Abstract

At each channel use, the complex quadrature spatial modulation (CQSM) transmits two signal symbols drawn from two disjoint modulation sets. The indices of the antennas from which symbols are transmitted also carry information. In the improved CQSM (ICQSM), an additional antenna is used to transmit the second signal symbol only when the indices of the antennas to be used for transmission are equal. Conventionally, the second modulation set is a rotated version of the first, where the rotation angle is optimized such that the average unconditional error probability (AUP) is reduced. In this paper, we propose a low-complexity method to design the PSK modulation sets based on reducing the AUP. After introducing min-BER and max-, two exhaustive search methods, we analytically show that the AUP depends on Euclidean distance between transmitted vectors, which in turn depends on the power of signal symbols, the Euclidean distance between the symbols of each modulation set, and the Euclidean distance between the symbols of the two sets. The optimal rotation angle is analytically derived for any modulation order and the radii of the modulation sets are optimized such that AUP is reduced for a wide range of system configurations. The simulation results show that more than 3 dB of power gain is achieved in the case of 16PSK, where higher gains are achieved for higher modulation orders. These gains are achieved at no computational cost because the optimization does not depend on the channel realization.

1. Introduction

Index modulation (IM) uses the indices of a given resource of the communication system to convey information to the receiver [1]. These indices represent distinct antennas, subcarriers [2], spreading codes [3], or polarities [4]. A multimode orthogonal frequency division multiplexing (OFDM) with IM was proposed in [5], where all the available subcarriers are utilized for transmission and multimodes, that is, signal constellations, are used. IM is also applied to cluster-based wireless sensor networks in [6]. Spatial modulation (SM) is a special case of IM in which information is transmitted not only through signal symbols but also through the index(es) of the antenna(s) from which symbols are transmitted or to which they are designated. This multiple-input multiple-output (MIMO) gain is achieved while maintaining the transmitter cost as moderate as that of a single-input system [1, 7, 8]. In the conventional spatial modulation (SM), a single signal symbol, drawn from an M-ary phase shift-keying (PSK) or quadrature amplitude modulation (QAM), is transmitted from a single antenna whose index also carries information [9]. SM is generalized in [10], where a combination of transmit antennas are used to transmit the same signal symbol. The goal of this generalization is to reduce the number of required transmit antennas to achieve a given spectral efficiency. Also, the constraint of the log-two number of transmit antennas imposed in the conventional SM is relaxed.

A precoding-aided SM (PSM) system was proposed in [11] and generalized in [12]. PSM precodes the single signal symbol using a channel-based precoder such that the symbol is received only by the designated receive antenna whose index carries information. The precoding matrix is generated based on certain performance criterion such as zero-forcing or minimum mean square error [13]. A macrodiversity PSM (MD-PSM) scheme is proposed in [14], where two base stations simultaneously communicate with a single user through decentralized precoding.

Quadrature SM (QSM) was proposed in [15], where the spatial constellation was extended to the in-phase and quadrature dimensions. The real part of the single signal symbol is transmitted from the in-phase spatial dimension and the imaginary part from the quadrature spatial dimension. Compared to SM, QSM increases the achieved spectral efficiency by the number of bits transmitted over one spatial dimension. In [16], a differential QSM (DQSM) is proposed to render the recovery of the transmitted information at the receiver possible without the channel knowledge.

In complex QSM (CQSM), two signal symbols are transmitted over the in-phase and quadrature dimensions, leading to further improvement in the spectral efficiency [17]. To alleviate the ambiguity at the receiver, the two signal symbols are drawn from two disjoint modulation sets. When both symbols are transmitted from the same antenna, their sum belongs to the Minkowski sum of the two sets from which the two symbols are drawn. The modulation sets are therefore optimized to minimize the unconditional average bit error rate. The second modulation set is a rotated version of the first. The optimization in [17] reduces to finding the optimal rotation angle that minimizes the average unconditional bit error.

In [18], an improved CQSM (ICQSM) system was proposed, where, compared to CQSM, the transmitter is equipped with an additional antenna. This antenna is used only when the two spatial symbols are equal; that is, both signal symbols are transmitted from the same antenna in CQSM. In this case, the first symbol is transmitted from its designated antenna and the second from the additional antenna. This strategy tremendously reduces the density of the signal symbols in the Euclidean space, leading to a gain of as much as 20 dB of signal-to-noise ratio (SNR) in the case of 64PSK. Same as in the case of CQSM, the second modulation set is a rotated version of the first.

To reduce the bit error rate (BER), the design of the modulation set or the combination of the number of transmit antennas and the size of modulation set of the SM scheme is of interest. In [19], a bit-to-symbol mapping is proposed assuming channel knowledge at the transmitter, where the performance of the proposed scheme outperforms an earlier scheme proposed in [20]. In [21], the optimization of the number of transmit antennas and the size of the modulation set was conducted with and without the channel knowledge at the transmitter. The effect of the full or partial knowledge of the channel at the transmitter on the design of the spatial modulation schemes was investigated in [22].

The goal of this paper is to design the two modulation sets of the ICQSM to improve the average error performance. We assume that both base stations utilize phase shift keying modulations with the same order [23]. Against the related literature, the contributions of this paper are as follows:1.We introduce two exhaustive search methods of the optimal/suboptimal modulation sets: (1) minimum BER (min-BER), where the modulation sets are optimized to minimize the instantaneous error rate for a given channel realization, and (2) max-, where the minimum Euclidean distance among the received vectors for a given channel realization is maximized. The first method is optimal and the second is suboptimal. Both methods, as shown below, require high computational complexity and the optimal modulation sets should be updated once the channel changes.

The rest of this paper is organized as follows. In Section 2, the system model and related works are introduced. In Section 3, we introduce the optimal min-BER and the suboptimal max- methods, and propose a low-complexity optimization method that reduces the unconditional average BER. Simulation results and related discussions are given in Section 4. Conclusions are drawn in Section 5.

2. System Model and Related Works

2.1. System Model

Consider a MIMO system with transmit and receive antennas. In the sequel, is assumed to be a power of two; hence is an integer. The received vector is given bywhere and are the channel matrix and the noise vector whose elements are i.i.d. centered circularly-symmetric complex Gaussian and have a variance of one and , respectively, and is the transmitted vector.

2.2. Spatial Modulation

In SM, a single symbol , with , is transmitted from the th transmit antenna, where both and carry information and is the number of bits carried by each signal symbol [9]. The received vector is given bywhere is the transmitted vector and and are the th columns of and the identity matrix, respectively. The spectral efficiency of SM is () bits/s/Hz.

2.3. Quadrature Spatial Modulation

In QSM, at each channel use, a single signal symbol is transmitted and the spatial constellation is extended into in-phase and quadrature dimensions. The real part of the signal symbol is transmitted on the in-phase dimension; the imaginary part is transmitted on the quadrature dimension [15]. Let and be the real and imaginary parts of ; then the received vector is given bywhere , , and carry information. Based on (3), the spectral efficiency of QSM is () bits/s/Hz.

2.4. Complex Quadrature Spatial Modulation

Instead of transmitting a single signal symbol, CQSM transmits two symbols on the two spatial dimensions [17]. Let and be two signal symbols transmitted from the th and th transmit antennas; then the received vector is given byNote that , , , and carry information to the receiver. Therefore, the spectral efficiency of CQSM is bits/s/Hz. Based on (4), we make the following remarks: 1.Since the index of the antenna from which each symbol is transmitted is random, the symbols and should be distinguishable so that the receiver can recover them. This implies that and should be disjoint.2.If , one transmitted symbol still belongs to and the second belongs to .3.If , the system in (4) reduces to where , with , and denotes the Minkowski sum. The size of is . The full modulation set at the transmitter becomes . For instance, if and are 16PSK sets, . While CQSM outperforms QSM for the same spectral efficiency, the emergence of increases the density of symbols in the Euclidean space, leading to a reduction in the Euclidean distance among symbols and hence to a degradation in the error performance.

In CQSM, the set is derived from as follows:where the optimal value of the rotation angle minimizes the bit error rate.

2.5. Improved CQSM

ICQSM is proposed to tremendously reduce the size of of CQSM, resulting in an improved error performance. In ICQSM, the transmitter is equipped with an additional antenna that is used to transmit the second signal symbol only when . Therefore, the received vector is given by

where . As a consequence of this strategy, the transmitted vector always contains exactly two nonzero elements and . Therefore, the modulation set at the transmitter becomes . Assuming the base station is using 64PSK modulation sets for both and , and symbols in the case of ICQSM and CQSM, respectively. This huge reduction in the size of and the density of signal symbols in the Euclidean space increases the Euclidean distance between the signal symbols, leading to an improved bit error performance. For instance, ICQSM outperforms CQSM by more than 20 dB, assuming 64PSK modulation sets [18].

An Example. Let and let the message to be transmitted be . The message is split into four parts as follows:

where the size of and is equal to and that of and is equal to . The bit vectors and modulate the two signal symbols and , respectively, where and . The bit vectors and modulate the indices of the antennas, and , from which and are transmitted, respectively. In this example, .

In the case of CQSM, the transmitted vector is given byNote that includes the single nonzero element at the 6th position. The nonzero element in this case belongs to the Minkowski sum of the two sets and , denoted by , where and . The size of the Minkowski sum of the two sets is the product of the size of the two sets. This increase in the number of signal symbols in the Euclidean space leads to degradation in error rate.

To overcome this problem, the transmitter in the ICQSM system is equipped with an additional antenna that is used to transmit only when . Therefore, the transmitted vector is given byAccordingly, there are always two active transmit antennas transmitting two signal symbols; the first belongs to and the second to . This reduces the number of signal symbols in the Euclidean space, leading to an improvement in the error rate while keeping the receiver complexity intact. When , the additional antenna is muted and the remaining elements of are equivalent for both CQSM and ICQSM systems. It is worth mentioning that ICQSM with an additional antenna outperforms CQSM in terms of the BER, while the latter system uses more antennas in total (the sum of the transmit and receive antennas).

3. Proposed Constellation Design

Let , be two transmitted vectors, and the corresponding received vectors are and , respectively. The conditional pairwise error probability (PEP) of the maximum-likelihood receiver is given by [24, 25]whereis the Gaussian tail probability function or simply the -function, and and are the squared Euclidean distance at the transmitter and the receiver, respectively.

3.1. max-: Maximizing the Minimum Euclidean Distance

The probability of the conditional error of the ML receiver is approximated by the sum over all possible PEPs, which is given bywhere .

At high values of SNR, (13) simplifies to

where

and is the number of neighbor points. The -function in (14) is a monotonically decreasing function of . Therefore, the error may be decreased by maximizing . The modulation sets are therefore chosen such that

The optimization in (16) depends on the instantaneous channel realization and the symbol difference (). The ICQSM symbol is defined as follows:

Therefore, for given and modulation sets and , the computation of requires precomputing and . Assuming , computing these two terms requires real multiplications. For each pair of symbols , computing the norm in (16) requires real multiplications, leading to a total computational complexity of

For instance, for and , evaluating (16) for each pair of (, ) requires 134,187,520 real multiplications. In this method, only the squared minimum distance of the linear transformation is considered as the search metric. Therefore, even though this method is suboptimal in terms of the BER performance, its computational complexity is very high. Furthermore, this computation should be performed for each channel realization.

3.2. min-BER: Minimizing the Conditional Error Rate

Based on (11) and referring to [26], the upper bound of the conditional BER is formulated as follows:where is the Hamming distance between and , that is, the number of errors associated with the event . The optimal modulation sets are therefore given by

Evaluating (20) for each pair requires (1) computing , (2) evaluating the -function times, and (3) real multiplications for multiplying the -function by . The first term is computed in (18). The exact computation of the -function is complex as shown in (12). Alternatively, several approximations of the -function, which vary in terms of computational complexity and accuracy, were proposed in [27, 28]. In [28], the -function is approximated using a second-order exponential model as follows:

where the real numbers , , and are the fitting parameters. Evaluating (21) requires 3 real multiplications and an evaluation of the exponential function, which in turn requires real multiplications, with denoting the number of digits of precision (Theorem 10.1 in [29]). For , , leading to a total of 42 real multiplications to evaluate (21). This leads to a total computational complexity of

The search for the optimal modulation sets depends on the instantaneous channel realization . Assuming that the optimization is performed over a moderate number of ten combinations (), the search requires ten times the number of real multiplications of (22). This turns the search unfeasible for large and .

A method to reduce the computational complexity is explained as follows. The -function in (19) is monotonically decreasing; that is, if . For instance, . This implies that a greedy algorithm can be used to exclude the computation of the -function for arguments larger than a given threshold. This proposal is of interest at high values of SNR as the denominator of the argument of in (19) becomes small. However, the computational complexity of this algorithm is still high.

3.3. Reducing the Unconditional Error Rate

The performance gains achieved through choosing the modulation sets that minimize the receive Euclidean distance in (16) or those that minimize the BER in (20) come at a high computational cost. Since both search problems depend on the channel realization , computations should track the change in the channel and the transmitter should have perfect knowledge of the channel state information. Suboptimal modulation sets are obtained through assuming the unconditional BER as the cost function for the search problem on hand. The unconditional BER is obtained by averaging the upper bound in (19) over , resulting in

where the unconditional PEP is given by:andAssuming that the elements of are independent and circularly-symmetric Gaussian with mean and variance of zero and one, respectively, is given by

The asymptotic unconditional PEP, obtained by taking the Taylor series of (24) and ignoring the higher order terms, is given bywhere is the signal-to-noise ratio (SNR) [15].

It turns out that, for large values of , the unconditional average BER in (27) depends only on and . While the effect of on the error rate is evident, the effect of requires further explanation. The value of depends on the mutual relationship between , , , and . For these four parameters, there exist 64 relational conditions. Among these conditions, only fifteen are not contradictory, which lead to the following seven different values of :

Let be the frequency of the th condition leading to the th value of . The expressions of these frequencies are given as follows: