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College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P. R. China

Abstract: Space shift keying (SSK) is a spectrally efficient and low-complexity technique that only uses antenna index to convey information. Combining SSK with cooperative communication, the transmission reliability of SSK system can be improved effectively. In this paper, considering imperfect channel information, the performance of cooperative SSK system with amplify-and-forward (AF) relaying protocol is investigated, and the effect of estimation error on the performance is analyzed. According to the performance analysis, the probability density function and moment generating function of effective signal-to-noise ratio are derived, respectively. Using these results, the closed-form expression of average bit error rate (BER) can be achieved. Meanwhile, the asymptotically approximated BER and the corresponding diversity order analysis are presented for the performance evaluation. By computer simulations, the validness of the presented theoretical analysis is verified, and the theoretical BERs with different estimation errors are shown to be close to those of the corresponding simulations.

Key words: space shift keying; amplify-and-forward; channel-estimation error; bit error rate; diversity order

0 Introduction

Cooperative relaying, as a promising technology to improve spectral efficiency and link reliability, has attracted much attentions[1,2]. In a cooperative communication system, the relay aids to the transmission of two nodes with different protocols and brings about diversity gain. Amplify-and-forward (AF) is one of the relaying protocols, where the relay amplifies and transfers the signals sent by the source to the destination. For its simple implementation, AF has been widely employed in cooperative communication.

Space shift keying (SSK) activates only one antenna to transmit information at any time slot, so that the need for transmit antenna synchronization and inter-channel interference can be eliminated[3,4]. In SSK, the active transmit antenna index is the unique way to convey information. Thereby, the optimal detection at the receiver only needs to detect antenna indices. For lack of the symbol detection, the receiver complexity of SSK is decreased[5].

Considering the low complexity of SSK and the diversity gain brought by cooperative relaying, the combination of SSK and cooperative communication has been proposed to further enhance the performance[6-9]. In Ref. [6], the cooperative SSK system with AF and decode-and-forward relaying was presented, and the average bit error rate (BER) expressions are derived. Based on the conventional AF protocol, Refs. [6, 7] proposed an opportunistic AF relaying scheme for the cooperative SSK system, where the best relay is chosen to transmit signals. Spatial modulation (SM) is an extension of SSK that simultaneously exploits antenna indices and constellation symbols to convey information[8]. In Refs. [9, 10], SM is introduced into the cooperative system, and the corresponding BER performance is analyzed. The above works all assume that the channel state information (CSI) is accurately estimated at the receiver, while in particular the influence of channel-estimation error needs to be considered. The performance of cooperative DF and cognitive radio system with SM are studied under the assumption that the imperfect CSI is available at the relay and destination[11,12]. However, these literatures focus on the SM system with DF and only consider single antenna at the destination for convenient analysis. There are few works addressing the study on the performance of the AF aided SSK (AF-SSK) system, especially in the presence of imperfect channel estimation.

Therefore, in this paper, we will study the performance of the AF-SSK system with multiple receive antennas under imperfect channel estimation, and analyse the effect of channel-estimation error on the performance of AF-SSK system. With the performance analysis, the probability density functions (PDF) and the moment generating functions (MGF) of effective signal-to-noise ratio (SNR), as well as the pairwise error probability (PEP) are derived. Thereby the closed-form expression of average BER is achieved by a union upper bound. Moreover, the asymptotically approximated BER at high SNR is also analyzed. Using this approximation, the diversity gain of this system is further derived for the performance evaluation. Simulation shows that the theoretical BER results can match the simulated values well for different channel-estimation errors, which verifies the correctness of theoretical analysis, and can provide the effective evaluation for the system performance.

1 System Model

The AF-SSK system combining cooperative AF relaying and SSK is shown as Fig.1, which consists of a source withNttransmit antennas, a single-antenna relay and a destination withNrreceive antennas. At the source, the SSK mapper utilizes log2Ntbits to determine the active transmit antenna indexi,i∈[1,Nt]. At the destination, with the channel estimate, the optimal detection algorithm based on the maximum likelihood (ML) principle is employed. The output of SSK mapper is given by[13]

xi=[0 0 … 1 … 0]T

(1)

Fig.1 Cooperative AF-SSK system model

wherexiis anNt-dimensional symbol vector, and the unique nonzero element is in thei-th row, which means that thei-th antenna is activated. Based on the basic idea of cooperative communication, the transmission process is divided into two phases. In the first phase, the source transmits the signal to the relay and the destination. The received signals of the relay and the destination are respectively given by

(2)

(3)

wherePsis the transmission power of the source,hsraNt×1 channel vector of source-to-relay link, andHsdaNr×Ntchannel matrix of source-to-destination link. In the second phase, the relay amplifies the received signalysrand transfers it to the destination, thus the received signal at the destination is expressed as

yrd=hrd(Aysr)+nrd

(4)

(5)

Substituting Eq. (5) into Eqs. (3) and (4), the received signalsysdandyrdcan be respectively expressed as

(6)

(7)

(8)

(9)

With the received signals and the channel estimates at the destination, the detection algorithm based on the ML principle is written as[13]

(10)

2 Performance Analysis of AF-SSK System

In this section, the performance of cooperative AF-SSK system in the presence of imperfect channel information will be analyzed, and the closed-form expression of average BER will be derived. Using the detection algorithm in Eq.(10), the antenna index is detected by minimizingDj, which is expressed as

(11)

If thei-th antenna is activated at the source and the estimate of antenna index at the destination isj,j≠i, then we can obtain

(12)

If the detection is correct (i.e,j=i), then Eq. (12) can be simplified as

(13)

(14)

where

(15)

(16)

(17)

fγsd(γ)fγsrd(γ)dγsrddγsd=

(18)

(19)

whereMγsd(·) andMγsrd(·) are MGFs ofγsdandγsrd, respectively.

Lett=sinθand using the transformation of variables, Eq. (19) can be rewritten as

(20)

whereφu=cos((2u-1)π/(2Np)), andNpis the order of the Chebyshev polynomial[16].

Based on the definitions in Eq. (17), the PDFs ofγsd,γsrandγrdover Rayleigh channel are respectively given by

(21)

(22)

(23)

Correspondingly, the cumulative distribution function (CDF) ofγsris obtained by

(24)

With the PDF ofγsd, the corresponding MGF can be derived as

(25)

Using Eqs. (22) and (24), the CDF ofγsrdis given by

(26)

whereKv(·) is thev-th order modified Bessel function of the second kind[17]. Then the MGF ofγsrdis derived as

Mγsrd(s)=sL{Fγsrd(γ)}=

(27)

Substituting Eqs. (25) and (27) into Eq. (20), the average PEP can be written as

(28)

(Nt/2)PEP(i→j)

(29)

Substituting Eq. (28) into Eq. (29), the closed-form BER expression of the AF-SSK system with channel estimation error can be derived as

(30)

3 Asymptotic BER Analysis and Diversity Gain

In this section, the asymptotic performance of the AF-SSK system under large SNR is analyzed, and asymptotically approximated expression of average BER is derived. With this approximation, the diversity gain of the system is attained.

Using the series representation ofKv(·)(which is Eq.(8.446) in Ref.[17]),Kv(2x) with smallxcan be approximated as

(31)

whereψ(·)is the psi function[17].

(32)

With the Laplace transform, the MGF ofγsrdis approximated as

(33)

(34)

With (3.621.3) and (4.387.4) in Ref.[17], we can obtain

(35)

ForNr=1, the average PEP expression of Eq.(35) under high SNR can be rewritten as

PEP(i→j)≈

(36)

Thus the approximate average BER is obtained by

(37)

ForNr≥2, the average PEP can be further simplified as

PEP(i→j)≈

(38)

Then the average BER is approximated as

(39)

With Eqs. (37) and (39), the asymptotically approximated BER expressions forNr=1 andNr≥2 are attained, respectively. It is shown that they have values close to the corresponding simulations under high SNR.

(40)

(41)

With the results above, we can evaluate the diversity orderGd, which is an important error performance indicator.Gdis defined as the slope of average BER curve for average SNR approaching infinity, and can be derived as[19]

(42)

Therefore, with the estimation error modeled as the decreasing function of SNR, the AF-SSK system will obtain the diversity order ofNr+1. As a result, the system performance will become better as theNrincreases.

4 Simulation Results

Fig.2 Average BER of AF-SSK system with different receive antennas (Nt=2)

Fig.3 illustrates the BER performance of AF-SSK system with different receive antennas in the presence of imperfect estimations, whereNr=1, 2, andNt=4. As shown in Fig.3, the results similar to Fig.2 can be found. Namely, the system withNr=2 has better performance than that withNr=1 because the former has larger diversity order than the latter. Moreover, the theoretical BERs can match the corresponding simulated ones, only small performance gap is observed at very low SNR. Besides, the approximated BERs are also close to the simulations at large SNR. All these results further show that the derived theoretical expressions are valid, and can provide good performance evaluation for the AF-SSK system under imperfect channel information.

Fig.3 Average BER of AF-SSK system with different receive antennas (Nt=4)

Fig.4 Average BER of AF-SSK system with different estimation error variances

5 Conclusions

Considering imperfect estimation information in practice, we have investigated the performance of cooperative AF-SSK system over Rayleigh fading channel. In the presence of estimation errors, the PDFs and MGFs of the effective SNR are respectively derived for the performance analysis. Based on this, closed-form average BER and the corresponding asymptotical BER are also deduced. Using the asymptotical BER, the diversity order is analyzed. As a result, the AF-SSK system can achieve the diversity order ofNr+1. Simulation results illustrate that the derived theoretical BERs can match well with the corresponding simulations for imperfect estimation information, and thus the correctness and effectiveness of theoretical analysis are validated. Thereby, the system performance can be evaluated well. Considering that the SM is the extension of SSK, in the future works, we will study the performance of the cooperative SM system with AF in the presence of channel estimation error so that the corresponding system performance can be well analyzed.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Nos. 61601220, 61172077), the Foundation of Graduate Innovation Center in NUAA (No. kfjj20170410), the Fundamental Research Funds for the Central Universities, the Open Research Fund of National Mobile Communications Research Laboratory of Southeast University (No. 2017D03), and the Six Talent Peaks Project of Jiangsu Province (No. 2015-DZXX-007).