In this paper performance of modulation diversity with multiple-input multiple-output transmission is studied over flat Nakagami-m fading channels with arbitrary fading parameter m. In the system, orthogonal space-time block coding and maximal ratio combining like combiner are used for transmission and reception, respectively. Exact pairwise error probability expression is derived to observe performance of the system. Moreover, in order to obtain the diversity order of the system, asymptotic pairwise error probability expression is also derived. Optimum rotation angles are analytically obtained for binary and quadrature phase shift keying modulations. Theoretical results are validated by Monte Carlo simulations.
Diversity techniques are useful to combat the effects of fading that have disruptive effects on the performance of the wireless communication systems. Antenna, time and frequency diversity or combinations of these techniques improve the error performance of communication systems significantly. Antenna diversity can be implemented at the transmitter and/or receiver side [1]. Space Time Block Coding (STBC) [2] and [3] is the most common transmit diversity technique, which does not require channel state information at the transmitter. Orthogonal STBCs (OSTBC) have a simple decoding scheme and achieve full diversity order [2] and [3]. Additionally, Maximal Ratio Combining (MRC) is the optimum receive diversity technique, which maximizes the received Signal-to-Noise Ratio (SNR) [1].
A different kind of diversity technique from antenna diversity is signal space diversity, also known as modulation diversity, which uses rotated signal constellation and component interleaving at the transmitter, and component de-interleaving at the receiver [4] and [5]. In the signal constellation of conventional modulation techniques such as M-ary Phase Shift Keying (M-PSK), any symbol always has one common in-phase or quadrature component with the other symbols (as shown in Fig. 1 for Quadrature Phase Shift Keying (QPSK)). For this reason, component interleaving alone does not improve the error performance of the system considerably. However, by using rotated signal constellation, in which there is no common component between any two symbols (as shown in Fig. 1), together with component interleaving, which has components affected by independent fading coefficients, diversity gain can be obtained [6] and [7].
Error performance of OSTBC has been investigated in [8] and exact Symbol Error Probability (SEP) expressions have been derived for Independent Identically Distributed (i.i.d.) Nakagami-m channels. In addition, in [8] error performance of OSTBC has been studied for i.i.d. keyhole Nakagami-m fading channels by numerical integration. Asymptotic SEP performance of OSTBC has been examined in [9] for Independent Non-identically Distributed (i.n.d.) Nakagami-m fading channels.
Modulation diversity with Alamouti scheme has first been presented in [10]. Authors of [10] have investigated the performance of the system by simulations for Rayleigh fading channels and obtained optimum rotation angle for QPSK. Performance of modulation diversity with Alamouti scheme has also been investigated by simulations in [11] in the presence of channel estimation errors for Rayleigh fading channels. In [12], the system proposed in [10] has been studied based on asymptotic Pairwise Error Probability (PEP) approach over Rayleigh fading channels. In [13], authors have considered modulation diversity with Orthogonal Frequency Division Multiplexing (OFDM) and OSTBC transmission studied by simulations for two; three and four transmit antennas and Rayleigh fading channels. In [14], modulation diversity has been adapted to Vertical Bell Labs Layered Space-Time (VBLAST) with OFDM system, and the authors have investigated performance of the system with simulations over Rayleigh fading channels. In [15], authors have studied modulation diversity with bit interleaved coded modulation, spatial division multiplexing and OFDM, and obtained performance results via simulations over Rayleigh fading channels. In [16], authors have introduced a system in which modulation diversity and VBLAST are used and the components of the n-dimensional signal constellation are transmitted over n-transmit antennas by using Space-Time (ST) encoder. Modulation diversity has been simulated with coded Multiple-Input Multiple-Output (MIMO) system in which components of rotated n-dimensional signal constellation are transmitted over n-transmit antennas by using ST encoder in [17]. In [18], ST coding from Coordinate Interleaved Orthogonal Design (CIOD) has been designed and its performance has been analyzed in detail over Nakagami-m fading channels in [19].
In this paper, performance of modulation diversity technique with OSTBC transmission (i.e., space-time block codes from orthogonal design but not CIOD) is investigated for Nakagami-m fading channels. In the system, OSTBC and MRC like combiner are considered as transmission and reception techniques, respectively. We obtain exact PEP expression of the system for flat Nakagami-m fading channels with arbitrary fading parameter m. In addition, we derive the asymptotic PEP expression to obtain the diversity order of the system, and validate theoretical results by Monte Carlo simulations. Optimum rotation angles of Binary Phase Shift Keying (BPSK) and QPSK modulations are analytically obtained.
The remainder of this paper is organized as follows. In Section 2, the system model is described. Performance analysis is presented in Section 3. In Section 4, numerical and simulation results are given. Finally, conclusions are drawn in Section 5.
In this paper, performance of OSTBC system with modulation diversity has been analyzed over Nakagami-m fading channels for arbitrary m and numbers of transmit and receive antennas. Exact and closed-form PEP expression, which provides a union bound for SEP, has been derived. Besides, by deriving an asymptotic PEP, diversity order of the system has been obtained. Theoretical SEP results are validated by Monte-Carlo simulations. Results show that by employing modulation diversity, the diversity order of OSTBC system is doubled, and substantial performance improvement is obtained. In addition, the optimum rotation angle changes with the diversity order for QPSK while it is fixed for BPSK.
