Zero-Padded Orthogonal Frequency Division Multiplexing with Index Modulation Using Multiple Constellation Alphabets

Orthogonal frequency-division multiplexing (OFDM) with index modulation has been emerged as a promising technique for next-generation networks, where the specific activation pattern of the OFDM subcarriers is capable of conveying additional information implicitly without additional energy consumption, hence improving the system energy efficiency. In this paper, zero-padded tri-mode OFDM with index modulation (ZTM-OFDM-IM) is proposed, which is capable of achieving high spectral and energy efficiency. In ZTM-OFDM-IM, subcarriers are partitioned into subblocks. In each OFDM subblock, only a fraction of subcarriers are modulated by two distinguishable constellation alphabets, while the others remain empty, which reduces the energy consumption. With such an arrangement, extra information bits can be carried by the subcarrier activation pattern. At the receiver, a maximum-likelihood (ML) detector as well as a reduced-complexity two-stage log-likelihood ratio (LLR) detector are developed for signal demodulation. Theoretical performance analysis based on Euclidean distance metric and pairwise error probability are conducted. Besides, in order to attain better BER performance, the design strategies for the two distinguishable constellation alphabets are investigated, which is followed by brief discussions on generalization methods for the proposed ZTM-OFDM-IM. It is demonstrated via Monte Carlo simulations that ZTM-OFDM-IM is capable of enhancing the BER performance compared with the conventional OFDM and other conventional index-modulated OFDM benchmarks, which provides high-rate data transmission with low energy consumption. Moreover, the LLR detection of ZTM-OFDM-IM only suffers from slight performance loss in comparison with the ML detection and achieves significant complexity reduction.

SECTION I. Introduction

Orthogonal frequency division multiplexing (OFDM) has emerged as a promising modulation scheme in wireless communications, due to its distinctive advantages such as high-rate transmission, simple one-tap equalization and robustness to inter-symbol interference (ISI) caused by a dispersive channel [1], [2]. These merits motivate the application of OFDM to many broadband wireless standards, such as the third-generation partnership project (3GPP), long-term evolution (LTE), the HIPERLAN/2 standard, the wireless fidelity (Wi-Fi), and the worldwide interoperability for microwave access (WiMAX) [1], [3]–​[7].

In 5G networks, the explosive growth in the mobile data services and the popularization of smart devices have necessitated the requirement for high spectral and energy efficiency. Following this trend, the novel index modulation technique has been introduced to OFDM for high rate transmission and low energy consumption. More specifically, for classical index modulation schemes, only part of the subcarriers are activated for modulation, loaded with amplitude/phase modulation (APM) schemes, e.g., quadrature amplitude modulation (QAM) or phase shift keying (PSK). Aside from the transmitted symbol bits, additional information bits (termed as index bits) can be conveyed by the indices of the activated subcarriers in an implicit manner, leading to possible throughput gain. Moreover, since no extra energy consumption is required to transmit these index bits, index modulation is capable of enhancing the energy efficiency of OFDM [8]. In [9], the subcarrier-index modulated OFDM (SIM-OFDM) was proposed by modulating only a fraction of subcarriers, whose activated statuses were determined by an on-off keying (OOK) data stream. Despite the fact that SIM-OFDM can achieve energy efficiency gain, it suffers from unstable data rate and bit error propagation. To address this issue, the authors proposed an enhanced version of SIM-OFDM (ESIM-OFDM) by splitting the subcarriers into pairs [10]. Within each pair, only a single subcarrier was activated, whose index carried one binary bit. However, the frame structure of ESIM-OFDM is not flexible enough so that the diversity gain brought by index modulation is limited. In order to improve its flexibility, Basar et al. [11] proposed OFDM with index modulation (OFDM-IM), where subcarriers were partitioned into subblocks. In each subblock, only a fraction of subcarriers were modulated, carrying additional information bits through their indices. In OFDM-IM, the number of activated subcarriers for each subblock, the OFDM subblock size and the used constellations are very flexible, and can be adjusted for better BER performance, which have been discussed in several literature based on theoretical analysis [12]–​[14]. However, in spite of its high energy efficiency, the frequency resources are wasted in OFDM-IM, since only part of the subcarriers are employed for data transmission, whilst the others remain empty. This may lead to data rate loss compared to the conventional OFDM. To overcome the throughput limit, several improved OFDM-IM schemes have been proposed [15]–​[18]. In [15], two generalized OFDM-IM schemes were developed, named as GIM-OFDM1 and GIM-OFDM2. Explicitly, in OFDM-GIM1, the number of activated subcarriers for each OFDM subblock is variable, leading to more possible realizations of OFDM subblocks. Besides, in OFDM-GIM2, index modulation is performed on both the in-phase and quadrature (I/Q) components of the subcarriers independently, harvesting twice the index bits. To attain even higher spectral efficiency, GIM-OFDM2 was further enhanced by performing index modulation on I/Q components jointly [16], [17]. Moreover, various number of activated subcarriers and constellation alphabets were employed for different OFDM subblocks in [18], which achieved enhanced data rate. In [19] and [20], the subcarrier-level interleaving techniques were applied to OFDM-IM in order to eliminate the channel correlation, attaining extra diversity gain. Due to the aforementioned advantages of OFDM-IM, it has been applied to different fields. For instance, OFDM-IM was incorporated into the multiple-input multiple-output (MIMO) systems in [21] and [22], which achieved performance gain over other classical MIMO systems. OFDM-IM has also been adopted in acoustic communications [23], visible light communications [24], [25], as well as vehicle-to-vehicle and vehicle-to-infrastructure applications [26], all presenting superiority over its conventional counterparts. Despite the distinctive merits of OFDM-IM at low spectral efficiency, it was indicated in [27] that the performance gain of OFDM-IM over conventional OFDM became negligible when high-order constellations were employed, for the reason that, the number of index bits was proportional to the logarithm of the subblock size, which was less dominant when the APM constellation order gets larger. Moreover, at high data rate, the OFDM-IM may even suffer from signal-to-noise ratio (SNR) degradation in comparison with conventional OFDM. To break this performance limit, dual-mode index modulation aided OFDM (DM-OFDM) [28] has been proposed, where all the subcarriers are modulated by two distinguishable constellation alphabets, and the subcarrier indices also carry additional information. Since the spectrum is fully utilized, whilst the index modulation property is maintained, DM-OFDM is capable of attaining considerable performance gain over OFDM-IM and conventional OFDM. As a continuous work, Mao et al. [29] developed generalized DM-OFDM, where the number of subcarriers modulated by either constellation alphabet was alterable for each OFDM subblock, leading to additional diversity gain.

From the aforementioned descriptions, DM-OFDM is capable of enhancing the data rate of OFDM-IM but with the sacrifice of inevitable energy efficiency loss. Therefore, it can be inferred that, there is an interesting trade-off between the spectral and energy efficiency, which in other words, means that merely activating a fraction of subcarriers for modulation with two differentiable constellation alphabets may contribute to better BER performance compared with pure OFDM-IM and DM-OFDM. Following this philosophy, in this paper, zero-padded tri-mode OFDM with index modulation (ZTM-OFDM-IM) is proposed, where subcarriers are split into subblocks, and in each OFDM subblock, only part of the subcarriers are modulated by two different constellation alphabets, whilst the others remain empty. With such arrangement, information bits can be not only transmitted by the APM symbols, but also conveyed by the subcarrier activation pattern. At the receiver, both an optimal maximum-likelihood (ML) detector and a two-stage log-likelihood ratio (LLR) detector with reduced complexity are used for demodulation. Theoretical analysis based on the minimum Euclidean distance and pairwise error probability (PEP) approximation is conducted. Besides, constellation design strategy for ZTM-OFDM-IM is described, which is followed by brief discussions on several generalizations of ZTM-OFDM-IM. Simulation results demonstrate that, the proposed ZTM-OFDM-IM can attain performance gain over OFDM, OFDM-IM and DM-OFDM under both the frequency-selective Rayleigh fading channel and the additive white Gaussian noise (AWGN) channel. Moreover, the two-stage LLR detector only suffers from slight SNR degradation compared to the optimal ML detection, and is capable of reducing the computational complexity significantly.

The remainder of the paper is organized as follows. Section II gives a brief illustration of the transceiver model of OFDM-IM and DM-OFDM schemes. Afterward, Section III introduces the principle of the proposed ZTM-OFDM-IM, which is followed by the corresponding performance analysis based on minimum Euclidean distance and PEP approximation in Section IV. After that, the constellation design strategy and several generalization algorithms for ZTM-OFDM-IM are discussed in Section V. In section VI, the BER performance evaluation via Monte Carlo simulations is conducted, and Section VII draws the conclusion.

SECTION II. System Model of Typical Index Modulated OFDM Schemes

In this section, the transceivers of two typical index modulated OFDM schemes, namely, OFDM-IM and DM-OFDM are illustrated.

Consider a general structure of OFDM-IM and DM-OFDM. At the transmitter, $N$ symbols $\mathbf {X}=\left [{X_{1},X_{2},\cdots ,X_{N} }\right ]$ modulated on the $N$ subcarriers are partitioned into $g$ OFDM subblocks of length $l$ , which is given by $$\begin{equation} \mathbf {X}^{(p)}=\left [{ X_{(p-1)l+1},X_{(p-1)l+2},\cdots ,X_{pl} }\right ] \quad 1\leq p\leq g. \end{equation}$$ In each subblock, subcarriers are modulated by two differentiable constellation alphabets $\mathcal {M}_{A}$ and $\mathcal {M}_{B}$ ($\mathcal {M}_{A}\wedge \mathcal {M}_{B}=\varnothing$ ) with sizes of $M_{A}$ and $M_{B}$ , corresponding to two subcarrier index sets $\mathcal {I}_{A}$ and $\mathcal {I}_{B}$ , respectively. For OFDM-IM, $\mathcal {M}_{A}$ or $\mathcal {M}_{B}$ equals $\{0\}$ , whilst for DM-OFDM, $\mathcal {M}_{A}$ and $\mathcal {M}_{B}$ are two arbitrary distinguishable constellation alphabets. From this perspective, OFDM-IM can be seen as a special case of DM-OFDM, where either of the constellation alphabet is $\{0\}$ . Therefore, only the DM-OFDM architecture is introduced here for brevity. Aside from the information bits carried by the data symbols, additional index bits can be conveyed by $\mathcal {I}_{A}$ or $\mathcal {I}_{B}$ . Therefore, by assuming that $k$ subcarriers are modulated by $\mathcal {M}_{A}$ in each OFDM subblock, the number of transmitted bits per subblock for DM-OFDM is given by $$\begin{equation} n_{\text {DM}}=k\log _{2}M_{A}+(l-k)\log _{2}M_{B}+\left \lfloor{ \log _{2} \binom {l}{k} }\right \rfloor , \end{equation}$$ which is also applicable for OFDM-IM by setting $\mathcal {M}_{A}$ or $\mathcal {M}_{B}$ as {0}. After mapping, an $N$ -point inverse fast fourier transform (IFFT) is performed on $\mathbf {X}$ to generate the time-domain signals $\mathbf {x}=\left [{x_{1},x_{2},\cdots ,x_{N} }\right ]$ . To combat the ISI effects, the cyclic prefix (CP) is imposed, and the resultant signals are transmitted by antennas after passing through a parallel-to-serial converter (P/S) and a digital-to-analog converter (D/A). In this paper, the AWGN channel and the frequency-selective Rayleigh fading channel are both considered for data transmission. Explicitly, for the Rayleigh fading channel, the $L$ elements of the time-domain channel impulse response, denoted as $\mathbf {h}_{t}=\left [{ h_{1},h_{2},\cdots ,h_{L} }\right ]$ , can be modeled as circularly symmetric complex Gaussian random variables following $\mathcal {CN}\left ({ 0,1/L }\right )$ , where $L$ is assumed to be no larger than the CP length $L_{\text {CP}}$ .

From the principles of DM-OFDM and OFDM-IM, it can be seen that, the energy-efficient OFDM-IM inevitably causes a waste of frequency resources due to the intentionally unused subcarriers, whilst the high-rate classical DM-OFDM (DM-OFDM with two non-zero constellation alphabets) suffers from considerable energy consumption. Since the system capacity is determined by both the spectral efficiency and the average bit energy from the Shannon’s formula, there is a trade-off between DM-OFDM and OFDM-IM to approach the capacity limit, in other words, to achieve better BER performance. This yields the philosophy of ZTM-OFDM-IM, which will be discussed below.

SECTION III. Principle of ZTM-OFDM-IM

A. Transmitter Model for ZTM-OFDM-IM

The transmitter of ZTM-OFDM-IM is illustrated in Fig. 1, where the number of subcarriers equals $N$ . $m$ information bits are divided into $g$ parallel data streams of $n_{\text {ZTM}}$ bits, transmitted by $g$ OFDM subblocks of length $l=N/g$ , respectively. Each parallel data stream is processed by one index selector and two constellation mappers Mapper A and Mapper B with two distinguishable alphabets $\mathcal {M}_{A}$ and $\mathcal {M}_{B}$ . More specifically, the $n_{\text {ZTM}}$ bits are split into 2 sub-streams of $n_{1}$ (symbol bits) and $n_{2}$ (index bits) bits. Afterward, the $n_{1}$ symbol bits are imposed into Mapper A and Mapper B, generating the transmitted data symbols. Meanwhile, $n_{2}$ index bits are fed into the index selector to determine the subcarrier activation pattern for the corresponding OFDM subblock, in other words, the indices of subcarriers modulated by $\mathcal {M}_{A}$ , subcarriers modulated by $\mathcal {M}_{B}$ , and the empty subcarriers. According to the obtained subcarrier activation pattern, the data symbols are modulated onto the corresponding activated subcarriers in each OFDM subblock. For different subblocks of the proposed ZTM-OFDM-IM, the numbers of subcarriers modulated by $\mathcal {M}_{A}$ and $\mathcal {M}_{B}$ are assumed to be the same for brevity. Therefore, by assuming that $k_{1}$ and $k_{2}$ subcarriers ($k=k_{1}+k_{2}$ ) are modulated by $\mathcal {M}_{A}$ and $\mathcal {M}_{B}$ respectively for each subblock, whilst the other $(l-k)$ subcarriers remain empty, $n_{1}$ and $n_{2}$ can be calculated as $$\begin{equation} n_{1}=k_{1}\log _{2}M_{A}+k_{2}\log _{2}M_{B}, \end{equation}$$ and $$\begin{equation} n_{2}=\left \lfloor{ \log _{2} \left ({\binom {l}{k}\times \binom {k}{k_{1}} }\right )}\right \rfloor . \end{equation}$$ Therefore, $n_{\text {ZTM}}$ is calculated as $$\begin{align} n_{\text {ZTM}}=&k_{1}\log _{2}M_{A}+k_{2}\log _{2}M_{B}\notag \\&\quad \qquad \qquad +\left \lfloor{ \log _{2} \left ({\binom {l}{k}\times \binom {k}{k_{1}} }\right )}\right \rfloor .\qquad \end{align}$$ The principle of ZTM-OFDM-IM can be exemplified by Table 1, where $l=4$ , $k_{1}=k_{2}=1$ , $k=2$ , and $S_{A}$ and $S_{B}$ denote two arbitrary elements of $\mathcal {M}_{A}$ and $\mathcal {M}_{B}$ . In the second column, the indices of activated subcarriers are presented, whose first and second elements denote the subcarrier indices corresponding to $S_{A}$ and $S_{B}$ respectively. It can be seen that, for each OFDM subblock, the indices of subcarriers modulated by $\mathcal {M}_{A}$ or $\mathcal {M}_{B}$ depend on the $n_{2}=3$ index bits.

TABLE 1 A Look-up Table of ZTM-OFDM-IM With $(k_{1},k_{2},k,l)=(1,1,2,4)$

FIGURE 1.

ZTM-OFDM-IM transmitter diagram.

Assume that $\mathcal {M}_{A}$ and $\mathcal {M}_{B}$ are employed in ZTM-OFDM-IM and DM-OFDM, and $M$ -ary QAM ($M$ -QAM) is utilized in OFDM-IM and OFDM. Besides, for DM-OFDM, $k$ subcarriers are modulated by $\mathcal {M}_{A}$ in each subblock, whilst the other $(l-k)$ subcarriers are modulated by $\mathcal {M}_{B}$ . For OFDM-IM, only $k$ subcarriers are employed in each subblock. With such arrangement, the spectral efficiencies of the proposed ZTM-OFDM-IM, DM-OFDM, OFDM-IM, and conventional OFDM can be derived as $$\begin{align} \eta _{\text {ZTM}}=&\frac {N\left ({\log _{2}M_{A}^{k_{1}}+\log _{2}M_{B}^{k_{2}}+\left \lfloor{ \log _{2} \left ({\binom {l}{k}\times \binom {k}{k_{1}} }\right )}\right \rfloor }\right )}{(N+L_{\text {CP}})l},\notag \\ \\ \eta _{\text {DM}}=&\frac {N\left ({\log _{2}M_{A}^{k}+\log _{2}M_{B}^{l-k}+\left \lfloor{ \log _{2}\binom {l}{k}}\right \rfloor }\right )}{(N+L_{\text {CP}})l}, \\ \eta _{\text {IM}}=&\frac {N\left ({\log _{2}M^{k}+\left \lfloor{ \log _{2} \binom {l}{k} }\right \rfloor }\right )}{(N+L_{\text {CP}})l}, \end{align}$$ and $$\begin{equation} \eta _{\text {OFDM}}=\frac {N\log _{2}M}{N+L_{\text {CP}}}. \end{equation}$$ ZTM-OFDM-IM can be seen as a trade-off between the spectrum-efficient DM-OFDM and the energy-efficient OFDM-IM. It is seen that the proposed ZTM-OFDM-IM is capable of enhancing the energy efficiency compared to DM-OFDM, since the former employs only a fraction of subcarriers and conveys more index bits than the latter. Moreover, ZTM-OFDM-IM can achieve spectral efficiency gain over OFDM-IM and conventional OFDM. For instance, $(k_{1},k_{2},k,l)$ is set to (2, 1, 3, 4) for ZTM-OFDM-IM, and $N$ and $L_{\text {CP}}$ are assumed as 128 and 16. When quadrature PSK (QPSK) is employed for modulation, i.e., $M_{A}=M_{B}=M=4$ , $\eta _{\text {ZTM}}$ , $\eta _{\text {IM}}$ and $\eta _{\text {OFDM}}$ are calculated to be 2 bit/s/Hz, 1.78 bit/s/Hz, and 1.78 bit/s/Hz, where 0.22 bit/s/Hz spectral efficiency gain is attained by the proposed ZTM-OFDM-IM over OFDM-IM and conventional OFDM.

B. ML Detection for ZTM-OFDM-IM

After data transmission through the frequency-selective Rayleigh fading channel (as illustrated in Section II), an $N$ -point fast Fourier transform (FFT) operation is performed on the detected signals at the receiver, generating $\mathbf {R}=\left [{ R_{1},R_{2},\cdots ,R_{N} }\right ]$ with $g$ OFDM subblocks after the removal of CP. By denoting the $p$ th received subblock as $\mathbf {R}^{(p)}=\left [{ R _{(p-1)l+1},R _{(p-1)l+2},\cdots ,R _{pl}}\right ]$ , it can be formulated as $$\begin{equation} \mathbf {R}^{(p)}=\text {diag}\{\mathbf {X}^{(p)}\}\mathbf {H}_{p}+\mathbf {Z}_{p},\quad p=1,2,\cdots ,g. \end{equation}$$ In (10), $\text {diag}\{\mathbf {X}^{(p)}\}$ is a diagonal matrix whose prime-diagonal elements are $\mathbf {X}^{(p)}$ , and $\mathbf {Z}^{(p)}=\left [{ Z _{(p-1)l+1},Z _{(p-1)l+2},\cdots ,Z _{pl}}\right ]$ denotes the AWGN components with its elements following $\mathcal {CN}(0,N_{0})$ , where $N_{0}$ stands for the average noise energy in frequency domain. $\mathbf {H}_{p}=\left [{ H _{(p-1)l+1},H _{(p-1)l+2},\cdots ,H _{pl} }\right ]$ represents the frequency-domain channel coefficients. To obtain $H_{i}$ for $i=1,2,\cdots ,N$ , an FFT operation is performed on the $N$ -dimensional vector $\mathbf {h}_{0}=[h_{1},h_{2},\cdots ,h_{L},0,\cdots ,0]^{T}$ generated by appending zeros to $\mathbf {h}_{t}$ , which can be formulated as $$\begin{equation} \mathbf {H}=\left [{ H_{1},H_{2},\cdots ,H_{N} }\right ]^{T}=\frac {1}{\sqrt {N}}\text {FFT}(\mathbf {h}_{0}). \end{equation}$$ For signal demodulation, the channel state information (CSI) has to be acquired using pilots or training sequences. For simplicity, perfect channel estimation is assumed at the receiver [11].

For signal demodulation, an ML detector is utilized by minimizing the Euclidean distance between the estimation and the received subblock, where all the possible realizations of the OFDM subblocks are considered for detection, and the estimation of the $p$ th subblock can be determined by $$\begin{equation} \hat {\mathbf {X}}^{(p)}=\underset {\hat {\mathbf {X}}^{(p)} \in \mathcal {X}}{\arg }\min \sum _{i=1}^{l}\left |{ R_{(p-1)l+i}-H_{(p-1)l+i}\hat {X}^{(p)}_{i} }\right |^{2},\quad \end{equation}$$ where $\mathcal {X}=\{\tilde {\mathbf {X}}^{(1)},\tilde {\mathbf {X}}^{(2)},\cdots ,\tilde {\mathbf {X}}^{(2^{n_{\text {ZTM}}})}\}$ consists of all the possible realizations of OFDM subblocks, and $\hat {X}^{(p)}_{i}$ represents the $i$ th element of $\hat {\mathbf {X}}^{(p)}$ . After detection, information bits can be demodulated subblock by subblock with a simply look-up table that maps the subblock realization to its corresponding binary bits. According to (12), the computational complexity of the ML detector in terms of complex multiplications yields $\mathcal {O}(2^{n_{2}}M_{A}^{k_{1}}M_{B}^{k-k_{1}})$ , which implies that the ML detector is undisirable for practical implementation, since its complexity increases exponentially with $n_{2}$ , $M_{A}$ , $M_{B}$ and $k$ .

C. Reduced-Complexity Two-Stage LLR Detection for ZTM-OFDM-IM

To address the high-complexity issue, a two-stage LLR detector is proposed for ZTM-OFDM-IM. However, unlike the LLR detector employed in DM-OFDM [28], for each subblock, the index pattern (the subcarrier activation pattern) cannot be directly obtained by one-tap LLR comparison, since there are equivalently three differentiable alphabets used for modulation, i.e., $\mathcal {M}_{A}$ , $\mathcal {M}_{B}$ , and {0}. Hence, a two-stage LLR detection algorithm is proposed, where the first-stage detection distinguishes modulated subcarriers from the empty ones in each OFDM subblock, whilst the second-stage detection obtains the subcarrier indices corresponding to $\mathcal {M}_{A}$ and $\mathcal {M}_{B}$ by LLR comparison. More specifically, at the first stage, for each subcarrier, the logarithm of the ratio between the a posteriori probabilities of the event whether the subcarrier is modulated or empty is calculated, which is formulated as $$\begin{equation} \gamma _\alpha =\ln \left ({ \frac {\sum _{i=1}^{M_{A}+M_{B}}P\left ({ X_\alpha =S_{C,i}|R_\alpha }\right )}{P\left ({ X_\alpha =0|R_\alpha }\right )}}\right ), \end{equation}$$ where $\alpha =1,2,\cdots ,N$ , and $S_{C,i}$ is the $i$ th element of $\mathcal {M}_{C}=\mathcal {M}_{A}\cup \mathcal {M}_{B}$ . It is implied that the subcarrier is more likely to be modulated if the LLR is positive, whilst it is more probably empty if the LLR is negative. Therefore, the index sets of the modulated subcarriers and the empty subcarriers $\mathcal {I}_{C}$ and $\mathcal {I}_{0}$ can be obtained by the LLR comparison subcarrier by subcarrier, illustrated as $$\begin{equation} \alpha \in \begin{cases} \mathcal {I}_{C}, &\quad \gamma _{\alpha }\geq 0; \\ \mathcal {I}_{0}, & \quad \gamma _{\alpha }<0. \end{cases} \end{equation}$$ Afterwards, $\mathcal {I}_{C}$ and $\mathcal {I}_{0}$ are input into the second stage of the detector, where the subcarriers in $\mathcal {I}_{C}$ modulated by $\mathcal {M}_{A}$ and $\mathcal {M}_{B}$ are distinguished by similar LLR calculation as the 1st stage, which is given by $$\begin{equation} \gamma ^{*}_\beta =\ln \left ({ \frac {\sum _{i=1}^{M_{A}}P\left ({ X_\beta =S_{A,i}|R_\beta }\right )}{\sum _{j=1}^{M_{B}}P\left ({ X_\beta =S_{B,j}|R_\beta }\right )}}\right ), \end{equation}$$ where $\beta \in \mathcal {I}_{C}$ , and $S_{A,i}$ and $S_{B,j}$ are the $i$ th and $j$ th elements of $\mathcal {M}_{A}$ and $\mathcal {M}_{B}$ , respectively. Then the subcarrier index subsets corresponding to $\mathcal {M}_{A}$ and $\mathcal {M}_{B}$ , denoted as $\mathcal {I}^{*}_{A}$ and $\mathcal {I}^{*}_{B}$ , can be obtained in a straightforward way similar to (14). According to the estimated $\{\mathcal {I}^{*}_{A},\mathcal {I}^{*}_{B},\mathcal {I}_{0}\}$ , which consists of the detected index pattern for each OFDM subblock, the index bits can be demodulated by employing a look-up table which maps the index pattern to the binary bits. Meanwhile, two demappers of $\mathcal {M}_{A}$ and $\mathcal {M}_{B}$ are utilized for demodulating the subcarriers in $\mathcal {I}^{*}_{A}$ and $\mathcal {I}^{*}_{B}$ , respectively.

Remark 1:

When the subcarrier activation pattern for each OFDM subblock is determined at the transmitter, meaning that $k_{1}$ and $k_{2}$ are fixed, the number of positive LLR values for each OFDM subblock at the first stage should be equal to $k$ , whilst the negative ones amount to $(l-k)$ . For the second stage, the positive and negative LLR values should be $k_{1}$ and $k_{2}$ . For both stages, the index pattern under the aforementioned constraints is termed as legal index pattern. However, under highly noisy condition, there may exist illegal index pattern, where the positive LLR values calculated by (13) and (15) are no longer equal to $k$ and $k_{1}$ respectively. Under such scenario, we take the first-stage detection for instance to describe one feasible decision method. Explicitly, in each OFDM subblock, the $k$ subcarriers with the largest $k$ LLR values are considered to be occupied, whilst the others are assumed to be empty. This method helps to promote the robustness of LLR detection.

According to the Bayes’ formula, (13) and (15) can be further derived as $$\begin{align} \gamma _\alpha=&\ln \left ({ \frac {k}{(M_{A}+M_{B})(l-k)} }\right )+\frac {R_\alpha ^{2}}{N_{0}}\notag \\&\qquad +\,\ln \left ({ \sum _{i=1}^{M_{A}+M_{B}}\exp \left({-\frac {\left |{ R_\alpha -H_\alpha S_{C,i} }\right |^{2}}{N_{0}}}\right) }\right ),\qquad \end{align}$$ and $$\begin{align} \gamma ^{*}_\beta=&\ln \left ({ \frac {M_{B}k_{1}}{N_{A}k_{2}} }\right )+\ln \left ({ \sum _{i=1}^{M_{A}}\exp \left({-\frac {\left |{ R_\beta -H_\beta S_{A,i} }\right |^{2}}{N_{0}}}\right) }\right )\notag \\&\qquad \quad -\,\ln \left ({ \sum _{j=1}^{M_{B}}\exp \left({-\frac {\left |{ R_\beta -H_\beta S_{B,j} }\right |^{2}}{N_{0}}}\right) }\right ).\qquad \end{align}$$ From (16) and (17), it can be seen that the detection complexities of first and second stages are both equal to $\mathcal {O}(l(M_{A}+M_{B}))$ per subblock in terms of complex multiplications. Moreover, the complexity of the demapping operation for each OFDM subblock is $\mathcal {O}(k_{1}M_{A}+k_{2}M_{B})$ . Therefore, the computational complexity of the proposed two-stage LLR detector yields $\mathcal {O}(l(M_{A}+M_{B}))$ per subblock in terms of complex multiplications, which is dramatically reduced compared to the ML detector.

SECTION IV. Performance Analysis for ZTM-OFDM-IM

A. Minimum Euclidean Distance Analysis

When the ML detector is adopted, the BER performance of the proposed ZTM-OFDM-IM depends on the minimum Euclidean distance between different possible realizations of the OFDM subblocks. Specifically, the system BER becomes larger when the minimum Euclidean distance is decreased, and vice versa. The Euclidean distance between any two subblocks can be represented by $$\begin{equation} d_{(i,j)}=\frac {1}{\sqrt {E_{b}}}\left \|{\tilde {\mathbf {X}}^{(i)}-\tilde {\mathbf {X}}^{(j)} }\right \|^{2}_{2}, \end{equation}$$ where $1\leq i \leq j \leq 2^{n_{\text {ZTM}}}$ , and $\left \|{ \cdot }\right \|_{2}$ represents the 2-norm. $E_{b}$ is defined as the average bit energy of the system, which is utilized for normalization. Then the minimum Euclidean distance metric can be formulated as $$\begin{equation} d_{\text {min}}=\underset {1\leq i \neq j \leq 2^{n_{\text {ZTM}}}}{\min }d_{(i,j)}, \end{equation}$$ which can be directly applied to OFDM-IM and DM-OFDM, since joint-ML detection is also employed in these systems in a subblock-by-subblock manner.

B. Average PEP Analysis

The average PEP (APEP) calculation can be utilized to acquire the asymptotically tight upper bound for the performance of ZTM-OFDM-IM. During the analysis, only one single OFDM subblock needs to be considered, for the reason that the PEP events in different subblocks are mutually independent [11], [28]. Firstly, by considering the $p$ th subblock $\mathbf {X}^{(p)}$ , the conditional PEP (CPEP) of the event that $\mathbf {X}^{(p)}$ is mistakenly detected as $\hat {\mathbf {X}}^{(p)}$ can be expressed by $$\begin{equation} \Pr \left ({ \mathbf {X}^{(p)}\rightarrow \hat {\mathbf {X}}^{(p)}| \mathbf {H}_{p} }\right )=Q\left ({ \sqrt {\frac {\zeta }{2N_{0}}} }\right ), \end{equation}$$ where $Q(\cdot )$ denotes the Gaussian Q-function, and $$\begin{equation} \zeta =\left \|{ \text {diag}\left \{{ \mathbf {X}^{(p)}- \hat {\mathbf {X}}^{(p)} }\right \}\mathbf {H}_{p}}\right \|_{2}^{2}=\left \|{ \mathbf {D}\mathbf {H}_{p}}\right \|_{2}^{2}. \end{equation}$$ Here $\mathbf {D}= \text {diag}\left \{{ \mathbf {X}^{(p)}- \hat {\mathbf {X}}^{(p)} }\right \}$ . Afterward, the unconditional PEP (UPEP) can be calculated by $$\begin{equation} \Pr \left ({ \mathbf {X}^{(p)}\rightarrow \hat {\mathbf {X}}^{(p)} }\right )=E_{\mathbf {H}_{p}}\left \{{Q\left ({ \sqrt {\frac {\zeta }{2N_{0}}} }\right )}\right \}, \end{equation}$$ where $E_{\mathbf {H}_{p}}\left \{{\cdot }\right \}$ represents the expectation operation with respect to the channel coefficients $\mathbf {H}_{p}$ . To attain the closed-form expression of UPEP, approximations are often applied to the Gaussian Q-function [11], [28]. In [30], several approximation methods have been reviewed. For instance, a tight approximative expression is proposed in [31], given by $$\begin{align} Q(x)\approx&\frac {1}{\sqrt {2\pi }\left ({ 0.661x+0.339\sqrt {x^{2}+5.510} }\right )}e^{-x^{2}/2},\notag \\[-2pt]&\qquad \qquad \qquad \qquad \qquad \qquad \qquad x>0, \end{align}$$ whose form is too complicated for implementations. For the simplicity of further derivations, exponential sums (Prony approximations) are utilized to obtain an upper bound of Q-function, which is illustrated in [32] and [33] as $$\begin{equation} Q(x)\approx \frac {1}{12}e^{-x^{2}/2}+\frac {1}{4}e^{-2x^{2}/3}\quad x>0.5, \end{equation}$$ which has been adopted in [11], [17], and [28]. To enhance the accuracy, two more approximative expressions for Q-function are provided in [34] as $$\begin{equation} Q(x)\approx 0.208e^{-0.971x^{2}}+0.147e^{-0.525x^{2}}\quad x>0, \end{equation}$$ and $$\begin{align} Q(x)\approx&0.168e^{-0.876x^{2}}+0.144e^{-0.525x^{2}}+0.002e^{-0.603x^{2}}\notag \\[-2pt]&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad x>0. \end{align}$$ In this paper, to achieve tighter upper BER bound, (26) is applied to (22), which can be expressed by $$\begin{align}&\hspace {-1.1pc}\Pr \left ({ \mathbf {X}^{(p)}\rightarrow \hat {\mathbf {X}}^{(p)} }\right )\notag \\&\approx E_{\mathbf {H}_{p}} \big \{0.168\exp \left({ \frac {-0.876\zeta }{2N_{0}}}\right)\notag \\&\quad +\,0.144\exp \left({ \frac {-0.525\zeta }{2N_{0}}}\right)+0.002\exp \left({ \frac {-0.603\zeta }{2N_{0}}}\right) \big \}\qquad \end{align}$$ Afterward, according to the spectral theorem [35], (27) is further derived as [11] $$\begin{align} \Pr \left ({ \mathbf {X}^{(p)}\rightarrow \hat {\mathbf {X}}^{(p)} }\right )\approx&\frac {1}{1/0.168\det \left ({ \mathbf {I}_{L} +\frac {0.876}{2N_{0}}\boldsymbol {\Theta }\boldsymbol {\Gamma }}\right )}\notag \\&+\frac {1}{1/0.144\det \left ({ \mathbf {I}_{L} +\frac {0.525}{2N_{0}}\boldsymbol {\Theta }\boldsymbol {\Gamma }}\right )}\notag \\&+\frac {1}{1/0.002\det \left ({ \mathbf {I}_{L} +\frac {0.603}{2N_{0}}\boldsymbol {\Theta }\boldsymbol {\Gamma }}\right )},\qquad \end{align}$$ where $\mathbf {I}_{L}$ denotes the $L$ -dimensional identity matrix, $\boldsymbol {\Theta }=E\left \{{\mathbf {H}_{p}\mathbf {H}_{p}^{H} }\right \}$ , and $\boldsymbol {\Gamma }=\mathbf {D}^{H}\mathbf {D}$ . After obtaining the UPEP for different combinations of $\mathbf {X}^{(p)}$ and $\hat {\mathbf {X}}^{(p)}$ , the APEP can be derived as $$\begin{align} {\Pr }_{\text {avg}}\approx&\frac {1}{n_{\text {ZTM}}2^{n_{\text {ZTM}}}}\underset {\mathbf {X}^{(p)}\neq \hat {\mathbf {X}}^{(p)}}{\sum }\Pr \left ({ \mathbf {X}^{(p)}\rightarrow \hat {\mathbf {X}}^{(p)} }\right )\notag \\&\qquad \qquad \qquad \qquad \qquad \qquad \!\! e\left ({ \mathbf {X}^{(p)},\hat {\mathbf {X}}^{(p)} }\right ),\qquad \end{align}$$ where $e\left ({ \mathbf {X}^{(p)},\hat {\mathbf {X}}^{(p)} }\right )$ denotes the number of bit errors in the event that $\mathbf {X}^{(p)}$ is detected as $\hat {\mathbf {X}}^{(p)}$ . Finally, ${\Pr }_{\text {avg}}$ is utilized to approximate the system BER of ZTM-OFDM-IM as an upper bound.

Remark 2:

Due to the approximation of the Gaussian Q-function in (22), there may be deviations between the theoretical APEP and the simulated BER, since there exist approximative errors especially when the value of $x$ is beyond the fittest region of the Q-function approximation, e.g., $x>0.5$ for (24). Therefore, when a low-rate ZTM-OFDM-IM system is investigated, which means that the number of possible realizations of OFDM subblocks $2^{n_{\text {ZTM}}}$ is relatively small, the theoretical APEP result may fit the BER curve at high-SNR region quite well in that less approximative operations are utilized. On the contrary, for large-scale ZTM-OFDM-IM system with high-order constellations and more flexible index patterns, much more PEP events have to be considered, causing inevitable estimation errors due to the increased error-inducing operations.

SECTION V. Performance Enhancement for ZTM-OFDM-IM

In this section, to enhance the overall performance of ZTM-OFDM-IM, the constellation design strategy is firstly discussed, then several possible generalizations of ZTM-OFDM-IM are investigated.

A. Constellation Design Strategy

When ML detection is employed, the BER performance of the proposed ZTM-OFDM-IM depends on the minimum Euclidean distance between subblocks, in other words, the multi-dimensional constellations. Therefore, the constellation design is crucial in order to optimize the system performance, which should obey the following empirical criterions.

• Criterion 1:

  In order to obtain the index pattern for each OFDM subblock, the two constellation alphabets have to be differentiable, i.e., $\mathcal {M}_{A}\wedge \mathcal {M}_{B}=\varnothing$ . Besides, the zero element is not included in these two sets to distinguish modulated and empty subcarriers.

• Criterion 2:

  To achieve better BER performance, the minimum Euclidean distance between subblocks needs to be maximized. Therefore, in constellation design, the minimum Euclidean distance between subblocks with different index patterns, denoted as $d_{\text {inter}}$ , should not be smaller than the counterpart between constellation points within each alphabet, denoted as $d_{\text {min},A}$ and $d_{\text {min},B}$ for $\mathcal {M}_{A}$ and $\mathcal {M}_{B}$ respectively.

Based on these criterions, one feasible combinations of two constellation alphabets is illustrated in Fig. 2, where $(N_{A},N_{B})=(4,8)$ , $(k_{1},k_{2},k,l)=(2,1,3,4)$ , $\mathcal {M}_{A}=\left \{{ -1-\mathsf {j},-1+\mathsf {j},1-\mathsf {j},1+\mathsf {j} }\right \}$ , and $\mathcal {M}_{B}=\{ (1+\sqrt {2})+\mathsf {j},(1+\sqrt {2})-\mathsf {j},1+(1+\sqrt {2})\mathsf {j},1-(1+\sqrt {2})\mathsf {j},-1+(1+\sqrt {2})\mathsf {j},-1-(1+\sqrt {2})\mathsf {j},-(1+\sqrt {2})\mathsf {j},-(1+\sqrt {2})-\mathsf {j} \}$ . From calculations, we have $d_{\text {min}}=d_{\text {inter}}=d_{\text {min},A}=d_{\text {min},B}$ with normalized average bit energy. As for the general case, if taking QAM constellations into account, unlike DM-OFDM, where an $(N_{A}+N_{B})$ -QAM constellation set is simply separated into two subsets $\mathcal {M}_{A}$ and $\mathcal {M}_{B}$ , the QAM constellation design for ZTM-OFDM-IM is more complicated, since it is influenced by the values of $(k_{1},k_{2},k,l)$ under the constraints of Criterion 1 and Criterion 2. For instance, when $(k_{1},k_{2},k,l)=(2,1,3,4)$ , by assuming that $\mathcal {M}_{A}$ is surrounded by $\mathcal {M}_{B}$ for simplicity, the feasible constellation alphabets should follow that: 1). $d_{\text {min},A}=d_{\text {min},B}=d$ ($d$ is a positive constant); 2). The minimum Euclidean distance between constellations of $\mathcal {M}_{A}$ and those of $\mathcal {M}_{B}$ is set as $\frac {\sqrt {2}}{2}d$ ; 3). The minimum Euclidean distance between constellations of $\mathcal {M}_{A}$ as well as those of $\mathcal {M}_{B}$ and the origin is set as $\frac {\sqrt {2}}{2}d$ .

FIGURE 2.

ZTM-OFDM-IM constellation diagram with $N_{A}=4$ and $N_{B}=8$ .

B. Generalizations for ZTM-OFDM-IM

To enhance the throughput of the proposed ZTM-OFDM-IM, several possible generalizations can be applied, which can be set aside as our future work.

Firstly, to fully utilize the flexibility of the activation index pattern, the critical parameters $(k_{1},k_{2},k,l)$ are allowed to be alterable for different subblocks, which yields much larger set of subblock realizations. Explicitly, if these parameters can be arbitrarily set provided that $1\leq k \leq l$ , $0\leq k_{1},k_{2}\leq k$ and $k_{1}+k_{2}=k$ , then the number of OFDM subblock realizations is derived as $\sum _{k=1}^{l}\sum _{k_{1}=0}^{k}\binom {l}{k}\binom {k}{k_{1}}M_{A}^{k_{1}}M_{B}^{k_{2}}$ , and the transmitted bits per OFDM subblock can be formulated as $$\begin{equation} n_{\text {GZTM1}} = \left \lfloor{ \log _{2}\left ({ \sum _{k=1}^{l}\sum _{k_{1}=0}^{k}\binom {l}{k}\binom {k}{k_{1}}M_{A}^{k_{1}}M_{B}^{k_{2}} }\right ) }\right \rfloor .\quad \end{equation}$$ By comparing (5) with (30), it is indicated that this generalization is capable of significantly enhancing the spectral efficiency of ZTM-OFDM-IM. Nevertheless, the detection complexity is inevitably augmented. Take the sub-optimal LLR-based detector for instance. Unlike classical ZTM-OFDM-IM, where only a two-stage LLR detection and corresponding demapping are required for demodulation, the generalized scheme has to perform the same detection operations as ZTM-OFDM-IM for each possible combination of $(k_{1},k_{2},k,l)$ , generating its corresponding solution respectively. Afterward, ML detection is utilized to acquire the best solution among all the resultant candidates.

Another generalization strategy is to adopt multiple constellation alphabets. More specifically, a fraction of subcarriers are activated to be modulated by multiple (≥ 3) constellation alphabets, whilst the others keep empty, which leads to additional diversity gain. By assuming $\kappa$ ($3\leq \kappa \leq l-1$ ) constellation modes with the sizes of $[M_{1},M_{2},\cdots ,M_{\kappa }]$ , the numbers of corresponding modulated subcarriers are denoted as $[k_{1},k_{2},\cdots ,k_\kappa ]$ , and $k$ and $l$ are fixed, the number of information bits carried by each OFDM subblock can be derived as $$\begin{align} n_{\text {GZTM2}}=\log _{2}\left ({\prod _{i=1}^{\kappa }M_{i}^{k_{i}}}\right )+\left \lfloor{ \log _{2}\prod _{i=1}^{\kappa }\binom {l-\sum _{j=0}^{i-1}k_{j}}{k_{i}} }\right \rfloor ,\notag \\[4pt] {}\end{align}$$ where $k_{0}$ is set to 0. With the aid of such enhancement, the modified system is capable of significantly improving the throughput. However, due to the usage of multiple constellation alphabets, constellation design becomes more difficult in order to ensure the BER performance, in other words, to keep the normalized minimum Euclidean distance from decreasing.

SECTION VI. Numerical Results

The performance of the proposed ZTM-OFDM-IM is compared to conventional OFDM and other index modulated OFDM schemes including DM-OFDM and OFDM-IM, under both the frequency-selective Rayleigh fading channel and the AWGN channel assumptions. Besides, to further enlarge the Euclidean distances between modulated subblocks and to combat the highly-correlated channel fading induced by the frequency-selective Rayleigh fading channel, the subcarrier-level interleaving technique [19], [20], [29] is also applied to ZTM-OFDM-IM, DM-OFDM and OFDM-IM, where the corresponding performances are compared. In simulations, $E_{b}/N_{0}$ denotes the SNR per bit. The number of subcarriers $N$ , the length of CP $L_{\text {CP}}$ , and the length of channel fading coefficients $L$ are set to 128, 16, and 10, respectively. Moreover, the subblock size is 4, and the other critical parameters are listed as follows. ZTM-OFDM-IM: In each subblock, 2 subcarriers are modulated by $\mathcal {M}_{A}$ , and 1 subcarrier is modulated by $\mathcal {M}_{B}$ ; DM-OFDM: In each subblock, 2 subcarriers are modulated by $\mathcal {M}_{A}$ , whilst the others are modulated by $\mathcal {M}_{B}$ ; OFDM-IM: In each subblock, 2 subcarriers are modulated.

The BER performance of the proposed ZTM-OFDM-IM is compared with conventional OFDM using ML detection under both the Rayleigh fading channel and the AWGN channel in Fig. 3, where two different binary PSK (BPSK) schemes are employed in ZTM-OFDM-IM, whilst BPSK is utilized in OFDM, and non-interleaved systems are considered. The spectral efficiencies of ZTM-OFDM-IM and OFDM equal 1.33 and 0.89 bit/s/Hz. It can be seen that, aside from the 0.44 bit/s/Hz spectral efficiency gain over OFDM, the proposed ZTM-OFDM-IM achieves about 2 dB performance enhancement over its conventional counterpart at the BER of 10⁻³ for the Rayleigh fading channel, and only slight performance loss of ZTM-OFDM-IM is observed under AWGN channel, for the reason that $E_{b}$ is reduced for higher spectral efficiency, leading to smaller required $E_{b}/N_{0}$ .

FIGURE 3.

Performance comparison between the proposed ZTM-OFDM-IM and conventional OFDM under the frequency-selective Rayleigh fading channel and the AWGN channel.

Figure 4 illustrates the performance comparison between ZTM-OFDM-IM, DM-OFDM and OFDM-IM using ML detection under different channel conditions, where interleaving is not applied. To attain the spectral efficiency of 2.22 bit/s/Hz, constellations in Fig. 2 are used in ZTM-OFDM-IM, whilst two QPSK schemes in [28, Fig. 2] and 16-QAM are utilized in DM-OFDM and OFDM-IM. The $d_{\text {min}}$ metric for ZTM-OFDM-IM, DM-OFDM and OFDM-IM are equal to 1.812, 1.371, and 1.333, respectively, which implies that the proposed scheme is capable of enhancing the BER performance compared to its conventional counterparts. From Fig. 4 it can be seen that, ZTM-OFDM-IM achieves 1 and 2 dB performance gains over OFDM-IM at the BER of 10⁻³ under the Rayleigh fading channel and the AWGN channel, respectively. Although the performances of ZTM-OFDM-IM and DM-OFDM are almost the same under the Rayleigh fading channel, the former outperforms the latter by 1 dB under the AWGN channel when the BER equals 10⁻³, which is in accordance with the aforementioned theoretical results based on $d_{\text {min}}$ . Besides, It is seen that, at BER of 10⁻³, ZTM-OFDM-IM attains 0.22 bit/s/Hz gain over DM-OFDM with the same BER performance under AWGN channel and only slight performance loss under Rayleigh fading channel, validating the superiority of the proposed ZTM-OFDM-IM over DM-OFDM and OFDM-IM.

FIGURE 4.

Performance comparison between the proposed ZTM-OFDM-IM, DM-OFDM and OFDM-IM without interleaving under the frequency-selective Rayleigh fading channel and AWGN channel assumptions.

In Fig. 5, performance comparison between the proposed ZTM-OFDM-IM, DM-OFDM and OFDM-IM with the aid of interleaving technique under different channel conditions is illustrated, where the spectral efficiency is set to 2.22 bit/s/Hz. By comparing Fig. 4 and Fig. 5, it can be seen that interleaving technique is capable of enhancing the BER performance of the three index modulated OFDM systems under frequency-selective Rayleigh fading channel, due to the mitigation of the channel correlation. Besides, from Fig. 5, it is shown that the proposed ZTM-OFDM-IM achieves about 0.5 dB and 2 dB performance gains over DM-OFDM and OFDM-IM under both channel conditions, which demonstrates the distinctive advantages of the proposed ZTM-OFDM-IM over other index modulated OFDM counterparts.

FIGURE 5.

Performance comparison between the proposed ZTM-OFDM-IM, DM-OFDM and OFDM-IM with interleaving under the frequency-selective Rayleigh fading channel and AWGN channel assumptions.

In Fig. 6, the performances of the proposed ML detector and the reduced-complexity two-stage LLR detector for the proposed ZTM-OFDM-IM are evaluated under the frequency-selective Rayleigh fading channel and the AWGN channel, where the spectral efficiency is set to 2.22 bit/s/Hz, and interleaving is not used. Under both channel assumptions, despite the dramatic decrease of the computational complexity brought by LLR detection, there is only slight performance loss in comparison with ML detection. Therefore, it is validated that the reduced-complexity two-stage LLR detector is more suitable for practical implementations.

FIGURE 6.

Performance comparison between ML-based and LLR-based ZTM-OFDM-IM systems under the frequency-selective Rayleigh fading channel and the AWGN channel.

Moreover, in Fig. 7, the simulated BER performances of the proposed ZTM-OFDM-IM are compared to the corresponding APEP results under the frequency-selective Rayleigh fading channel at different spectral efficiencies, where interleaving is not considered. For both spectral efficiencies of 2.22 and 1.33 bit/s/Hz, at low-SNR region, there is inevitable difference between the theoretical and simulated results due to the approximative errors. However, as SNR becomes larger, the deviation is gradually decreased, and the APEP and simulated BER results fit quite well at the high-SNR region, which validates the accuracy of our theoretical BER approximation.

FIGURE 7.

Comparison between the theoretical APEP analysis and the simulated BER performance of the proposed ZTM-OFDM-IM at 2.22 bit/s/Hz under the frequency-selective Rayleigh fading channel and the AWGN channel.

SECTION VII. Conclusions

In this paper, ZTM-OFDM-IM is proposed to improve the BER performance of conventional index modulated OFDM schemes such as DM-OFDM and OFDM-IM. In the proposed scheme, OFDM subcarriers are partitioned into subblocks, and only a fraction of subcarriers in each OFDM subblock are modulated by two distinguishable constellation alphabets, whilst the others remain empty. The proposed ZTM-OFDM-IM is capable of enhancing the spectral efficiency and the energy efficiency compared to its conventional counterparts, leading to significant performance gain. Besides, an optimal ML detector and a reduced-complexity two-stage LLR detector are proposed for demodulation. Furthermore, the constellation design strategy and several generalization ideas for the proposed ZTM-OFDM-IM are briefly discussed. Besides, theoretical analysis based on minimum Euclidean distance and APEP calculation is conducted to investigate the BER performance of ZTM-OFDM-IM. Theoretical and simulative results demonstrate that the proposed ZTM-OFDM-IM can achieve performance gain over conventional OFDM and other index modulated OFDM schemes with and without the subcarrier-level interleaving technique, and the reduced-complexity LLR detector only suffers from slight performance loss compared to the optimal ML detector.