Transmit antenna subset selection for highrate MIMOOFDM systems in the presence of nonlinear power amplifiers
 Ngoc Phuc Le^{1}Email author,
 Farzad Safaei^{1} and
 Le Chung Tran^{1}
https://doi.org/10.1186/16871499201427
© Le et al.; licensee Springer. 2014
Received: 9 September 2013
Accepted: 20 January 2014
Published: 13 February 2014
Abstract
The deployment of antenna subset selection on a persubcarrier basis in MIMOOFDM systems could improve the system performance and/or increase data rates. This paper investigates this technique for the MIMOOFDM systems suffering nonlinear distortions due to highpower amplifiers. At first, some problems pertaining to the implementation of the conventional persubcarrier antenna selection approach, including power imbalance across transmit antennas and noncausality of antenna selection criteria, are identified. Next, an optimal selection scheme is devised by means of linear optimization to overcome those drawbacks. This scheme optimally allocates data subcarriers under a constraint that all antennas have the same number of data symbols. The formulated optimization problem to realize the constrained scheme could be applied to the systems with an arbitrary number of multiplexed data streams and with different antenna selection criteria. Finally, a reduced complexity strategy that requires smaller feedback information and lower computational effort for solving the optimization problem is developed. The efficacy of the constrained antenna selection approach over the conventional selection approach is analyzed directly in nonlinear fading channels. Simulation results demonstrate that a significant improvement in terms of error performance could be achieved in the proposed system with a constrained selection compared to its counterpart.
Keywords
1 Introduction
Recent years have seen a great demand for very fast data speeds in wireless multimedia applications. One of the most attractive techniques that could deliver highrate transmission is multiinput multioutput orthogonal frequency division multiplexing (MIMOOFDM) [1]. The major benefits of this technique resulting from OFDM include high spectral efficiency and robustness against intersymbol interference (ISI) in multipath fading channels [1]. Simultaneously, an increased capacity and/or diversity gain could be achieved with MIMO [2, 3]. Among various MIMO schemes, antenna selection appears to be promising for OFDM wireless systems. This is mainly due to lowcost implementation and small amount of feedback information required, in comparison with other precoding methods [4]. In addition, this scheme is shown to be effective in equivalent isotropic radiated power (EIRP)restricted systems, such as ultrawideband (UWB) [5, 6].
Many research works have considered the application of antenna selection in OFDM systems, e.g., in [7–13]. In general, they can be categorized into two approaches: bulk selection (i.e., choosing the same antennas for all subcarriers) [7–10] and persubcarrier selection (i.e., selecting antenna on each subcarrier basis) [10–13]. The main benefit of the latter over the former is that a much larger coding gain can be achieved by exploiting the frequencyselective nature of the channels [10]. Thus, persubcarrier selection is very attractive for wideband communications. However, as the conventional persubcarrier selection method selects antennas independently for each subcarrier, a large number of data symbols may be allocated to some particular antennas. The input signal powers of the highpower amplifiers (HPAs) associated with these antennas might be very large, whereas those at the other antennas might be small. As a result, the HPAs on some antennas may operate in their inefficient power regions due to the small average powers of the input signals. Meanwhile, on the other antennas, nonlinear distortions, including inband and outofband distortions, occur when the very large signal powers pass through the HPAs. The inband distortion degrades error performance and system capacity [14], whereas the outofband distortion arising from the spectral broadening effect of the HPAs interferes the systems operating in the adjacent channels [15, 16].
It is obvious that the imbalance allocation of data subcarriers associated with the conventional persubcarrier antenna selection scheme reduces the potential benefits of the antenna selection OFDM systems. One possible approach to deal with this problem is selecting transmit antennas under a constraint that the number of data subcarriers allocated to each antenna is equal. As a balance constraint is required, the constrained selection (i.e., powerbalance selection) scheme should retain the benefits in terms of error performance or capacity as large as possible. Some research works have studied the constrained selection approach in the literature, such as [11–13]. In [11, 12], allocation algorithms were developed to realize the constrained selection scheme. Meanwhile, the authors in [13] considered linear optimization to devise their constrained selection scheme. It was shown that the selection scheme based on optimization could offer a better performance than the suboptimal solutions in [11, 12]. However, the formulated optimization problem in [13] is only applicable to OFDM systems where one antenna is active on each subcarrier. More importantly, to the best of our knowledge, all the existing works about constrained antenna selection, e.g., [11–13], only consider the effects of nonlinear HPAs on the system performance by means of simulations for demonstration purposes. This approach obviously has some limitations as it does not fully give an insight into the system characteristics. In particular, the question about whether antenna selection criteria originally derived in linear channels are still effective in nonlinear channels has not been addressed. This issue is of importance as the occurrence of nonlinear distortions may have impacts on the antenna selection criteria. Besides, the benefits in terms of error performance and/or capacity of the powerbalance selection over the conventional scheme have not been analyzed directly for the systems suffering nonlinear distortions due to HPAs. It is clearly worth performing such an analysis, given that the efficacy of powerbalance selection over its counterpart comes from the HPA nonlinearity. In addition, [11–13] only considered antenna selection schemes where data are transmitted from one antenna on each subcarrier. Thus, the achieved spectral efficiency was limited. To fulfill the expectation of delivering very fast data speeds for future wireless applications, antenna subset selection, where multiple data symbols are transmitted simultaneously from multiple antennas on each subcarrier, should be investigated.
 1.
A noncausal problem associated with the implementation of conventional persubcarrier antenna selection in MIMOOFDM systems suffering nonlinear distortions is identified for the first time. The noncausality arises because the impacts of nonlinear HPAs on transmitted data symbols need to be known in order to select a proper antenna subset for each subcarrier. Meanwhile, the calculations of these impacts require the total number of data subcarriers assigned on each antenna to be known.
 2.
An efficient constrained antenna subset selection scheme is proposed for MIMOOFDM systems to overcome the drawbacks of the conventional scheme. The proposed scheme is realized based on a linear optimization problem that is formulated in systems with an arbitrary number of multiplexed data streams. Although the formulated optimization problem introduces an additional complexity in the proposed scheme, it can be solved efficiently by wellknown linear programming methods.
 3.
A reduced complexity strategy that simultaneously requires a smaller number of feedback bits and lower computational effort to solve the optimization problem is proposed by exploiting the channel correlation between adjacent OFDM subcarriers.
 4.
The efficacy of the constrained antenna selection approach over the conventional approach is analyzed directly in the nonlinear fading channels. Specifically, we show that the average mean squared error (MSE) and the average signaltonoiseplusdistortion ratio (SNDR) in the proposed system with a constrained selection are better than those in its counterpart. Numerical results are also provided to verify the analyses and demonstrate the improvement in terms of error performance in the proposed system.
The remainder of the paper is organized as follows. In Section 2, an antenna selection MIMOOFDM system model with nonlinear HPAs is described. In Section 3, persubcarrier antenna subset selection criterion is investigated in the systems suffering nonlinear distortions. In Section 4, an optimization problem for data subcarrier allocation with a power balancing is formulated. Performance analysis is carried out in Section 5. Simulation results are provided in Section 6. Finally, Section 7 concludes the paper.
1.1 Notation
Throughout this paper, a bold letter denotes a vector or a matrix, whereas an italic letter denotes a variable. (.)^{ * }, (.)^{ T }, (.)^{ H }, (.)^{−1}, ⊗, E{.}, and tr{.} denote complex conjugation, transpose, Hermitian transpose, inverse, the Kronecker product, expectation, and the trace of a matrix, respectively. I_{ n } indicates the n × n identity matrix, and 1_{ K } is a K × 1 vector of ones. diag(a) is the n × n diagonal matrix whose elements are the elements of vector a. ℜ indicates the set of real numbers.
2 Antenna subset selection MIMOOFDM systems with nonlinear HPAs
2.1 Transmitter
where P_{o,sat} is the output saturation power level of HPAs; s_{ i }(n) and ∠s_{ i }(n) denote the magnitude and phase of s_{ i }(n), respectively. Also, it is assumed that P_{o,sat} = P_{i,sat}, where P_{i,sat} is the input saturation power level.
2.2 Receiver
In the above equations, ${h}_{j,i}^{k}$ indicates the channel coefficient between the i th transmit antenna and the j th receive antenna. d_{ i }^{ k } denotes the frequencydomain distortion noise at the i th transmit antenna. Also, y_{ j }^{ k } and n_{ j }^{ k } denote the received signal and the thermal noise at the j th receive antenna, respectively. The effective channel matrix ${\underset{\xaf}{\mathbf{H}}}_{k}$, the effective scale factor $\underset{\xaf}{\mathbf{\alpha}}=\mathrm{diag}\left(\phantom{\rule{0.2em}{0ex}}\left[{\underset{\xaf}{\alpha}}_{1}\phantom{\rule{0.24em}{0ex}}{\underset{\xaf}{\alpha}}_{2}\dots {\underset{\xaf}{\alpha}}_{{n}_{\mathrm{D}}}\right]\phantom{\rule{0.2em}{0ex}}\right)$, and the effective distortion noise ${\underset{\xaf}{\mathbf{d}}}_{k}={\left[{\underset{\xaf}{d}}_{1}^{k}\phantom{\rule{0.12em}{0ex}}{\underset{\xaf}{d}}_{2}^{k}\dots {\underset{\xaf}{d}}_{{n}_{\mathrm{D}}}^{k}\right]}^{\phantom{\rule{0.12em}{0ex}}T}$ are obtained by eliminating the columns of H_{ k }, the rows of α, and the elements of d_{ k } that correspond to the unselected transmit antennas, respectively. The distortion noise d_{ i }^{ k } can be modeled as a zeromean complex Gaussian random variable with variance ${\sigma}_{{d}_{i}}^{2}={\sigma}_{{\eta}_{i}}^{2}$ (i.e., ${\sigma}_{{d}_{i}}^{2}$ is equal to that of the timedomain distortion noise). Note that as clipping is performed on the Nyquistrate samples, all the subcarriers on the i th antenna experience the same attenuation α_{ i } and the variance ${\sigma}_{{d}_{i}}^{2}$[17]. Thus, the factors of α and $\underset{\xaf}{\mathbf{\alpha}}$, the variance of d, denoted as ${\mathbf{\sigma}}_{d}^{2}=\mathrm{diag}\left(\phantom{\rule{0.2em}{0ex}}\left[{\sigma}_{{d}_{1}}^{2}{\sigma}_{{d}_{2}}^{2}\dots {\sigma}_{{d}_{{n}_{\mathrm{T}}}}^{2}\phantom{\rule{0.2em}{0ex}}\right]\right)$, and the variance of $\underset{\xaf}{\mathbf{d}}$, denoted as ${\mathbf{\sigma}}_{\underset{\xaf}{d}}^{2}=\mathrm{diag}\left(\phantom{\rule{0.2em}{0ex}}\left[{\sigma}_{{\underset{\xaf}{d}}_{1}}^{2}\phantom{\rule{0.12em}{0ex}}\phantom{\rule{0.24em}{0ex}}{\sigma}_{{\underset{\xaf}{d}}_{2}}^{2}\dots {\sigma}_{{\underset{\xaf}{d}}_{{n}_{\mathrm{D}}}}^{2}\right]\phantom{\rule{0.2em}{0ex}}\right)$, are the same for all subcarriers. Here, the indices k associated with α_{ i } and ${\sigma}_{{d}_{i}}^{2}$ are dropped for simplicity. The thermal noise is modeled as a Gaussian random variable with zero mean and $E\left\{{\mathbf{n}}_{k}{\mathbf{n}}_{k}^{H}\right\}={\sigma}_{n}^{2}{\mathbf{I}}_{{n}_{\mathrm{R}}}.$ Also, it is assumed that persubcarrier power loading is not an option due to the limited feedback rate and the strict regulation of a power spectral mask, such as in UWB systems.
where ${\mathbf{G}}_{k}={\underset{\xaf}{\mathbf{H}}}_{k}\underset{\xaf}{\mathbf{\alpha}}$, and G_{ k }^{+} = (G_{ k }^{ H }G_{ k })^{−1}G_{ k }^{ H } denotes the MoorePenrose pseudoinverse of a matrix G_{ k }. It can be seen from (13) that the estimated symbols consist of the desired component q_{ k }, the distortion noise after equalization ${\underset{\xaf}{\mathbf{\alpha}}}^{1}{\underset{\xaf}{\mathbf{d}}}_{k}$, and the thermal noise after equalization G_{ k }^{+}n_{ k }. Note that to characterize the impacts of nonlinear distortions on the system performance, many other physical layer impairments, such as channel estimation error or I/Q imbalance, were not taken into consideration in this paper. For the case of existing errors in channel estimation, the readers are referred to [22], where the performance of a MIMO system in the presence of both nonlinear distortions and channel estimation errors is investigated. Although [22] does not consider antenna selection OFDM systems, the obtained results are useful for analyzing this system.
3 Persubcarrier antenna subset selection criteria in the presence of nonlinear HPAs
3.1 Persubcarrier antenna subset selection criteria
Antenna subsets
γ  Γ_{γ} 

1  {1,2} 
2  {1,3} 
3  {1,4} 
4  {2,3} 
5  {2,4} 
6  {3,4} 
 1.If the same number of data subcarriers is allocated to all transmit antennas, the OFDM symbols in all antennas experience the same distortion characteristics (cf. (3) to (5)). Therefore, (15) can be simplified to$\begin{array}{ll}{\mathbf{\Gamma}}_{\gamma}\left(k\right)& =arg\underset{\gamma =1,\dots ,\Gamma}{min}{\sigma}_{n}^{2}\phantom{\rule{0.12em}{0ex}}\mathrm{tr}\left\{{\left({\mathbf{G}}_{k}^{H}{\mathbf{G}}_{k}\right)}^{1}\right\}\\ =arg\underset{\gamma =1,\dots ,\Gamma}{min}\mathrm{tr}\left\{{\left({\underset{\xaf}{\mathbf{H}}}_{k}^{H}{\underset{\xaf}{\mathbf{H}}}_{k}\right)}^{1}\right\},\end{array}$(16)
 2.
On the other hand, if the above condition is not satisfied, the persubcarrier antenna selection criteria, e.g., MMSE criterion in (15), cannot be realized due to a noncausal problem. The noncausality arises because the selection of antenna subset for each subcarrier, i.e., calculating a metric MSE _{ γ } ^{ k }, requires the values $\underset{\xaf}{\mathbf{\alpha}}$ and ${\mathbf{\sigma}}_{\underset{\xaf}{\tilde{d}}}^{2}$. Meanwhile, the calculations of these two values require the total number of data subcarriers assigned on each antenna to be known. To realize the persubcarrier antenna selection, the criterion in (16) could be applied. However, as shown in (14) and (15), when the impacts of nonlinear HPAs are ignored, the selected antenna subset may not be the one that could obtain minimum MSE. Thus, the optimality of the selection criterion in terms of minimum MSE might not be fully achieved.
Although only the MMSE criterion is considered in this paper, we note that the noncausal problem occurs with all persubcarrier antenna selection criteria in the OFDM systems suffering nonlinear distortions.
3.2 Feedback considerations
where [H_{ k }]_{ i,j } denotes the (i,j)th entry of the matrix H_{ k }, φ_{ t } (where t = 0, 1, …, T − 1) denotes the normalized channel power delay profile, i.e., $\sum _{t=0}^{T1}{\phi}_{t}^{2}}=1$, and δ(.) is the Kroneckerdelta function. It can be seen from (18) that the frequency correlation coefficients depend on the difference between subcarriers (k_{1}k_{2}), rather than on the subcarriers themselves. Thus, given ${\rho}_{{k}_{1}{k}_{2}}$, we can estimate (k_{1}k_{2}). In other words, the number of subcarriers in one cluster (i.e., the value of L) can be estimated, given the level of crosscorrelation among the subcarriers within a cluster. The study of optimal designs regarding feedback reduction (e.g., deriving an optimal value of L with respect to error performancefeedback rate tradeoff) is beyond the scope of this paper. The readers are referred to, e.g., [25, 26], for this topic of research.
4 Optimization formulation for data subcarrier allocation with power balancing
 1.
A selection variable (i.e., optimization variable) in [13] was defined based on an antenna basis. When n _{D} > 1, a similar definition of a selection variable will result in binary nonlinear optimization problems. This is clearly not favorable from a practical viewpoint. As shown later in this section, binary linear optimization could be obtained by defining a selection variable based on a subset basis.
 2.
Only a system with full feedback was considered in [13]. In OFDM systems with large number of subcarriers, not only a large amount of feedback information is required but also the complexity to solve the optimization problem increases. Thus, it is of interest to formulate linear optimization working in conjunction with feedback reduction.
In the following, linear optimization problems are formulated for both full feedback and reduced feedback systems with an arbitrary number of data streams n_{D} ≥ 1.
4.1 Optimization formulation
For instance, if n_{T} = 4, n_{D} = 2, and K = 12, then ${\lambda}_{\gamma}=\frac{12}{6}=2,$ ∀γ = 1, 2, …, 6. As all subsets are chosen twice, from Table 1, we know that each antenna has six data symbols (cf. Figure 3b).
The optimization problem is now a minimization of the cost function (19) subject to two constraints, (20) and (21). Note that in the system without power balancing, a problem of subcarrier allocation is equivalent to minimizing (19), subject to the constraint (20) only.
It is obvious that (27) has a canonical form of a binary linear optimization problem. Moreover, this binary optimization problem can be relaxed to a linear programming (LP) problem that has a solution z ∈ {0, 1}^{KΓ × 1} (see Appendix 1). As a result, the optimization problem in (27) can be solved efficiently by wellknown linear programming methods, such as simplex methods or interior point methods [27]. When n_{D} = 1, the formulated problem in (27) is identical to the one in [13]. In addition, it is worth noting that as the optimization problem in (27) has been formulated in a way of minimizing the cost, a negative sign has to be included in the cost metric if capacity or SNR is used.
4.2 Optimization in the system with reduced feedback
 1.
The number of variables is ΓK/L, i.e., z ∈ {0, 1}^{(KΓ/L) × 1}.
 2.
A cost vector is c ∈ ℜ^{(K Γ/L) × 1} and its elements are ${c}_{\gamma}^{m}$.
 3.
Matrix A and vector a in the constraint will need to be modified accordingly.
With respect to the complexity of the proposed selection scheme, we note that the complexity to solve linear optimization using interior point methods can be reduced to O([(ΓK/L)^{3}/ln(ΓK/L)]ζ), where O(.) denotes an order of complexity, and ζ is the bit size of the optimization problem [28]. Therefore, solving the optimization associated with reduced feedback (i.e., L > 1) will require much lower computational effort compared to that on a subcarrier basis (i.e., L = 1). As a result, the proposed system with this combined strategy could enjoy both small feedback overhead and low complexity for optimization.
5 Performance analysis
In Section 4, a linear optimization problem has been formulated to realize an optimal (constrained) selection scheme from a viewpoint of minimum MSE. In this section, we analyze the effectiveness of this selection scheme by showing that in the presence of nonlinear distortions, the average MSE, as well as the average SNDR, in the proposed system is better than that in the conventional system. Without loss of generality, it is assumed that all HPAs have the input saturation level of P_{i,sat} and operate with an input backoff of $\mathrm{IBO}={P}_{\mathrm{i},\mathrm{sat}}/{\sigma}_{\overline{K}}^{2}$. In the conventional (unconstrained) system, the power backoff is required on the antennas where the numbers of the allocated data subcarriers are larger than $\overline{K}$, i.e., ${K}_{i}>\overline{K}$, to avoid error floor and other deleterious effects. This is equivalent to scaling the amplitudes of the signals on these antennas by a factor ${\beta}_{i}=\sqrt{{\sigma}_{\overline{K}}^{2}/{\sigma}_{{K}_{i}}^{2}}<1$. Meanwhile, the powers of the signals on the other antennas, i.e., ${K}_{i}\le \overline{K}$, are not scaled up due to an EIRP restriction as well as the complexity of power loading.
where V denotes a set of antennas in which the number of allocated data subcarriers on these antennas are smaller than or equal to $\overline{K}$, and $\overline{\mathbf{V}}$ is a set of the remaining antennas.
It can be seen from (38) that the change in the average MSE when implementing balanced allocation compared to the case of unbalanced allocation comes from I_{ V }, ${I}_{\overline{\mathbf{V}}}$, and I_{ Δ }, where

I_{ V } is a kind of MSE penalty that is associated with data subcarriers on the antennas where ${K}_{i}<\overline{K}$. It can be seen from (4) and (5) that when K_{ i } increases, α_{ i } decreases and ${\sigma}_{{\eta}_{i}}^{2}$ increases. Thus, the value of the function F(u, ${\underset{\xaf}{\mathbf{H}}}_{k}$, K_{ i }), defined in (32), increases when K_{ i } increases. Consequently, the value of I_{ V } in (39) is always negative (i.e., I_{ V } < 0).

${I}_{\overline{\mathbf{V}}}$ is a MSE benefit that is associated with data subcarriers on the antennas where ${K}_{i}>\overline{K},$i = Ω_{ k }(u). As the scale factor β_{ i }^{2} < 1, it is clear that ${I}_{\overline{\mathbf{V}}}>0$. The more data subcarriers are allocated to some particular antennas, the smaller the value ${\beta}_{i}^{2}={\sigma}_{\overline{K}}^{2}/{\sigma}_{{K}_{i}}^{2}=\overline{K}/{K}_{i}$ is required, and thus, ${I}_{\overline{\mathbf{V}}}$ becomes larger.

I_{ Δ } is a kind of MSE penalty that is incurred because the chosen effective channel matrices in the constrained system are different from the ones in the unconstrained system. Note that I_{ Δ } < 0 because Δ > 0 as mentioned before.
It is important to note that for a given system with defined HPAs in terms of nonlinear characteristics, only I_{ Δ } among the three components depends on the effective channel matrices ${\underset{\xaf}{\overline{\mathbf{H}}}}_{k},k=0,1,\dots ,K1.$ Therefore, while different balanced selection schemes introduce different changes in the average MSE, the difference in the average MSE indeed comes from the difference in I_{ Δ }. From this observation, it is clear that to make the value Θ, the difference in the average MSE between the unconstrained and constrained systems, become as positive as possible, the constrained selection method should result in the cost penalty Δ as small as possible. We note that the formulated optimization in (27) could achieve the minimum possible value of the total cost. Hence, with the definition of Δ as shown in (36), it is expected that the proposed constrained selection scheme based on linear optimization will guarantee the minimum achievable value of Δ. In addition, an upper bound of the expected value of the cost penalty is derived in Appendix 2. Based on the obtained bound, it is observed that for fixed values of n_{T} and n_{D}, the cost penalty becomes smaller when the number of receive antennas n_{R} increases.
then the proposed system can achieve PAPR reduction. The reason is that while the average power across antennas is similar in both systems, i.e., $\left(1/{n}_{\mathrm{T}}K\right){\displaystyle \sum _{n=0}^{K1}\phantom{\rule{0.12em}{0ex}}{\displaystyle \sum _{i=1}^{{n}_{\mathrm{T}}}\phantom{\rule{0.12em}{0ex}}\left{s}_{i}\left(n\right)\right{}^{2}}}={n}_{\mathrm{D}}{\sigma}^{2}/{n}_{\mathrm{T}}$, the proposed system can achieve the peakpower reduction as mentioned before. Note that all the analyses in this section hold for both full feedback and reduced feedback systems.
6 Performance evaluations
Simulation parameters
Parameter  Value 

Sampling frequency  528 MHz 
FFT size  128 
Number of samples in zeropadded suffix (ZPS)  37 
Modulation scheme  Modified dual carrier modulation (MDCM) 
Channel code  LDPC code defined in [31, Table 6.31], code rate 3/4, decoder 10 iterations 
IEEE 802.15.3a channel model  CM1 
Appendices
Appendix 1: linear relaxation of the binary optimization in (27)
As matrix A, defined in (26), is totally unimodular (i.e., every square submatrix of A has determinant +1, −1, or 0), it follows from [33] (also in [13, Proposition 1]) that B is also a totally unimodular matrix. On the other hand, vector b, defined in (48), is an integer vector. Therefore, the solution obtained by solving the LP relaxation using known programming methods is integral [33]. In other words, the optimal solution of the LP relaxation is also optimal for the original problem in (27).
Appendix 2: upper bound of an expected value of cost penalty
where σ_{ w }^{2} is the variance of $\mathrm{tr}\left\{{\left({\underset{\xaf}{\mathbf{H}}}_{k}^{H}{\underset{\xaf}{\mathbf{H}}}_{k}\right)}^{1}\right\}$ that is assumed to be the same for all matrices ${\underset{\xaf}{\mathbf{H}}}_{k}.$
7 Conclusions
In this paper, a persubcarrier antenna subset selection MIMOOFDM system in the presence of nonlinear HPAs has been investigated. We have shown that the implementation of the conventional persubcarrier selection in such a system suffers from the problem of performance degradation due to the large power backoff (resulting from an unequal allocation of data subcarriers across antennas) as well as the noncausality associated with the selection criteria. To overcome these drawbacks, we have proposed an optimal constrained selection scheme that can equally allocate data subcarriers among transmit antennas by means of linear optimization. The optimization problem to realize the proposed scheme is formulated in the system with an arbitrary number of multiplexed data streams. Moreover, it can be solved efficiently by existing methods. In addition, the reduced complexity strategy that requires less feedback information and lower computational effort for solving the optimization problem has been developed. We have analyzed the efficacy of the constrained antenna selection approach over the conventional approach directly in the nonlinear fading channels. The analysis could provide an insight into the system characteristics, i.e., the impacts of nonlinear HPAs on the performance of the antenna selection OFDM system. The simulation results show that a significant improvement in terms of error performance could be achieved in the system with a constrained antenna selection compared to its counterpart.
Declarations
Acknowledgements
The authors would like to thank the anonymous reviewers for their helpful comments.
Authors’ Affiliations
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