Robust Walsh–Hadamard transform-based spatial modulation

Abstract

This paper presents a combination of spatial modulation (SM) and Walsh–Hadamard transform (WHT) for achieving robustness against wireless fading channels. The WHT operation merge together SM transmitting symbols, where the number of symbols depends on the order of WHT matrix. A WHT matrix of order two merges two transmitting symbols by their sum and difference, therefore both the symbols mutually carry their information which results in a diversity gain. A bad noise sample might corrupt completely a symbol, while transmitting the combined symbols allows to average out the noise and have a lower noise variance in average. This kind of approach has been used for generalized frequency division multiplexing (GFDM) and space–time block coding (STBC) with WHT matrix of order two, and thus achieves a diversity gain of two. Unlike, the conventional communication schemes like GFDM and STBC, the use of WHT in SM is not straight forward due to the two information carrying domains, i.e. antenna index, and transmitted symbols. A simple WHT of the SM symbols causes a degraded bit error rate (BER) performance due to the fact that the WHT of these symbols results in some transmit symbols having zero energy. To overcome this problem, we use the phase shifting trick. The required phase shift depends on the order of the WHT matrix and M-ary phase shift keying (M-PSK) modulation scheme. Before the WHT of the transmit symbols, the phase of each constellation point is shifted by a certain angle that is different for each symbol and thus overcomes the problem of transmit symbols having zero energy after the WHT operation. The proposed scheme is evaluated under maximum received ratio combining (MRRC) and optimal Maximum likelihood (ML) detector. The complexities of both MRRC and ML are calculated for WHT–SM. Extensive simulations are presented for the proposed WHT–SM system with and with-out the phase shifting approach. It is shown, how the WHT matrix order and modulation level M effects the performance of WHT–SM system with-out the phase shift approach. The impact of high diversity gains is discussed and shown in the simulation results. The proposed scheme achieves a superior BER performance compared to the conventional SM scheme.

Introduction

A high spectral efficiency is the main requirement of future radio networks. Limited spectrum resources have forced the research community to formulate new techniques and methods to cope with the future requirements. Multiple-input–multiple-output (MIMO) technology is a key solution to this problem (see [1] and the references therein for MIMO). The main disadvantag of any MIMO scheme is the high complexity, Inter-Channel Interference (ICI) which arise due to the superposition of multiple transmit sequences transmitted by multiple antennas [2]. Another drawback of MIMO systems is the Inter–Antenna Synchronization (IAS) [3]. The transceiver design for such schemes is a hectic job as the number of receiving antennas must be greater than the number of transmitting antennas [2]. This limits the application of MIMO systems in hand-held devices due to device processing capabilities, significant energy requirements and the physical size of these devices. To overcome these issues one ground breaking work on MIMO is presented in [3] and is known as spatial modulation (SM). SM boosts the spectral efficiency by introducing a new information carrying source known as spatial domain or transmitted antenna index. In SM, the transmitted symbols and the active transmitting antenna index jointly carries the useful information bits, resulting in an improved spectral efficiency [3]. In SM only one transmitted antenna is activated for data transmission at a particular transmitting time. The active antenna is decided by the incoming information bits by splitting them into antenna index selector bits and signaling bits. The receiver of SM can successfully decode the information bits by estimating the transmitting antenna index and symbols. Due to the single Radio Frequency (RF) chain, SM does not encounters the problems of ICI and IAS and thus the complexity of SM receivers decreases. Different version of SM are proposed after [3]. To further improve the spectral efficiency of SM based systems, a scheme known as generalized spatial modulation (GSM) is given in [4], which is inspired by the concept of generalized space shift keying (GSSK) [5]. In GSM multiple transmitting antennas are activated. The combination of these antennas also carries useful information bits. Unlike SM, where the total number of transmitting antennas are given by $2^x$, where x is a positive integer, GSSK can make use of any number of transmitting antennas. A combination of space–time block coding (STBC) and SM is presented in [6] and is known as space–time block coded spatial modulation (STBC-SM). Differential spatial modulation is presented in [7], [8]. A Trellis coding based spatial given in [9]. The concept of SM is applied to optical communication, the work is presented in [10]. In every kind of SM scheme two information carrying domains, i.e., the transmitted signal and the space/antenna index is used. Which renders the detection of the SM signals, complex. A maximum received ratio combining (MRRC) based detection technique is used in [3]. An optimal maximum likelihood (ML) based detection technique is provided in [11]. To further reduce the complexity of SM receivers a sphere decoding technique are proposed in [12], [13], [14].

Walsh–Hadamard transform (WHT) [15] is a well-known transform used in image processing and communication systems. Orthogonal rows of the WHT matrix are used as orthogonal variable spreading factors (OVSFs) in communication systems like code division multiple access (CDMA) [16]. A robust WHT-based generalized frequency division multiplexing (GFDM) is presented in [17]. The input symbols mapped to each subcarrier are initially transformed by WHT with a transformation level of two. The proposed scheme achieves a superior BER performance compared to conventional GFDM systems. An Alamouti scheme with WHT is developed in [18], where the transmit symbols are first WHT transformed and merged together to achieve diversity. The proposed joint WHT and Alamouti scheme achieves a better BER performance than the conventional Alamouti scheme.

Even though WHT is used in this paper, our approach is different from [17] and [18]. In our proposed scheme, the WHT is used for merging two or more transmit symbols of the SM to achieve a diversity gain. A simple WHT of the SM symbols results in a degraded BER performance; therefore, a phase shifting solution is proposed to solve the problem. The phase of each transmit symbol is shifted by a certain angle followed by the WHT; thereby, each constellation point mutually carries the information of the other constellation points. To recover the original constellation points, the received symbols are again transformed by WHT followed by an inverse phase shift of each symbol to recover the original transmit constellation points. The proposed scheme is compared with the conventional SM. For a fair comparison, a MRRC and ML based detection techniques are used for detection of both the antenna index and the symbol estimation. The complexities of both these detectors are calculated and discussed. A WHT of the input symbols achieves a diversity, this is addressed in [17] for WHT matrix of order two, to the authors' best knowledge, the effect of higher order WHT matrices on the diversity gain has not been studies yet in any communication system. The application of WHT to SM is also not reported yet. We investigate WHT-based SM and discuss the effect of higher order WHT matrices on the BER performance of SM.

The rest of the paper is organized as follows. Section 2 presents the proposed system model for WHT-based SM and explains the core concepts of Walsh–Hadamard transform, transmitter modeling and the significance of phase shift, detection of WHT–SM signals, and the complexities calculation of MRRC and ML based detection methods for WHT–SM. Simulation results are presented in Section 3. Finally Section 4 concludes this paper.

Notations:

We notate all equations and symbols in italics. A WHT matrix of order K is denoted by ${\mathrm{\Omega }}_K$ and the channel information by H. The Hermitian conjugate transpose is represented by ${(.)}^H$ and the transpose by ${(.)}^T$. The term ${\mathbf{A}}_{a,b}$ represents a matrix with a rows and b columns.