Vol. 4, No. 9, September 2019
Abstract—As we are entering the 5G era, high demand is made of wireless communication. Consistent effort has been ongoing in multiple-input multiple-output (MIMO) systems, which provide correlation on temporal and spatial domain, to meet the high throughput demand. To handle the characteristic nature of wireless channel effectively and improve the system performance, this paper considers the combination of diversity and equalization. Space-Time trellis code is combined with single-carrier modulation using two- choice equalization techniques, namely: minimum mean squared error (MMSE) equalizer and orthogonal triangular (QR) detection. MMSE gives an optimal balance between noise enhancement and net inter-symbol interference (ISI) in the transmitted signal. Use of these equalizers provides the platform of investigating the bit error rate (BER) and the pairwise error probability (PEP) at the receiver, as well as the effect of cyclic prefix reduction on the receivers. It was found that the MMSE receiver outperforms the QR receiver in terms of BER, while in terms of PEP; the QR receiver outperforms the MMSE receiver. When a cyclic prefix reduction test was carried out on both receivers, it yields a significant reduction in BER of both receivers but has no significant effect on the overall performance.
Index Terms—Diversity, ISI, MMSE Equalizer, QR Detection, Space-Time Coding.
I. INTRODUCTION
The choice of receiver architecture determines the outcome of signals received in multiple input multiple output (MIMO) system. MIMO uses multiple antennae at the base station and serves multiple terminals over the same resourced time-frequency. In wireless communication, signals are sent through space that constantly exhibit fading characteristics, causing ISI (inter-symbol interference) [1], thereby resulting in receiver architectures incorporating equalization techniques, which when properly configured help in reducing bit error rates, as well as obtaining a higher throughput and spectral efficiency. Equalization techniques grow in complexity as the presence of ISI increases [2, 3].
Equalization is encapsulated in the modulation techniques 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 systems. OFDM based on MIMO (MIMO-OFDM), operating on the principle of orthogonality, is used to mitigate the channel’s frequency selectivity. Transmission of signals follows in parallel scheme in sub-channels at
Published on September 27, 2019.
I. A. Adebanjo is with the Federal University of Technology, Akure, Nigeria (e-mail: [email protected]).
Y. O. Olasoji is with the Federal University of Technology, Akure, Nigeria (e-mail: [email protected]).
M.O. Kolawole is with Jolade Strategic Environmental and engineering Consults, Melbourne, Australia (e-mail: [email protected])
different frequencies. Several works have been carried out on the drawbacks of OFDM, - peak-to-average power ratio (PAPR), receiver’s time and frequency offset, leading to increased cost, higher power consumption, and high error rate [4, 6]. These drawbacks are easily addressed by precoding, clipping and filtering techniques [5]. Aside the facts that these drawbacks are addressed, single modulation has found to be appealing in the uplink transmission in MIMO systems (because of its single carrier modulation); it has lesser envelope variation [7]. Single-carrier frequency domain equalization (SC-FDE) has comparable performance to OFDM, as well as similarity in complexity of operation: the transmit structure of SC-FDE makes it viable for use in uplink transmission in mobile communication [8], but the receiver structure is quite similar to OFDM since the core of the receiver is greatly dependent on the equalization technique. Several research studies have been carried out on equalization techniques in single and multi-carrier modes of transmission, but central to both modes is the effectiveness of the equalizer algorithm used. As the wireless channel constantly pose danger due to its inherent fading attribute, its strength varies with time, frequency and space, and as high demand is placed on efficient data delivery; several schemes of signal processing modulation are combined to provide an appreciable route of quality service delivery.
In this paper, space-time coding in trellis modulation is combined with single-carrier modulation. For the choice of equalizers, the MMSE equalizer and QR detection were used. MMSE forms the basic building blocks of most known linear equalizers and gives an optimal balance between noise enhancement and net ISI in the transmitted signal [9, 10]. QR detection is an equalization technique used in Vertical Bell Layered Space-Time (V-BLAST) architecture that factorizes the channel matrix into unitary and upper triangular matrices, making MIMO system to become a causal system [11]. Comparison of the bit error rate (BER) performance was done for uplink transmission for both 3G and 4G architectures using the QR detection and MMSE in single-carrier modulation in space-coding.
II. SYSTEM CONCEPTUAL MODEL
Figure 1 shows the System model having NT-input NR- output MIMO channel model given as
y=Hx+η (1)
where y is the received vector signal, H is the scattering complex Rayleigh flat fading MIMO channel matrix, x is transmitted signal vector, and η is the additive noise.
Mapped and parallel combination of data streams are encoded in space and in time by the STTC encoder.
Space-Time Trellis Coding with Equalization
Ibukunoluwa A. Adebanjo, Yekeen O. Olasoji, and Michael O. Kolawole
Vol. 4, No. 9, September 2019 Assuming x(t) was obtained at the output of the encoder, it
forms the input to the single carrier transmit block.
STTC
encoder CP insertion P/S
Mapping Input Data
x(t)
H
r(t) S/P Y (f)=Ƒ(R (f))
FFT CP removal
IFFT FDE STTC decoder y (t) s(t)
s(t)
Fig. 1. System model [9].
Cyclic-Prefix (CP) is appended in single carrier transmission to the transmitted data to combat inter-symbol interference (ISI) and the serial conversion is sent through the wireless channel. Whilst the appended CP can also make the received symbol periodic it must not be too large, otherwise transmission efficiency is reduced [9]. The circulant channel is decomposed into singular values [12],
𝐻 = 𝑈𝐻𝑆𝑉 (2)
where 𝑈 ∈ 𝐶𝑁𝑟×𝑁𝑡 and 𝑉 ∈ 𝐶𝑁𝑡×𝑁𝑡 are unitary matrices of the left and right singular vectors of H, S is a diagonal matrix having non-negative singular values of H; the diagonal matrix transmits the transformed transmitted singular vector V. The decomposed matrix U reverses the transformation at the receive side.
The received vector after CP removal is
ȓ𝑡𝑗(𝑡) = ∑𝑛𝑖=1𝐻𝑖𝑗(𝑡)𝑥𝑡𝑖(𝑡) + 𝜂𝑡𝑗 (3) where j is the number of receive antennas, 𝜂𝑡𝑗 is the additive white Gaussian noise, 𝑥𝑡𝑖(𝑡) is the transmitted signal from i number of transmit antenna, 𝐻𝑖𝑗(𝑡) is the complex channel coefficient, with 1≤ 𝑖 ≤ 𝑁𝑡and 1≤ 𝑗 ≤ 𝑁𝑅.
Equation (3) was received in circular convolution, otherwise written as,
Ȓ(𝑗) = 𝐻1(𝑗)𝑋1(𝑗) + 𝐻2(𝑗)𝑋2(𝑗) + ⋯ + 𝐻𝑛(𝑗)𝑋𝑛(𝑗) + 𝜂(𝑗) (4) Applying FFT operation on the received vector, we have
𝑌(𝑓) = ℱ[𝑅(𝑓)] (5)
A. Equalization- MMSE Receiver
Using the rule of orthogonality, since the number of transmit antenna equals the number of receive antenna, the MMSE equalizer coefficient (in matrix form) is given as [13]
𝑤 = (𝐻𝐻H + 𝜌−1𝐼)−1𝐻𝐻 (6)
where 𝐻𝐻 is the Hermitian matrix of the channel matrix, H is the channel matrix, I is the identity matrix and 𝜌 is the transmission signal-to-noise ratio.
Then channel output is given by
𝑦̂ = 𝑤𝐻𝑀𝑆𝐸𝑥 + 𝜂 (7)
where 𝐻𝑀𝑆𝐸 is the equalized channel matrix.
Therefore, equation (7) becomes,
ŷ = (𝐻𝐻H + 𝜌−1𝐼)−1𝐻𝐻𝐻𝑀𝑆𝐸𝑥 + 𝜂 (8) From equation (2), we have
ŷ = (𝐻𝐻H + 𝜌−1𝐼)−1𝐻𝐻𝑈𝐻𝑆𝑈 X + η (9) According to Hermitian theory, the circulant matrix H of the form (ℎ𝑘) = (ℎ𝑘−𝑗+1) has eigenvalues that are grouping of the coefficients of the channel, having zero mean [13].
The SINR is usually the yardstick of evaluating the MMSE output. Following [14], the SINR denoted as 𝛾 is
𝛾 = 1
(𝐼+𝜌𝐻𝐻𝐻)−1− 1 (10)
Since the channel matrix is circulant, the SINR can be expressed in terms of the eigenvalues 𝜆𝑘 as
𝛾 =1 1
𝐿∑ 1
1+𝜌|𝜆𝑘|2 𝐿𝑘=1
− 1 (11)
where L is the block length, and k = 1, …, v+1, v is the channel memory length.
The eigenvalues of H are given by [15]
∑𝑛−1𝑤𝑛𝑗𝑘ℎ𝑖𝑗
𝑗=0 k = 0,…, n-1 (12)
where 𝑤𝑛= 𝑒2𝜋𝑖⁄𝑛, ℎ𝑖𝑗 is the channel coefficients. The eigenvalues of H contain the DFT of the first row and also the inverse DFT of the eigenvalues is the first row of the circulant H matrix [16].
If H is square and non-singular, then 𝐻+= 𝐻−1, where 𝐻+ is the Moore-Penrose pseudo inverse of the channel matrix, then, there will be an inverse of the channel matrix and since H is circulant, therefore the inverse is circulant, making H a simple matrix [15, 17]. The output of the MMSE detector is the input of the Viterbi decoder which decodes STTC.
B. Equalization- QR Detection
The channel matrix H given in Equation (2) is decomposed into
𝐻 = 𝑄𝑅 (13)
where Q denotes unitary matrix that is satisfied with 𝑄𝐻𝑄 = 𝑄𝑄𝐻= 𝐼, where I is an identity matrix and R is the upper triangular matrix. The Schur algorithm [18] can be applied since H is assumed to be full-ranked because nT = nR; therefore, H can be expressed as
𝐻𝑇𝐻 = 𝑅𝑇𝑄𝑇𝑄𝑅 = 𝑅𝑇𝑅 (14)
Following the expression given in Equation (3), and by modification with the transform of Q, the received signal becomes,
Vol. 4, No. 9, September 2019
ŷ = 𝑄𝐻𝑄𝐻𝑅𝑆 + 𝑄𝐻𝑛 (15) According to [19], the Grammian matrix of H is a complex Hermitian Toeplitz matrix
𝐵 = 𝐻∗𝐻, where = {𝑏𝑖,𝑗}𝑖.𝑗𝑛= 0, 𝑏𝑖,𝑗 = 𝑟𝑘𝑚𝑜𝑑 𝑛, and 𝑟𝑘 is obtained by the cyclic convolution of the first column of H with itself. Obtaining deductions from Equations (3) and (12),
𝑦 = 𝑄𝐻𝑟 = 𝑅𝑆 + 𝑄𝐻𝑛 (16)
III. SIMULATION PARAMETERS
The simulation parameters employed in the conceptual frame of the research is provided in Table 1.
TABLEI:SIMULATION PARAMETERS
MIMO Channels 3GPP ITU Pedestrian
A (3G)
Extended Pedestrian A (4G)
Fading Distribution Rayleigh Rayleigh
FFT Size 512 512
Channel Bandwidth 5 MHz 5 MHz
Cyclic Prefix Length 2,10,20,30,40 128
Modulation Scheme QPSK QPSK
Antenna Configuration 2 x 2 2 x 2
Channel Coding None None
Channel Estimation and Equalization
Minimum Mean Square Error (MMSE) and Orthogonal Triangular (QR) Detection
IV. RESULTS AND DISCUSSION
As observed in Figure 2, the BER performance obtained by the different equalizer output depends relatively on the SNR (signal-to-noise ratio) of interest. At SNR of 24 dB, the QR equalizer gives an approximate increase of 10−6 over MMSE equalizer at 10−4. Also, Figure 3 gives the pairwise error probability (PEP) performance curve of MMSE and QR equalizer. Minimum PEP results in space- time code when the Euclidean distance is maximized. The diversity order played a key role in obtaining the PEP. For the space-time trellis coding (STTC), PEP gap was maintained evenly. At 10−4 the PEP obtained for STTC has a gain of 4 dB over STTC-MMSE and 2 dB over STTC-QR.
Yet, at SNR of 24 dB, STTC-MMSE gives an approximate PEP of 10−6 while STTC and STTC-QR gives approximate values of 10−4 and 10−5, respectively.
The operation of having to decode the transmitted symbol in the time domain could explain the gain. The probability that the decoder would select an erroneous signal was low due to the fact that equalization and the FFT/IFFT operations were carried out before the STTC decoding.
Figure 4 gives a graphical comparison of the Bit Error Rate performance of the 3G and 4G architecture. It can be observed that space-time trellis coding came be adapted to the 4G architecture. In the 3G architecture, the maximum BER is at SNR of 24dB, while the 4G, the maximum BER is at SNR of 28dB. For the 4G architecture, obtaining an approximate BER of 10−6 at such a high SNR than 3G
architecture is possible when coding, equalization is combined for a 4G network.
Fig. 2. Bit Error Rate performance comparison of MMSE and QR Equalizer.
Fig. 3. Pairwise Error Probability (PEP) performance of MMSE and QR Equalizer
Fig. 4. Comparison of Bit Error performance of 3G and 4G architecture using MMSE and QR Equalizers
V. CYCLIC PREFIX REDUCTION ANALYSIS FOR THE 3G ARCHITECTURE
The 3GPP LTE (long time evolution) system has two levels of CP length: 4.7μs and 16.6μs. An attempt is made to study the effect of CP reduction in the access network performance. A comparison was made in the receiver choice of a STTC-SCFDE system. The CP length was reduced from 8μs to 0.4μs and the results are observed in Figures 5 and 6. There was a significant reduction in BER of both MMSE and QR receivers but has no significant effect on the overall performance. It appears that for both
Vol. 4, No. 9, September 2019 equalizers, cyclic prefix of 20 samples (i.e. 4 μs) got an
average response. Aside this, the reduction of the cyclic prefix from 40 samples to 2 samples gives no significant change. As observed, a conclusion can be inferred that space-time coding can effectively handle the effect of ISI in wireless communication, and consequently, a reduction in cyclic prefix does not affect the overall performance of the system.
Fig. 5. Cyclic Prefix reduction for QR Equalizer
Fig. 6. Cyclic Prefix reduction for MMSE Equalizer
VI. CONCLUSION
This paper has used space-time coding in trellis modulation combined with single-carrier modulation and two-choice equalizers—QR and MMSE—to obtain a higher throughput and spectral efficiency. MMSE gives an optimal balance between noise enhancement and net ISI in the transmitted signal. There are advantages and disadvantages in the use of these equalizers: MMSE receiver outperforms the QR receiver in terms of bit error rate (BER); while in terms of pairwise error probability, the QR receiver outperforms the MMSE receiver. When a cyclic prefix reduction test was conducted, there was a significant reduction in BER of both MMSE and QR receivers but has no significant effect on the overall performance. Results obtained in employing a 4G architecture depicts the viability of combining coding and equalization methods in combating inherent fading present in the wireless channel.
REFERENCES
[1] R. Kashyap, and J. Bagga, J, “Equalization Techniques for MIMO Systems in Wireless Communication: A Review”. International Journal of Engineering and Advanced Technology (IJEAT), vol.3, pp. 260-264, 2014
[2] J. Ketonen, “Equalization and Channel estimation algorithms and implementations for Cellular MIMO-OFDM downlink”. Doctoral dissertation, University of Oulu Graduate School, 2017
[3] E.A. Lee, and D.G. Messerschmitt, “Digital communication”, Springer Science and Business Media, 2012.
[4] E. Dahlman, S. Parkvall, J. Sköld, and P. Beming, “3G Evolution:
HSPA and LTE for Mobile Broadband”, Academic Press, 2007.
[5] T. Hwang, C. Yang, G. Wu, S. Li, and G. Y. Li, “OFDM and its wireless applications: a survey”, IEEE transactions on Vehicular Technology, vol. 58, pp. 1673-1694, 2009.
[6] T.J. Willink, and P. H. Wittke, “Optimization and performance evaluation of multicarrier transmission”, IEEE Transactions on Information Theory, vol. 43, pp. 426-440, 1997.
[7] D. Falconer, S. L. Ariyavisitakul, A. Benyamin-Seeyar, and B.
Eidson, “Frequency domain equalization for single-carrier broadband wireless systems”, IEEE Communications Magazine, vol. 40, pp. 58- 66, 2002.
[8] T. N. Yune, D. Y. Seol, D. Kim and G. H. Im, “Single-carrier frequency domain equalization for broadband cooperative communications”. Cooperative Communications for Improved Wireless Network Transmission: Framework for Virtual Antenna Array Applications: Framework for Virtual Antenna Array Applications, pp. 399, 2010.
[9] I. A. Adebanjo, Y. O. Olasoji, and M. O. Kolawole, “Single carrier frequency domain equalization with space-time trellis codes”, Communications and Networks, vol. 9, pp. 164-171, 2017.
[10] Y. Jiang, M. K. Varanasi, and J. Li, “Performance analysis of ZF and MMSE equalizers for MIMO systems: an in-depth study of the high SNR regime”, IEEE Transactions on Information Theory, vol. 57, pp.
2008-2026, 2011.
[11] A. H. Mehana, and A. Nosratinia, “Diversity of MMSE MIMO receivers”, IEEE Transactions on Information Theory, vol. 58, pp.
6788-6805, 2012.
[12] A. Tajer, and A. Nosratinia, A. “Diversity order of MMSE single- carrier frequency domain linear equalization”, IEEE Global Telecommunications Conference, pp.1524-1528, 2007
[13] M. W. Meckes, “Some results on random circulant matrices”. High Dimensional Probability V: The Luminy Volume, 213-223, IMS Collections 5, Institute of Mathematical Statistics, Beachwood, OH, 2009.
[14] A. H. Mehana, and A. Nosratinia, “Single-carrier frequency-domain equalizer with multi-antenna transmit diversity”, IEEE Transactions on Wireless Communications, vol. 12, pp. 388-397, 2013.
[15] P. J. Davis, “Circulant matrices”, 2nd Edition, American Mathematical Society, Wiley, 2012.
[16] R. M. Gray, “Toeplitz and circulant matrices: A review”. in Communications and Information Theory, vol. 2 no. 3, pp.155-239, 2006.
[17] H. Jafarkhani, “Space-Time Coding, Theory and Practice”, 1st edition, Cambridge University Press, 2005
[18] D. Kressner, “Numerical methods for structured matrix factorizations”, Ph.D. thesis, Technical University of Chemnitz, Germany, 2001.
[19] C. Demeure, and L. Scharf, “Fast algorithms to QR factor circulant matrices, Acoustics, Speech, and Signal Processing”, ICASSP-89., International Conference on, IEEE, pp. 1123-1126, 1989.
[20] W. Liu, X. Liao, and Y. Bai, “SC-FDE Transmission with Length- Adaptive Cyclic Prefix Aided by Delay Spread Estimation”, Wireless Personal Communications, vol. 79, pp. 2059-2067, 2014.
[21] H. Witschnig, T. Mayer, A. Springer, A. Koppler, L. Maurer, M.
Huemer, and R. Weigel, “A different look on cyclic prefix for SC/FDE”, Personal, Indoor and Mobile Radio Communications, The 13th IEEE International Symposium, 824-828.
Adebanjo I. A. obtained B. Eng. and M.Eng degrees in Electrical and Electronics Engineering from Federal University of Technology, Akure, Nigeria in 2010 and 2016 respectively.
Her research interests include, digital signal processing, electronics, communication systems artificial intelligence in optical communication. She is currently pursuing her doctoral degree in free space optical communication at the Federal University of Technology, Akure, Nigeria.
She is a member of Optical Society of America (OSA) and NSE. She is a registered engineer with COREN.
Vol. 4, No. 9, September 2019 Olasoji Y.O. obtained M.Sc. degree in
Communication Engineering from Technical University, Burno, Czech Republic, in 1990 and Ph.D degree from the Federal University of Technology, Akure, in 2011. His research interests include digital signal processing, electronics, radio wave propagation, free space optical communication. He currently teaches in the department of Electrical and Electronics Engineering at both undergraduate and postgraduate levels in Federal University of Technology, Akure, Nigeria. He is a member of Nigeria Society of Engineers (NSE) and a registered Engineer.
Kolawole M. O. obtained PhD (UNSW) in electrical engineering and is Professor of Communication Engineering. He is an experienced project and research leader with knowledge across a broad range of business and technology environments. He has overseen a number of operational innovations. He holds 2 patents. He consults and leads research in three broad areas: Communications and Networking, Signal and Image Processing, and Remote sensing and Space systems. He is the author of four books: (i) Satellite Communication Engineering (www.crcnetbase.com/isbn/9780203910283); (ii) Radar
Systems, Peak Detection and Tracking
(https://www.elsevier.com/books/radar-systems-peak-detection-and- tracking/kolawol...); (iii) A Course in Telecommunication Engineering (www.schandgroup.com); (iv) Basic Electrical Engineering (ISBN: 978- 978-50084-7-0). He has also published over 60-refereed scientific papers and 24 technical/client reports, and a reviewer for a number of scientific journals. He is an active invited speaker to both trade and academic audiences in his areas of expertise. He plays clarinet and saxophone, and enjoys composing, arranging, and listening to music.
