Dual-Mode Index Modulation Aided OFDM With Constellation Power Allocation and Low-Complexity Detector Design

Dual-mode index modulation aided orthogonal frequency division multiplexing (DM-OFDM) is recently proposed, which modulates all subcarriers to eliminate the limits of spectrum efficiency (SE) in OFDM with index modulation (OFDM-IM). In DM-OFDM, the subcarriers within each subblock are divided into two groups, which are modulated by two distinguishable constellation alphabets drawn from the inner and the outer constellation points of a given M-ary QAM constellation. In this paper, a new DM-OFDM scheme, called DM-OFDM with constellation power allocation (DM-OFDM-CPA) is proposed, where the two groups of subcarriers within each subblock are set at different power levels and modulated by different M-ary PSK symbols, leading to the improvement of the system performance at low-order modulation. At the receiver, a low-complexity maximum likelihood (LC-ML) detector and two reduced search complexity detectors, based on energy detection and log-likelihood ratio (LLR) criterion, respectively, are employed for demodulation. Then, the bit error rate (BER) analyses based on pairwise error probability are provided for the proposed DM-OFDM-CPA, and the power ratio between the two groups is optimized to maximize achievable BER performance for a given signal to noise ratio. The simulation results confirm that at a given SE, the proposed DM-OFDM-CPA achieves a significantly better BER performance than the existing OFDM-IM. When the modulation order is lower than 16, it outperforms the classic DM-OFDM. The results also show that two reduced search complexity detectors, especially the LLR detector, provide similar performance to the LC-ML detector in the DM-OFDM-CPA.

SECTION I. Introduction

The next-generation wireless networks are expected to provide high-speed internet access anywhere and anytime. The development of intelligent terminals is doubtlessly accelerating the process and creating new data traffic demands, such as mobile video and gaming, environmental monitoring, geological exploration. The exponentially growing demand of data traffic has triggered continuous expansion of network infrastructures and needs dramatic improvement of spectrum efficiency (SE). At the same time, energy efficiency (EE) becomes another important issue as the growth of data traffic also leads to dramatic increase of energy consumption [1]. Therefore, SE and EE are two important factors considered in the design of an excellent wireless network and their improvements become research hotspot recently [2]–​[5].

Orthogonal Frequency Division Multiplexing (OFDM), which can effectively mitigate the inter-symbol interference (ISI) caused by a frequency selectivity channel with a high SE and low complexity, has been widely studied in recent years and adopted in many wireless standards, such as 802.11a/g Wi-Fi, 802.16 WiMAX, and long-term evolution (LTE). In [6], spatial modulation (SM), a promising multiple input multiple output (MIMO) transmission technique, was extended to the frequency domain (FD) of OFDM system for the first time. The resulting system, called as subcarrier index modulation OFDM (SIM-OFDM), employs the subcarrier index to carry information by on-off keying (OOK) modulation and therefore improves EE and bit error ratio (BER) performance. However, in SIM-OFDM, an incorrect detection of a subcarrier state not only leads to incorrect demodulation of the current subcarrier but also to incorrect demodulation of other subcarriers. In [7], an enhanced SIM-OFDM (ESIM-OFDM) was put forward, which takes one bit to control the state of two adjacent subcarriers in which only one subcarrier is activated every time. Unfortunately, ESIM-OFDM has to adopt higher order symbol modulation to achieve the same SE as that of the classic OFDM, since only half of the subcarriers are active. Then, Basar et al. [8] proposed a generalization scheme of ESIM-OFDM, OFDM with Index Modulation (OFDM-IM). In OFDM-IM, the state of two or more subcarriers is used to convey the information and the number of active subcarriers can be adjusted to balance the trade-off between system performance and SE.

Recently, some researches of OFDM-IM have been followed up in many literatures [9]–​[16]. In [9], a tight BER upper bound of OFDM-IM was presented, and the interleaving technology that can obtain additional diversity gain was introduced for OFDM-IM to improve its BER performance in [10] and [11]. In [12] and [13], the EE and SE of OFDM-IM were analyzed and the number of active subcarriers was optimized. In [14], the achievable throughput of OFDM-IM was investigated, which provides the guidance for the system design of OFDM-IM. In [15] and [16], a new spectrum-efficient OFDM-IM scheme allowing each OFDM subblock to use a various number of active subcarriers was proposed to enhance the SE. Additionally, in [15], Fan presented an OFDM with in-phase/quadrature index modulation (OFDM-I/Q-IM) scheme to transmit more information, which employs both the in-phase and the quadrature components of signals for index modulation. Besides, a novel MIMO-OFDM-IM scheme in [17] combining OFDM-IM with MIMO transmission technology was presented, which obtains a considerable performance gain over the classic MIMO-OFDM. Following [17], Zheng proposed two types of low-complexity detectors for MIMO-OFDM-IM in [18]. However, in all aforementioned studies, some subcarriers are deactivated and do not carry any information.

Against this background, T. Mao proposed a dual-mode index modulation aided OFDM (DM-OFDM) [19], where the “inactive” subcarriers of OFDM-IM are allowed to transmit modulated symbols drawn from a secondary constellation, which enhances the SE of the existing OFDM-IM techniques. In this scheme, the two distinguishable constellation alphabets are respectively designed with the inner and the outer constellation points of a given $M$ -ary QAM constellation. It is demonstrated that DM-OFDM achieves a considerably better BER performance than OFDM-IM at the same SE. Furthermore, T. Mao presented a generalized DM-OFDM (GDM-OFDM) scheme [20] to obtain a higher SE, where the number of subcarriers modulated by the same constellation alphabet is variable. Recently, in [21], a novel multiple-mode OFDM-IM (MM-OFDM-IM) has been proposed, where the modulated symbols are drawn from two or more distinguishable constellation alphabets according to a given MQAM or MPSK constellation and the full permutations of these distinguishable modes are utilized for IM purposes. The MM-OFDM-IM further enhances the SE, but also increases largely the complexity of the detector at the receiver as the increase of the modes.

In this paper, a new dual-mode OFDM-IM scheme termed dual-mode index modulation aided OFDM with constellation power allocation (DM-OFDM-CPA) is proposed. As the DM-OFDM, the DM-OFDM-CPA also modulates all subcarriers to eliminate the limits of SE in OFDM-IM. In DM-OFDM-CPA, the subcarriers are partitioned into several subblocks and then each subblock is divided into two groups according to an incoming bit sequence. Unlike the DM-OFDM, the DM-OFDM-CPA allocates different powers to two different groups of subcarriers and then modulates them by different PSK constellation alphabets. Besides, the interleaving technology is introduced for DM-OFDM-CPA to attain additional diversity gain in Rayleigh fading channels. At the receiver, three detection schemes are employed to demodulate the signals, including a low-complexity maximum likelihood (LC-ML) detector, an energy-detection (ED) detector and a log-likelihood ratio (LLR) detector. Compared with the LC-ML detector, the ED and the LLR detector have a lower search complexity. Moreover, the theoretical BER performance analysis based on pairwise error probability (PEP) is provided, and the power ratio between the high-power subcarrier and the low-power subcarrier is optimized to maximize the BER performance of DM-OFDM-CPA. Simulation results demonstrate that DM-OFDM-CPA is capable of achieving a significantly better BER performance than OFDM-IM at a given SE. Meanwhile, it outperforms DM-OFDM when the modulation order is lower than sixteen.

The rest of this paper is organized as follows. Section II describes the system model of DM-OFDM-CPA in detail and Section III gives the corresponding LC-ML, ED and LLR detectors for demodulation at the receiver. In Section IV, our theoretical performance analysis of the proposed DM-OFDM-CPA is presented and the power ratio is optimized. Monte Carlo simulations for quantifying the BER performance of the proposed DM-OFDM-CPA are given in Section V. Finally, Section VI concludes the paper.

SECTION II. DM-OFDM-CPA System Model

The block diagram of the DM-OFDM-CPA baseband system structure is depicted in Fig. 1. For the transmission of each OFDM symbol, a total number of $m$ incoming data bits are first partitioned by a bit splitter into $G$ groups, each consisting of $p$ bits, i.e., $p = m/G$ bits. Each group of $p$ -bits is then mapped to an OFDM subblock of length $n$ , where $n = N/G$ subcarriers and $N$ is the total number of subcarriers of an OFDM symbol. As the DM-OFDM, the DM-OFDM-CPA modulates all the subcarriers each subblock. For each subblock $g$ , the incoming $p$ -bits are divided into two parts. As shown in Fig.1, the first $p_{1}$ -bits are fed to the index selector to divide the indices of the subblock into two index subsets, denoted as $I_{h}^{g}$ and $I_{l}^{g}$ . Unlike the DM-OFDM, the subcarriers corresponding to $I_{h}^{g}$ and $I_{l}^{g}$ are set at two different power levels, denoted as $P_{h }$ and $P_{l}$ for $P_{h }>P_{l}$ . The remaining $p_{2}$ -bits are mapped onto the $M$ -ary Mapper to determine their data symbols with different $M$ -ary PSK modulation.

FIGURE 1.

Block diagram of the DM-OFDM-CPA baseband system structure.

In the index selector, the $p_{1}$ -bits are first expressed as a combination of $k$ high-power subcarriers chosen from $n$ subcarriers, where $I_{h}^{g}$ are independently given by $I_{h}^{g} =\{i_{h}^{g} \left ({ 1 }\right ),\ldots ,i_{h}^{g} \left ({ k }\right )\}$ , $i_{h}^{g} \left ({ r }\right )\in \{1,\ldots ,n\}$ for $r=1,\ldots ,k$ and $i_{h}^{g} \left ({ {r_{1} } }\right )\ne i_{h}^{g} \left ({ {r_{2} } }\right )$ if $r_{1} \ne r_{2}$ . Therefore, $p_{1 }$ can be calculated according to

$$\begin{equation} p_{1} =\left \lfloor{ {\log _{2} \left ({ {C(n,k)} }\right )} }\right \rfloor \!, \end{equation}$$ where $C(n,k)$ and $\left \lfloor{ {\ast } }\right \rfloor$ denote the binomial coefficient and the floor function, respectively. This selection procedure can be efficiently implemented by a look-up table or the combinatorial method proposed in [8]. Once $I_{h}^{g}$ is known, $I_{l}^{g}$ is also determined, which is the complement of $I_{h}^{g}$ with respect to the set $\left \{{ {\left .{ {1,\ldots ,n} }\right \} } }\right .$ . Clearly, there are a total of $c=2^{p_{1} }$ distinctive index patterns for $I_{h}^{g}$ and $I_{l}^{g}$ . A mapping table example of index selector for $n=4$ , $k=2$ is presented in Table 1. According to (1), $I_{h}^{g}$ is determined by the two index bits and four index patterns are used, which are [1,2], [2,3], [3,4] and [1,4], corresponding to the bits values of [0,0], [0,1], [1,0] and [1,1] and to $I_{l}^{g}$ of [3,4], [1,4], [1,2] and [2,3], respectively.

TABLE 1 Mapping Table of Index Selector When $n=4$ , $k =2$

As shown in Fig. 2, the high and low power subcarriers are modulated by $M_{h}$ -ary and $M_{l}$ -ary PSK symbols. Therefore, $p_{2 }$ can be calculated by:

$$\begin{equation} p_{2} =k\log _{2} \left ({ {M_{h} } }\right )+(n-k)\log _{2} \left ({ {M_{l} } }\right ). \end{equation}$$ In order to adapt to their transmission power, we have $M_{h} \ge M_{l}$ . In the $M$ -ary Mapper, firstly, the $p_{2}$ -bits are divided into two parts, which are respectively fed to a $M_{h}$ -ary mapper and a $M_{l}$ -ary mapper for different symbol modulation. The output symbols are respectively given by $S_{h}^{g} =[s_{h}^{g} \left ({ 1 }\right ),\ldots ,s_{h}^{g} \left ({ k }\right )]$ where $s_{h}^{g} \left ({ r }\right )\in S_{h}$ , $E[\left |{ {s_{h}^{g} \left ({ r }\right )} }\right |^{2}]=1$ for $r=1,\ldots ,k$ , and $S_{l}^{g} =[s_{l}^{g} \left ({ 1 }\right ),\ldots ,s_{l}^{g} \left ({ {n-k} }\right )]$ , where $s_{l}^{g} \left ({ q }\right )\in S_{l}$ , $E[\left |{ {s_{l}^{g} \left ({ q }\right )} }\right |^{2}]=1$ for $q=1,\ldots ,n-k$ . $S_{h}$ and $S_{l}$ are the constellation point sets of the $M_{h}$ -ary mapper and the $M_{l}$ -ary mapper, respectively. Then, their power allocations are implemented and the output symbols of the $M$ -ary mapper are given by $X_{h}^{g} =P_{h} \times S_{h}^{g}$ and $X_{l}^{g} =P_{l} \times S_{l}^{g}$ . After obtaining $\left ({ {I_{h}^{g},I_{l}^{g} } }\right )$ and $\left ({ {X_{h}^{g},X_{l}^{g} } }\right )$ , we can get the $g$ -th subblock $$\begin{equation} X^{g}=[X^{g}(1),X^{g}(2),\ldots ,X^{g}(n)]^{T} \end{equation}$$ where $X^{g}(\beta )\in \{X_{h},X_{l} \}$ , $\beta =1,\ldots ,n$ , $X_{h} =P_{h} \times S_{h}$ , and $X_{l} =P_{l} \times S_{l}$ . With all the subblocks, the $N\times 1$ OFDM block is then built up by $$\begin{equation} X=[X(1),X(2),\ldots ,X(N)]^{T}. \end{equation}$$ Before that, a depth-$G$ block interleaver [10] is used in the OFDM Block Creator to reduce the correlation of channel coefficients in FD, and therefore we have $X^{g}(\beta )=X(G(\beta -1)+g)$ . Next, the inverse FFT operation, adding cyclic prefix (CP), parallel to serial (P/S) and digital-to-analog (D/A) conversion are implemented to generate the transmit signals.

FIGURE 2.

Block diagram of the $M$ -ary Mapper and Demapper in DM-OFDM-CPA.

The signals are then sent through a frequency-selective Rayleigh fading (FSRF) channel. The channel impulse response (CIR) is given by $h=[h(1),h(2),\ldots ,h(L)]^{T}$ , where $L$ is the length of the CIR and $h\left ({ l }\right )$ for $l=1,\ldots ,L$ is a circularly symmetric complex Gaussian random variable with the distribution of $CN(0,1/L)$ . Assuming $L$ is no higher than the CP length $N_{g}$ , the output of the FFT algorithm at the receiver $Y(\alpha )$ is given by

$$\begin{equation} Y(\alpha )=X(\alpha )H(\alpha )+W(\alpha ),\quad \alpha =1,\ldots ,N \end{equation}$$ where $H(\alpha )$ and $W(\alpha )$ are the channel fading coefficients and the noise samples in the FD. The distribution of $W(\alpha )$ is $CN(0,N_{0,F} )$ , where $N_{0,F}$ is the noise variance in the FD. As shown in Fig. 1, the received signal $Y(\alpha )$ is then sent to the detector to detect the index combinations of the high-power and low-power subcarriers and the corresponding modulated symbols. The detector will be discussed in detail in section III. Finally, for each subblock, the estimated index combinations and modulated symbols are respectively fed into the index demapper and the $M$ -ary demapper presented in Fig. 2, to obtain the transmitted $p_{1 }$ and $p_{2 }$ bits information.

According to (1) and (2), the SE of the proposed DM-OFDM-CPA can be given by

$$\begin{align} \eta _{SE}^{DP}=&(\left \lfloor{ {\log _{2} \left ({ {C(n,k)} }\right )} }\right \rfloor +k\log _{2} \left ({ {M_{h} } }\right )+(n-k)\log _{2} \left ({ {M_{l} } }\right ))\notag \\&\qquad \qquad \qquad \qquad \qquad \qquad \ast G/(N+N_{g} ). \end{align}$$ Compared to the OFDM-IM with $M_{h}$ -ary QAM modulation [8], DM-OFDM-CPA offers a $(G(n-k)\log _{2} \left ({ {M_{l} } }\right ))/(N+N_{g} )$ higher SE. For example, when $N=128$ , $N_{g}=16$ , $n=4$ , $k=2$ , $M_{h}=8$ , $M_{l}=4$ , the growth ratio is up to 50%. In the paper, the signal-to-noise ratio (SNR) is defined as $\gamma =E_{b} /N_{0}$ , where $E_{b}$ is the average transmitted energy per bit.

SECTION III. Proposed Detector

In the section, we investigate three different types of detectors, LC-ML, ED, LLR detector for our DM-OFDM-CPA scheme. As the OFDM-IM scheme, the detection of each subblock can be separately carried out. For the $g$ -th subblock, the proposed detectors are described in detail in the following.

A. LC-ML Detector

In ML detection, all possible subblock realizations need to be considered by searching for all possible subcarrier index combinations $\left ({ {I_{h}^{g} ,I_{l}^{g} } }\right )$ and the modulated symbols $\left ({ {X_{h}^{g} ,X_{l}^{g} } }\right )$ , and making a joint decision on $\left ({ {I_{h}^{g} ,I_{l}^{g} } }\right )$ and $\left ({ {X_{h}^{g} ,X_{l}^{g} } }\right )$ by minimizing the following metric:

$$\begin{align}&\hspace {-2pc}\left ({ {{\hat {I}_{h}^{g} } ,{\hat {I}_{l}^{g} } ,{\hat {X}_{h}^{g} ,} {\hat {X}_{l}^{g} }} }\right )\notag \\=&\arg \min \limits _{I_{h}^{g} ,I_{l}^{g} ,X_{h}^{g} ,X_{l}^{g} } \sum \limits _{r=1}^{k} {\left |{ {Y^{g}(i_{h}^{g} \left ({ r }\right ))-H^{g}(i_{h}^{g} \left ({ r }\right ))X_{h}^{g} \left ({ r }\right )} }\right |^{2}}\notag \\&+\,\sum \limits _{q=1}^{n-k} {\left |{ {Y^{g}(i_{l}^{g} \left ({ q }\right ))-H^{g}(i_{l}^{g} \left ({ q }\right ))X_{l}^{g} \left ({ q }\right )} }\right |^{2}} \end{align}$$ where $Y^{g}(\beta )=Y(G(\beta -1)+g)$ and $H^{g}(\beta )=H(G(\beta -1)+g)$ for $\beta =1,\ldots ,n$ , are respectively the received signals and the channel fading coefficients in the FD for the $g$ -th subblock. It can be seen from (7) that the calculation complexity of the ML detector in terms of the number of multiplication operations is up to $o\left ({ {6cM_{h}^{k}M_{l}^{n-k}} }\right )$ , which exponentially increases with $M_{h }$ and $M_{l}$ . Therefore, the traditional ML detector is impractical to implement for large $n$ , $k$ , $M_{h }$ and $M_{l}$ . To reduce its complexity, we propose a low-complexity ML (LC-ML) detector by considering the detection of $\left ({ {I_{h}^{g} ,I_{l}^{g} } }\right )$ and $\left ({ {X_{h}^{g} ,X_{l}^{g} } }\right )$ , individually. The following are the details of the detector.

Firstly, assuming that each subcarrier is modulated by a $M_{h}$ -ary PSK symbol, the modulated symbol carried on the subcarrier is detected by

$$\begin{equation} {\hat {X}_{nh}^{g} } \left ({ \beta }\right )=\min \limits _{X_{nh}^{g} \left ({ \beta }\right )\in X_{h} } \left |{ {Y^{g}(\beta )-H^{g}(\beta )X_{nh}^{g} \left ({ \beta }\right )} }\right |^{2}. \end{equation}$$ If the subcarrier is modulated by a $M_{l}$ -ary PSK symbol, the similar detection is carried out and ${\hat {X}_{nl}^{g} } \left ({ \beta }\right )$ is given by $$\begin{equation} {\hat {X}_{nl}^{g} } \left ({ \beta }\right )=\min \limits _{X_{nl}^{g} \left ({ \beta }\right )\in X_{l} } \left |{ {Y^{g}(\beta )-H^{g}(\beta )X_{nl}^{g} \left ({ \beta }\right )} }\right |^{2}. \end{equation}$$ Then, according to (7), $\left ({ {I_{h}^{g} ,I_{l}^{g} } }\right )$ can be estimated as: $$\begin{align}&\hspace {-2pc}\left ({ {{\hat {I}_{h}^{g} },{\hat {I}_{l}^{g} }} }\right ) \notag \\=&\arg \min \limits _{I_{h}^{g} ,I_{l}^{g} } \sum \limits _{r=1}^{k} {\left |{ {Y^{g}(i_{h}^{g} \left ({ r }\right ))-H^{g}(i_{h}^{g} \left ({ r }\right )){\hat {X}_{nh}^{g} } \left ({ {i_{h}^{g} \left ({ r }\right )} }\right )} }\right |^{2}} \notag \\&+\,\sum \limits _{q=1}^{n-k} {\left |{ {Y^{g}(i_{l}^{g} \left ({ q }\right ))-H^{g}(i_{l}^{g} \left ({ q }\right )){\hat {X}_{nl}^{g} } \left ({ {i_{l}^{g} \left ({ q }\right )} }\right )} }\right |^{2}} \notag \\=&\arg \min \limits _{I_{h}^{g} ,I_{l}^{g} } \sum \limits _{r=1}^{k} {SEY_{h}^{g} (i_{h}^{g} \left ({ r }\right ))} +\sum \limits _{q=1}^{n-k} {SEY_{l}^{g} (i_{l}^{g} \left ({ q }\right ))} \end{align}$$ where $$\begin{equation} SEY_{h}^{g} (\beta )=\left |{ {Y^{g}(\beta )-H^{g}(\beta ){\hat {X}_{nh}^{g} } \left ({ \beta }\right )} }\right |^{2} \end{equation}$$ and $$\begin{equation} SEY_{l}^{g} (\beta )=\left |{ {Y^{g}(\beta )-H^{g}(\beta ){\hat {X}_{nl}^{g} } \left ({ \beta }\right )} }\right |^{2} \end{equation}$$ that are calculated in (8) and (9), respectively.

Finally, the correspondent modulated symbols can be estimated from

$$\begin{align} {\hat {X}_{h}^{g} \left ({ r }\right )}=&{\hat {X}_{nh}^{g} } \left ({ {{\hat {I}_{h}^{g} \left ({ r }\right )} } }\right ),\quad r=1,\ldots ,k, \\ {\hat {X}_{l}^{g} \left ({ q }\right )}=&{\hat {X}_{nl}^{g} } \left ({ {{\hat {I}_{l}^{g} \left ({ q }\right )}} }\right ),\quad q=1,\ldots ,n-k. \end{align}$$

From (8), (9) and (10), the LC-ML detector significantly reduces the calculation complexity. As shown in Table 2, the number of multiplication operations is only $o\left ({ {6n(M_{h} +M_{l} )} }\right )$ , which is linear with $M_{h}$ and $M_{l}$ . However, it has a search complexity of $o\left ({ {n(M_{h} +M_{l} )+c} }\right )$ and would become impractical for a larger $p_{1}$ as $c$ grows exponentially with it. To solve this problem, we propose the ED and LLR detector.

TABLE 2 Complexity of the Detector

B. ED Detector

At the transmitter, the subcarriers corresponding to $I_{h}^{g}$ and $I_{l}^{g}$ are assigned the power $P_{h }$ and $P_{l}$ , respectively. Therefore, the average power of the $\beta$ -th subcarrier in the $g$ -th OFDM subblock at the receiver can be written as:

$$\begin{align} E[\left |{ {Y^{g}(\beta )} }\right |^{2}]=&E[\left |{ {H^{g}(\beta )X^{g}\left ({ \beta }\right )+W^{g}\left ({ \beta }\right )} }\right |^{2}]\notag \\=&\begin{cases} {P_{h} \left |{ {H^{g}(\beta )} }\right |^{2}+N_{0,F} , \beta \in I_{h}^{g} } \\ {P_{l} \left |{ {H^{g}(\beta )} }\right |^{2}+N_{0,F} , \beta \in I_{l}^{g} }. \end{cases} \end{align}$$ From (15), we have $$\begin{equation} E[ED_{X}^{g} (\beta )]=\begin{cases} {P_{h} , \beta \in I_{h}^{g} } \\ {P_{l} , \beta \in I_{l}^{g} }, \end{cases} \end{equation}$$ where $$\begin{equation} ED_{X}^{g} (\beta )={\left ({ {\left |{ {Y^{g}(\beta )} }\right |^{2}-N_{0,F} } }\right )} / {\left |{ {H^{g}(\beta )} }\right |^{2}} \end{equation}$$ is the instantaneous power of the received symbol. After calculation of all the $n$ values of (17), $I_{h}^{g}$ can be estimated by picking up $k$ indices that have maximum values of (17) and $I_{l}^{g}$ is also determined.

Then, the modulated symbol can be obtained from

$$\begin{align} {\hat {X}^{g}\left ({ \beta }\right )}=\begin{cases} \min \limits _{X^{g}\left ({ \beta }\right )\in X_{h} } \left |{ {Y^{g}(\beta )-H^{g}(\beta )X^{g}\left ({ \beta }\right )} }\right |^{2},& \beta \in {\hat {I}_{h}^{g} } \\ \min \limits _{X^{g}\left ({ \beta }\right )\in X_{l} } \left |{ {Y^{g}(\beta )-H^{g}(\beta )X^{g}\left ({ \beta }\right )} }\right |^{2},& \beta \in {\hat {I}_{l}^{g} }. \end{cases}\notag \\ {}\end{align}$$ where $\hat {I}_{h}^{g}$ and $\hat {I}_{l}^{g}$ are the estimated values of $I_{h}^{g}$ and $I_{l}^{g}$ , respectively.

From the above detection, the ED detector has a search complexity of $o\left ({ {kM_{h} +(n-k)M_{l} +n\log n} }\right )$ , which is greatly smaller than that of the LC-ML detector for large $p_{1}$ . In addition, it also decreases the calculation complexity that is shown in Table 2.

However, the ED detector may determine wrong ${\hat {I}_{h}^{g} }$ and ${\hat {I}_{l}^{g} }$ , due to the impact of noise, especially in low SNR. In order to achieve a better performance, the following improvements are made for the ED detector:

1. The $k$ indices having maximum values of (17) are used to initialize ${\hat {I}_{h}^{g} }$ , while the rest form a new set, denoted as ${\hat {I}_{l,0}^{g} }$ ;

2. The following ML detection is carried out to demodulate all the subcarriers of ${\hat {I}_{h}^{g} }$ :

   $$\begin{align} \begin{cases} {{\hat {X}_{nh}^{g} }\left ({ \beta }\right )=\min \limits _{X^{g}\left ({ \beta }\right )\in X_{h} } \left |{ {Y^{g}(\beta )-H^{g}(\beta )X^{g}\left ({ \beta }\right )} }\right |^{2}} \\ {{\hat {X}_{nl}^{g} }\left ({ \beta }\right )=\min \limits _{X^{g}\left ({ \beta }\right )\in X_{l} } \left |{ {Y^{g}(\beta )-H^{g}(\beta )X^{g}\left ({ \beta }\right )} }\right |^{2}} \\ {{\hat {X}^{g}}\!\left ({ \beta }\right )=\min \limits _{X^{g}\!\left ({ \beta }\right )\in \{{\hat {X}_{nh}^{g} } \!\left ({ \beta \!}\right ),{\hat {X}_{nl}^{g} }\left ({ \beta }\right )\}} \!\left |{ {Y^{g}(\beta )-H^{g}(\beta )X^{g}\!\left ({ \beta \!}\right )} \!}\right |^{2}}. \end{cases}\!\!\!\!\notag \!\!\!\!\\ {}\end{align}$$

3. If ${\hat {X}^{g}} \left ({ \beta }\right )\in X_{h}$ for all the subcarriers of ${\hat {I}_{h}^{g} }$ , go to (f); Otherwise, go to (d).

4. If all the subcarriers of ${\hat {I}_{l,0}^{g} }$ have been detected by (19), go to (f); Otherwise, go to (e).

5. Replace the subcarriers of ${\hat {I}_{h}^{g} }$ for ${\hat {X}^{g}} \left ({ \beta }\right )\in X_{l}$ with the subcarriers of ${\hat {I}_{l,0}^{g} }$ and update ${\hat {I}_{h}^{g} }$ . Then, demodulate the updated subcarriers by using (19) and go to (c).

6. Determine ${\hat {I}_{l}^{g} }$ with the complement of ${\hat {I}_{h}^{g} }$ regarding the set $\left \{{ {\left .{ {1,\ldots ,n} }\right \} } }\right .$ and demodulate all the subcarriers in the $g$ -th subblock by using (18).

As shown in Table 2, the improved ED detector slightly increases calculation complexity and search complexity of energy detection. The increase can be neglected since they are still linear with $M_{h}$ and $M_{l}$ .

C. LLR Detector

In DM-OFDM-CPA, each subcarrier is modulated by either a $M_{h}$ -ary symbol with the power $P_{h}$ or a $M_{l}$ -ary symbol with the power $P_{l}$ , so the modulation mode of each subcarrier can be obtained by calculating the logarithm of the ratio between the posteriori probabilities of the subcarrier being modulated by the $M_{h}$ -ary symbol and by the $M_{l}$ -ary symbol, which is formulated as

$$\begin{equation} \lambda (\beta )=\ln \frac {\sum \nolimits _{\chi =1}^{M_{h} } {P(X^{g}\left ({ \beta }\right )=X_{h,\chi } \left |{ {Y^{g}(\beta )} }\right .)} }{\sum \nolimits _{\gamma =1}^{M_{l} } {P(X^{g}\left ({ \beta }\right )=X_{l,\gamma } \left |{ {Y^{g}(\beta )} }\right .)} } \end{equation}$$ where $\beta =1,\ldots ,n$ , $X_{h,\chi } \in X_{h}$ and $X_{l,\gamma } \in X_{l}$ . Since $\sum \nolimits _{\chi =1}^{M_{h} } {P(X^{g}\left ({ \beta }\right )=X_{h,\chi } )} =k / n$ and $\sum \nolimits _{\gamma =1}^{M_{l} } {P(X^{g}\left ({ \beta }\right )=X_{l,\gamma } )} ={(n-k)}/ n$ , using Bayes rule, (20) can be rewritten as $$\begin{align}&\hspace {-1.8pc}\lambda \left ({ \beta }\right )=\ln \left ({ k }\right )-\ln (n-k) \notag \\&+\,\ln \left ({ {\sum \limits _{\chi =1}^{M_{h} } {\exp \left ({ {-\frac {1}{N_{0,F} }\left |{ {Y^{g}(\beta )-H^{g}(\beta )X_{h,\chi } } }\right |^{2}} }\right )} } }\right ) \notag \\&-\,\ln \left ({ {\sum \limits _{\gamma =1}^{M_{l} } {\exp \left ({ {-\frac {1}{N_{0,F} }\left |{ {Y^{g}(\beta )-H^{g}(\beta )X_{l,\gamma } } }\right |^{2}} }\right )} } }\right ).\notag \\ {}\end{align}$$

In order to prevent numerical overflow, the Jacobian logarithm [8] is used for calculating the last two polynomials in (21). From (20), a larger $\lambda ~(\beta )$ value means that it is more probable that the $\beta$ -th subcarrier is modulated by the $M_{h}$ -ary symbol. Therefore, after calculation of all the $n$ LLR values, the LLR detector decides on $k$ indices of ${\hat {I}_{h}^{g} }$ having maximum LLR values. Then, the remaining $n-k$ indices determine ${\hat {I}_{l}^{g} }$ . As the ED detector, the correspondent modulated symbols can be estimated by using (18).

Obviously, the LLR detector has the same search complexity as the ED detector. As for the calculation complexity, it requires a large number of additional division and logarithm/exponential operations besides the same number of multiplication operations as the LC-ML detector, which needs to be considered in practical application.

To reduce the errors of the estimation for $I_{h}^{g}$ and $I_{l}^{g}$ , a) ~ d) in III-B also can be used to improve the LLR detector at the cost of a slight increase of search complexity.

SECTION IV. Performance Analysis of DMIM-OFDM-PCA and Power Parameter Optimization

A. Performance Analysis

In this subsection, we evaluate the BER performance of DM-OFDM-CPA systems based on PEP assuming the optimal ML detection. Since the PEP events in different subblocks are identical [8], it is sufficient that the PEP events of a single subblock are investigated to estimate the overall system performance. According to (5), the received signals for the g-th subblock can be rewritten as:

$$\begin{equation} Y^{g}=X_{D}^{g} H^{g}+W^{g} \end{equation}$$ where $Y^{g}=[Y^{g}(1),Y^{g}(2),\ldots ,Y^{g}(n)]^{T}$ , $X_{D}^{g}$ is an $n\times n$ diagonal matrix with its diagonal elements given by the elements of $X^{g}$ , $H^{g}=[H^{g}(1),H^{g}(2),\ldots ,H^{g}(n)]^{T}$ and $W^{g}=[W^{g}(1),W^{g}(2),\ldots ,W^{g}(n)]^{T}$ , $W^{g}(\beta )=W(G(\beta -1)+g)$ .

The conditional PEP (CPEP), which is defined as the probability of transmitting $X_{D}^{g}$ and deciding on ${\hat {X}_{D}^{g} }$ conditioned on $H^{g}$ , is given as [8]

$$\begin{equation} \Pr (X_{D}^{g} \to {\hat {X}_{D}^{g} }\left |{ {H^{g}} }\right .)=Q\left({\sqrt {\frac {\delta }{2N_{0} }} }\right) \end{equation}$$ where $Q(\ast )$ is the Gaussian Q-function and $$\begin{equation} \delta =\left \|{ {(X_{D}^{g} - {\hat {X}_{D}^{g} } )H^{g}} }\right \|=(H^{g})^{H}AH^{g} \end{equation}$$ in which $A=(X_{D}^{g} - {\hat {X}_{D}^{g} } )^{H}(X_{D}^{g} - {\hat {X}_{D}^{g} })$ . According to [24], (23) can be approximated as $$\begin{align} \Pr (X_{D}^{g} \to {\hat {X}_{D}^{g} }\left |{ {H^{g}} }\right .)\cong \frac {1}{12}\exp \left({-\frac {\delta }{4N_{0} }}\right)+\frac {1}{4}\exp \left({-\frac {\delta }{3N_{0} }}\right).\quad \notag \\ {}\end{align}$$

Thus, the corresponding unconditional PEP (UPEP) can be obtained by

$$\begin{align} \Pr (X_{D}^{g} \to {\hat {X}_{D}^{g} })=&E_{H^{g}} \{\Pr (X_{D}^{g} \to {\hat {X}_{D}^{g} }\left |{ {H^{g}} }\right .)\} \notag \\\cong&E_{H^{g}} \left\{{\frac {1}{12}\exp \left({-\frac {\delta }{4N_{0} }}\right)+\frac {1}{4}\exp \left({-\frac {\delta }{3N_{0} }}\right)}\right\}\notag \\ {}\end{align}$$ According to [8], (26) can be calculated as $$\begin{align} \Pr (X_{D}^{g} \to {\hat {X}_{D}^{g} } )\cong \frac {1/12}{\det (I_{n} +{KA} /}+\frac {1/4}{\det (I_{n} +{KA} / {(3N_{0} )})} \notag \\ {}\end{align}$$ where $I_{n}$ denotes the $n\times n$ identity matrix and $K=E\{H^{g}(H^{g})^{H}\}$ is the covariance matrix of $H^{g}$ .

After the calculation of UPEP, the BER of DM-OFDM-CPA systems, which is approximately equal to the average bit error probability (ABEP) [19], can be estimated by

$$\begin{align} P_{e}\approx&P_{b} \notag \\\approx&\frac {1}{pn_{X} }\sum \limits _{X_{D}^{g} } {\sum \limits _{{\hat {X}_{D}^{g} } \ne X_{D}^{g} } {\Pr (X_{D}^{g} \to {\hat {X}_{D}^{g} })e({\hat {X}_{D}^{g} } ,X_{D}^{g} )} } \end{align}$$ where $n_{X} =2^{p}$ is the number of the possible realization of $X_{D}^{g}$ and $e({\hat {X}_{D}^{g} },X_{D}^{g} )$ denotes the number of bit errors for the corresponding pairwise error events. Since the interleaving technique is used in DM-OFDM-CPA, the spacing of any two subcarriers within a subcarrier group is usually lager than the coherence bandwidth of a practical channel. According to [14, Proposition 4], we have $K\approx I_{n}$ . Thus, (28) can be approximated as $$\begin{align} P_{e}\approx&\frac {1}{pn_{X} }\sum \limits _{X_{D}^{g} } \sum \limits _{{\hat {X}_{D}^{g} } \ne X_{D}^{g} } \big ( \frac {1/12}{\mathop {\prod }\limits _{\beta =1}^{n}~(1{+\Delta X^{g}(\beta )} /(4N_{0}))}\notag \\&+\,\frac {1/4}{\mathop {\prod }\limits _{\beta =1}^{n} (1{+\Delta X^{g}(\beta )} / (3N_{0}))}\big )e({\hat {X}_{D}^{g} } ,X_{D}^{g} ) \end{align}$$ where $$\begin{equation} \Delta X^{g}(\beta )=\left |{ {X^{g}(\beta )-{\hat {X}^{g}}(\beta )} }\right |^{2}\!. \end{equation}$$

B. Power Parameter Optimization

In DM-OFDM-CPA, as shown in Fig. 2, the subcarriers of $I_{h}^{g}$ and $I_{l}^{g}$ are respectively allocated different transmit power, $P_{h }$ and $P_{l}$ . When SNR is determined, i.e., the total power is fixed, the power ratio $R_{p} =P_{h} /P_{l}$ decides on their power allocation. To maximize the transmission performance of the DM-OFDM-CPA system, we carry out a numerical simulation to identify the optimum power ratio.

The simulated results are shown in Fig. 3. It describes the variation of the BER performance of the DM-OFDM-CPA system as a function of $R_{p}$ for different value of SNR = 10, 20, 30dB, when $N=128$ , $N_{g}=16$ , $n=4$ , $k=2$ , $L = 10$ . For a given SNR, the variation plots a concave curve. When $R_{p}$ is very low, the $M_{h}$ -ary and $M_{l}$ -ary PSK constellation points are too close to be detected. This causes a high BER to occur. When $R_{p}$ is very high, the $M_{h}$ -ary PSK constellation points are largely separated from the $M_{l}$ -ary PSK constellation points, but this reduces the $M_{l}$ -ary PSK constellation sizes because the total signal power is fixed. This causes a BER increase with increasing $R_{p}$ . Therefore, as shown in Fig. 3, for a given SNR, there is always an optimum $R_{p}$ , denoted as $R_{p}^{aq}$ , which can maximize the BER performance. In Fig. 3 (a), 8PSK and QPSK are used for the symbol modulation of $I_{h}^{g}$ and $I_{l}^{g}$ , respectively. It can be observed that $R_{p}^{aq}$ decreases as SNR increases. In Fig. 3 (b), the system adopts 16PSK and 8PSK modulation and the corresponding $R_{p}^{aq}$ also shows the same decreasing tendency. However, for a given SNR, $R_{p}^{aq}$ is always smaller than that of 8PSK/QPSK modulation. It should be attributed to the increase of the proportion of the $M_{l}$ -ary PSK modulation information.

FIGURE 3.

BER vs $\text{R}_{\mathrm {p}}$ in DM-OFDM-CPA When SNR = 10dB, 20dB, 30dB. (a) $M_{h}=8$ and $M_{l}=4$ , (b) $M_{h}=16$ and $M_{l}=8$ .

SECTION V. Simulation Results

To compare our proposed DM-OFDM-CPA scheme with the other OFDM-IM schemes [8], the BER performance of these schemes were evaluated via Monte Carlo simulations. In all simulations, the number of subcarriers is set to $N=128$ and the length of CP is 16. The frequency-selective channel [8] that has a CIR length of $L=10$ is used, and the channel fading coefficients and the noise variance are assumed to be perfectly estimated at the receiver. To make a fair comparison, the interleaving technique is used for all the schemes. In DM-OFDM-CPA, different $R_{p}^{aq}$ values are adopted for different SNRs to obtain the optimum performance.

In Figs. 4–​6, we compare the BER performance of the proposed DM-OFDM-CPA, DM-OFDM and OFDM-IM and ESIM-OFDM for three different SE values, 2.22bps/Hz, 3.11 bps/Hz and 4 bps/Hz when $n=4$ , $k=2$ . Among the comparison, they both adopt the ML detection method for demodulation at the receiver. To reduce the calculation, the proposed LC-ML detector in the DM-OFDM-CPA is also applied to DM-OFDM and OFDM-IM. In the DM-OFDM-CPA, the same orders are adopted for the modulation of the high-power and the low-power subcarriers to facilitate their comparison, i.e.,

FIGURE 4.

BER Performance Comparison between DM-OFDM-CPA, DM-OFDM, OFDM-IM and ESIM-OFDM with the spectral efficiency of 2.22bps/Hz under frequency-selective Rayleigh channel when $n=4$ , $k=2$ .

FIGURE 5.

BER Performance Comparison between DM-OFDM-CPA, DM-OFDM, OFDM-IM and ESIM-OFDM with the spectral efficiency of 3.11bps/Hz under frequency-selective Rayleigh channel when $n=4$ , $k=2$ .

FIGURE 6.

BER Performance Comparison between DM-OFDM-CPA, DM-OFDM, OFDM-IM and ESIM-OFDM with the spectral efficiency of 4bps/Hz under frequency-selective Rayleigh channel when $n=4$ , $k=2$ .

$M_{h} =M_{l}$ . As seen from Fig. 4–​6, the ESIM-OFDM and the OFDM-IM have almost similar BER performance. Compared to them, the proposed DM-OFDM-CPA and the DM-OFDM, which modulate all the subcarriers, both provide significant improvements in performance. In Fig. 4, at the BER level of 10⁻³, the gains of the improvement are respectively up to 3dB and 1dB. This is because that the higher order modulation, 16QAM, has to be employed in ESIM-OFDM and OFDM-IM for obtaining the same SE as DM-OFDM-CPA, which is more sensitive both to noise and to interference. Similarly, in Fig. 5 and Fig. 6, the ESIM-OFDM and the OFDM-IM have to adopt 64QAM and 256QAM to reach the SE of 3.11bps/Hz and 4bps/Hz, and the proposed DM-OFDM-CPA and the DM-OFDM obtain larger gains. Besides, the proposed DM-OFDM-CPA using the LC-ML detector and the DM-OFDM using the LC-ML detector have the same calculation complexity and it is always lower than that of the OFDM-IM using the LC-ML detector and the ESIM-OFDM using LC-ML. Furthermore, it is seen from Fig. 4 and Fig. 5 that at the BER level of 10⁻³, the proposed DM-OFDM-CPA attains 2 dB and 1 dB gain over the classic DM-OFDM. However, in Fig. 6, the proposed DM-OFDM-CPA has about 1dB loss at the BER level of 10⁻³. This should be attributed to the fact that $M$ -ary QAM modulation has better performance than $M$ -ary PSK when M is larger [14]. That is to say, if the modulation order is lower than 16, the proposed DM-OFDM-CPA performs better than the classic DM-OFDM.

In Fig. 7, the theoretical BER analysis based on PEP is compared with the simulated BER performance for the DM-OFDM-CPA with the spectral efficiency of 2.22bps/Hz under the frequency-selective Rayleigh fading channel. The theoretical BER values are calculated from (29). As seen from Fig. 7, at low SNR the theoretical analysis is very inaccurate, and the theoretical BER values are gradually close to the simulated values with increasing SNR. When the SNR is above 20 dB, the proposed theoretical analysis can provide accurate BER estimation.

FIGURE 7.

BER Performance Comparison between the theoretical BER analysis based on PEP and the simulated BER performance for DM-OFDM-CPA with the spectral efficiency of 2.22bps/Hz under frequency-selective Rayleigh channel when $n=4$ , $k=2$ .

Additionally, we investigate the BER performance of DM-OFDM-CPA with different detectors including the LC-ML, the ED, the LLR, the improved ED and the improved LLR detector. Fig. 8 shows the results of DM-OFDM-CPA with 8PSK/QPSK modulation for $n=4$ , $k=2$ . There is no doubt that the optimal detector, the LC-ML detector, shows the best performance. Compared with the LC-ML, the ED and LLR detector have some losses in BER performance, but they both impose a lower search complexity, specifically $o\left ({ {28} }\right )$ in comparison to $o\left ({ {52} }\right )$ for each OFDM subblock. As shown in Fig. 8, the ED detector has about 2.8 dB loss at the BER level of 10⁻³, while the LLR detector has a slight performance loss at low-to-medium SNR region. When the SNR is above 25 dB, the performance of the LLR detector becomes indistinguishable from that of the LC-ML detector. It is seen from Fig. 8 that the improved ED detector and the improved LLR detector decrease their own loss though slightly increasing the complexity as shown in Table 2. The improved ED detector gets 1dB gain over the ED detector, while the performance of the improved LLR detector is almost the same as the LC-ML detector. Therefore, the improved ED and LLR detector can perform well in practice, especially for a large $c$ . In addition, compared with the improved ED, the improved LLR detector obtains better performance, but the higher calculation complexity, especially a large number of additional division and logarithm/exponential operations, also needs to be considered in practical application.

FIGURE 8.

BER Performance Comparison between LC-ML, ED, Improved ED, LLR and Improved LLR detector in DM-OFDM-CPA with 8PSK/QPSK modulation under frequency-selective Rayleigh channel when $n=4$ , $k=2$ .

Finally, in Fig. 9, we compare the BER performance of DM-OFDM-CPA with 8PSK/QPSK modulation, DM-OFDM with QPSK and OFDM-IM with 8QAM modulation when $n=8$ and $k=3$ . It is known from (1) that $c$ has $2^{p_{1} }=2^{5}=32$ . Among the comparison, three different detectors, including the improved ED, the LLR and the improved LLR detector, are implemented for the DM-OFDM-CPA to reduce the search complexity, and the DM-OFDM and the OFDM-IM adopt the corresponding LLR detectors in [8] and [19]. As shown in Fig. 9, the DM-OFDM-CPA with LLR detector performs better than the classic DM-OFDM with LLR detector at high SNR region and at low-to-medium SNR region the performance is almost the same, despite the fact that the DM-OFDM-CPA achieves 20.3% higher SE. Compared with the OFDM-IM, the DM-OFDM-CPA shows 14.6% higher SE. Meanwhile, At the BER level of 10⁻³, the DM-OFDM-CPA with LLR detector obtains 2.5 dB gains over the OFDM-IM with LLR detector. Furthermore, the improved LLR detector further enhances the BER performance of DM-OFDM-CPA, especially at low-to-medium SNR region. As for the DM-OFDM-CPA with the improved ED, the same BER performance is shown at medium-high SNR region as the OFDM-IM with LLR detector.

FIGURE 9.

BER Performance Comparison between DM-OFDM-CPA with 8PSK/QPSK, DM-OFDM with QPSK and OFDM-IM with 8QAM modulation under frequency-selective Rayleigh channel when $n=8$ , $k=3$ .

SECTION VI. Conclusion

A new DM-OFDM scheme, called DM-OFDM-CPA, has been proposed in this paper. As the classic DM-OFDM, the DM-OFDM-CPA modulates all subcarriers to eliminate the limits of SE in OFDM-IM and partitions each subblock into two groups at the transmitter. Specifically, the two groups of subcarriers are set at different power levels and modulated by different $M$ -ary PSK symbols, respectively. At the receiver, three detection schemes have been employed to demodulate the received signal including a LC-ML detector, an ED detector and a LLR detector. Meanwhile, we have provided some additional measures to improve the performance of the ED and the LLR detector at the cost of a slight increase in complexity. Furthermore, the theoretical performance analysis has been performed based on PEP to estimate the BER of DM-OFDM-CPA. Then, the influence of power parameters on BER performance has been analyzed and for a given SNR, there is always an optimum power ratio that maximizes achievable transmission performance. The Monte Carlo simulation results have confirmed that the proposed DM-OFDM-CPA is capable of achieving a better BER performance than the classic DM-OFDM for a given SE when the modulation order is lower than sixteen. The results have also demonstrated that compared with the LC-ML detector, both ED detector and LLR detector produce some performance losses in DM-OFDM-CPA though reducing the search complexity, while the improved LLR detector provides similar performance to the LC-ML detector.

In this paper, we restrict our work on the transceiver design and the performance optimization of dual-mode OFDM-IM. In the future study, we will consider three or more modes OFDM-IM based on the power allocation of different constellation alphabets to obtain higher SE and better performance. Moreover, we will also investigate the optimization of the power allocation strategy, further improving the BER performance.