Research Article  Open Access
Yongzhi Li, Cheng Tao, Yapeng Li, Liu Liu, Tao Zhou, "Investigation of Sphere Decoder and Channel Tracking Algorithms for MediaBased Modulation over TimeSelective Channels", Wireless Communications and Mobile Computing, vol. 2017, Article ID 2509824, 11 pages, 2017. https://doi.org/10.1155/2017/2509824
Investigation of Sphere Decoder and Channel Tracking Algorithms for MediaBased Modulation over TimeSelective Channels
Abstract
The performance of mediabased modulation (MBM) systems, where additional information can be conveyed by the indices of the channel states created by RF mirrors, over timeselective channels is investigated. By transforming the MBM system model into a traditional MIMO system model, we first propose a reduced complexity sphere decoder algorithm. Then two channel tracking algorithms, which are based on least mean square adaptive filter and recursive leastsquares adaptive filter, are employed in order to combat the performance loss caused by the timevarying channels. Numerical results show that the proposed sphere decoder and these two channel tracking algorithms perform well in MBM systems.
1. Introduction
Recently, mediabased modulation (MBM) [1–3] has been proposed as a new technology to improve the spectral efficiency. Compared to the traditional modulation, where the information is embedded in the RF source (namely, the sourcebased modulation (SBM) as referred in [1]), additional information can be embedded into the indices of the active channel states in MBM systems. This can be achieved, for example, by placing a number of mirrors around each transmit antenna and indexing the mirror configuration.
MBM can be regarded as an instance of index modulation (or channel modulation as described in [4]), which has attracted significant attention recently due to the advantages in significantly improving the spectral and energy efficiency compared to the traditional digital modulation technique. For index modulation techniques, additional information bits can be embedded into the indices of the building blocks of the considered communication systems [5]. Although the concepts of antenna selection technique [6, 7] and reconfigurable antennas [8, 9] are very similar to the concept of index modulation, the main difference is that the antenna selection and reconfigurable antennas techniques do not index the activated antennas indices and the antenna patterns to convey additional information bits. One application of index modulation is the space shift keying (SSK) [10]. In SSK, the transmitter only activates one out of all the transmit antennas to convey the information bit, and hence the spectral efficiency scales only logarithmically in the number of transmit antennas. This implies that the SSK should exponentially increase the number of transmit antennas in order to linearly improve the spectral efficiency, which will also exponentially increase the system hardware cost in terms of both economic and energy consumption. However, these drawbacks for SSK can be significantly reduced in the MBM system, since the spectral efficiency in MBM only scales linearly with the number of RF mirrors, while the number of transmit antennas remains fixed.
For MBM technique, it has been proved in [2] that a MBM over static multipath channel asymptotically achieves the capacity of parallel AWGN channels, where for each unit of energy over the single transmit antenna, the effective energy for each AWGN channel is the statistical average of channel fading. It is confirmed in [3] that the performances of SIMOMBM are better than the SBM in some aspects and a significant coding gain can be obtained if single parity check symbol code is applied to a SIMOMBM. The authors in [11] presented the layered MIMO mediabased modulation (LMIMOMBM), which overcomes the recovery complexity issues at the cost of losing symbol error rate performance. By combining the spacetime block coding and MBM, the concept of spacetime channel modulation scheme is introduced in [4]. It is shown in [4] that proposed schemes achieve considerably better error performance than the existing MBM systems. Reference [12] combined MBM and space modulation technique to further improve the error performance of MIMOMBM systems. More importantly, a simple and convenient model, which is named generalized spatial modulationMBM (GSMMBM), is proposed in [12]. Based on [12], the intricate MBM model originally described in [1] can be reformed as an explicit MIMO system model, which allows us to use existing MIMO algorithm to handle the newly raised problem in MBM systems.
There have been a variety of existing works on the performance of MBM systems. However, the performance of MBM systems over timeselective channels has not been previously treated. Thus, in this paper, we investigate the performance of the MBM systems over timeselected channels by using the GSMMBM model. Based on this model, traditional MIMO decoding algorithms can be used in MBM systems. Although the maximum likelihood (ML) decoder is optimal, the algorithm complexity is prohibited in practice. While the authors in [3] proposed a novel detection algorithm for SIMOMBM based on sphere decoder to improve the processing speed of the receiver, we note that there is still room to reduce the complexity of sphere decoder. Therefore, in our work, we first present a reduced complexity sphere decoder for the MIMOMBM systems. As illustrated later, our proposed decoder has almost the same performance as ML decoder. Then the symbol error rate (SER) performance of MIMOMBM over timeselective channels is investigated using the proposed decoder method and two tracking algorithms called least mean square (LMS) MBM and recursive leastsquares (RLS) MBM are proposed to improve the performance of MIMOMBM. Numerical simulations verify that the proposed channel tracking algorithms can perform well in MBM systems.
2. System Model
In this section, we consider a MIMO system used with MBM technique and assume that there are bits information to be transmitted per channel use, where bits are the MBM information, bits are the SBM information, and and represent the number of transmit and receive antennas, respectively. In such system, each transmit unit can create independent channel states by turning on/off RF mirrors. By employing the GSMMIMO model described in [12], the received signal at the receiver can be expressed aswhere and represent the overall channel matrix and the overall data symbol vector, respectively. More precisely, the overall channel matrix and the overall data symbol vector are given bywhere with denoting the data symbol modulated by bits and with denoting the th channel state at the th transmit antenna, respectively. is the Gaussian noise, where we assume that each element has zeromean and finite variance . Note that the system model in (1) is a special case of the GSMMBM model by assuming that all the transmit antennas are activated. Thus, in the following, we refer to the system model in (1) as MIMOMBM.
Next, we will take an example to illustrate how to use the equivalent system model to realize the process of MIMOMBM. For simplicity, we take the example as illustrated in Figure 1, where a MIMOMBM system with , , , and is considered. According to (1)(2), the received signal can be expressed asThen different transmit units can send data symbol using the rule in Table 1. If the th transmit unit sends MBM information 0 and SBM information , which implies that the th channel is used to convey the SBM data symbol , then the whole transmit data of the th transmit unit can be represented as . If the th transmit unit sends MBM information 1 and SBM information , which implies that the th channel is used to convey the SBM data symbol , then the whole transmit data of the th transmit unit can be represented as .

Intuitively, if the receiver can perfectly recover the transmit data , then the MBM information can be obtained by using the same rule as shown in Table 1. In this way, both the MBM information and SBM information are demodulated at the receiver.
3. Reduced Complexity Sphere Decoder for MIMOMBM
By using the generalized system model in (1), the receiver can recover the transmit data symbol using existing decoder methods, that is, the ML decoder. However, the complexity of ML decoder is very high, especially when the constellation size is large. For completeness, we provide the computational complexity of ML decoder in the sequel.
At the receiver side, the ML decoder is considered to detect the signal aswhere is the set of all possible ’s. We then evaluate the complexity of ML decoder using the number of add/multiply/comparisons operations [3]:where , , , is the constellation size, and is the number of receive antennas. Here we take the quadrature amplitude modulation (QAM) for example. We can easily see that the complexity of ML decoder is prohibitive if the whole constellation size of MBM is large. Nevertheless, we note that the complexity of comparison operation is relatively lower than that of add/multiply operation. Therefore, we aim at reducing the number of add/multiply operations at the cost of increasing a few more comparison operations in order to reduce the complexity of the recovery process. In particular, we will propose a modified sphere decoder (SD) for MBM to achieve the same performance as the ML decoder.
The considered SD for MBM, which is called SDMBM algorithm later in this paper, is a modified version of the SD algorithm presented in [13]. Assume that is the sphere radius and the SD can be expressed aswhich means that SD performs a ML search only through those points in the sphere with radius . If is sufficiently large, then the solution of (6) will be the same as the solution of (4). If a candidate is within the search radius , then the following equation must be true:where is the th row of . In other words, a candidate can be rejected from further calculation if any partial sums are out of the sphere. This property reduces the amount of work required to evaluate candidates when the constellation size is large.
In fact, if we find a point in the sphere, we can update the radius with the Euclidean distance of that point to further reduce the complexity. But the initial sphere radius is not easy to find because if the radius is too large, there will be too many points to be searched; if the radius is too small, there will be no point to be found. The authors in [7] give a conventional method of deciding initial radius:where is the variance of noise, is the number of receive antennas, and is a constant number based on experience.
If we set the radius properly, we can always find the optimal solution and reduce the complexity. If we set the radius small occasionally, no point is in the sphere after the search and hence we need to enlarge the radius and do the search again. In this case, fortunately, since many points have been searched last time, we do not need to start over; we only need to start from where we stopped at last time and hence each point is searched no more than once. In this way, the numbers of add and multiply operations are reduced only at the cost of a few more comparisons. In what follows, we provide the full procedure of SDMIMOMBM: let be the constellation size of MBM, let be a set containing all possible transmit signals, , let be the position of search along path , and let be the Euclidean distance along path Then the algorithm can be formulated as Algorithm 1.

The advantage of the modified SDMIMOMBM is that it always can find a point in the sphere and search every point along each path no more than once, which reduces the complexity of the decoding process significantly.
4. TimeSelective Channels Model for MIMOMBM
It is known that a moving terminal, especially for the terminal on the highspeed railway [14], will induce large Doppler shift effect, which is the cause of the timeselectivity in the fastfading channel. Therefore, in this section, we use an improved sumofsinusoids statistical simulation model of Rayleigh fading channels to model timeselective channels property and study the performance of MIMOMBM. The normalized lowpass fading process of a new statistical sumofsinusoids simulation model is defined by [15]where , , , , and are statistically independent and uniformly distributed in the range of for all , and is the number of sinusoids. Thus, the channel matrix of MIMOMBM at time instant can be represented aswhere denotes the th channel coefficient between the th transmit unit and the th receive antenna at th time instant and is described as (9). We use autocorrelation to model the variation of channel factors with time [16]:where is the 0thorder Bessel function of the first kind and is the maximum Doppler shift. The coefficient relies on and .
5. Channel Tracking for MIMOMBM
In this section, we discuss how to obtain the channel matrix at the receiver and improve the performance of the decoding process. During the channel estimation phase, the transmitter sends pilots by scanning through the whole constellation space. The accuracy of estimated channel matrix depends on the length of pilots [17]. The initial estimated channel matrix can be expressed as [18]where is the exact channel matrix, is the estimation error matrix, and is the scale of estimation error. The estimated channel matrix is used to recover the data at the receiver. Intuitively, for timeselective channels, the symbol error rate will be larger than that in timeinvariant channel. Therefore, it is important to use channel tracking to update channel matrix in order to improve the system performance.
5.1. Least Mean Square (LMS) Algorithm for MIMOMBM
The adaptive LMS algorithm is widely used in practice due to its simplicity, computational efficiency, and good performance under a variety of operating conditions. It is based on minimum mean squared error (MMSE) criterion and is introduced in [19] in the form of vectors updating problem. We use it to track the channel matrix of MIMOMBM in timeselective channels and develop an algorithm to improve the performance of recovery process at the receiver side. We assume that the channel matrix is obtained after the pilot training at time instant . We also assume that the receiver employs the channel estimate to detect the transmit data symbol at time Then the estimation error vector iswhere is the received signal at time So the mean squared error (MSE) can be expressed asThen, substituting (15) into (16) and calculating the derivative of (16) yieldSince our goal is to minimize MSE of (16), we update the channel matrix using (18):where is the step size. According to [19], in order to simplify the selection of step size , we employ the normalized LMSMBM algorithm to update the channel matrix:where and After obtaining the channel matrix , the receiver recalculates with and as the decoded signal.
The idea of LMSMBM algorithm is that the receiver updates the estimated channel matrix to decode signal and reduce MSE. The implementation of the adaptive algorithm starts with a training transmission, which is used to acquire initial estimates. Then we use LMSMBM algorithm to estimate channel matrix and detect the transmit data symbol. The steps of LMSMBM algorithm are summarized as follows: Step 1: initial , , and parameter Step 2: recovering with estimated channel matrix and received signal : Step 3: calculating estimation error : Step 4: updating the channel matrix : Step 5: recalculating with the updated channel matrix ; then and go to Step :, , and are the estimated symbol, the received signal, and the estimated channel matrix at time , respectively. is the step size parameter of updating process and ; is the time of data transmission after pilot transmission. The calculation complexity is listed at the end of each step.
5.2. Recursive Least Square (RLS) Algorithm for MIMOMBM
The complexity of RLS algorithm [19] is a little higher than that of LMS algorithm because it considers the minimization of the total weighted squared error, which depends on all received signals and all transmitted symbols before time instant and the channel matrix at time . Therefore, the channel matrix at time index is chosen to minimize the cost function:where is the forgetting factor and We try to obtain channel matrix to reduce the weighted squared error and calculate the derivative of (20):whereFrom (22), we haveSubstituting and into (21), we obtainwhereBy assuming that is invertible and multiplying both sides of (24) by , we can obtainThe step size factor is defined asSimilarly, substituting and into (21), we obtainwhereBy assuming that is invertible and multiplying both sides of (28) by , we can obtainThe step size factor is defined asCombining (30) with (29), we getwhereSimilarly, combining (26), (30), and (32), we obtainwhich shows that the two adaptation gains have the same direction but different mold. And the major problem is how to update step size factor or By definingwe have such an updatingwhich is proven in [19] and omitted here. The update of channel matrix can be achieved by using (27), (31), (33), (34), (35), (36), and (26).
We use this algorithm called RLSMBM to track the channels of generalized mediabased modulation and improve the performance of symbol error rate. It is summarized as follows: Step 1: initial , , parameter , and Step 2: recovering with received signal and estimated channel matrix : Step 3: calculating step size parameter , , and : Step 4: calculating estimation error : Step 5: updating the channel matrix : Step 6: recalculating with the updated channel matrix and : Step 7: updating ; then and go to Step 2: and are the estimated symbol, the received signal, and the estimated channel matrix at time , respectively. is the forgetting factor and ; is a large constant initial parameter and is the data transmission time after pilot transmission.
5.3. Complexity Analysis for LMSMBM and RLSMBM
The aim of this subsection is to present the computational complexity of the proposed LMSMBM and RLSMBM channel tracking algorithms. The LMSMBM and RLSMBM algorithms are shown in Sections 5.1 and 5.2, respectively, with the complexity corresponding to each step of the algorithms. As we can see, the overall order of complexity of LMSMBM is given byAs for RLSMBM, the overall order of complexity is given byIn practice, we assume that because the SER performance loss of MIMOMBM is huge when is large. Therefore, if , , which means that the complexities of LMSMBM and RLSMBM are the same. If , and . Therefore, the computational complexities of LMSMBM and RLSMBM are equal.
6. Numerical Simulations
In this section, we present simulations to evaluate the performance of proposed sphere decoder for MIMOMBM and the tracking ability of LMSMBM and RLSMBM for MIMOMBM in timeselective channels. The timeinvariant channel vectors corresponding to each transmit unit are generated with complex i.i.d. Gaussian random components of unit variance, while the timeselective channel vectors of each transmit unit are generated according to Section 4. Energy per bit is defined as the sum of signal energies of transmit units divided by the total number of bits per channel use. Thus the signaltonoise rate (SNR) is defined as [20]where is the signal energy of each transmit unit and is the number of bits per channel use. Other default values of various parameters are listed in Table 2. We employ average symbol error rate (SER) and mean square error (MSE) of channel estimation as performance metrics to evaluate performances of MIMOMBM. MSE of channel estimation is defined as [21]

6.1. Performance of Sphere Decoder for MIMOMBM
In this subsection, we present the number of add/multiply/comparison operations to evaluate performance of the proposed SDMIMOMBM developed in Section 3. Assume that the empirical parameter [22] and the whole process is under timeinvariant Rayleigh fading channels. We consider the case with transmit units, each transmit unit with 1 bit to be sent, and the number of receive antennas .
Figure 2 compares the SER performances of the proposed SD decoder and ML decoder for MIMOMBM. We can readily see that SDMIMOMBM and ML decoder have an identical performance. Therefore, in the following, all the figures regarding the SER performance are plotted by using the SDMIMOMBM algorithm. The radio of computational complexity between the sphere decoder and the ML decoder, which is defined asis depicted in Figure 3. We can see that a higher reduction in computational complexity is achieved for high SNR values for both decoders; compared to traditional SD decoder, a higher reduction is achieved for more receive antennas, which means that the larger the constellation size is, the more reduction can be achieved for SDMIMOMBM; the computation complexity of SDMIMOMBM is slightly higher than that of traditional SD for MIMOMBM when the number of receive antennas is 2 due to the small constellation size. This is because the SDMIMOMBM decoder needs a few more comparison operations.
6.2. Performance of MIMOMBM in TimeSelective Channels
In this subsection, we evaluate performances of MIMOMBM in both timeinvariant channels and timeselective channels. We consider the case with transmit units, each transmit unit with 2 bits to be sent, and the number of receive antennas . The receiver is assumed to have perfect knowledge of the channel matrix during the pilot estimation phase.
The average SER performance of MIMOMBM versus SNR with different channels and different receive antennas is shown in Figure 4. As for timeselective channels, we assume that the maximum Doppler shift is 100 Hz and the maximum variation of channel factors is 0.96. We can see the following: compared with the timeinvariant channel case, the SER performance loss of MIMOMBM in timeselective channels is negligible at low SNR; the SER performance of timeselective channel saturates to a constant value in the high SNR region, which implies that higher SNR cannot improve the SER performance for timeselective channels; as the number of receive antennas grows large, the SER performance can be significantly improved, since the constellation size also grows large, which results in improving the accuracy of transmission.
Figure 5 shows the average SER performance of MIMOMBM versus the maximum variation of channel factor for timeinvariant and timeselective channels with receive antennas and The range of is from 0.84 to 1 and the change of is only based on the change of maximum Doppler shift. We can see that as grows larger, the change of channel becomes slower and the SER performance goes better. When is close to 1, the SER performance of MIMOMBM in timeselective channels is getting close to that in timeinvariant channels.
6.3. Performances of LMSMBM and RLSMBM
In this subsection, we evaluate the performance of channel tracking algorithms discussed in Section 5 and the performances of MIMOMBM in timeselective channels with LMSMBM and RLSMBM. For simplicity, we assume that the number of receive antennas , the length of pilot sequences is , the step size parameter of LMSMBM is 0.1, and the initial parameter and forgetting parameter of RLSMBM are 100 and 0.99, respectively.
Figure 6 shows the average SER of MIMOMBM versus the SNR over timeinvariant and timeselective channels with and without channel tracking algorithms. We can see that the SER performance of MIMOMBM in timeselective channels without channel tracking algorithm is the worst. The SER with LMSMBM algorithm and RLSMBM algorithm both outperform the SER without channel tracking algorithm. However, RLSMBM performs better than the LMSMBM. For example, when the target SER is , the SNR degradation of the one with LMSMBM is 2.3 dB and the SNR degradation of the one with RLSMBM is 2.7 dB. But there is still room compared to the one in timeinvariant channels because the SNR degradation is 5.8 dB compared to the one in timeselective channels without channel tracking.
The MSE of MIMOMBM versus the SNR with different channel tracking algorithms is plotted in Figure 7. We can see that when the SNR is low, the curves of LMSMBM and RLSMBM are overlapped and both of them outperform pilotaided estimation. When the SNR is high, RLSMBM performs better than LMSMBM. Figure 8 shows the MSE versus the maximum variation of channel factors at We can see that when , the channel becomes timeinvariant. However, the MSEs of LMSMBM and RLSMBM increase with the decreasing of , which implies that the faster the channel changes, the larger the estimation error is, even if the channel tracking algorithms are used. Figure 9 shows the SER versus the maximum variation of channel factors. We can see that when the target SER is , the coefficient of LMSMBM and RLSMBM estimated channels can be about 0.959 and 0.885, respectively.
LMSMBM and RLSMBM algorithms are only carried out once during the channel tracking process in Figures 6–9. Actually, system can improve the reliability by implementing channel tracking algorithms more times at each symbol. Furthermore, during the pilotaided channel estimation, the length of pilots affects the accuracy of channel tracking. Apparently, better SER performance can be obtained if the length of pilots is long. However, since using longer pilots reduces the spectral efficiency of system, it is very important to choose the reasonable length of pilots. MBM system needs to transmit pilots by scanning through all the channel states, which makes MBM need at least pilots.
Figure 10 gives the performance of LMSMBM and RLSMBM with pilots, respectively. We can see that as the number of pilots gets larger, the performance of LMSMBM and RLSMBM gets better. As for LMSMBM, when the target SER is , the corresponding SNRs for MBM with , , and pilots are 9.9 dB, 8.5 dB, and 8 dB, respectively. As for RLSMBM, when the target SER is , the corresponding SNRs for MBM with , , and pilots are 9.5 dB, 8.2 dB, and 7.8 dB, respectively. Therefore, longer pilot training length results in a better performance of LMSMBM and RLSMBM.
7. Conclusion
In this paper, by employing the GSMMBM model, a reduced complexity sphere decoder and two channel tracking algorithms called LMSMBM and RLSMBM in timeselective channels based on the MIMOMBM model are proposed. The proposed sphere decoder performs as well as ML decoder in SER performance and reduces the complexity of the decoding process, especially when the constellation size is large. Two channel tracking algorithms based on least mean square adaptive filter and recursive leastsquares adaptive filter called LMSMBM and RLSMBM are proposed. The complexities of these two algorithms are equal to each other on a macro level, but RLSMBM is more complicated than LMSMBM in some details. Numerical simulations verify that although the two channel tracking algorithms improve systems’ SER performance significantly, RLSMBM outperforms the LMSMBM. Meanwhile, the length of pilots affects the performance of LMSMBM and RLSMBM. Longer pilot training length results in a better performance. The details about the performances of the proposed sphere decoder and two tracking algorithms were illustrated by simulations.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The research was supported in part by the NSFC Project (Grant no. 61471027), Fundamental Research Funds for the Central Universities (Grant 2017JBM306), Beijing Nova Programme (no. Z161100004916068), the Open Research Fund of National Mobile Communications Research Laboratory, Southeast University (Grant no. 2017D01), and Beijing Science and Technology Talents Program (2016023).
References
 A. K. Khandani, “Mediabased modulation: a new approach to wireless transmission,” in Proceedings of the 2013 IEEE International Symposium on Information Theory, ISIT 2013, pp. 3050–3054, July 2013. View at: Publisher Site  Google Scholar
 A. K. Khandani, “Mediabased modulation: Converting static Rayleigh fading to AWGN,” in Proceedings of the 2014 IEEE International Symposium on Information Theory, ISIT 2014, pp. 1549–1553, June 2014. View at: Publisher Site  Google Scholar
 K. Vamaraju, Single input multiple output media based modulation [Ph.D. thesis], University of Waterloo, 2015.
 E. Basar and I. Altunbas, “Spacetime channel modulation,” IEEE Transactions on Vehicular Technology, vol. 66, no. 8, pp. 7609–7614, 2017. View at: Publisher Site  Google Scholar
 E. Basar, “Index modulation techniques for 5G wireless networks,” IEEE Communications Magazine, vol. 54, no. 7, pp. 168–175, 2016. View at: Publisher Site  Google Scholar
 S. Sanayei and A. Nosratinia, “Antenna selection in MIMO systems,” IEEE Communications Magazine, vol. 42, no. 10, pp. 68–73, 2004. View at: Publisher Site  Google Scholar
 R. W. Heath Jr., S. Sandhu, and A. Paulraj, “Antenna selection for spatial multiplexing systems with linear receivers,” IEEE Communications Letters, vol. 5, no. 4, pp. 142–144, 2001. View at: Publisher Site  Google Scholar
 C. G. Christodoulou, Y. Tawk, S. A. Lane, and S. R. Erwin, “Reconfigurable antennas for wireless and space applications,” Proceedings of the IEEE, vol. 100, no. 7, pp. 2250–2261, 2012. View at: Publisher Site  Google Scholar
 J. Costantine, Y. Tawk, S. E. Barbin, and C. G. Christodoulou, “Reconfigurable antennas: design and applications,” Proceedings of the IEEE, vol. 103, no. 3, pp. 424–437, 2015. View at: Publisher Site  Google Scholar
 J. Jeganathan, A. Ghrayeb, L. Szczecinski, and A. Ceron, “Space shift keying modulation for MIMO channels,” IEEE Transactions on Wireless Communications, vol. 8, no. 7, pp. 3692–3703, 2009. View at: Publisher Site  Google Scholar
 C. Xu and M. Zhao, “Mutual information and secrecy enhancement analysis of mediabased modulation,” in Proceedings of the 8th International Conference on Wireless Communications and Signal Processing, WCSP 2016, p. 5, 1, October 2016. View at: Publisher Site  Google Scholar
 Y. Naresh and A. Chockalingam, “On mediabased modulation using RF mirrors,” IEEE Transactions on Vehicular Technology, vol. 66, no. 6, pp. 4967–4983, 2017. View at: Publisher Site  Google Scholar
 E. Viterbo and J. Boutros, “A universal lattice code decoder for fading channels,” Institute of Electrical and Electronics Engineers. Transactions on Information Theory, vol. 45, no. 5, pp. 1639–1642, 1999. View at: Publisher Site  Google Scholar  MathSciNet
 T. Zhou, C. Tao, S. Salous, L. Liu, and Z. Tan, “Channel sounding for highspeed railway communication systems,” IEEE Communications Magazine, vol. 53, no. 10, pp. 70–77, 2015. View at: Publisher Site  Google Scholar
 Y. R. Zheng and C. Xiao, “Improved models for the generation of multiple uncorrelated Rayleigh fading waveforms,” IEEE Communications Letters, vol. 6, no. 6, pp. 256–258, 2002. View at: Publisher Site  Google Scholar
 Z. Liu, X. Ma, and G. B. Giannakis, “Spacetime coding and Kalman filtering for timeselective fading channels,” IEEE Transactions on Communications, vol. 50, no. 2, pp. 183–186, 2002. View at: Publisher Site  Google Scholar
 B. Hassibi and B. M. Hochwald, “How much training is needed in multipleantenna wireless links?” IEEE Transactions on Information Theory, vol. 49, no. 4, pp. 951–963, 2003. View at: Publisher Site  Google Scholar
 R. Couillet and M. Debbah, Random Matrix Methods for Wireless Communications, Cambridge University Press, Cambridge, UK, 2011. View at: Publisher Site  MathSciNet
 D. G. Manolakis, V. K. Ingle, and S. M. Kogon, Statistical and Adaptive Signal Processing: Spectral Estimation, Signal Modeling, Adaptive Filtering, and Array Processing, McGrawHill, Boston, Mass, USA, 2008.
 K. A. K. Seifi E and M. Atamanesh, “Mediabased MIMO: a new frontier in wireless communications,” https://arxiv.org/pdf/1507.07516.pdf. View at: Google Scholar
 Y. Li, “Simplified channel estimation for OFDM systems with multiple transmit antennas,” IEEE Transactions on Wireless Communications, vol. 1, no. 1, pp. 67–75, 2002. View at: Publisher Site  Google Scholar
 O. Xia, H. Hu, and H. Wang, “Reduced initial searching radius for sphere decoder,” in Proceedings of the 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, PIMRC'07, pp. 1–4, September 2007. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2017 Yongzhi Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.