Performance Analysis of Spatial Modulation-Multiple Input Multiple Output with Transmit Antenna Selection over Nakagami-m Fading Channels

Abstract— In this paper, the outage performance analysis of spatial modulation multiple input multiple output (SM-MIMO) system and transmit antenna selection (TAS) in Nakagami-m fading environment is proposed. The SM uses both the antenna indices as well as the signal constellation to transmit information bits. Further, TAS is employed in the proposed SM-MIMO to maximize the receive signal power. The closed form expression for the outage probability of the SM MIMO system with TAS for Nakagami-m channel is derived. Analytical results show that the outage performance of the proposed system with TAS is significantly better than SM without TAS. Keyword Spatial Modulation, Multiple input multiple output, Transmit antenna selection, Nakagami fading channels.

In the recent studies on SM and SSK techniques, bit error rates and outage performance are analysed for more general fading channels [12]- [14], such as Nakagamim channels. There is no diversity gain for systems without antenna selection even if m is varied. In [14] it is shown that to obtain diversity gain in Nakagamim channels, the use of TAS techniques in the classical SM and SSK systems can be done.
In this paper the outage probability of SM MIMO with TAS in Nakagami-m fading channels is analyzed for arbitrary value of m. A closed form analytical term for the outage probability is derived. Analytical results show that the outage performance of the proposed system significantly improved by TAS compared to conventional system of SM without TAS.
The rest of the paper is organized as follows. In Section II, the system model considered is described. In Section III, the closed-form outage probability expression of the system is obtained. The closed-form expression for a special condition of Nakagamim channel is also obtained. Comparisons of computer simulation and analytical results are given in Section IV. Finally, Section VI concludes the paper.
II. SYSTEM MODEL Consider a point to point communication system model between Node A and Node B. It is assumed that both the nodes are equipped with t M transmit antennas and r M receive antennas as shown in the Fig. 1. In this proposed work s M transmit antennas are selected which maximize the received SNR at the receiver from the available t M antennas. The information bits are mapped to the selected s M antennas and modulated symbols. s M should be a power of two since mapping is based on the binary information bits in SM. It is assumed that perfect channel state information (CSI) is available at the receiver and the transmitter doesn't require prior knowledge of the channel. At the receiver, there are r M receive antennas and maximum ratio combiner (MRC) is used. The received signal vector y is given as Where s is the information symbol transmitted from transmitter S 1 , w is the additive Gaussian noise vector. Also, vector h  is the th ), so h  is the channel coefficient vector between the th  transmit antenna and receive antennas and it can be given as , , Here, complex channel coefficient , The PDF of Nakagamim distribution is given by where m is the fading parameter,  .
 is gamma function [17, (8.310.1)]. Also,  is defined In the proposed work 1   is assumed. The phase of the channel coefficients is statistically between   0, 2 of the random variables which are independent of each other and are uniformly distributed.
In proposed technique, antenna selection is based on the maximization of norms of the channel coefficients. Norm square of the channel coefficients for the th  transmit antenna can be given as where j A is the chi-squared distributed random variable with 2 r mM degrees of freedom. The norms of channel coefficients between transmit and receive antennas are ordered in an ascending order. s M transmit antennas corresponding to the largest channel norms are selected from t M transmit antennas and SM is applied to the selected transmit antennas. Bandwidth efficiency of the system, in terms of bits/s/Hz, where R b data rate is, B is bandwidth. Also, G represents the symbol constellation size. As seen, the bandwidth efficiency increases as the number of selected antennas s M and/or constellation size G increases.

III. OUTAGE PROBABILITY ANALYSIS
In this section, analytical expressions for the outage probability are derived for the proposed network. The outage probability that the instantaneous end-to-end received SNR   end  is lower than a predetermined is the PDF of the instantaneous received SNR. In order to simplify the analysis, we assume that the antenna index error at the receiver is negligible. It is known that this is a good approach to determine receiver performance of the SM systems when the constellation size G is sufficiently more than the numbers of transmit antennas [7]. Therefore th  is defined as Also the CDF is calculated as [ Using Binomial expansion Substituting the above equation (11) the PDF of instantaneous SNR is given as The outage probability for a SM-MIMO system with TAS is given as The above integral (14) is calculated using [17, (3.944.3)], and the outage probability expression for SM-MIMO with TAS system is given as  is the lower incomplete gamma function. The closed-form outage probability for the Rayleigh channel ( 1 m  ) is given as follows for The outage probability of the SM MIMO systems without TAS in the Nakagamim fading channels, The above integral (17) is calculated using [17, (3.944.3)], the outage probability for a SM-MIMO system without TAS is given as IV. NUMERICAL RESULT In this section, the outage performance of the proposed SM-MIMO with TAS and SM-MIMO without TAS are compared using the analytical expressions derived in Section III. The list of parameters for the outage analysis is listed in Table I. The outage performance of the proposed SM-MIMO point to point network is shown in Fig. 2 2 10  for the proposed system, the minimum SNR requirement is around 3 dB when 0.5 m  . As the fading parameter is increased to 1, 2 and 3 the minimum SNR requirement is around -2 dB, -3 dB and -2.75 dB respectively. Whereas in the case of conventional SM MIMO system without TAS the minimum SNR requirement is 22 dB. A significant improvement in the outage probability is observed for the proposed system over conventional SM MIMO system without TAS by increasing the fading parameter.

V. CONCLUSION
In this paper the outage performance of the SM MIMO with TAS in point-to-point network is proposed to maximise the received signal power. A closed form analytical expression is derived for the end-to-end outage probability of the proposed system in Nakagamim fading channel environment. The outage performance of the proposed network shows a significant improvement over the conventional SM without TAS at the rate of increase of antennas at the source nodes and also by varying m .