APPLICATION OF DIGITAL CELLULAR RADIO FOR MOBILE LOCATION ESTIMATION IIUM Engineering Journal, Vol. 19, No. 2, 2018 Alafia et al. 105 A STUDY ON LOW-COMPLEXITY TRANSMIT ANTENNA SELECTION FOR GENERALIZED SPATIAL MODULATION ADEWALE ALAFIA*, SIMEON AJOSE AND AGBOTINAME IMOIZE Department of Electrical and Electronics Engineering, Faculty of Engineering, University of Lagos, Akoka Lagos, Nigeria. * Corresponding author: adealafia@yahoo.com (Received: 19th Feb 2018; Accepted: 20th July 2018; Published on-line: 1st Dec 2018) https://doi.org/10.31436/iiumej.v19.i2.899 ABSTRACT: Generalized spatial modulation (GSM) maps its information to the index of the transmit antenna combination, making simultaneous transmission of multiple symbol possible. However, SM outperform GSM scheme in terms of error performance of the same data rate, due to average power effect. Transmit and receive diversity or the combination of both allow huge improvement in mimo systems in terms of error performance. In this paper, we investigate a near optimal low-complexity Euclidean distance antenna selection (LC-EDAS) technique in GSM system, to further improve the performance of the conventional GSM system. The LC-EDAS technique independently search across signal and spatial dimension to eliminate the worse channel prior to transmission. Secondly, we investigate a sub-optimal low-complexity transmit antenna selection (LCTAS) in the GSM system to further reduce the computational complexity (CC) imposed by LC-EDAS. The Monte Carlo simulation results obtained reveals a trade- off between the GSM scheme with LC-EDAS and GSM scheme with sub-optimal transmit antenna selection in terms of error performance and CC. ABSTRAK: Modulasi Spatial Keseluruhan (GSM) menghubung informasi kepada indeks kombinasi antena yang dipancarkan, membuatkan pemancaran keseluruhan simbol dapat dilakukan. Walau bagaimanapun, SM lebih bagus daripada skim GSM pada prestasi kesilapan pada kadar data yang sama, kerana kesan purata kuasa. Kepelbagaian penghantaran dan penerimaan ataupun kombinasi keduanya memberi pembaharuan yang lebih besar dalam sistem mimo pada prestasi kesalahan. Penyelidikan ini akan mengkaji optima terdekat Euclidean kurang rumit, melalui teknik (LC-EDAS) pilihan jarak antenna dalam sistem GSM, bagi menambah prestasi sistem GSM sedia ada. Teknik LC-EDAS secara sendiri mencari signal dan dimensi separa bagi mengurangkan saluran lebih teruk semasa penghantaran. Kedua, kami mengkaji sub-optima proses pemilihan kurang rumit penyebaran antena (LCTAS) dalam sistem GSM bagi mengurangkan kerumitan pengiraan (CC) yang dikenakan oleh LC-EDAS. Keputusan simulasi Monte Carlo yang diperoleh menunjukkan timbangan antara skim GSM dan LC-EDAS dan skim GSM bersama sub- optima proses pemilihan penyebaran antena berdasarkan kesilapan prestasi dan CC. KEYWORDS: multi-input multiple-output; computational complexity; spatial modulation; Euclidean distance antenna selection; transmit antenna selection 1. INTRODUCTION In the past decade, the wireless communication sector has been categorized as the fastest growing sector in the communication industry. This is due to the need for IIUM Engineering Journal, Vol. 19, No. 2, 2018 Alafia et al. 106 improvement in next generation wireless devices in terms of speed, reliability, and throughput [1]. This led to multiple-input multiple-output (MIMO) systems [2]. MIMO systems exhibits great potential with respect to high throughput and improved reliability in wireless communication [2]. The benefit of employing a MIMO system lies on the fact that an improved error performance and high throughput coupled with improved diversity gain can be achieved. However, in MIMO systems, the simultaneous transmission of data requires the need for inter-antenna synchronization (IAS) and the channel will experience an inter-channel interference (ICI) at the receiver [1,2]. Hence, it is clear that the dominant consideration in wireless technology is to employ a system that is able to mitigate these impairments. Lately, the use of multiple antennas has been investigated in numerous papers as the future of wireless communication [2]. This led to the development of an improved MIMO system called spatial modulation (SM) [3], which employs its spatial dimension to convey additional information improving the major drawback of conventional MIMO systems. However, the SM system also has a drawback of high complexity, due to the number of required transmit antennas to transmit high data streams [3]. This brought about the generalized spatial modulation system (GSM) [4]. The GSM system [4-6], is an SM-based scheme that maps its information bits to the index of the transmit antenna combination, thus reducing the required number of transmit antennas for high data streams. However, SM outperforms the GSM system in terms of error performance. The attractive features of SM and GSM include avoidance of ICI and IAS, which formed the major limitation of conventional MIMO systems. GSM exhibits significant improvement when compared to a conventional SM in terms of CC [3,5]. The constraint of a large number of transmit antennas required in SM was improved on in GSM [4, 5]. The complexity of the GSM system is still not practically implementable. In literature [7,8], incorporating a transmit antenna selection (TAS) into a MIMO system can further enhance the performance of the system. This is evident in [9,12], where SM was introduced as an improved MIMO scheme. However, practical realization was still a problem due to high CC imposed by the system. In [9], a maximum ratio combining (MRC), was introduced to an SM system, to initially estimate the index of the transmit antenna, which was later employed to detect the transmitted symbol. The result obtained exhibited an improved error performance and the imposed CC was greatly reduced. Similarly, in [10], TAS was employed in SM to improve the performance of the system, employing a channel amplitude technique to eliminate the worst channel at every transmission instant. Transmit and receive diversity or a combination of both in spatial multiplexing allow huge improvement in terms of error performance [7]. Employing TAS in selecting subset of antennas combination has proven to be extremely beneficial for link initialisation and maintenance [10,11]. Likewise, employing TAS has shown to increase the achieved diversity gain [12]. The selection criterion proposed in [13] was based on the Shannon capacity, and as such, yielded optimal capacity gain. In addition, in [14], a selection criterion that minimised the probability of symbol error rate (SER) for spatial multiplexing systems is investigated. In [12], the availability of additional antennas is revealed as an inexpensive way of exploiting diversity advantage. This was corroborated in [15], whilst further proving that the diversity of space shift keying (SSK) can be improved by increasing the number of surplus transmit antennas. Furthermore, TAS, which maximised the minimum Euclidean distance (ED) of a received constellation to yield an optimal performance, considers ED as file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_1 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_2 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_2 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_1 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_2 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_2 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_3 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_3 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_4 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_4 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_3 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_5 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_4 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_5 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_7 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_8 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_9 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_10 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_7 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_10 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_11 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_12 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_13 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_14 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_12 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_15 IIUM Engineering Journal, Vol. 19, No. 2, 2018 Alafia et al. 107 a function of both the received constellation and the channel. As a result, the ED criterion selects an optimal antenna subset in terms of the minimum error rate [16]. This is evident in [17], where an optimal performance was achieved in the SM system by employing a decision metric that maximized the minimum ED among the transmit vectors. This scheme also increased the diversity order of SM system but at a cost of high complexity. In this study, a low-complexity Euclidean distance antenna selection (LC-EDAS) is introduced to the GSM system to further improve the performance of the system by splitting the exhaustive search of EDAS into signal, spatial, and joint evaluation. This allowed an independent search across signal, spatial, and joint (signal and channel) to reduce the CC imposed on the system similar to [18]. Secondly, we investigate a sub-optimal transmit antenna selection in the GSM system. Employing channel amplitude and antenna correlation to further reduce the high CC imposed by LC-EDAS. The Monte Carlo simulation result obtained reveals a trade-off in terms of performance and CC between the GSM scheme with LC-EDAS and GSM scheme with sub-optimal transmit antenna selection. The structure of the remainder of the paper is as follows: in section 2, the system model for the proposed scheme is presented. In section 3, the numerical analysis of the system is presented. In section 4, the CC of the system is presented. In section 5, a summary of findings of the proposed scheme is presented while conclusions are drawn in section 6. 2. SYSTEM MODEL OF THE PROPOSED SCHEME In the GSM system, information bits are mapped into the transmit antenna combination, making the transmission of two symbols possible at the same time. Thus, reducing the required number of transmit antennas to transmit a high data rate. 2.1 Transmission Model of GSM System GSM Mapper Spatial bits Spatial bits G S M T ra n s m itte r R e c e iv e r H 1 2 1 2 Input bits Output bits. . . . . . TN GSM Detector RN Fig. 1: System model for a conventional GSM system [4]. Figure 1 reveals a conventional GSM system, which is equipped with 𝑁𝑇 transmit antennas and 𝑁𝑅 receive antennas, respectively. The spectral efficiency of the GSM system is π‘š = log2(𝑀) + ⌊log2 ( 𝑁𝑇 2 )βŒ‹ b/s/Hz. For example, consider a GSM system with a configuration setting of 4 Γ— 4, 4-QAM. The spectral efficiency yielded will be 4 b/s/Hz, in which the first two bits are used to select the active antenna combination and the last two bits are used to select the constellation symbol π‘₯π‘ž to be transmitted via a Rayleigh fading channel 𝑯 of dimension 𝑁𝑅 Γ— 𝑁𝑇, where 𝑁𝑅 and 𝑁𝑇 is the number of transmit and receive antennas, respectively. In the presence of an additive white Gaussian noise (AWGN) 𝒏. In file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_16 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_17 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_18 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_4 IIUM Engineering Journal, Vol. 19, No. 2, 2018 Alafia et al. 108 this paper, we considered same symbol to be transmitted at both transmit antenna combination at the same time. The received signal vector π’š becomes: π’š = √ 𝜌 πœ‡β„ 𝑯π‘₯π‘ž + 𝒏 (1) 𝜌 πœ‡β„ is the average signal-to-noise ratio (SNR) and π‘₯π‘ž = [π‘₯1 π‘₯2]. The transmitted signal can be detected optimally, employing maximum likelihood (ML) such that all possibilities are estimated across all the signal space similar to SM. This can be expressed as [4]: [𝑗, π‘₯π‘ž] = argmin β„“1,β„“2,π‘₯1 π‘ž ,π‘₯2 π‘ž (β€–π’š βˆ’ √ 𝜌 πœ‡β„ (𝑯π‘₯π‘ž)β€– 𝐹 2 ) (2) 2.2 Transmit Antenna Selection for GSM System Figure 2 reveals a conventional GSM system, equipped with 𝑁𝑇 transmit antennas and 𝑁𝑅 receive antennas, respectively, together with an antenna selection module and a feedback link. Fig. 2: System model for the proposed LC-GSM system. In literature [15,17], it has been stated that TAS can improve the error performance of a spatial multiplexing system. In [19], EDAS was employed to select the transmit antenna combination in GSM system. The error performance achieved in the scheme is superior but the CC imposed on the system is high. However, in [18], LC-EDAS was investigated in SM, imposing a much lower CC and exhibiting a performance near to that of the EDAS SM scheme of [16]. The GSM system exhibits less error performance compared to SM due to the average power effect, but required less transmit antenna to achieve a high data rate. This motivates us to investigate LC-EDAS in the GSM system to further improve the performance of the system. Secondly, a low-complexity TAS (LCTAS), employing the combination of channel amplitude and antenna correlation to eliminate the worse channel at every transmission GSM Transmitter GSM Receiver H Antenna Selection Channel state information Perfect feedback Link 1 1 2 2 . . . . . . TN RN file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_4 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_15 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_17 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_19 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_18 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_16 IIUM Engineering Journal, Vol. 19, No. 2, 2018 Alafia et al. 109 instant in order to maximize transmit diversity of the GSM system to attain a performance similar to SM system is investigated in the GSM system. The CC imposed by this scheme is lower than LC-EDAS approach but the performance achieved is also lower than that of the LC-EDAS system. The Monte Carlo simulation results reveal the trade-off between CC and error performance with LC-EDAS and LCTAS. 2.2.1 LC-EDAS for GSM System In [18], the instantaneous SER for each candidate subsets was computed into signal, spatial, and joint minimum squared Euclidean distance denoted as πΈπ·π‘ π‘–π‘”π‘›π‘Žπ‘™ 2(β„“) (𝑯), πΈπ·π‘ π‘π‘Žπ‘‘π‘–π‘Žπ‘™ 2(β„“) (𝑯) and πΈπ·π½π‘œπ‘–π‘›π‘‘ 2(β„“) (𝑯), respectively. The receiver employs the channel estimates to select a subset β„“ ∈ 1: 𝑁𝑠, 𝑁𝑠 = ( π‘π‘‡π‘œπ‘‘π‘Žπ‘™ 2 ) and π‘π‘‡π‘œπ‘‘π‘Žπ‘™ > 𝑁𝑇. We employ a similar approach to [18], to compute the three minimum squared Euclidean distance for signal, spatial and joint, respectively for the GSM system, maximizing the minimum Euclidean distance employing singular value decomposition for the joint evaluation. The close form of expression of the formulated πΈπ·πΊπ‘†π‘€βˆ’π‘ π‘–π‘”π‘›π‘Žπ‘™ 2(β„“) (𝑯), πΈπ·πΊπ‘†π‘€βˆ’π‘ π‘π‘Žπ‘‘π‘–π‘Žπ‘™ 2(β„“) (𝑯) and πΈπ·πΊπ‘†π‘€βˆ’π½π‘œπ‘–π‘›π‘‘ 2(β„“) (𝑯) proceeds as: πΈπ·πΊπ‘†π‘€βˆ’π‘ π‘–π‘”π‘›π‘Žπ‘™ 2(β„“) (𝑯) = 𝜌 2 min π‘˜βˆˆ1:π‘π‘‡π‘œπ‘‘π‘Žπ‘™ β€–π’‰π‘˜1 β„“ + π’‰π‘˜2 β„“ βˆ’ οΏ½Μ‚οΏ½π‘˜1 β„“ βˆ’ οΏ½Μ‚οΏ½π‘˜2 β„“ β€– 𝐹 2 min π‘₯1 π‘ž ,π‘₯2 π‘ž β‰ οΏ½Μ‚οΏ½1 π‘ž ,οΏ½Μ‚οΏ½2 π‘ž π‘₯1 π‘ž ,π‘₯2 π‘ž ∈𝝌 |π‘₯1 π‘ž + π‘₯2 π‘ž βˆ’ οΏ½Μ‚οΏ½1 π‘ž βˆ’ οΏ½Μ‚οΏ½2 π‘ž | 2 (3) πΈπ·πΊπ‘†π‘€βˆ’π‘ π‘π‘Žπ‘‘π‘–π‘Žπ‘™ 2(β„“) (𝑯) = 𝜌 2 min π‘˜βˆˆ1:π‘π‘‡π‘œπ‘‘π‘Žπ‘™ β€–π’‰π‘˜1 β„“ + π’‰π‘˜2 β„“ βˆ’ οΏ½Μ‚οΏ½π‘˜1 β„“ βˆ’ οΏ½Μ‚οΏ½π‘˜2 β„“ β€– 𝐹 2 min π‘₯π‘žβˆˆπŒ |π‘₯1 + π‘₯2| 2 (4) πΈπ·πΊπ‘†π‘€βˆ’π½π‘œπ‘–π‘›π‘‘ 2(β„“) (𝑯) = 𝜌 2 min π‘˜βˆˆ1:π‘π‘‡π‘œπ‘‘π‘Žπ‘™ π‘₯1 π‘ž ,π‘₯2 π‘ž β‰ οΏ½Μ‚οΏ½1 π‘ž ,οΏ½Μ‚οΏ½2 π‘ž π‘₯1 π‘ž ,π‘₯2 π‘ž ∈𝝌 β€–π’‰π‘˜1 β„“ π‘₯1 π‘ž + π’‰π‘˜2 β„“ π‘₯2 π‘ž βˆ’ οΏ½Μ‚οΏ½π‘˜1 β„“ οΏ½Μ‚οΏ½1 π‘ž βˆ’ οΏ½Μ‚οΏ½π‘˜2 β„“ οΏ½Μ‚οΏ½2 π‘ž β€– 𝐹 2 β‰₯ 𝜌 2 min π‘˜βˆˆ1:π‘π‘‡π‘œπ‘‘π‘Žπ‘™ 𝜎 π‘˜,οΏ½Μ‚οΏ½ 2 min π‘₯1 π‘ž ,π‘₯2 π‘ž β‰ οΏ½Μ‚οΏ½1 π‘ž ,οΏ½Μ‚οΏ½2 π‘ž π‘₯1 π‘ž ,π‘₯2 π‘ž ∈𝝌 β€– β€– [ π‘₯1 π‘ž π‘₯2 π‘ž οΏ½Μ‚οΏ½1 π‘ž οΏ½Μ‚οΏ½2 π‘ž ] β€– β€– 𝐹 2 (5) where min π‘˜βˆˆ1:π‘π‘‡π‘œπ‘‘π‘Žπ‘™ 𝜎 π‘˜,οΏ½Μ‚οΏ½ 2 is the minimum squared singular value decomposition of the channel matrix [π’‰π‘˜1 β„“ + π’‰π‘˜2 β„“ βˆ’ οΏ½Μ‚οΏ½π‘˜1 β„“ + οΏ½Μ‚οΏ½π‘˜2 β„“ ] and β„“ ∈ [1: 𝑁𝑇]. The channel matrix 𝑯 has entries, which is modelled as an independent and identically distributed (iid) complex Gaussian random variables with 𝐢𝑁(0,1). The next sub-section presents the algorithm for a LC-EDAS for GSM system. Algorithm 1 Step 1: Construct an 𝑁𝑅 Γ— π‘π‘‡π‘œπ‘‘π‘Žπ‘™ dimension channel matrix, where π‘π‘‡π‘œπ‘‘π‘Žπ‘™ > 𝑁𝑇. file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_18 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_18 IIUM Engineering Journal, Vol. 19, No. 2, 2018 Alafia et al. 110 Step 2: Compute the minimum squared Euclidean distance for πΈπ·πΊπ‘†π‘€βˆ’π‘ π‘–π‘”π‘›π‘Žπ‘™ 2(β„“) (𝑯), πΈπ·πΊπ‘†π‘€βˆ’π‘ π‘π‘Žπ‘‘π‘–π‘Žπ‘™ 2(β„“) (𝑯) and πΈπ·πΊπ‘†π‘€βˆ’π½π‘œπ‘–π‘›π‘‘ 2(β„“) (𝑯), respectively. Employing Eq. (3-5), considering all possible combination of 𝑁𝑠 = ( π‘π‘‡π‘œπ‘‘π‘Žπ‘™ 2 ). Step 3: Choose the minimum instantaneous minimum square Euclidean distance for each condition, i.e. 𝐸𝐷𝐺𝑆𝑀 2(β„“) (𝑯) = min{πΈπ·πΊπ‘†π‘€βˆ’π‘ π‘–π‘”π‘›π‘Žπ‘™ 2(β„“) (𝑯), πΈπ·πΊπ‘†π‘€βˆ’π‘ π‘π‘Žπ‘‘π‘–π‘Žπ‘™ 2(β„“) (𝑯), πΈπ·πΊπ‘†π‘€βˆ’π½π‘œπ‘–π‘›π‘‘ 2(β„“) (𝑯)} (6) Step 4: Select the antenna subset that has the largest instantaneous minimum squared Euclidean distance among the candidates in step 3, using: ℓ𝑠𝑒𝑙𝑒𝑐𝑑𝑒𝑑 = max β„“=1βˆˆπ‘π‘  {𝐸𝐷𝐺𝑆𝑀 2(β„“) (𝑯)} (7) where 𝑁𝑠 is the possible antenna pairs in the corresponding vectors in the channel matrix 𝑯. Thus, the selected transmit antenna will be employed for transmission at the next transmission instant. 2.2.2 TAS Based on Channel Amplitude and Antenna Correlation for the GSM System Employing the concept of channel amplitude with a decision of β€œthe larger the amplitude of the channel, the better it is” as investigated in [8], and an antenna correlation, a concept was investigated in [20], transmit antennas were discarded based on high correlation. Employing both channel amplitude and antenna correlation, which are sub- optimal techniques will impose a very low CC [15]. This motivates the application of this technique to the GSM system. Algorithm 2 presents the step/procedure for the implementation of the channel amplitude and antenna correlation technique. Algorithm 2 Step 1: Construct an 𝑁𝑅 Γ— π‘π‘‡π‘œπ‘‘π‘Žπ‘™ dimension channel matrix, where π‘π‘‡π‘œπ‘‘π‘Žπ‘™ is the total number of transmit antennas available, which is greater than 𝑁𝑇. 𝑯 = [𝒉1 𝒉2 𝒉3 … π’‰π‘π‘‡π‘œπ‘‘π‘Žπ‘™] (8) Step 2: Compute the channel amplitude using ‖𝑯‖𝐹 for the above channel constructed in Step 1 and sort in descending order. Step 3: Choose 𝑁𝑇 + 1 such that the dimension matrix becomes 𝑁𝑅 Γ— (𝑁𝑇 + 1), i.e. 𝑯 = [𝒉1 𝒉2 … 𝒉𝑁𝑇+1] (9) Step 4: Calculate the angle of correlation of the channel combination similar to [21] and [22] using: πœƒ = arccos ( |π’‰π‘Ž 𝐻 𝒉𝑏| β€–π’‰π‘Žβ€–πΉβ€–π’‰π‘β€–πΉ ) (10) Step 5: Arrange the angle of correlation in vectors and eliminate the smallest angle (highest correlation) from the channel 𝑁𝑇 + 1 computed in Step 4. file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_8 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_20 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_15 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_21 file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_22 IIUM Engineering Journal, Vol. 19, No. 2, 2018 Alafia et al. 111 The number of transmit antennas left will be employed for the next transmission instant in the GSM system, maximizing transmit diversity in the GSM system. 2.2.3 TAS for EDAS GSM TAS based on EDAS, selects the subset of transmit antennas, which maximises the minimum ED between all transmit antenna vectors, achieving an optimal performance. EDAS for GSM (EDAS-QSM) is based on the nearest neighbour approximation, the PEP of GSM for a given channel 𝑯 is: 𝑃𝑒 β‰… πœ† βˆ™ 𝑄 (√ 1 2 min π‘₯1β‰ π‘₯2∈𝝌 ‖𝑯′(π‘₯1 βˆ’ π‘₯2)‖𝐹 2 ) = πœ† β‹… 𝑄 (√ 1 2 π‘‘π‘šπ‘–π‘› 2 (𝑯′)) (11) where πœ† is the number of neighbor points and π‘‘π‘šπ‘–π‘› 2 (𝑯′) = min π‘₯π‘ž1β‰ π‘₯π‘ž2∈𝝌 ‖𝑯′(π‘₯1 βˆ’ π‘₯2)‖𝐹 2 is the squared minimum Euclidean distance between the pair of neighbor symbols. In maximizing the minimum Euclidean distance, the selected antennas can be chosen using: π‘‘π‘šπ‘–π‘› 2 (𝑯′) = min π‘₯1β‰ π‘₯2∈𝝌 β„“βˆˆ[1:𝑁𝑠] π‘₯1β‰ π‘₯2 π‘žβˆˆ1:𝑀 ‖𝒉ℓ1π‘₯1 + 𝒉ℓ2π‘₯2 βˆ’ 𝒉ℓ̂1π‘₯1Μ‚ βˆ’ 𝒉ℓ̂2π‘₯2̂‖𝐹 2 (12) 3. NUMERICAL ANALYSIS The Monte Carlo simulation results obtained from this research work are presented in this section. The notation 𝑁𝑅 Γ— 𝑁𝑇 is used to denote the number of receive antennas and transmit antennas, respectively. This algorithm was adopted from the literature [9,11,20]. The proposed system was compared to the conventional GSM system to validate the relationship between open loop and close loop systems. In Fig. 3, the system is equipped with four transmit antennas 𝑁𝑇 and four receive antennas 𝑁𝑅, employing 4-QAM modulation orders, i.e. 4 Γ— 4 4-QAM. It was observed that the LC-EDAS-GSM system achieves an SNR gain of approximately 4.5 dB over the conventional GSM system in terms of error performance at a BER of 10βˆ’5. This is due to the exhaustive search employed by the LC-EDAS-GSM. In addition, in Fig. 3, the LCTAS-GSM system exhibits an error performance gain of approximately 1 dB over a conventional GSM system. Likewise, the LC-EDAS-GSM system reveals a gain of approximately 3.5 dB when compared to the LCTAS-GSM system. But at a cost of higher CC due to the exhaustive search approach that was adopted by the LC-EDAS-GSM technique. The total number of transmit antenna π‘π‘‡π‘œπ‘‘π‘Žπ‘™ available is 6, in which 4 were chosen for transmission i.e. 𝑁𝑇 = 4. IIUM Engineering Journal, Vol. 19, No. 2, 2018 Alafia et al. 112 Fig. 3: BER performance comparison of GSM, LCTAS-GSM and LC-EDAS-GSM of 4 b/s/Hz. Figure 4, depicts a similar configuration to Fig. 3, a gain of approximately 6 dB was achieved with respect to the LC-EDAS-GSM system when compared to the conventional GSM system. This is due to the higher modulation order that was employed coupled with the high number of available transmit antennas, as GSM systems are said to exhibit a better performance with a high number of transmit antennas at high SNR region. Likewise, considering the LCTAS-GSM system, which achieves an SNR gain of 2 dB over the conventional system, with a lower CC when compared to LC-EDAS-GSM system. The various variations achieved under different configurations are tabulated in Table 1. Table 1: SNR gain achieved with respect to LC-EDAS-GSM system Scheme 4 b/s/Hz 6 b/s/Hz 7 b/s/Hz GSM 4.5 dB 6 dB 5 dB LCTAS-GSM 3.5 dB 3 dB 4 dB EDAS-GSM 1 dB 1.2 dB 1 dB 0 2 4 6 8 10 12 14 16 18 20 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 Es/No, (dB) B E R 4 b/s/Hz GSM (4x4 4QAM) LCTAS-GSM (4x4 4QAM Ntotal=6) EDAS-GSM (4x4 4QAM Ntotal=6) LC-EDAS-GSM (4x4 4QAM Ntotal=6) IIUM Engineering Journal, Vol. 19, No. 2, 2018 Alafia et al. 113 Fig. 4: BER performance comparison of GSM, LCTAS-GSM, LC-EDAS-GSM of 6 b/s/Hz. In Fig. 5, a gain of approximately 5 dB was achieved with respect to the LC-EDAS- GSM system when compared to the conventional GSM system. Likewise, considering the LCTAS-GSM system, which achieves an SNR gain of 1 dB over the conventional system, with a lower CC when compared to the LC-EDAS-GSM system. The total number of transmit antennas π‘π‘‡π‘œπ‘‘π‘Žπ‘™ available is 8, in which 5 was employed for transmission i.e. 𝑁𝑇 = 5. The SNR gain achieved is tabulated in Table 1 with respect to the LC-EDAS-GSM system. 5. COMPUTATIONAL COMPLEXITY A trade-off exists between the LC-EDAS-GSM and LCTAS-GSM in terms of CC and error performance. The CC for each TAS algorithm employed is formulated based on the floating points of operation (flops) approach similar to [10]. The CC imposed by the LC- EDAS is: I. Signal: Evaluating the Forbenius norm in (3) requires (2𝑁𝑅 βˆ’ 1) flops, across ( π‘π‘‡π‘œπ‘‘π‘Žπ‘™ 2 ) in 𝑁𝑠 times. 0 5 10 15 20 25 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 Es/No, (dB) B E R 6 b/s/Hz GSM (4x4 16QAM) LCTAS-GSM (4x4 16QAM Ntotal=6) EDAS-GSM (4x4 16QAM Ntotal=6) LC-EDAS-GSM (4x4 16QAM Ntotal=6) file:///C:/Users/Prof.Dr.Hamzah/Desktop/IIUM-EJ%20Nov%202018/899-Article%20Text-4844-1-6-20181011-LE.docx%23_ENREF_10 IIUM Engineering Journal, Vol. 19, No. 2, 2018 Alafia et al. 114 Fig. 5: BER performance comparison of GSM, LCTAS-GSM, LC-EDAS-GSM of 7 b/s/Hz. II. Spatial: The CC imposed by the spatial evaluation in Eq. (4) requires (2𝑁𝑅 βˆ’ 1) flops to compute the square Forbenius norm, across ( π‘π‘‡π‘œπ‘‘π‘Žπ‘™ 2 ) in 𝑁𝑠 times. III. Joint: The joint evaluation requires (2𝑁𝑅 βˆ’ 1) flops in Eq. (5) and there are 4 multiplications, 1 addition, and 2 subtractions that took place within the Forbenius norm, across ( π‘π‘‡π‘œπ‘‘π‘Žπ‘™ 2 ) in 𝑁𝑠 times. The number of flops required is (9𝑁𝑅 βˆ’ 1) ( π‘π‘‡π‘œπ‘‘π‘Žπ‘™ 2 ) 𝑁𝑠. The overall CC imposed on the system by LC-EDAS-GSM is: π›ΏπΏπΆβˆ’πΈπ·π΄π‘†βˆ’πΊπ‘†π‘€ = (13𝑁𝑅 βˆ’ 4) ( π‘π‘‡π‘œπ‘‘π‘Žπ‘™ 2 ) 𝑁𝑠 (13) Similarly, algorithm 2 of the LCTAS-GSM approach requires (2𝑁𝑅 βˆ’ 1) flops across π‘π‘‡π‘œπ‘‘π‘Žπ‘™ in Step 2. Likewise, Step 4 requires 2𝑁𝑅 + 2 flops for ( 𝑁𝑇 + 1 2 ) combinations. Therefore, the overall CC imposed by the LCTAS-GSM is: 0 5 10 15 20 25 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 Es/No, (dB) B E R 7 b/s/Hz GSM (5x4 16QAM) LCTAS-GSM (5x4 16QAM Ntotal=6) EDAS-GSM (5x4 16QAM Ntotal=8) LC-EDAS-GSM (5x4 16QAM Ntotal=8) IIUM Engineering Journal, Vol. 19, No. 2, 2018 Alafia et al. 115 π›ΏπΏπΆπ‘‡π΄π‘†βˆ’πΊπ‘†π‘€ = π‘π‘‡π‘œπ‘‘π‘Žπ‘™(2𝑁𝑅 βˆ’ 1) + (2𝑁𝑅 + 2 ) ( 𝑁𝑇 + 1 2 ) (14) The exhaustive search requires ED to be evaluated for all symbol combinations and 4𝑀(4𝑀 βˆ’ 1) flops is required for the EDAS-GSM technique across ( π‘π‘‡π‘œπ‘‘π‘Žπ‘™ 2 ) 𝑁𝑠, imposing an overall CC of 4𝑀(4𝑀 βˆ’ 1)𝑁𝑇(2𝑁𝑅 βˆ’ 1) + 64 ( π‘π‘‡π‘œπ‘‘π‘Žπ‘™ 2 ) 𝑁𝑠 flops similar to [10]. The numerical comparison in terms of complexity between both systems is computed in Table 2. Table 2: Numerical Comparison of Computational Complexity of LC-EDAS-GSM and LCTAS-GSM Configuration LC-EDAS-GSM LCTAS-GSM EDAS-GSM 𝑴 = πŸ’ π‘π‘‡π‘œπ‘‘π‘Žπ‘™ = 6 𝑁𝑇 = 4 𝑁𝑅 = 4 10,800 142 21,120 𝑴 = πŸπŸ” π‘π‘‡π‘œπ‘‘π‘Žπ‘™ = 6 𝑁𝑇 = 4 𝑡𝑹 = πŸ’ 10,800 142 127,296 𝑴 = πŸ’ π‘π‘‡π‘œπ‘‘π‘Žπ‘™ = 6 (LCTAS) π‘π‘‡π‘œπ‘‘π‘Žπ‘™ = 8 (LC- EDAS) 𝑁𝑇 = 5 𝑡𝑹 = πŸ’ 75,264 192 108,752 5. SUMMARY OF FINDINGS In this research, it is once more validated that TAS can further improve the error performance of a spatial multiplexing system. The results in this paper conform to the theory stated in the literature that TAS, a closed loop system, can maximize transmit diversity. In Table 1, LC-EDAS-GSM achieves up to approximately 5 dB and 4 dB SNR gain at a bit error rate (BER) of 10βˆ’πŸ“ of 4 b/s/Hz configuration setting when compared to LCTAS-GSM and the conventional GSM system, respectively. In addition, EDAS-GSM outperforms other TAS techniques employed, due to the exhaustive search. However, the system gain is thus reduced as the spectral efficiency increases, due to the high modulation order employed, as seen in Fig. 3 and Fig. 4, respectively. Likewise, Table 2 presents the numerical comparison of both TAS techniques, in which LCTAS-GSM imposed a lower CC with the same configuration settings in LC-EDAS and EDAS-GSM imposed a high CC when compared to other techniques employed. This is because the EDAS CC is a function of the modulation order 𝑴, 𝑡𝑻𝒐𝒕𝒂𝒍, and 𝑡𝑹 while LCTAS and LC-EDAS are functions of just 𝑡𝑻 and 𝑡𝑹 6. CONCLUSION This research demonstrated the applicability of transmit antenna selection in improving the reliability/error performance of a MIMO based scheme. 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