the unconditional PEP (UPEP) of the OFDM-IM scheme can be obtained by (31) As seen from this result, the diversity order of the system is (23) determined by , which is upper bounded according to the rank inequality [20] by ,where .On where and .Let the other hand we have when the receiver correctly .Since for our scheme, we use detects all of the active indices and makes a single decision error the spectral theorem [20] to calculate the expectation above on out of -ary symbols. It can be shown that can take values deﬁning and ,where from the interval . is an diagonal matrix. Considering Remark 2 (Effects of Varying Values): Keepinginmind and the p.d.f. of given by that the diversity order of the system is determined by the worst case PEP scenario when , we conclude that the distance (24) spectrum of the system improves by increasing the values of , which is a new phenomenon special to the OFDM-IM concept the UPEP can be calculated as as opposed to classical OFDM. As a secondary factor, considering the inequality for the eigenvalues of the products of positive semideﬁnite Her- mitian matrices [21] where ,we also conclude that the UPEP decreases with increasing values of . In Table II, for varying and values, corresponding to the and values, are calculated for , (25) whereweassumedthat if .As seen from Table II, and values increase with increasing (26) values of , i.e., increasing frequency selectivity of the fading channel. However, although this a factor which improves the PEP distance spectrum of the OFDM-IM scheme, it does not (27) considerably affect the error performance for high SNR values because the worst case PEP event with dominates the system performance in the high SNR regime. In other words, (28) , , 2, does not change for different values when 5542 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 22, NOVEMBER 15, 2013

. We also observe from Table II that and values In other words, the receiver uses the decision metric of the per- also increase with the increasing values of ; nevertheless, it is fect CSI (P-CSI) case by simply replacing by . For this case, not possible to make a fair comparison for this case because the (20) can be rewritten as spectral efﬁciency of the system also increases (due to the increasing number of bits transmitted in the index domain) with (35) increasing values of . On the other hand, we observe from Table II that the effect of the increasing values on the error where . Considering (34) and performance can be more explicit for larger values in the (35), the CPEP of the OFDM-IM scheme can be calculated as low-to-mid SNR regime due to the larger variation on values. follows: Remark 3 (Improving the Diversity Order): To improve the diversity order of the system, we can by-pass the -ary modulation by setting and only transmit data via the indices of the active subcarriers at the expense of reducing the bit rate, sincewealwaysguarantee in the absence of the symbol errors. Remark 4 (Generalization): Although the CPEP expression given in (21) and therefore, the main results presented above (36) are valid for ML detector of the OFDM-IM scheme, as we will show in the sequel, the error performance of the near-ML LLR It can be shown that the decision variable is Gaussian dis- detector is almost identical as that of the ML detector; therefore, tributed with without loss of generality, the presented results can be assumed to be valid for OFDM-IM schemes employing LLR detectors. After the evaluation of the UPEP from (28), the ABEP of the OFDM-IM can be evaluated by

(32) which yields the following CPEP expression: where is the number of the possible realizations of and represents the number of bit errors for the corre- (37) sponding pairwise error event.

V. OFDM-IM UNDER REALISTIC CHANNEL CONDITIONS In order to obtain the UPEP, the CPEP expression given in In this section, we analyze the OFDM-IM scheme under re- (37) should be averaged over the multivariate complex Gaussian alistic channel conditions such as imperfect CSI and very high p.d.f. of which is given by [23] mobility by providing analytical tools to determine the error performance and proposing different implementation techniques. (38) A. OFDM-IM Under Channel Estimation Errors where and .How- In this subsection, we analyze the effects of channel estima- ever, due to the complexity of (37), this operation is not an easy tion errors on the error performance of the OFDM-IM scheme. task, therefore, we consider the following upper bound for the In practical systems, the channel estimator at the receiver pro- CPEP of the OFDM-IM scheme: vides an estimate of the vector of the channel coefﬁcients as [22]

(33) (39) where represents the vector of channel estimation errors which is independent of , and has the covariance matrix which is obtained by using the inequality . In this work, we assume that and are related via , i.e., the power of the estimation error decreases (40) with increasing SNR. Under channel estimation errors, the receiver uses the mismatched ML decoder by processing the re- where both inequalities hold if all active subcarrier indices ceived signal vector given by (20) to detect the corresponding have been correctly detected. In other words, the actual CPEP data vector as expression given in (37) simplifies to the CPEP expression given in (39) for the error events corresponding to the erro- (34) neous detection of only -ary symbols in a given subblock BAŞAR et al.: ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING WITH INDEX MODULATION 5543 which are chosen from a constant envelope constellation as padded by zeros. In particular, we assume that the first and last -PSK since elements of the main OFDM block are not used for data transmission, where is the size of the zero padding. Con- sidering that the first and the last elements of the main OFDM block have been padded with zeros, we define the mean- ingful received signal vector, OFDM block and the channel matrix, respectively, as follows: , ,and ,where . Due to the structure of the modified channel matrix given where , and in (42), different OFDM subblocks interfere with each other, ,andonly out of ’s have non-zero values. unlike in the model of Section II; therefore, algorithms are re- Using (22) and (38), the UPEP upper bound of the OFDM-IM quired to detect the active indices as well as the corresponding scheme with channel estimation errors can be calculated as constellation symbols. MMSE equalization can be considered follows: as an efficient solution to the detection problem of (42) because the interference between the different OFDM subblocks can be easily eliminated by MMSE equalization. However, as we will show in the sequel, MMSE detection can diminish the effective- ness of the transmission of the additional bits in the spatial domain by eliminating the effect of the channel matrix completely on the transmitted OFDM block. On the other hand, considering the banded structure of the channel matrix , different inter- (41) ference unaware or aware detection algorithms can be implemented for the OFDM-IM scheme. In the following, we propose where , and different detection methods for the OFDM-IM scheme under . After calculation of the UPEP, the mobility: ABEP of the OFDM-IM can be evaluated by using (32). 1) MMSE Detector: The MMSE detector chooses the matrix In order to obtain a different approximation to the actual (called the equalizer matrix) that minimizes the cost func- UPEP, one can consider the UPEP in the high-SNR regime tion . The resulting detection statististic be- in which the worst case error events and their multiplicities comes dominate the error performance. Similar to the P-CSI case, according to (41), the diversity order of the system is determined (43) by , which is upper bounded by ,i.e., the system performance is dominated by the error events cor- where is the MMSE equalized signal, is the av- responding to the erroneous detection of only -ary symbols erage SNR at the frequency domain, and is in a given subblock. Therefore, for this type of error events the number of available subcarriers. After MMSE equalization, the inequality given in (39) holds and the actual UPEP can be which eliminates the ICI caused by the time selective channels evaluated by (41), and then the ABEP can be calculated by with high mobility, one can consider either of the reduced-com- using (32) considering only the corresponding worst case error plexity ML or LLR detectors to determine the active indices and events. On the other hand, the calculation of the UPEP for corresponding constellation symbols depending on the system general -ary signal constellations is left for future work. configuration. As an example, for the case of the LLR decoder, (12) can be used for detection with the assumption of B. OFDM-IM Under Very High Mobility for all due to the effect of the MMSE equalization. For the It is well known that OFDM eliminates intersymbol inter- reduced-complexity ML decoder, the calculated LLR values are ference and simply uses a one-tap equalizer to compensate for used with the look-up table to determine the corresponding LLR multiplative channel distortion in quasi-static and frequency-se- sums. lective channels. However, in fading channels with very high 2) Submatrix Detector: This interference unaware detector mobilities, the time variation of the channel over one OFDM assumes that ,where has the following symbol period results in a loss of subchannel orthogonality structure: which leads to ICI. The received signal in the frequency domain can be expressed as [24] . . . . (44) (42) . .. where which the equivalent channel matrix in the time domain and the FFT matrix, and it is no where is an longer diagonal due to the ICI. Unlike the system model given matrix that corresponds to the subblock , . in Section II, we assume that some of the available subcarriers In other words, this detector does not consider the interference are not allocated for data transmission, i.e., the OFDM block is between different subblocks. Therefore, for each subblock, the 5544 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 22, NOVEMBER 15, 2013 receiver makes a joint decision on the active indices and the constellation symbols by minimizing the following metric:

(45) where is the corresponding received signal vector of length for OFDM-IM subblock ,whichhas different realizations. Therefore, unlike the MMSE detector, the decoding complexity of this detector grows exponentially with increasing values of . 3) Block Cancellation Detector: The block cancellation detector applies the same procedures as those of the submatrix detector; however, after the detection of , this detector updates the received signal vector by eliminating the interference of from the remaining subblocks by Fig. 2. Comparison of ML, reduced-complexity ML and LLR detectors for OFDM-IM with different configurations. (46) where .Inotherwords, reduced-complexity ML decoder calculates the corresponding after the detection of , its effect on the received signal vector LLR sum values for each possible index combination from the is totally eliminated by the update equation given by (46) under predefined look-up table and then decides on the most likely the assumption that . combination. On the right hand side of Fig. 2, we compared the 4) SP Detector: The signal-power (SP) detector for the BER performance of two completely different , OFDM-IM scheme applies the same detection and cancelation OFDM-IM schemes. The , scheme, which uses techniques as the block cancelation detector; however, first, it a reduced complexity ML decoder, employs a look-up table of calculates the SP values for all subblocks via size 64, composed of lexicographically ordered 4 combinations of 8. On the other hand, the , scheme with an LLR (47) decoder, does not employ a look-up table, instead it uses the combinatorial method. As seen from Fig. 2, interestingly, the and then sorts these SP values, starting from the subblock with LLR detector, which is actually a near-ML detector, achieves the highest SP, and proceeding towards the subblock with the the same BER performance as that of the ML detector. This can lowest SP. In other words, the SP detector starts with the detec- be attributed to the fact that the percentage of the number of tion of the subblock with the highest SP; after the detection of unused combinations out of the all this subblock, it updates the received signal vector using (46) possible index combinations is relatively low andsoon. for , selection (8.6%). Therefore, we conclude that it is not very likely for the receiver to decide to a catastrophic set VI. SIMULATION RESULTS of active indices for this case. On the other hand, this percentage In this section, we present simulation results for the is equal to 36.3% and 10.7% for , and , OFDM-IM scheme with different configurations and make OFDM-IM schemes respectively. Note that, it is not comparisons with classical OFDM, ESIM-OFDM [14] and possible to implement a look-up table, therefore an ML decoder, the ICI self-cancellation OFDM scheme [25]. The BER perfor these schemes to make comparisons. formance of these schemes was evaluated via Monte Carlo In Fig. 3, we compared the BER performance of different simulations. In what follows, we investigate the error per- OFDM-IM schemes with classical OFDM for BPSK. As seen formance of OFDM-IM under ideal and realistic channel from Fig. 3, at a BER value of our new scheme with , conditions. achieves approximately 6 dB better BER performance than classical OFDM operating at the same spectral efficiency. A. Performance of OFDM-IM Under Ideal Channel This significant improvement in BER performance can be ex- Conditions plained by the improved distance spectrum of the OFDM-IM In this subsection, we investigate the error performance of scheme, where higher diversity orders are obtained for the bits OFDM-IM in the presence of frequency selective channels only carried by the active indices. For comparison, the theoretical as described in Section II. In all simulations, we assumed the curve obtained from (32) is also depicted on the same figure following system parameters: , and . for the , scheme, which uses an ML decoder. As In Fig. 2, we compare the BER performance of the ML, the seen from Fig. 3, the theoretical curve becomes very tight with reduced-complexity ML and the LLR detectors for OFDM-IM the computer simulation curve as the SNR increases. For higher with the same system parameters for BPSK. As seen from the values of , we employ the combinatorial method for the index left hand side of Fig. 2, for the , scheme, the ML mapping and demapping operations with the LLR decoder. We and reduced-complexity ML decoders exhibit exactly the same observe that despite their increased data rates, the OFDM-IM BER performance, as expected. As mentioned previously, the schemes with , and , ,exhibitBER BAŞAR et al.: ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING WITH INDEX MODULATION 5545

Fig. 4. The effect of varying values on BER performance for , and , OFDM-IM schemes. Fig. 3. BER performance of OFDM-IM with different conﬁgurations, BPSK.

noticeable performance improvement over classical OFDM performance close to the low-rate , OFDM scheme. when both schemes operate at the same spectral efficiency This can be explained by the fact that for high SNR, the error since ESIM-OFDM requires higher order modulation. performance of the OFDM-IM scheme is dominated by the PEP events with aswediscussedintheSectionIV.Finallywe B. Performance of OFDM-IM Under Channel Estimation observe from Fig. 3 that OFDM-IM achieves significantly better Errors BER performance than classical OFDM at high SNR values due In Fig. 8, we present computer simulation results for the im- to the improved error performance of the bits transmitted in the perfect CSI case. As a reference, we consider the BER perfor- index domain, which is more effective at high SNR. mance of the , OFDM-IM scheme using BPSK for In Fig. 3, we also show the BER performance of the . In Fig. 8, the theoretical upper bound calculated from OFDM-IM scheme which does not employ -ary modulation (41) is also given.For comparison, as mentioned in Section V-A, ( , , LLR, w/o ), and relies on the transmis- we also present the theoretical curve obtained by the calcula- sion of data via subcarrier indices only. As seen from Fig. 3, tion of the UPEP values, considering only the worst case error this scheme achieves a diversity order of two, as proved in events. As seen from Fig. 8, both approximations become very Section IV, and exhibits the best BER performance for high tight with increasing SNR values and can be used to predict the SNR values with a slight decrease in the spectral efficiency BER behavior of the OFDM-IM scheme under channel estima- compared to classical OFDM employing BPSK modulation. tion errors. In Fig. 4, we investigate the effect of varying fading channel tap lengths on the error performance of the OFDM-IM schemes C. Performance of OFDM-IM Under Mobility Conditions using BPSK with , and , in the In this subsection, we present computer simulation results light of our analysis presented in Remark 2 of Section IV. As for the OFDM-IM scheme operating under realistic channel seen from this figure, increasing values create a separation in mobility conditions. Our simulation parameters are given in BER performance especially in the mid SNR region, while the Table III. A multipath wireless channel having an exponentially differences in error performance curves become smaller in the decaying power delay profile with the normalized powers is high SNR region as expected, due to the identical worst case assumed [26]. PEP events. On the other hand, we observe that increasing In Fig. 9, we compare the BER performance of three different values have a bigger effect on the BER performance for the OFDM-IM schemes employing various detectors with the clas- , scheme due to the larger variation in values sical OFDM and the ICI self-cancellation OFDM scheme pro- of Table II for this configuration. posed in [25] for a mobile terminal moving at a speed of In Figs. 5–7, we compare the BER performance of the . OFDM and OFDM-IM schemes use BPSK while the proposed scheme with classical OFDM and ESIM-OFDM for ICI self-cancellation scheme uses QPSK and applies precoding. three different spectral efficiency values (0.8889, 1.7778 and For the , OFDM-IM scheme, four different type of 2.6667 bits/s/Hz, respectively). We consider ML and LLR detectors presented in Section V-B are used. For the higher rate detectors for ESIM-OFDM and OFDM-IM, respectively. As , and , OFDM-IM schemes, which seen from Figs. 5–7, the proposed scheme provides significant rely on the combinatorial method to determine the active in- improvements in error performance compared to ESIM-OFDM dices, the received signal vector is processed by the MMSE de- and OFDM operating at the same spectral efficiency. Fur- tector and then an LLR detector is employed. On the other hand, thermore, we observe that ESIM-OFDM cannot provide the classical OFDM scheme and ICI self-cancellation OFDM 5546 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 22, NOVEMBER 15, 2013

Fig. 5. Performance of classical OFDM, ESIM-OFDM and OFDM-IM for Fig. 7. Performance of classical OFDM, ESIM-OFDM and OFDM-IM for 0.8889 bits/s/Hz. 2.6667 bits/s/Hz.

Fig. 6. Performance of classical OFDM, ESIM-OFDM and OFDM-IM for Fig. 8. BER performance of the , OFDM-IM scheme with im- 1.7778 bits/s/Hz. perfect CSI, BPSK, .

TABLE III scheme employ an MMSE detector. As seen from Fig. 9, com- SIMULATION PARAMETERS pared to classical OFDM, OFDM-IM cannot exhibit exceptional performance with the MMSE detector, since this detector, which works as an equalizer, does not improve the detection process of the OFDM-IM scheme, which relies on the fluctuations between the channel coefficients to determine the indices of the active subcarriers. In other words, the MMSE equalizer eliminates the effect of the channel coefficients, and therefore, the receiver makes decisions by simply calculating the Euclidean distance between the constellation points and the received signal. On the other hand, the interference unaware submatrix receiver tends to values, which causes an error floor. Although considering the in- error floor just after reaching to the BER value of . This can terference, the performance of the block cancellation detector is be explained by the fact that this detector does not take into ac- also dominated by the error floor as seen from Fig. 9; however, count the inference between different subblocks; therefore, the it pulls downs the error floor to lower BER values compared performance is limited by this interference with increasing SNR to the submatrix receiver. Meanwhile the SP detector provides BAŞAR et al.: ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING WITH INDEX MODULATION 5547

Fig. 9. Performance of OFDM-IM for a mobile terminal moving at a speed of , BPSK. Fig. 11. Performance of OFDM-IM for a mobile terminal moving at a speed of , QPSK.

We observe from Fig. 10 that the OFDM-IM scheme with the SP detector provides a significant improvement (around 12 dB) compared to classical OFDM if the target BER is , while for much lower target BER values, one may consider the OFDM-IM scheme with the MMSE detector, which is invulnerable to the error floor. In Fig. 11, we compare the BER performance of OFDM-IM using QPSK and employing the MMSE and LLR detector with classical OFDM using QPSK and ESIM-OFDM using 8-QAM for a mobile terminal moving at a speed of . As seen from Fig. 11, for the same spectral efficiency, the proposed scheme achieves better BER performance than the classical OFDM and ESIM-OFDM, while the higher rate , OFDM-IM scheme achieves better BER performance than classical OFDM with increasing SNR. Fig. 10. Performance of OFDM-IM for a mobile terminal moving at a speed of , BPSK. D. Complexity Comparisons In this subsection, we compare the computational complexity the best error performance by completely eliminating the error of the proposed method with the reference systems. Without floor in the considered BER regime. For a BER value of , mobility, the detection complexity (in terms of complex opera- the SP detector provides approximately 9 dB better BER perfor- tions) of the LLR and the reduced-complexity ML detectors em- mance than classical OFDM. We also observe that the proposed ployed in the proposed scheme is the same as that of the classical scheme with the SP detector achieves better BER performance OFDM and ESIM-OFDM systems, and, as shown in Sections II than the ICI self-cancellation OFDM scheme at high SNR. In and III, it is per subcarrier. On the other hand, in the same figure, we also show the BER curves of higher rate the case of high mobility, while the detection complexity of the , and , OFDM-IM schemes. Inter- MMSE+LLR detector of the proposed scheme is the same as the estingly, our , OFDM-IM scheme achieves better classical OFDM, ESIM-OFDM and ICI self-cancelation OFDM BER performance than classical OFDM with increasing SNR schemes , the detection complexity of the ML based values even if transmitting 22 additional bits per OFDM block. detectors (submatrix, block cancelation and SP detectors) of the In Fig. 10, we extend our simulations to the proposed scheme grows exponentially with increasing values case. As seen from Fig. 10, by increasing the mobile terminal of , as shown in Section V, i.e., it is per subblock, velocity, the detectors that do not use the MMSE equalization, which is higher than the reference systems. Consequently, this tend to have error floors for lower BER values compared to detector can be used only for smaller values of ,suchas . the case, and the ICI self-cancellation OFDM On the other hand, for higher values of , the MMSE+LLR de- scheme cannot compete with the proposed scheme any more. tector is the potential technique to employ in practice. 5548 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 22, NOVEMBER 15, 2013

VII. CONCLUSION AND FUTURE WORK [10]J.Fu,C.Hou,W.Xiang,L.Yan,andY.Hou,“Generalisedspatial modulation with multiple active transmit antennas,” in Proc. IEEE A novel multicarrier scheme called OFDM with index GLOBECOM Workshops, Dec. 2010, pp. 839–844. modulation, which uses the indices of the active subcarriers to [11] E. Başar,Ü.Aygölü,E.Panayırcı, and H. V. Poor, “Performance of transmit data, has been proposed in this paper. In this scheme, spatial modulation in the presence of channel estimation errors,” IEEE Commun. Lett., vol. 16, no. 2, pp. 176–179, Feb. 2012. inspired by the recently proposed SM concept, the incoming [12] E. Başar, Ü. Aygölü, E. Panayırcı, and H. V. Poor, “Super-orthogonal information bits are transmitted in a unique fashion to im- trellis-coded spatial modulation,” IET Commun., vol. 6, no. 17, pp. prove the error performance as well as to increase spectral 2922–2932, Nov. 2012. [13] R. Abu-alhiga and H. Haas, “Subcarrier-index modulation OFDM,” in efficiency. 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Ümit Aygölü (M’90) received his B.S., M.S. and called NEWCOM (Network of Excellent on Wireless Communications) and Ph.D. degrees, all in electrical engineering, from WIMAGIC Strep project representing Kadir Has University. Dr. Panayırcı Istanbul Technical University, Istanbul, Turkey, was an Editor for IEEE TRANSACTIONS ON COMMUNICATIONS in the areas of in 1978, 1984 and 1989, respectively. He was a Synchronization and Equalizations in 1995–1999. He served as a Member of Research Assistant from 1980 to 1986 and a Lecturer IEEE Fellow Committee in 2005–2008. He was the Technical Program Chair from 1986 to 1989 at Yildiz Technical University, of ICC-2006 and PIMRC-2010 both held in Istanbul, Turkey. Presently he is Istanbul, Turkey. In 1989, he became an Assistant head of the Turkish Scientific Commission on Signals and Systems of URSI Professor at Istanbul Technical University, where (International Union of Radio Science). he became an Associate Professor and Professor, in 1992 and 1999, respectively. His current research interests include the theory and applications of combined channel coding and modulation techniques, MIMO systems, space-time H. Vincent Poor (S’72–M’77–SM’82–F’87) re- coding and cooperative communication. ceived the Ph.D. degree in EECS from Princeton University in 1977. From 1977 until 1990, he was on the faculty of the University of Illinois at Urbana-Champaign. Since 1990 he has been on Erdal Panayırcı (S’73–M’80–SM’91–F’03) re- the faculty at Princeton, where he is the Michael ceived the Diploma Engineering degree in Electrical Henry Strater University Professor of Electrical Engineering from Istanbul Technical University, Engineering and Dean of the School of Engineering Istanbul, Turkey and the Ph.D. degree in Electrical and Applied Science. Dr. Poor’s research interests Engineering and System Science from Michigan are in the areas of stochastic analysis, statistical State University, USA. Until 1998 he has been with signal processing, and information theory, and their the Faculty of Electrical and Electronics Engineering applications in wireless networks and related fields such as social networks at the Istanbul Technical University, where he was and smart grid. Among his publications in these areas the recent books Smart a Professor and Head of the Telecommunications Grid Communications and Networking (Cambridge, 2012) and Principles of Chair. Currently, he is Professor of Electrical En- Cognitive Radio (Cambridge, 2013). gineering and Head of the Electronics Engineering Dr. Poor is a member of the National Academy of Engineering and the Department at Kadir Has University, Istanbul, Turkey. Dr. Panayırcı’s recent National Academy of Sciences, a Fellow of the American Academy of Arts and research interests include communication theory, synchronization, advanced Sciences, an International Fellow of the Royal Academy of Engineering (U.K.), signal processing techniques and their applications to wireless communica- and a Corresponding Fellow of the Royal Society of Edinburgh. He is also a tions, coded modulation and interference cancelation with array processing. Fellow of the Institute of Mathematical Statistics, the Acoustical Society of He published extensively in leading scientific journals and international America, and other organizations. He received the Technical Achievement and conferences. He has co-authored the book Principles of Integrated Maritime Society Awards of the SPS in 2007 and 2011, respectively. Recent recognition Surveillance Systems (Boston, Kluwer Academic Publishers, 2000). of his work includes the 2010 IET Ambrose Fleming Medal, the 2011 IEEE Dr. Panayırcı spent the academic year 2008–2009, in the Department Eric E. Sumner Award, and honorary doctorates from Aalborg University, of Electrical Engineering, Princeton University, New Jersey, USA. He has the Hong Kong University of Science and Technology, and the University of been the principal coordinator of a 6th and 7th Frame European project Edinburgh.