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Quadrature Spatial Modulation Decoding Complexity: Study and Reduction Ibrahim Al-Nahhal, Octavia A. Dobre, Senior Member, IEEE, and Salama S. Ikki, Member, IEEE
Abstract This letter presents the computational complexity reduction of the maximum likelihood-quadrature spatial modulation (QSM-ML) decoder as compared with the conventional SM-ML. Furthermore, a novel reduced-complexity (RC) sphere decoder algorithm, especially designed for QSM decoders, is proposed. It is shown that the QSM-RC algorithm achieves the optimum QSM-ML bit error ratio performance. Using Monte Carlo simulations and mathematical analysis, at the same spectral efficiency and with notable superior performance, it is shown that the QSM-ML and QSM-RC decoders provide at least 50% and up to 96% reduction in the number of visited nodes, respectively, compared to the SM-ML decoder.
Index Terms Multiple-input multiple-output (MIMO), spatial modulation (SM), quadrature spatial modulation (QSM), sphere decoder (SD), complexity analysis.
I. I NTRODUCTION Low complexity represents an important requirement for the next generation of wireless systems [1]. In practical applications, reducing the computational complexity of algorithms or This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) through its Discovery program. O. A. Dobre and I. Al-Nahhal are with the Faculty of Engineering and Applied Science, Memorial University, 300 Prince Phillip Dr., St. John’s, NL, A1B 3X5, Canada (e-mail: {odobre, ioalnahhal}@mun.ca). S. Ikki is with the Department of Electrical Engineering, Lakehead University, Thunder Bay, ON P7B 5E1, Canada (e-mail:
[email protected]).
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systems is of utmost importance, while maintaining the performance within acceptable limits. Quadrature spatial modulation (QSM) is a promising technique [2], which employs the inphase and quadrature dimensions to improve the throughput of the conventional SM [3]. At the receive-side, the optimum maximum likelihood (ML) detector [2] jointly estimates the two active transmit antenna indices conveying the in-phase and quadrature pulse amplitude modulations (PAMs) as well as the PAMs. The detection process requires a high running cost. Recently, low-complexity decoding algorithms have been proposed for SM [4]-[6], and surveyed in [7]. In [4], [5], the sphere decoder (SD) [8] is applied to SM by employing a pruned radius (threshold) which depends on the number of receive antennas and noise variance. Estimation of the noise variance can be done either based on pilots/preamble or blindly [9]. The former leads to a loss of spectral efficiency, while the accuracy in the latter case depends on the data length. Furthermore, the noise variance estimation is required when the channel environment changes. In [6], a low-complexity algorithm has been proposed for SM. However, the algorithm requires an exhaustive pre-processing step to calculate the pseudo inverse of the channel matrix entries; additionally, the optimum bit error rate (BER) performance is not attained. In [10], the issue of the exhaustive pre-processing required in [6] is overcome by exploiting the sparsity property of large-scale QSM and the minimum mean square equalization. However, this does not hold for low/moderate-scale QSM systems. Furthermore, the algorithm in [10] requires estimation of the noise variance and does not provide the optimum BER performance. In previous work [2], [10], the authors have not noticed that the QSM not only improves the conventional SM BER performance, but it also reduces the decoding complexity. In this letter, the contribution is twofold: 1) The QSM-ML decoding complexity is studied: it is shown that the QSM-ML reduces the complexity of the SM-ML decoder with at least half, at the same spectral efficiency. 2) A novel low-complexity SD algorithm is proposed for QSM systems; the proposed algorithm provides: a) the optimum ML BER performance; b) simple radius which requires neither estimation of the noise variance nor exhaustive pre-processing; and c) more reduction in the complexity when compared with the existing algorithms, if they are directly applied to QSM. Analytical results are obtained for the complexity of the algorithms, and are confirmed with the Monte Carlo simulation.
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II. T HE QSM S YSTEM M ODEL Consider an Nr ×Nt MIMO system, which employs QSM [2], with Nt and Nr as the number of = transmit and receive antennas, respectively. The complex-valued transmitted symbol st = s< t +st = is divided into two real-valued PAMs, s< t and st , where st ∈ {s1 , · · · , sMQSM }, with MQSM as the
modulation order of the corresponding quadrature amplitude modulation (QAM). The transmitted symbol is delivered through an Nr × Nt flat fading channel, H ∈ CNr ×Nt . The noisy received vector, y ∈ CNr ×1 , is expressed as s= y = hn
