Spectrally Efficient Frequency Division Multiplexing with Index-Modulated Non-Orthogonal Subcarriers

POSTPRINT (ACCEPTED VERSION), DOI: 10.1109/LWC.2018.2867869 1

Spectrally Efﬁcient Frequency Division Multiplexing With Index-Modulated Non-Orthogonal Subcarriers Miyu Nakao, Student Member, IEEE, and Shinya Sugiura, Senior Member, IEEE

Abstract—In this letter, we propose a novel index-modulated typically limited to as few as tens of non-orthogonal subcar- non-orthogonal spectrally efficient frequency-division multiplex- riers. ing (SEFDM) scheme, in order to attain a higher bandwidth Index modulation (IM) [10] is a recent promising technique efficiency than the conventional orthogonal frequency-division multiplexing (OFDM) and SEFDM counterparts. More specifi- that is capable of achieving specific merits over multiplexing. cally, the recent index modulation concept is amalgamated with The IM principle is based on the activation of a subset of SEFDM, for the sake of reducing the effects of intercarrier multiple communication resources in the space, time, and interference while benefiting from SEFDM’s increased spectral frequency domains, where the combination of activated in- efficiency. We also formulate a low-complexity log-likelihood ratio dices is used for conveying information bits, in addition to (LLR)-based detection algorithm, which allows the proposed SEFDM to operate in the configuration of an arbitrarily high conventional modulated symbols. Most recently, in [11], time- number of subcarriers. Our simulation results demonstrate that domain IM is combined with FTN signaling, where a high the proposed SEFDM scheme outperforms the conventional sparsity of IM symbols allows us to reduce the effects of FTN- SEFDM and OFDM, especially in a low-rate scenario. specific ISI while benefiting from the FTN-specific increased Index Terms—Faster-than-Nyquist, index modulation, interfer- bandwidth efficiency. Furthermore, in [12], a low-complexity ence, log-likelihood ratio, multicarrier, non-orthogonal subcarri- successive detection algorithm based on minimum mean- ers, frequency-division multiplexing. square error (MMSE) criterion and log-likelihood ratio (LLR) detection was proposed for index-modulated FTN signaling. In addition, an IM scheme operating in the frequency domain, I.INTRODUCTION i.e., subcarrier index modulation (SIM), was developed for HE concept of faster-than-Nyquist (FTN) signaling has improving the achievable performance of OFDM [13], while T gained significant attention as a means of boosting the in [14] multiple-mode index modulation was also applied to transmission rates of the next generation of communication SIM. systems, which is achievable without imposing additional Against the above-mentioned background, the novel contri- bandwidth and power consumptions [1]–[3]. In FTN signaling, butions of this letter are as follows. We are the first to pro- a symbol interval is set to lower than the one given by the first pose an improved SEFDM scheme where the non-orthogonal Nyquist criterion that guarantees the orthogonality of time- subcarriers are index-modulated for the sake of reducing the domain symbols, and hence the rate enhancement specific to effects of detrimental intercarrier interference while benefit- FTN signaling is attainable at the expense of introducing non- ing from the SEFDM-specific increased bandwidth efficiency. orthogonality between a block of symbols. Furthermore, we derive a low-complexity successive detection While FTN signaling is based on non-orthogonal symbol algorithm of MMSE filtering and LLR-based IM detection, packing in the time domain, its frequency-domain counterpart which allows us to operate the proposed scheme in a scenario was invented in the context of multicarrier systems [4], [5], of a practically high number of subcarriers. Our simulation where intercarrier interference is tolerated for boosting the results demonstrate the fundamental benefits of the proposed bandwidth efficiency, which is referred to as spectrally effi- SEFDM with IM (SEFDM-IM) scheme over the conventional cient frequency-division multiplexing (SEFDM). Furthermore, OFDM and SEFDM schemes. the SEFDM scheme was applied in optical [6] and satellite communication systems [7]. Moreover, in order to exploit non- II.SYSTEM MODEL orthogonal resource packing both in the time and frequency domains, a multicarrier FTN system was developed in [8], [9]. A. Transmitted Signal Model Note that in the literature, the beneficial scenario of SEFDM In the proposed SEFDM-IM scheme, N subcarriers are over orthogonal frequency-division multiplexing (OFDM) is divided into L groups, each containing M subcarriers, and hence we have the relationship of N = LM. The entire ⃝ c 2018 IEEE. Personal use of this material is permitted. Permission from frequency-domain transmission frame s ∈ CN is formulated IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional as purposes, creating new collective works, for resale or redistribution to servers T or lists, or reuse of any copyrighted component of this work in other works. s = [s0, s1, ··· , sN−1] (1) [ ]T M. Nakao is with the Department of Computer and Information Sciences, − Tokyo University of Agriculture and Technology, Kogani, Tokyo 184-8588, = s(0)T , s(1)T , ··· , s(L 1)T , (2) Japan. S. Sugiura is with the Institute of Industrial Science, University of Tokyo, Meguro-ku, Tokyo 153-8505, Japan (e-mail: [email protected]). where we have the frequency-domain symbols in the lth sub- (Corresponding author: Shinya Sugiura.) The present study was supported (l) (l) (l) ··· (l) T ∈ CM in part by the Japan Society for the Promotion of Science (JSPS) KAKENHI carrier group as follows: s = [s0 , s1 , , sM−1] . Grant Numbers 16KK0120, 17H03259, and 17K18871. In this letter, we assume an additive white Gaussian noise POSTPRINT (ACCEPTED VERSION), DOI: 10.1109/LWC.2018.2867869 2

(AWGN) channel, for the sake of simplicity. However, the where bn(t) is the nth orthonormal basis calculated based on proposed scheme may be readily applicable to an arbitrary the Gram–Schmidt orthonormalization method [4]. channel. Finally, the observation statistics r = [r0, ··· , rN−1] can be At the transmitter, B information bits B ∈ ZB are divided reformulated as into L groups, each containing b = b1 + b2 bits; hence, the r = Ms + n, (6) relationship B = bL holds. The b information bits b(l) ∈ Zb where M is the N ×N covariance matrix, whose pth-row and in the lth group are modulated onto M symbols s(l). More qth-column element is calculated by specifically, the first b1 bits out of the b information bits are ∫ T used for selecting the K out of M symbols, and then the √1 ∗ P mp,q = exp(j2πqαt/T )bp(t)dt. (7) selected K symbols are modulated onto -point amplitude T 0 and phase shift keying (APSK) symbols based on b2 bits. The Furthermore, n = [n0, ··· , nN−1] is the related noise matrix, remaining M −(K)symbols are set to zero. Therefore, we ⌊ M ⌋ P which is represented by have b1 = log2 K and b2 = K log2 . Finally, in order to ∫ T maintain the average transmission power per symbol√ at unity, √1 ∗ ni = n(t)bi (t)dt, (8) the K non-zero symbols are scaled by a factor of M/K. T 0 Each subcarrier of the conventional SEFDM and the pro- noting that the elements of n remains uncorrelated with each posed SEFDM-IM systems is allocated in a non-orthogonal other. manner, such that its separation in the frequency domain is smaller than that of the OFDM counterpart, in order to increase C. Error-Rate Bound the bandwidth efficiency. More specifically, the bandwidth In this section, we derive the analytical BER bound of compression factor is represented by α = ∆fT (α < 1), the proposed SEFDM-IM scheme, employing the optimal ML where ∆f is the minimum separation in the frequency domain detector. The conditional pairwise error probability, where the between the subcarriers, and T is the symbol duration in the symbol vector s is misdemodulated as s′, is given by [12] time domain. Note that the parameter α becomes unity in the √  conventional OFDM system. ∥ M(s − s′) ∥2 → ′  F  Hence, the transmission rate of the proposed SEFDM-IM( ) Pr(s s ) = Q (9) ⌊ M ⌋ 2N0 scheme R [bps/Hz] is formulated as R = (1/α)( log2 K + P K log2 )/M, where the coefficient 1/α corresponds to the where Q(·) is the Q-function. The analytical BER bound is effects( of) subcarrier packing in the frequency domain, while evaluated by ⌊ M ⌋ P log2 K and K log2 are the information bits carried by ∑ ∑ 1 1 the IM and the conventional APSK symbols, respectively. P ≤ d(s, s′)Pr(s → s′) (10) BER B · 2B The time-domain signal representation of the proposed s s′ SEFDM-IM scheme, which is transmitted to the receiver, is where d(s, s′) represents the Hamming distance between the given by binary version of s and s′. − 1 N∑1 x(t) = √ sn exp(j2πnαt/T ). (3) III.DETECTION ALGORITHMS T n=0 A. Conventional ML Detection The information bits estimated by the optimal maximum likelihood (ML) detection are described as B. Received Signal Model { } 2 Bˆ ML = arg min ∥r − Ms∥ , (11) The time-domain received signals y(t) are expressed as B y(t) = x(t) + n(t), (4) where ∥·∥ denotes the Euclidean norm. Although ML detection exhibits the attainable performance, the complexity of ML where n(t) is the related AWGN component, which follows detection grows exponentially with a linear increase in the the complex-valued Gaussian distribution CN (0,N ), N be- 0 0 number of subcarriers N. This implies that only a limited ing the noise variance. number of subcarriers N are tractable in the conventional ML The receiver consists of N correlators and a detector. In detection. order to eliminate the effects of colored noise, an orthonormalization operation is required at each correlator. More B. Conventional Successive MMSE and ML Detection specifically, the output of the nth receiver correlator is given In order to reduce the high complexity, which is excessive by ∫ T in the above-mentioned ML detection, we revisit successive ∗ ··· − rn = y(t)bn(t)dt (n = 0, ,N 1), (5) detection, based on linear MMSE, which is followed by ML 0 detection. More specifically, the symbols are first estimated by 1Note that in the SEFDM and SEFDM-IM schemes, although the transmis- the MMSE criterion, according to [15], [16] sion rate increases with the decrease of the α value, the resultant error-rate ( )−1 MH MMH performance typically deteriorates, due to the effects of increased inter-carrier ˆs = + N I r, (12) interference. MMSE M/K M/K 0 POSTPRINT (ACCEPTED VERSION), DOI: 10.1109/LWC.2018.2867869 3 where I is the identity matrix. MMSE filtering, we propose improved successive MMSE and Then, the information bits modulated on the lth subframe LLR detection, where we consider an increased number of are detected, based on an exhaustive ML search as follows: candidates when estimating the active subcarriers at the stage { } 2 of the LLR detection of Section III-C. This also results in an ˆ(l) (l) − (l) bMMSE-ML = arg min ˆsMMSE s , (13) increase in the search space of the activated APSK symbols, in b(l) comparison to that of Section III-B. For the sake of simplicity, (l) we only consider the scenario in which K = 1 subcarrier out where ˆsMMSE is the lth subframe of MMSE-filtered symbols (0) ··· (L−1) T of N = M subcarriers is activated. However, the method can ˆsMMSE = [ˆsMMSE, ,ˆsMMSE ] . While the complexity of the MMSE operation in (12) is O((N 3)), that of the ML- be readily extended to the generalized scenario with arbitrary based bit estimation in (13) is O 2b . Hence, the total parameters (K,M,N). To be more specific, instead of se- complexity imposed by detection of the whole frame does not lecting only the single subcarrier corresponding to the highest exponentially increase with a linear increase in the number LLR value as in Section III-C, we here use also one additional of subcarriers N, unlike the ML detection of Section III-A. subcarrier, the one having the second highest LLR value. Note that the original MMSE-based SEFDM symbol detection Then, the search space of low-complexity single-subcarrier- of [15] is not followed by subframe-based bit detection of based ML detection becomes doubled, in comparison to that (13), but by symbol-by-symbol bit detection, since the conven- of Section III-C. The additional computational cost of this O 2 tional SEFDM scheme, rather than the proposed SEFDM-IM detector over that of Section of III-C is (LM ) per block. scheme, was considered in [15]. IV. PERFORMANCE RESULTS C. Proposed Successive MMSE and LLR Detection In this section, we provide our simulation results, in order In order to further reduce the complexity associated with the to characterize the achievable performance of the proposed ML-based bit estimation of (13), the concept of LLR detection, SEFDM-IM scheme. The achievable BERs were calculated, which was originally developed for the OFDM-IM [17] and based on Monte Carlo simulations. We employed the SEFDM- SC-IM schemes [12], is exploited. More specifically, the LLR IM parameters (M,K, P) = (4, 1, 4).2 Furthermore, the value of the mth symbol in the lth subframe is calculated, conventional SEFDM and OFDM schemes were considered based on Bayes’ formula, as as the benchmark schemes. ∑ P−1 (l) S | (l) Fig. 1 shows the effects of bandwidth compression factor (l) i=0 P (sm = i sˆm ) γm = ln (14) α on the achievable BER performance in the conventional P (s(l) = 0|sˆ(l)) m m SEFDM and the proposed SEFDM-IM schemes. The number 2 ( ) (l) of subcarriers was set as N = 4, and we considered an Eb/N0 sm K of 5 dB. The BERs of both the ML and MMSE detectors were = ln − + M( K N0 ) plotted for the conventional SEFDM scheme, while all four of P− ( ) ∑1 1 2 the detectors presented in Section III, i.e., the conventional − (l) − S + ln exp sm i , (15) ML, the conventional successive MMSE ML, the proposed N0 i=0 successive MMSE-LLR, and the improved successive MMSE- where Si is the ith symbol of an P-sized APSK constellation. LLR detectors, were compared for the proposed SEFDM- The complexity imposed by the LLR detector of (15) is O(M), IM scheme. Observe in Fig. 1 that among the four curves noting that although it is possible to directly calculate the LLR associated with the proposed SEFDM-IM scheme, the pro- values from r, this imposes an excessively high complexity, posed improved successive MMSE-LLR detector exhibited the especially when the block size N is high. As mentioned in performance close to the optimal ML detector, while accom- [17], in order to prevent numerical overflow, the Jacobian plishing a significantly lower detection complexity than that logarithm is typically used in (15). of ML detection. Furthermore, it was found that the proposed (l) Having obtained the LLR value of γm , the receiver esti- successive MMSE-LLR detector exhibited nearly the same mates the K indices corresponding to the activated subcarriers, BER performance as the conventional successive MMSE-ML which are those having the K highest LLR values. Then, detector, which implies that introducing the LLR concept did K APSK symbols of the K estimated active subcarriers are not impose any substantial performance penalty. In addition, in separately demodulated, based on a low-complexity single- Fig. 1, we see that the proposed SEFDM-IM scheme exhibits a subcarrier-based ML search. clear performance gain over the conventional SEFDM scheme in the range of α ≥ 0.7.3

D. Improved Successive MMSE and LLR Detection 2In our extensive simulations, it was found that the SEFDM and the The successive detection presented in Section III-B suffers SEFDM-IM schemes typically exhibit the performance advantage over the OFDM counterpart for low-rate and low-K scenarios. Hence, we focused our from an unignorable performance penalty in comparison to attention only on a K = 1 scenario in this letter. the optimal ML detection. This is because the linear MMSE 3In order to elaborate a little further, we further compared the minimum operation of (12) is suboptimal, while the LLR detection Euclidean distances (MEDs) between the proposed SEFDM-IM scheme and the conventional SEFDM scheme, where the SEFDM-IM scheme exhibited a using (15) does not have any substantial performance loss. higher MED than the SEFDM scheme in the range of α ≥ 0.6 for the same In order to eliminate the performance penalty imposed by system parameters considered in Fig. 1. POSTPRINT (ACCEPTED VERSION), DOI: 10.1109/LWC.2018.2867869 4

100 100 SEFDM, ML SEFDM, MMSE SEFDM-IM, ML -1 SEFDM-IM, MMSE-ML 10 SEFDM-IM, MMSE-LLR SEFDM-IM, Improved MMSE-LLR 10-1 10-2

10-3 BER BER OFDM 10-2 10-4 SEFDM, ML SEFDM, MMSE SEFDM-IM, ML 10-5 SEFDM-IM, MMSE-LLR SEFDM-IM, Improved MMSE-LLR SEFDM, Theory SEFDM-IM, Theory 10-3 10-6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 7 8 9 10 11 12 α Eb/N0

Fig. 1. Achievable BER performance of the conventional SEFDM and Fig. 2. Achievable BER performances of the OFDM, conventional SEFDM, the proposed SEFDM-IM schemes in an AWGN channel. The number of and proposed SEFDM-IM schemes. The number of subcarriers was set as subcarriers was set as N = 4, and the SEFDM-IM parameters were N = 8, and the SEFDM-IM parameters were set as (M,K, P) = (4, 1, 4). set as (M,K, P) = (4, 1, 4), while BPSK modulation was used for the In addition, BPSK modulation was used for the OFDM and conventional conventional SEFDM scheme. Eb/N0 was set as 5 dB. SEFDM schemes.

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