Vector OFDM With Index Modulation

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SPECIAL SECTION ON INDEX MODULATION TECHNIQUES FOR NEXT-GENERATION WIRELESS NETWORKS

Received August 27, 2017, accepted September 19, 2017, date of publication September 26, 2017, date of current version October 25, 2017. Digital Object Identifier 10.1109/ACCESS.2017.2756080

Vector OFDM With Index Modulation YUN LIU1,2 , FEI JI2 , (Member, IEEE), MIAOWEN WEN DEHUAN WAN2 , AND BEIXIONG ZHENG2 1 School 2 School

2,

(Member, IEEE),

of Information Science and Technology, Zhongkai University of Agriculture and Engineering, Guangzhou 510225, China of Electronics and Information Engineering, South China University of Technology, Guangzhou 510641, China

Corresponding author: Miaowen Wen ([email protected]) This work was supported by the National Natural Foundation of China under Grant 61431005, Grant 61501531, Grant 61501190, and Grant 61672554, and by the Guangdong Provincial Research Project under Grant 2016A030308006, Grant 2015A030313602, and Grant 2014A030310389.

ABSTRACT The concept of index modulation (IM) has gained increased attention in recent years. As a member of the IM family, orthogonal frequency division multiplexing with index modulation (OFDM-IM), which is able to convey additional information bits using the indices of the active subcarriers, has emerged as an attractive physical layer transmission technique. Vector OFDM (V-OFDM) has inherently good peakto-average-power ratio (PAPR) feature and good bit error rate (BER) performance due to the nearly full diversity gains achieved by most of its sub-blocks. In this paper, in order to improve the BER performance of OFDM-IM and reduce the PAPR of the transmit signal, we propose an enhanced OFDM-IM scheme termed vector OFDM with index modulation (V-OFDM-IM). In addition to the structure of the transceiver, a complexity reduced maximum likelihood detector is presented. To further reduce the detection complexity, a near-optimal detector based on sequential Monte Carlo technique is introduced. The average bit error probability of the proposed scheme is derived in closed form over frequency-selective block Rayleigh-fading channel. Both the theoretical and the simulation-based results show that, under the same spectrum efficiency, V-OFDM-IM outperforms the conventional OFDM-IM and OFDM. INDEX TERMS Vector OFDM, OFDM-IM, index modulation, sequential Monte Carlo (SMC), performance analysis. I. INTRODUCTION

In recent years, there has been tremendous interest in index modulation (IM), which is a highly spectrum- and energy- efficient modulation technique. IM is able to convey additional information bits by activating a subset of the building blocks of the corresponding system with changing indices [1], [2]. Those information bits can be modulated onto the indices of many kinds of transmission entities, such as subcarriers [3]–[12], transmit antennas [13]–[16], spreading codes [17], [18], radio frequency (RF) mirrors [19], etc. The concept of IM was first used in multiple antenna systems and the resulting transmission scheme is termed spatial modulation (SM), where the IM is operating on the spatial/antenna dimension. In SM, a subset of the transmit antennas are activated, while the others keep silent. The data bits are split into two parts. One part is mapped into classical symbols on a two-dimension signal constellation (such as phase shift keying (PSK) and quadrature amplitude modulation (QAM)) and then sent by the active antennas. The other part is used to choose active antennas to be used. Compared with

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conventional multiple-input multiple-output (MIMO), SM has many advantages, such as reduced detection complexity, reduced interchannel interference, and relaxed interantenna synchronization requirement [15]. Since SM, the IM technique has been actively investigated in other systems, such as orthogonal frequency-division multiplexing (OFDM) systems. OFDM has become one of the most important digital communication techniques in the past decades and has been adopted in many commercial systems, such as 802.11X wireless local area networks (WLAN), digital video broadcasting (DVB), and LTE. OFDM achieves high-speed communication by splitting the frequency-selective broadband channel into a large number of parallel flat-fading subchannels. Consequently, at the receiver, the one-tap equalizer can be used for data detection with low complexity. For each symbol, by appending a cyclic prefix (or zero padding) longer than the delay spread of the channel impulse response, inter-symbol interference (ISI) free transmission can be achieved. OFDM with index modulation (OFDM-IM)

2169-3536 2017 IEEE. Translations and content mining are permitted for academic research only. Personal use is also permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

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Y. Liu et al.: Vector OFDM With Index Modulation

is a special IM scheme where a subset of subcarriers are activated and the indices of them are selected for carrying the additional information bits. With the same spectral efficiency, OFDM-IM can achieve better bit-error rate (BER) performance than that of conventional OFDM with comparable decoding complexity [3]. Furthermore, thanks to the inactive subcarriers, OFDM-IM outperforms conventional OFDM under channel estimation errors or motion-induced inter-carrier interference (ICI) [3]. Various generalizations or enhancements of OFDM-IM have been proposed in recent years. In [5] and [20], IM is performed independently on the in-phase and quadrature components of the subcarrier symbols. With such methods, a higher spectral efficiency than that of conventional OFDM-IM can be achieved. In [6], to achieve an additional diversity gain for the data bits conveyed by the active subcarriers, by combining OFDM-IM and space-time block codes with coordinate interleaving, the author proposed a coordinate interleaved OFDM-IM (CI-OFDM-IM), where the real and imaginary parts of the data symbols are divided and modulated on different subcarriers. In [7], the in-phase and quadrature dimensions are jointly explored for IM to get higher spectral efficiency, and the symbols of every two adjacent active subcarriers are linearly precoded to obtain a diversity gain of order two. In [21] and [22], by combing MIMO and OFDM-IM, the author proposed a transmission scheme termed MIMO-OFDM-IM, where the OFDM-IM blocks are transmitted independently on all the antennas as in the Vertical Bell Labs layered space-time (V-BLAST) scheme. At the receiver, these OFDM-IM blocks are separated by a minimum mean square error (MMSE) estimator with reduced computation complexity. In [23], at the cost of spectral efficiency, the repeated subcarrier-activating patterns are used to achieve transmit diversity for index bits. In [24], in order to obtain an additional transmit diversity gain, the authors proposed a modified scheme called OFDM-IM with transmit diversity (OFDM-IM-TD), where the data bits transmitted on the active subcarriers are also transmitted on the silent subcarriers using another signal constellation. Due to the nonlinearity of the power amplifier (PA) and the high peak-to-average ratio (PAPR) of OFDM, the transmit signal is often clipped in the time domain before being transmitted. This clipping operation introduces sidelobe splattering across the frequency-bins and leads to ICI. As demonstrated in [25], the PAPR of the conventional OFDM-IM is comparable to that of OFDM. Thus, OFDM-IM systems suffer the drawback of a high PAPR of the transmitted signal, especially in the situations where power-efficient nonlinear PA is more suitable for the transceiver than the expensive linear PA. It is well known that, under frequency selective fading channels, single-carrier frequency domain equalization (SC-FDE) is an alternative modulation scheme to OFDM with reduced PAPR [26]. However, the SC-FDE suffers from the lack of flexibility of bandwith and energy management. As a bridge connecting OFDM and SC-FDE, vector OFDM (V-OFDM) provides a general framework to 20136

strike a tradeoff between the PAPR and the resource management flexibility. Vector OFDM was originally designed to combat frequency null [27]. It can be seen as a general modulation scheme, subsuming OFDM and SC-FDE as two special cases. Compared to the conventional OFDM, owing to the much smaller size of the inverse fast Fourier transform (IFFT) in the modulator, vector OFDM has inherently better PAPR features [28]. As analyzed in [31] and [33], under a frequency-selective fading channel with independent subchannel gains, on most of the sub-blocks, vector OFDM can achieve nearly full diversity (of the order equal to the subblock length). Han et al. [36] proposed a novel constellationrotated Vector OFDM (CRV-OFDM) to attain full diversity gains on all of the sub-blocks at the price of higher transceiver complexity. To the best of our knowledge, the IM concept has not been used for vector OFDM in the previous literature. In this paper, we propose a Vector OFDM based index modulation scheme, termed vector OFDM with IM (V-OFDM-IM), where, in each subblock, a fixed number of subcarriers are activated, while the rest of the subcarriers keep silent, so that the indices of the active subcarriers can be used to convey IM bits. A complexity reduced maximum likelihood (ML) detector is proposed for the V-OFDM-IM receiver. To strike a better performance-complexity tradeoff, we also introduce a nearoptimal detector based on sequential Monte Carlo (SMC) technique. The average bit error probability over Rayleigh fading channels is derived in closed form. The theoretical results are verified by Monte Carlo simulations. It is shown via computation simulations that, under the same spectralefficiency, the proposed V-OFDM-IM is able to achieve better BER performance and lower PAPR than that of conventional OFDM-IM. The remainder of this paper is organized as follows. Section II describes the transceiver structure, mathematical model, and the detection algorithms of the V-OFDM-IM. In Section III, the performance analysis for V-OFDM-IM is provided. In Section IV, performance of V-OFDM-IM scheme is tested by computer simulations. This paper is concluded with remarks in Section V. For readability, some notations and variables used throughout this paper are listed in Table 1. II. SYSTEM DESCRIPTION

Let us first consider a conventional OFDM system operating over a frequency-selective fading channel. A total of N symbols, denoted as {X0 , X1 , · · · , XN −1 }, are transmitted on N subcarriers. The corresponding subcarrier frequency response is characterized by {H0 , H1 , · · · , HN −1 }. Let {Y0 , Y1 , · · · , YN −1 } and {W0 , W1 , · · · , WN −1 } be the received symbols and additive white Gaussian noise in the frequency domain. Assuming perfect synchronization and sufficient cyclic prefix (CP), we have the following inputoutput relationship [27], [31]: Yn = Hn Xn + Wn ,

n = 0, 1, · · · , N − 1.

(1)

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Y. Liu et al.: Vector OFDM With Index Modulation

TABLE 1. Notations.

ary quadrature amplitude modulation (M-QAM)). Therefore we have p = p1 + p2 . Specifically, for each subblock, p1 bits are used to select K active subcarriers out of M available subcarriers, while the remaining M −K subcarriers keep idle. The relationship between p1 and (M , K ) can be written as   p1 = log2 C (M , K ) . (2) Let us denote the selected indices for the active subcarriers on the lth subblock as Il = {il,1 , il,2 , · · · , il,K },

(3)

where il,k ∈ {1, 2, · · · , N } for k = 1, 2, · · · , K . The mapping from index bits to the indices of active subcarriers can be performed either using a look-up table, an equiprobable subcarrier activation method [32], or a combinatorial method [6], [13]. For example, a look-up table is presented in Table 2 for M = 4, and K = 2. Considering C(4, 2) = 6, we select 4 combinations out of 6 by discarding the other two. TABLE 2. A lookup table for N = 4, K = 2, and p1 = 2.

A. TRANSCEIVER STRUCTURE

The transceiver structure of V-OFDM-IM is depicted in Fig. 1. Similar to conventional OFDM-IM and V-OFDM, in V-OFDM-IM, the modulated symbols are processed blockby-block. Consider that there are m bits entering the transmitter. These bits are then split into L groups, each containing p bits. Each group of p bits is mapped to a frequency-domain subblock of length M , where M = N /L and N is the number of total subcarriers in the proposed system. Unlike conventional OFDM that maps all data bits to constellation points for all subcarriers, V-OFDM-IM divides each group of p-bits into two parts: the first part consists of p1 bits used for IM and the second one consists of p2 bits used for symbol modulation (e.g., M-ary phase shift keying (M-PSK) or MVOLUME 5, 2017

After the index selector, the remaining p2 = K log2 M bits are mapped onto K independent symbols from an M-ary signal constellation. As in conventional OFDM, we denote the obtained symbol on the nth subcarrier as Xn (Xn = 0 when the nth one is not activated by the index selector). Then, those symbols are processed by the way used in conventional V-OFDM systems [27], [31], [33], where the size of subblock is set to be M . Specifically, these IM-symbols, X0 , X1 , · · · , XN −1 , are first column-wise blocked into an M × L matrix X, whose (m, l)th entry, [X]m,l , is XlM +m . This matrix can also be written as X = [X0 , X1 , · · · , Xl , · · · , XL−1 ], where Xl is the lth IM subblock carrying the bits of the lth group within a block. Then, over each row of X ∈ CM ×L , an L-points IFFT is performed to get a matrix x ∈ CM ×L as x = XF−1 L ,

(4)

where F−1 IFFT matrix, whose L is an L-point normalized h i (k, n)-th entry is defined as F−1 = L −1/2 ej2πkn/L [31]. L k,n x is then used to generate the V-OFDM-IM signal in the time-domain. Let us rewrite x by its column vectors as x = [x0 , x1 , · · · , xL−1 ]. To avoid ISI, we pad the last P columns of x to itself as CP and get x˜ = [xL−P , xL−P+1 , · · · , xL−1 , x0 , x1 , · · · , xL−1 ]. Note that MP should be larger than the number of channel taps for ISI free transmission. After the parallel-to-serial (P/S) converter, the symbols of x˜ are transmitted serially into the channel column-by-column as {xTL−P , xTL−P+1 , · · · , xTL−1 , xT0 , xT1 , · · · , xTL−1 }. 20137

Y. Liu et al.: Vector OFDM With Index Modulation

FIGURE 1. Transceiver structure of the proposed vector OFDM index modulation.

At the receiver, after removing the CP, we get the ISI free samples within a V-OFDM-IM block in the time domain denoted as [y0 , y1 , · · · , yN −1 ]T . It is first blocked column-wisely into an M × L matrix y ∈ CM ×L , whose (m, l)-th entry is [y]m,l = ylM +m . Then we perform L-point fast Fourier transform (FFT) over each row of y and get a matrix Y ∈ CM ×L in the frequency domain as Y = yFL , where FL is the normalized L-point FFT matrix. The obtained matrix can be rewritten by its column vectors as Y = [Y0 , Y1 , · · · , Yl , · · · , YL−1 ], where the lth column vector, Yl , is then fed into the detector to get the bits carried by the lth subblock, i.e., Xl .

Yl as

B. MATHEMATICAL MODEL

ˆ is an M × M diagonal matrix defined as where H

In this subsection, under a quasi-static and frequencyselective fading channel, we discuss the relationship of the frequency-domain signals between the transmitter and the receiver, i.e., Xl and Yl , l = 0, 1, · · · , L − 1. First, we write the discrete-time baseband channel impulse response (CIR) as h = [h0 , h1 , · · · , hG−1 ], where hl stands for the lth channel tap, and G is the tap-number. The channel tap hl is assumed to be independently, and identically distributed (i.i.d.) with hl ∼ CN (0, 1/G). After appending N − G zeros to the CIR h, the N -point FFT is performed on it, yielding

Hn =

G−1 X

hl e−j2πl/N ,

n = 0, 1, . . . , N − 1.

(5)

l=0

As in [27], [29]–[31], and [33], assuming that the length of CP is sufficient for ISI free transmission, namely PM ≥ G, and the time/frequency synchronization is perfectly obtained at the receiver, we can write the frequency-domain subblock 20138

Yl = H l Xl + W l ,

l = 0, 1, · · · , L − 1,

(6)

l where Wl = [ω0l , ω1l , · · · , ωM −1 ] is the additive noise vector independent of the transmitted frequency-domain signal Xl . The entries of Wl are i.i.d. random variables and have the same statistical properties of the noise on the subl carriers of the original channel, i.e., ωm ∼ CN (0, N0 ), m = 1, 2, · · · , M . The equivalent channel matrix Hl can be formulated as (see [33] Eq. (4))

ˆ H l = UH l H l Ul ,

ˆ l = diag{Hl , Hl+L , · · · , Hl+(M −1)L }, H

(7)

(8)

and Ul ∈ CM ×M is a unitary matrix, whose entry in the sth row (s = 0, 1, · · · , M − 1) and mth column (m = 0, 1, . . . , M − 1) can be illustrated as   1 2π(l + sL)m . (9) [Ul ]s,m = √ exp −j N M It is easy to verify that Ul in (7) can be expressed as a more compact form Ul = FM 3l ,

(10)

where FM is the M -point FFT matrix, and 3l is a diagonal matrix defined as 3l = diag{1, γl , · · · , γlM −1 }

(11)

with γl = exp(−j2π l/N ). VOLUME 5, 2017

Y. Liu et al.: Vector OFDM With Index Modulation

C. DETECTION ALGORITHMS

In this subsection, assuming perfect channel state information at the receiver, we introduce two detection algorithms for V-OFDM-IM. The first one is a simplified ML detector for optimal demodulation. The second one is a sequential Monte Carlo (SMC) based detector which is able to achieve nearoptimal demodulation with reduced computation complexity.

CN (0, N0 ). The QL decomposition of the equivalent channel matrix in (16) can be written as  1/2 Hl H Hl = Ql Ll , (17) where Ql is a unitary matrix, and Ll is a lower tri1 ¯ angular matrix. By defining Zl = QH l Yl and substituting (16) and (17) into it, we have

1) ML DETECTOR

Zl = Ll Xl + Vl ,

Based on (6), the ML detection rule can be stated as ˜ l = arg min kYl − Hl Xl k2 . X Xl

(12)

In general, the required number of complex multiplications to compute Hl Xl is O(M 2 ). Considering the special structure of Hl presented in (7), here we seek an ML detection scheme with reduced computation complexity. Substituting (7) into (12), we have

2 ˜ l = arg min ˆ l Ul Xl X H

Yl − UH

l Xl

2

ˆ l Ul Xl = arg min Ul Yl − H (13)

. Xl

Let us define a code X0 l = Ul Xl corresponding to each possible Xl . Then the ML estimation of Xl can be written by

2 ˜ l = UH · arg min ˆ l X0 l X Y − H

U

l l l 0 Xl

=

UH l



Ul Yl − H ¯ l ◦ X0 l 2 , · arg min 0 Xl

2) NEAR-OPTIMAL DETECTOR

Clearly, the complexity of ML-based V-OFDM-IM grows exponentially with the total number of data bits conveyed in a subblock. Here, we introduce the deterministic SMC detector to achieve near-optimal performance with relatively low computational complexity [15], [37]. First, by performing channel-matched filtering and noisewhitening to the received subblock Yl in (6), we get  −1/2 1 ¯l = Y Hl H Hl Hl H Yl  1/2 ¯ l, = Hl H Hl Xl + W (16) H

−1/2

H

¯l = where W Hl Hl Hl Wl is an i.i.d. complex Gaussian vector whose element follows distribution VOLUME 5, 2017

namely,  1  1  1  l a1,1 Xl Vl Zl l   X2   V 2   Z 2   al a 2,2 2,1 l l    l       ..  +  ..  .  ..  =  . .. . . .  .   .   .   . . . XlM VlM ZlM alM ,1 alM ,2 · · · alM ,M | | {z } {z } | {z } | {z } Zl

Xl

Ll

Vl

(19) Since Ql is a unitary matrix, the complex Gaussian noise  T Vl = Vl1 , Vl2 , · · · , VlM in (18) is a vector composed by i.i.d. random variables with distribution Vli ∼ CN (0, N0 ), i = 1, 2, · · · , M . To compute P(Xl |Zl ) recursively, let us define probabilT 1  1 2 ¯ m |Z¯ m ), where X ¯m = ity distribution P(X Xl Xl · · · Xlm l l l   T 1 ¯m = Zl1 Zl2 · · · Zlm , and Xl and Zl are equal to and Z l ¯ M and Z¯ M , respectively. Using Bayes rule, we have X l l

(14)

¯ l ∈ CM ×1 is defined by where H   ¯ l = Hl , Hl+L , · · · , Hl+(M −1)L T , H (15)  ˆ l = diag H ¯ l . Since the Hadamard product V1 ◦ V2 is i.e., H the element by element matrix multiplication of V1 and V2 , ¯ l ◦ X0 l is the required number of multiplications to compute H only O(M ). Thus, for given Yl and Hl , compared to the ML detector of (12), the number of complex multiplications can be reduced from O((M 2 + M ) · 2p ) to O(2M · 2p + 2) by the equivalent ML detector in (14).

(18)

¯ m |Z¯ m ) ∝ p(Z¯ m |X ¯ m )P(X ¯ m) P(X l l l l l ¯ m )P(X ¯ m ). = p(Z¯ m−1 , Z m |X l

l

l

(20)

l

Since the entries of Vl in (19) are independent of each other, ¯ m , we have under given X l ¯ m ) = p(Z ¯ m−1 |X ¯ m )p(Z m |X ¯ m ). p(Z¯ m−1 , Zlm |X l l l l l l

(21)

Furthermore, from (19), we know that Z¯ m−1 is dependent of l ¯ m−1 but independent of X m . We then have X l l ¯ m ) = p(Z ¯ m−1 |X ¯ m−1 ). p(Z¯ m−1 |X l l l l

(22)

Based on (21) and (22), by using Bayes rule again, we can rewrite (20) as  ¯ m |Z ¯m P X l  l      ¯ m−1 )p(Z m |X ¯m P X ¯ m−1 P X m |X ¯ m−1 ∝ p Z¯ m−1 |X l

l

l

l

l

l

¯ m−1 |Z¯ m−1 )p(Z m |X ¯ m )P(X m |X ¯ m−1 ), ∝ P(X l l l l l l

l

(23)

¯ m ) and P(X m |X ¯ m−1 ) can be obtained by on-line where p(Zlm |X l l l calculation and an off-line look-up table respectively. Specif¯ m , Z m follows complex ically, as shown in (19), with known X l l P m l k Gaussian distribution with Zl ∼ CN ( m k=1 am,k Xl , N0 ). Thus, we have ¯ m) = p(Zlm |X l

1 e π N0

2 m P − N1 Zlm − alm,k Xlk 0 k=1

.

(24) 20139

Y. Liu et al.: Vector OFDM With Index Modulation

Furthermore, given the prior probabilities of all the possi¯ m−1 can be ble Xl , the probability of Xlm conditioned on X l calculated as ¯ m−1 ) = P(Xlm |X l

¯ m−1 , X m ) P(X l l . ¯ m−1 ) P(X

(25)

l

It should be noted that, on the inactive subcarriers, the transmitted signal is denoted as zero, i.e. the symbol 0 is seen as a specific point of the constellation. With (23), (24), and (25), we can apply the deterministic SMC to detect the ¯ 0) transmitted vector Xl recursively. We first initialize P(Xl1 |X l 0 0 1 ¯ ¯ and P(Xl |Zl ) as P(Xl ) and one respectively. In addition, for ¯ m, each m, 1 ≤ m ≤ M , to save the survivors of the possible X l we create a vector set Sm whose preset cardinality is denoted as |Sm |. Then, for each 0 < m ≤ M , from all of the possible transmitted symbols X¯ lm pairedh with the elements in Sm−1 , i ¯m = X ¯ m−1 ; X m with the highest we select the surviving X l l l importance weights, which are computed by ¯ m ) = $l (X ¯ m−1 )P(Z m |X ¯ m )P(Z¯ m−1 |X ¯ m−1 ). $l (X l l l l l l

III. PERFORMANCE ANALYSIS

In this section, assuming perfect channel estimation at the receiver, first, we analytically evaluate the average bit error probability of the V-OFDM-IM system under ML detection. Then, the different achievable diversity gains on different subblocks are discussed. A. AVERAGE BIT ERROR PROBABILITY

At the lth subblock time, if the IM vector Xcl is transmitted and it is erroneously detected as Xel , bit errors can be resulted over either active indices or constellation symbols. From (6), the pairwise error probability (PEP) conditioned on Hl can be expressed as [34] s  d 2 Xcl , Xel , Hl c e , (27) Pr(Xl → Xl |Hl ) = Q  2N0 where d Xcl , Xel , Hl is the Euclidean distance between two received vectors, Ycl and Yel , under the channel matrix Hl in the absence of noise, i.e., Ycl = Hl Xcl and Yel = Hl Xel . Thus, we have

2  d 2 Xcl , Xel , Hl = Hl (Xcl − Xel ) 2 . (28) 

Substituting (7) into (28) and considering Ul is a unitary matrix, we get

2 

ˆ c e d 2 Xcl , Xel , Hl = H (29) l Ul (Xl − Xl ) , 2

 2  ¯ l . d 2 Xcl , Xel , Hl = diag Ul (Xcl − Xel ) H 2

(30)

Here, we used the fact that, given two vectors V1 , V2 ∈ CM ×1 , diag(V1 )V2 and diag(V2 )V1 have the same squared Euclidean norms, i.e., kdiag(V1 )V2 k22 = kdiag(V2 )V1 k22 . Let us define a difference-vector Xdl = Xcl − Xel and a diagonal matrix Ed,l ∈ CM ×M as Ed,l = diag(Ul Xdl ).

(31)

Then, the squared Euclidean distance in (30) can be simplified as

 ¯ l 2 d 2 Xc , Xe , Hl = Ed,l H l

l

2

¯ H Td,l H ¯ l, =H l

(26)

By setting |Sm |, the number of surviving points in each step, we can make a good tradeoff between decoding complexity and performance for this algorithm. In particular, when all of the points are reserved in each step, the SMC algorithm is equivalent to the ML method.

20140

since given one complex vector V, multiplication by UH l

2 2

preserves its squared Euclidean norm, i.e., UH l V 2 = kVk2 . ˆ l is a diagonal matrix (see (8)), we can Observing that H rewrite the squared Euclidean distance as

(32)

where Td,l = Ed,l H Ed,l is a diagonal matrix, with nonnegative real numbers at the diagonal positions and zeros at the off-diagonal positions. On the other hand, the Q-function used in (27) can be approximated very well using [3], [35] Q(x) ∼ =

1 −x 2 /2 1 −2x 2 /3 e + e . 12 4

(33)

Thus, the unconditional PEP of the lth subblock can be obtained by averaging the conditional PEP with respect to the ¯ l as random channel vector H  s     2 Xc , Xe , H d l l l  . (34) Pr(Xcl → Xel ) = EHl Q    2N0 Substituting (32) and (33) into (34), we have   1 −q1 H¯ H Td,l H¯ l 1 −q2 H¯ H Td,l H¯ l l l Pr(Xcl → Xel ) ∼ e + e , = EHl 12 4 (35) where q1 = 1/(4N0 ) and q2 = 1/(3N0 ). We assume that the coherence bandwidth of the physical channel is smaller than L · 1F, where 1F is the frequency space between ¯ l (see (15)) two adjacent subcarriers. Then the elements of H can be seen as random variables independent of each other. ¯ l , as a comWe then model the equivalent channel-vector, H ¯ plex Gaussian random vector with E[Hl ] = 0M ×1 and ¯ lH ¯ H ] = IM , i.e., H ¯ l is an i.i.d. Rayleigh fading channel E[H l with normalized power gain. The probability density function ¯ l can be written as of H ¯ HH ¯l

¯ l ) = π −M e−Hl f (H

.

(36)

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Y. Liu et al.: Vector OFDM With Index Modulation

Thus the unconditional PEP can be calculated as Pr(Xcl → Xel ) Z     1 −M ∼ ¯ H IM + q1 Td,l H ¯ l dH ¯l exp −H π = l 12 ¯l H

1 + π −M 4

Z

    ¯ H IM + q2 Td,l H ¯ l dH ¯ l (37) exp −H l

¯l H

=

1/12 1/4 + . det IM + q1 Td,l det IM + q2 Td,l

(38)

As mentioned earlier, Td,l is a diagonal matrix with non-negative real numbers at the diagonal positions. Denoting the ordered non-zero diagonal elements of Td,l as λ0 > λ2 > · · · > λrank(Td,l )−1 , we can rewrite the unconditional PEP as 1/12 Pr(Xcl → Xel ) ∼ = QRd,l −1 m=0 (1 + q1 λm ) 1/4 + QR −1 , (39) d,l + q λ (1 ) 2 m m=0 where Rd,l = rank(Td,l ). Based on the evaluation of the PEP given in (39), the average bit error probability of the lth subblock can be evaluated by 1 X X ˆ l )e(Xl , X ˆ l ), Pr(Xl → X (40) Plb ≈ pnXl Xl X ˆ l 6=Xl

where nXl = 2p represents the number of possible realizaˆ l ) is the number of error bits when the tions of Xl and e(Xl , X ˆ l . Finally, the average transmitted vector Xl is detected as X bit error probability of the V-OFDM-IM can be approximated P l as Pb = L−1 l=0 Pb /L. B. DIVERSITY GAIN

From (39), at high SNR values (qi  1 for i = 1, 2), the unconditional PEP can be further approximated as  YRd,l −1 −1 Pr(Xcl → Xel ) ∼ λm = 12q1 Rd,l m=0

 + 4q2

Rd,l

YRd,l −1 m=0

λm

−1

.

(41)

According to this result, Rd.l characterizes the diversity order. Thus the diversity gain for each subblock l ∈ {0, 1, · · · , L − 1} in the system is Dl =

min

∀Xdl 6=0,Xdl ∈9

rank(Td,l ),

(42)

where 9 is a limited set formed by all the possible differencevectors Xdl . Finally, the diversity gain of the system, D, is determined by the minimum Dl across all sub-blocks of the system, i.e., D = min Dl . It can be inferred from (31) and (32) l that the different unitary matrices, Ul , result in different diversity gains of the subblocks. VOLUME 5, 2017

FIGURE 2. BER versus SNR, performance comparison between the conventional OFDM, OFDM-IM and the proposed V-OFDM-IM. The raw data rate is set to be 1 bit per channel use in each system.

IV. NUMERICAL RESULTS AND DISCUSSIONS

In this section, we validate our mathematical analysis using Monte Carlo simulation over frequency-selective fading channels with additive white complex Gaussian noise. Under the same spectral efficiency, the BER performance of the proposed V-OFDM-IM is compared with that of the conventional OFDM and OFDM-IM. In addition, the simulation channel is assumed to be unchanged within each OFDM/OFDM-IM/ V-OFDM-IM block, but vary independently among different blocks. For the sake of simplicity, we assume that, on the subchannels of the physical channel, the frequency-domain gain follows independent complex Gaussian distribution with zero mean and normalized variance. 107 blocks are used on each simulation run. Fig. 2 compares the BERs of the conventional OFDM, OFDM-IM, and the proposed V-OFDM-IM when the ML detection is employed. For a fair comparison, all systems are designed to have the same data rate (1 bit per channel use). Specifically, in OFDM, independent binary phase shift keying (BPSK) symbols are transmitted over the subchannels. In OFDM-IM and V-OFDM-IM, we let the subblock length M = 4, the number of active subcarriers K = 2. BPSK modulation is adopted for the active subcarriers. Thus, we have 2 IM-bits and 2 symbol-bits on each subblock. The number of subcarriers is set to be 128. The dash line in Fig. 2 denotes the results obtained by Monte Carlo simulations, while the solid line represents the theoretical upper bound on the BER. As can be seen from the figure, the theoretical upper bound is asymptotically tight at high SNR. For V-OFDM-IM, the performance under the SMC decoder is also provided by Monte Carlo simulations, where the maximum number of survivors in each step is set as 9. As shown in the figure, the proposed V-OFDM-IM significantly outperforms the conventional OFDM and OFDM-IM. The performance of ML detection is a bit better than that of SMC. 20141

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FIGURE 3. BER versus SNR, performance comparison between different sub-blocks of the proposed V-OFDM-IM. The raw data rate is set to be 1 bit per channel use.

Fig. 3 shows the BER performance comparison between different sub-blocks of the proposed V-OFDM-IM. Considering the derived theoretical upper bound is very tight at high SNR, here we just use the theoretical bound to make performance comparisons. The system parameters of each modulation scheme are set to be the same as those in Fig. 2. The solid lines with red color demonstrate the BER performance of different systems, while the dash lines represent the BERs of different sub-blocks in the V-OFDM-IM. As shown in the figure, the BER performance of different subblocks can be very different from each other. Specifically, compared to conventional OFDM and OFDM-IM, only coding gains are obtained on some subblocks, such as the 0th subblock, while both coding gains and diversity gains are obtained on the other subblocks. As analyzed earlier, the diversity gain Dl (see (42)) is highly dependent on the subblock index l. It can be also inferred from (7) that different unitary matrices, Ul , may result in very different equivalent channel matrices, over which the BER performance can be changing significantly with the subblock indices. In Fig. 4, we compare the PAPR performance of the proposed V-OFDM-IM with that of conventional OFDM-IM and OFDM. The PAPR of the transmit signal is computed from its discrete-time equivalent baseband signal xn as PAPR = Pmax /Pavg , where Pmax is max |xn |2 , and 0≤n≤N −1 P −1 2 |x | Pavg is N /N . The number of subcarriers N is set n n=0 to be 1024, and QPSK modulation is used for each system. Pr(PAPR > PAPR0 ) denotes the probability that the PAPR of a data block exceeds the given threshold PAPR0 . For OFDM-IM and V-OFDM-IM, the subblock length and the number of active subcarriers in each subblock are set to be M = 4, and 2, respectively. As can be seen from the figure, the PAPR performance of the conventional OFDM-IM is a bit better than that of conventional OFDM. This is because, 20142

FIGURE 4. PAPR performance comparison between the proposed V-OFDM-IM and the conventional OFDM-IM/OFDM, where the number of sub-carriers is set to be 1024, and QPSK modulation is adopted.

in OFDM-IM, the number of non-zero symbols in the frequency domain is half of that of OFDM. In addition, as shown in the figure, the proposed V-OFDM-IM significantly outperforms the other two schemes. This is because the IFFT/FFT points used in V-OFDM-IM are just a quarter of the size of that used in the other schemes, and only half of subcarriers are activated in each subblock. It is straightforward to show that one can further reduce the PAPR of the V-OFDM-IM by increasing the subblock length. V. CONCLUSIONS

In this paper, we have introduced the V-OFDM concept in the context of OFDM-IM to improve the BER performance and reduce the PAPR of the transmit signal. The proposed system achieves additional diversity gains on most of its subblocks. We have provided a simplified ML detector to achieve optimal demodulation with reduced complexity. As a tradeoff between complexity and performance, the SMC based detector has been introduced to achieve near-optimal demodulation. In the SMC algorithm, the metrics of the possible transmit vectors are computed recursively, and the complexity/performance of the demodulator can be changed by setting the number of surviving points at each step. Theoretical analysis and computer simulations have both shown that the proposed V-OFDM-IM outperforms the conventional OFDM-IM. Due to the much smaller size of the IFFT/FFT operations used in the transceiver, V-OFDM-IM has significantly better PAPR performance than that of conventional OFDM and OFDM-IM. REFERENCES [1] E. Basar, M. Wen, R. Mesleh, M. Di Renzo, Y. Xiao, and H. Haas, ‘‘Index modulation techniques for next-generation wireless networks,’’ IEEE Access, vol. 5, pp. 16693–16746, 2017. [2] M. Wen, X. Cheng, and L. Yang, Index Modulation for 5G Wireless Communications. Berlin, Germany: Springer, 2017. VOLUME 5, 2017

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YUN LIU received the B.S. degree in electronic and information engineering from Nanchang University, Nanchang, China, and the M.S. degree in radio physics from Sun Yat-sen University, Guangzhou, China, in 2004 and 2006, respectively. He is currently pursuing the Ph.D. degree with the South China University of Technology, Guangzhou. He is currently a Lecturer with the School of Information Science and Technology, Zhongkai University of Agriculture and Engineering, Guangzhou, China. His recent research interests include underwater acoustic communications and index modulation.

FEI JI (M’06) received the B.S. degree in applied electronic technologies from Northwestern Polytechnical University, Xi’an, China, and the M.S. degree in bioelectronics and the Ph.D. degrees in circuits and systems from the South China University of Technology, Guangzhou, China, in 1992, 1995, and 1998, respectively. She was a Visiting Scholar with the University of Waterloo, Canada, from 2009 to 2010. She was with the City University of Hong Kong as a Research Assistant from 2001 to 2002 and a Senior Research Associate from 2005 to 2005. She is currently a Professor with the School of Electronic and Information Engineering, South China University of Technology. Her research focuses on wireless communication systems and networking. She was the Registration Chair and the Technical Program Committee Member of the IEEE 2008 International Conference on Communication System. 20143

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MIAOWEN WEN (M’14) received the B.S. degree from Beijing Jiaotong University, Beijing, China, in 2009, and the Ph.D. degree from Peking University, Beijing, China, in 2014. From 2012 to 2013, he was a Visiting Student Research Collaborator with Princeton University, Princeton, NJ, USA. He is currently an Associate Professor with the South China University of Technology, Guangzhou, China. He has authored a book and over 70 papers in refereed journals and conference proceedings. His research interests include index modulation and nonorthogonal multiple access techniques. Dr. Wen received the Best Paper Award from the IEEE International Conference on Intelligent Transportation Systems Telecommunications in 2012, the IEEE International Conference on Intelligent Transportation Systems in 2014, and the IEEE International Conference on Computing, Networking and Communications in 2016. He received the Excellent Doctoral Dissertation Award from Peking University. He currently serves as an Associate Editor of the IEEE ACCESS, and on the Editorial Board of the EURASIP Journal on Wireless Communications and Networking, and the ETRI Journal.

BEIXIONG ZHENG received the B.S. degree from the South China University of Technology, Guangzhou, China, in 2013, where he is currently pursuing the Ph.D. degree. From 2015 to 2016, he was a Visiting Student Research Collaborator with Columbia University, New York, NY, USA. His recent research interests include spatial modulation, non-orthogonal multiple access, and pilot multiplexing techniques. He received the Best Paper Award from the IEEE International Conference on Computing, Networking and Communications 2016.

DEHUAN WAN received the B.S. degree from He’nan Normal University, Xinxiang, China, in 2007. He is currently pursuing the Ph.D. degree with the South China University of Technology, Guangzhou. His recent research interests include non-orthogonal multiple access and index modulation.

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