Low-Complexity Detection for Multiple-Mode OFDM with Index Modulation

Abstract

Multiple-mode orthogonal frequency division multiplexing with index modulation (MM-OFDM-IM) is a recently proposed multicarrier transmission technique, which transmits distinguishable constellations (modes) according to their full permutations. For an uncoded system, the maximum-likelihood (ML) detector achieves the optimal performance, but its computational complexity is extremely high due to a large number of symbol modes. Existing low-complexity detectors may output an illegal mode permutation, which leads to catastrophic errors in the demodulation. redIt is well known that sphere decoding algorithms can reduce the computational complexity effectively and achieve near-ML bit error rate (BER) performance. In this paper, employing a sphere-decoding-like algorithm, we propose a novel detector, which can achieve near-ML performance with considerably low computational complexity. For coded MM-OFDM-IM systems, we propose a simplified log-likelihood ratio (LLR) calculation algorithm to achieve near-optimal coded BER performance with considerably low computational complexity. Computer simulation and numerical results demonstrate that the proposed detectors can achieve outstanding performance at the cost of low computational complexity.

Introduction

Orthogonal frequency division multiplexing with index modulation (OFDM-IM) is a well known realization of IM in the frequency domain [1], [2]. As an improvement of classical OFDM, OFDM-IM can combat the frequency-selective fading caused by the multipath propagation in the wireless communication effectively [1], [2], [3], [4], [5], [6]. OFDM-IM can be used as an alternative of classical OFDM in many wireless communication standards, such as the 802.11a/g, 802.16, long-term evolution (LTE), and digital video broadcasting — terrestrial (DVB-T) standard for the European broadcast transmission of digital television and so on [7], [8], [9], [10], [11].

The concept of IM is first applied in multiple-input multiple-output (MIMO) systems, which is known as spatial modulation (SM) [12], [13], [14]. At each symbol time of SM, the information bits are conveyed by both the index of the activated antenna and the modulated symbol on it. In OFDM-IM, the indices of the activated subcarriers act as an information-carrying mechanism besides the ordinarily modulated symbols. Compared with classical OFDM, OFDM-IM achieves a lower bit error rate (BER) and a higher energy efficiency by exploiting the subcarriers’ indices as an additional information-carrying source [1], [2]. The spectral efficiency (SE) and the mutual information of OFDM-IM are analyzed in [15] and [16], which show that OFDM-IM has the potential to achieve a higher SE than classical OFDM. The optimal maximum-likelihood (ML) detection algorithm [1] and the low-complexity subcarrier-wise near-ML detector [17] are studied for OFDM-IM. In [18], MIMO-OFDM with index modulation (MIMO-OFDM-IM) is proposed to extend the frequency-domain IM into multiple antennas system. To reduce the detection complexity in MIMO systems, some minimum mean square error (MMSE) based detectors are proposed for MIMO-OFDM-IM. However, these detectors suffer from a severe BER performance loss compared with that of the ML detector [18], [19]. Based on the sequential Monte Carlo (SMC) theory, several detectors are proposed for achieving near-optimal BER performance with considerably low complexity [20], [21].

In (MIMO-) OFDM-IM, the inactive subcarriers themselves do not carry any information. To increase SE, a dual-mode OFDM-IM (DM-OFDM-IM) is proposed in [22], which replaces the zeros with symbols of another distinguishable symbol mode. However, the number of index-combinations of DM-OFDM-IM increases combinatorially as that of OFDM-IM. To fully exploit the potential of IM, a multiple-mode OFDM-IM (MM-OFDM-IM) is proposed in [23], which transmits symbols of different symbol modes on different subcarriers according to the permutations of symbol modes (PSMs) of each subblock. Since the number of PSMs increases factorially with the number of symbol modes, MM-OFDM-IM achieves a higher SE compared with classical OFDM-IM and DM-OFDM [22], [23].

MM-OFDM-IM has great potential to achieve higher energy-efficiency (EE), since the PSMs of each MM-OFDM-IM subblock conveys information bits without consuming additional energies. In MM-OFDM-IM, both the PSM and the transmitted symbols in the subblock need to be detected jointly. Thus, the detectors of MM-OFDM-IM are very complicated. The optimal ML detector for MM-OFDM-IM is studied in [23]. Its computational complexity is too high for practical systems when the number of symbol modes is large. To reduce the detection complexity, a low-complexity Viterbi-aided ML (Viterbi ML) detector and a subcarrier-wise log-likelihood ratio (SW-LLR) detector are proposed in [23]. Since the number of the used PSMs, i.e., the legal PSMs, is less than the total number of the PSMs, either the Viterbi ML detector or the SW-LLR detector will suffer from extremely high IM bit error rates when illegal PSMs are encountered.

In this paper, we study the design of low-complexity detectors for MM-OFDM-IM. Inspired by the channel decoding algorithms and the low-complexity detector in MIMO systems [24], [25], we regard the detection of estimate of each subblock as the process to find a path with optimal decision metric and legal PSM based on the most-likely transmitted symbols of each subcarriers. We propose a low-complexity sphere-decoding-like ML (SD-ML) detector for MM-OFDM-IM and build up a table of illegal paths for the SD algorithm to avoid illegal PSMs in the search process. In the SD-ML detector, we first calculate the most-likely transmitted symbol associated with each symbol mode for each subcarrier, and then detect the subblock based on the individual estimate of the symbol transmitted on each subcarrier. To fully exploit the calculation results at each subcarrier, we first concatenate the most-likely transmitted symbol with the optimal decision metric at each subcarrier to form the subblock and check the validity of the PSM. Thus, the SD algorithm is performed only when the obtained subblock has illegal PSM. Therefore, the computational complexity of the proposed SD-ML detector decreases as the SNR increases.

For coded MM-OFDM-IM, the soft value, such as the LLR of each bit, is needed for the soft decoder at the receiver. The maximum a posteriori-Log (MAP-Log) LLR calculation algorithm can achieve optimal BER performance [26]. However, it requires the likelihood probabilities of all possible transmitted symbol vectors, and the detector becomes computationally infeasible when the number of symbol modes is large. The Max-Log LLR calculation algorithm achieves extremely low computational complexity; however, it suffers from an unavoidable BER performance loss compared with that of the optimal MAP-Log LLR algorithm [27], [28], [29], [30]. Inspired by the SD algorithm [24], [25], [27], we propose a low-complexity LLR calculation algorithm, in which we calculate a radius associated with the SNR and reduce the reserved most likely symbols for each symbol mode step-by-step. Then, the LLR is calculated based on the reserved most likely PSMs and the symbols of each symbol mode. Finally, numerical simulation results demonstrate the superior BER performance and low computational complexity of the proposed algorithms.

The remainder of this paper is organized as follows. In Section 2, the system model of MM-OFDM-IM is presented. A low-complexity sphere-decoding-like ML detector is proposed in Section 3. In Section 4, we study the LLR calculation algorithms for MM-OFDM-IM and propose a low-complexity LLR calculation algorithm. The computer simulations and numerical results are provided in Section 5. Section 6 concludes the paper.

Notation: $\mathbf{X}$ denotes a matrix and $\mathbf{x}$ is a column-vector. ${\left(\cdot \right)}^T$ and ${\left(\cdot \right)}^H$ denote transposition and Hermitian transposition of a matrix or a vector, respectively. $\text{diag}\left\{\mathbf{x}\right\}$ returns a diagonal matrix whose diagonal elements are included in $\mathbf{x}$. ${\mathbf{I}}_{\mathbf{k}}$ denotes the $k\times k$ identity matrix. $x\sim \mathcal{CN}\left(0,{\sigma }_x^2\right)$ represents the distribution of a zero mean circularly symmetric complex Gaussian random variable $x$ with variance ${\sigma }_x^2$. $⌊\cdot ⌋$ is the integer floor operation and $0̸$ denotes the empty set. $\left|\cdot \right|$ denotes the absolute value of a complex vector. $\mathcal{S}$ denotes the complex symbol constellation of size $\bar{M}$ and $\mathcal{O}\left(\cdot \right)$ denotes the order of detection complexity with respect to the constellation size. $\mathrm{FFT}\left\{\cdot \right\}$ denotes the fast Fourier transform (FFT) operator.