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

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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 2 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) h iT 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 2 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 2 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) 2 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 Z T used for selecting the K out of M symbols, and then the p1 ∗ 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 b M c P which is represented by have b1 = log2 K and b2 = K log2 . Finally, in order to Z T maintain the average transmission power per symbolp at unity, p1 ∗ 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 s0, is given by [12] time domain. Note that the parameter α becomes unity in the 0s 1 conventional OFDM system. k M(s − s0) k2 ! 0 @ F A Hence, the transmission rate of the proposed SEFDM-IM( ) Pr(s s ) = Q (9) b M c 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 b M c P log2 K and K log2 are the information bits carried by X X 1 1 the IM and the conventional APSK symbols, respectively. P ≤ d(s; s0)Pr(s ! s0) (10) BER B · 2B The time-domain signal representation of the proposed s s0 SEFDM-IM scheme, which is transmitted to the receiver, is where d(s; s0) represents the Hamming distance between the given by binary version of s and s0. − 1 NX1 x(t) = p 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 n o 2 B^ ML = arg min kr − Msk ; (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 orthonor- malization operation is required at each correlator.
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