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Space-Time Trellis Coding with Equalization EJERS, European Journal of Engineering Research and Science Vol. 4, No. 9, September 2019 Space-Time Trellis Coding with Equalization Ibukunoluwa A. Adebanjo, Yekeen O. Olasoji, and Michael O. Kolawole different frequencies. Several works have been carried out Abstract—As we are entering the 5G era, high demand is on the drawbacks of OFDM, - peak-to-average power ratio made of wireless communication. Consistent effort has been (PAPR), receiver’s time and frequency offset, leading to ongoing in multiple-input multiple-output (MIMO) systems, increased cost, higher power consumption, and high error which provide correlation on temporal and spatial domain, to rate [4, 6]. These drawbacks are easily addressed by meet the high throughput demand. To handle the characteristic nature of wireless channel effectively and precoding, clipping and filtering techniques [5]. Aside the improve the system performance, this paper considers the facts that these drawbacks are addressed, single modulation combination of diversity and equalization. Space-Time trellis has found to be appealing in the uplink transmission in code is combined with single-carrier modulation using two- MIMO systems (because of its single carrier modulation); it choice equalization techniques, namely: minimum mean has lesser envelope variation [7]. Single-carrier frequency squared error (MMSE) equalizer and orthogonal triangular domain equalization (SC-FDE) has comparable (QR) detection. MMSE gives an optimal balance between noise enhancement and net inter-symbol interference (ISI) in performance to OFDM, as well as similarity in complexity the transmitted signal. Use of these equalizers provides the of operation: the transmit structure of SC-FDE makes it platform of investigating the bit error rate (BER) and the viable for use in uplink transmission in mobile pairwise error probability (PEP) at the receiver, as well as the communication [8], but the receiver structure is quite effect of cyclic prefix reduction on the receivers. It was found similar to OFDM since the core of the receiver is greatly that the MMSE receiver outperforms the QR receiver in terms dependent on the equalization technique. Several research of BER, while in terms of PEP; the QR receiver outperforms the MMSE receiver. When a cyclic prefix reduction test was studies have been carried out on equalization techniques in carried out on both receivers, it yields a significant reduction single and multi-carrier modes of transmission, but central in BER of both receivers but has no significant effect on the to both modes is the effectiveness of the equalizer overall performance. algorithm used. As the wireless channel constantly pose danger due to its inherent fading attribute, its strength varies Index Terms—Diversity, ISI, MMSE Equalizer, QR with time, frequency and space, and as high demand is Detection, Space-Time Coding. placed on efficient data delivery; several schemes of signal processing modulation are combined to provide an appreciable route of quality service delivery. I. INTRODUCTION In this paper, space-time coding in trellis modulation is The choice of receiver architecture determines the combined with single-carrier modulation. For the choice of outcome of signals received in multiple input multiple equalizers, the MMSE equalizer and QR detection were output (MIMO) system. MIMO uses multiple antennae at used. MMSE forms the basic building blocks of most the base station and serves multiple terminals over the same known linear equalizers and gives an optimal balance resourced time-frequency. In wireless communication, between noise enhancement and net ISI in the transmitted signals are sent through space that constantly exhibit fading signal [9, 10]. QR detection is an equalization technique characteristics, causing ISI (inter-symbol interference) [1], used in Vertical Bell Layered Space-Time (V-BLAST) thereby resulting in receiver architectures incorporating architecture that factorizes the channel matrix into unitary equalization techniques, which when properly configured and upper triangular matrices, making MIMO system to help in reducing bit error rates, as well as obtaining a higher become a causal system [11]. Comparison of the bit error throughput and spectral efficiency. Equalization techniques rate (BER) performance was done for uplink transmission grow in complexity as the presence of ISI increases [2, 3]. for both 3G and 4G architectures using the QR detection Equalization is encapsulated in the modulation techniques and MMSE in single-carrier modulation in space-coding. employed in MIMO system. The multicarrier modulation technique (e.g. orthogonal frequency-division multiplexing, OFDM) has readily found a place in its use in MIMO II. SYSTEM CONCEPTUAL MODEL systems. OFDM based on MIMO (MIMO-OFDM), operating on the principle of orthogonality, is used to Figure 1 shows the System model having NT-input NR- mitigate the channel’s frequency selectivity. Transmission output MIMO channel model given as of signals follows in parallel scheme in sub-channels at y=Hx+η (1) Published on September 27, 2019. where y is the received vector signal, H is the scattering I. A. Adebanjo is with the Federal University of Technology, Akure, complex Rayleigh flat fading MIMO channel matrix, x is Nigeria (e-mail: [email protected]). Y. O. Olasoji is with the Federal University of Technology, Akure, transmitted signal vector, and η is the additive noise. Nigeria (e-mail: [email protected]). Mapped and parallel combination of data streams are M.O. Kolawole is with Jolade Strategic Environmental and engineering encoded in space and in time by the STTC encoder. Consults, Melbourne, Australia (e-mail: [email protected]) DOI: http://dx.doi.org/10.24018/ejers.2019.4.9.1412 207 EJERS, European Journal of Engineering Research and Science Vol. 4, No. 9, September 2019 Assuming x(t) was obtained at the output of the encoder, it 푦̂ = 푤퐻푀푆퐸푥 + 휂 (7) forms the input to the single carrier transmit block. where 퐻푀푆퐸 is the equalized channel matrix. Therefore, equation (7) becomes, 퐻 −1 −1 퐻 H ŷ = (퐻 H + 휌 퐼) 퐻 퐻푀푆퐸푥 + 휂 (8) Input s(t) STTC x(t) Mapping CP insertion P/S From equation (2), we have Data encoder ŷ = (퐻퐻H + 휌−1퐼)−1퐻퐻푈퐻푆푈 X + η (9) Y (f)=Ƒ(R (f)) s(t) y (t) r(t) According to Hermitian theory, the circulant matrix H of STTC decoder IFFT FDE FFT CP removal S/P the form (ℎ푘) = (ℎ푘−푗+1) has eigenvalues that are grouping Fig. 1. System model [9]. of the coefficients of the channel, having zero mean [13]. The SINR is usually the yardstick of evaluating the MMSE Cyclic-Prefix (CP) is appended in single carrier output. Following [14], the SINR denoted as 훾 is transmission to the transmitted data to combat inter-symbol interference (ISI) and the serial conversion is sent through 1 훾 = − 1 (10) the wireless channel. Whilst the appended CP can also (퐼+휌퐻퐻퐻)−1 make the received symbol periodic it must not be too large, otherwise transmission efficiency is reduced [9]. The Since the channel matrix is circulant, the SINR can be circulant channel is decomposed into singular values [12], expressed in terms of the eigenvalues 휆푘 as 퐻 1 퐻 = 푈 푆푉 (2) 훾 = 1 1 − 1 (11) ∑퐿 퐿 푘=1 2 1+휌|휆푘| where 푈 ∈ 퐶푁푟×푁푡 and 푉 ∈ 퐶푁푡×푁푡 are unitary matrices of the left and right singular vectors of H, S is a diagonal where L is the block length, and k = 1, …, v+1, v is the matrix having non-negative singular values of H; the channel memory length. diagonal matrix transmits the transformed transmitted The eigenvalues of H are given by [15] singular vector V. The decomposed matrix U reverses the 푛−1 푗푘 transformation at the receive side. ∑푗=0 푤푛 ℎ푖푗 k = 0,…, n-1 (12) The received vector after CP removal is 2휋푖 ⁄푛 where 푤푛 = 푒 , ℎ푖푗 is the channel coefficients. The 푗 푛 푖 푗 ȓ푡 (푡) = ∑푖=1 퐻푖푗(푡)푥푡(푡) + 휂푡 (3) eigenvalues of H contain the DFT of the first row and also the inverse DFT of the eigenvalues is the first row of the 푗 where j is the number of receive antennas, 휂푡 is the additive circulant H matrix [16]. 푖 + −1 white Gaussian noise, 푥푡(푡) is the transmitted signal from i If H is square and non-singular, then 퐻 = 퐻 , where + number of transmit antenna, 퐻푖푗(푡) is the complex channel 퐻 is the Moore-Penrose pseudo inverse of the channel coefficient, with 1≤ 푖 ≤ 푁푡and 1≤ 푗 ≤ 푁푅. matrix, then, there will be an inverse of the channel matrix Equation (3) was received in circular convolution, and since H is circulant, therefore the inverse is circulant, otherwise written as, making H a simple matrix [15, 17]. The output of the MMSE detector is the input of the Viterbi decoder which Ȓ(푗) = 퐻1(푗)푋1(푗) + 퐻2(푗)푋2(푗) + ⋯ + 퐻푛(푗)푋푛(푗) + decodes STTC. 휂(푗) (4) B. Equalization- QR Detection Applying FFT operation on the received vector, we have The channel matrix H given in Equation (2) is 푌(푓) = ℱ[푅(푓)] (5) decomposed into 퐻 = 푄푅 (13) A. Equalization- MMSE Receiver Using the rule of orthogonality, since the number of where Q denotes unitary matrix that is satisfied with transmit antenna equals the number of receive antenna, the 푄퐻푄 = 푄푄퐻 = 퐼, where I is an identity matrix and R is the MMSE equalizer coefficient (in matrix form) is given as upper triangular matrix. The Schur algorithm [18] can be [13] applied since H is assumed to be full-ranked because nT = nR; therefore, H can be expressed as 푤 = (퐻퐻H + 휌−1퐼)−1퐻퐻 (6) 퐻푇퐻 = 푅푇푄푇푄푅 = 푅푇푅 (14) where 퐻퐻 is the Hermitian matrix of the channel matrix, H is the channel matrix, I is the identity matrix and 휌 is the Following the expression given in Equation (3), and by transmission signal-to-noise ratio.
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