Transmit antenna subset selection for high-rate MIMO-OFDM systems in the presence of nonlinear power amplifiers

The deployment of antenna subset selection on a per-subcarrier basis in MIMO-OFDM systems could improve the system performance and/or increase data rates. This paper investigates this technique for the MIMO-OFDM systems suffering nonlinear distortions due to high-power amplifiers. At first, some problems pertaining to the implementation of the conventional per-subcarrier 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.

an increased capacity and/or diversity gains 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 low-cost 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 EIRP (equivalent isotropic radiated power) restricted systems, such as Ultra-wideband (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 per-subcarrier 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 frequency-selective nature of the channels [10]. Thus, per-subcarrier selection is very attractive for wideband communications.
However, as the conventional per-subcarrier 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 high-power amplifiers (HPAs) associated with these antennas might be very large, whereas those at the other antennas might be small. This will affect the power efficiency of HPAs or make signals become distorted, which in turn reduces the potential benefits of the system [14].
To deal with the above issue, a power-balance selection approach has been considered in the literature, e.g., [11]- [13]. This approach selects antennas with a constraint that the number of data subcarriers allocated to each antenna is equal. It is crucial that constrained selection schemes incur a minimal loss of performance or capacity, compared to an unconstrained scheme. The constrained selection could be implemented by designing allocation algorithms, e.g., in [11], [12]. In [13], the authors considered linear optimization to devise a constrained selection scheme. This method could offer a better performance than suboptimal solutions, e.g., in [11]. However, the formulated problem in [13] is only applicable to antenna selection schemes 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 in simulations for demonstration purposes. This approach obviously has some limitations as it does not fully give an insight into the system's 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. Besides, the benefits in terms of error performance or capacity of the power-balance selection have not been analyzed directly for the systems suffering nonlinear distortions due to HPAs. In addition, those works only consider antenna selection schemes where data are transmitted from one antenna on each subcarrier. Thus, the achieved spectral efficiency is limited. To fulfill the expectation of delivering very fast data speeds, per-subcarrier antennas subset selection, where multiple data symbols are transmitted simultaneously from multiple antennas on each subcarrier, should be investigated.
Unlike the previous works, this paper proposes and analyses per-subcarrier antenna subset selection with power balancing for MIMO-OFDM systems in the presence of nonlinear distortions due to HPAs. The major contributions of this work could be summarized as follows: i) A non-causal problem associated with the implementation of conventional per-subcarrier antenna selection in MIMO-OFDM systems suffering nonlinear distortions is identified for the first time.
ii) A linear optimization problem is formulated to achieve an optimal solution for powerbalancing antenna selection in the systems with an arbitrary number of data streams.
iii) 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.
iv) The efficacy of the balance selection scheme over its counterpart is analyzed directly in the nonlinear fading channels. Specifically, we show that the average mean-squared error (MSE) and the average SNDR (signal-to-noise-plus-distortion ratio) in the proposed system 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 II, an antenna selection MIMO-OFDM system model with nonlinear HPAs is described. In Section III, per-subcarrier antenna subset selection criterion is investigated in the systems suffering nonlinear distortions. In Section IV, an optimization problem for data subcarrier allocation with a power balancing is formulated.
Performance analysis is carried out in Section V. Simulation results are provided in Section VI.
Finally, Section VII concludes the paper.
Notation: Throughout this paper, a bold letter denotes a vector or 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, 0 K is a K × 1 vector of zeros, 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.

A. Transmitter
We consider a MIMO-OFDM system with K subcarriers, n T transmit antennas, and n R receive antennas as shown in Fig.1. At the transmitter, the input data are demultiplexed into n D independent streams, where n D ≤ n T and n D ≤ n R . Each data bit stream is then mapped onto to be the symbols that the subcarrier allocation block takes at its u th input and outputs at its i th output, respectively.
The allocation block assigns the elements of q k = [q k 1 q k 2 ... q k n D ] T to n D selected antennas at the k th subcarrier, based on feedback information. As a result, only n D elements in a vector .. x k n T ] T are assigned values from q k , whereas the others are zeros. Here, it is assumed that E{q k q H k } = σ 2 I n D . The output sequences from the subcarrier allocation block are then fed into K-point IFFT (inverse fast Fourier transform) blocks. In this paper, the Nyquist sampling signal is considered. Thus, the discrete-time baseband OFDM signals can be expressed as In this work, nonlinear HPA with ideal predistortion, i.e., soft envelope limiter (SEL), is considered. The n th output sample from the SEL is given by 1 [15] 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.
For analytical tractability, we assume that the signals s i (n) are asymptotically independent and identically distributed (i.i.d.) Gaussian random variables. Note that this assumption, which is based on the central limit theorem [16], only holds when the number of data subcarriers on the 1 Although the inputs of HPAs are continuous-time signals in reality. For the purpose of analysis, it is reasonable to consider HPAs with a discrete-time baseband signal [15] i th antenna, denoted as K i , is large enough. By extending Bussgang's theorem [17] to complex Gaussian processes, the output of the nonlinear HPAs can be expressed as [15] where α i is a scale factor, and η i (n) represents time-domain distortion noise that is uncorrelated with s i (n). The factor α i and the variance σ 2 η i of η i (n) are, respectively, given by [15] and where σ 2 K i = σ 2 K i /K is the average input signal power of the HPA on the i th antenna, ϑ i = P i,sat /σ 2 K i is the clipping ratio, and erfc( In the system where the same number of data subcarriers is allocated to all antennas, we have K i = n D K/n T K, ∀i = 1, 2, ..., n T and σ 2 K i = n D σ 2 /n T σ 2 K , ∀i = 1, 2, ..., n T . An input power back-off (IBO) of the HPAs is defined as IBO = P i,sat /σ 2 K i . Also, all HPAs are assumed to have the same nonlinear behavior. The effects of the nonlinear characteristics of HPAs on data symbols are illustrated in Fig. 2.
This figure plots constellation diagrams of estimated 16-QAM symbols after FFT and channel equalization in the absence of thermal noise. It can be seen that signals become more distorted when data subcarriers are not evenly allocated across transmit antennas.

B. Receiver
At the receiver, the received signal at each antenna is fed into the FFT block after the GI (guard interval) is removed. The system model in the frequency domain corresponding to the k th subcarrier can be expressed as [18] where In the above equations, h k j,i indicates the channel coefficient between the i th transmit antenna and the j th receive antenna. d k i denotes the frequency-domain distortion noise at the i th transmit antenna. Also, y k j and n k j denote the received signal and the thermal noise at the j th receive antenna, respectively. The effective channel matrix H k , the effective scale factor α = diag([α 1 α 2 . . . α n D ]), and the effective distortion noise T are obtained by eliminating the columns of H k , the rows of α, and the elements of d k that are corresponding to the unselected transmit antennas, respectively. The distortion noise d k i can be modeled as a zeromean complex Gaussian random variable with variance σ 2 d i = σ 2 η i (i.e., σ 2 d i is equal to that of the time-domain distortion noise). Note that, as clipping is performed on the Nyquist-rate samples, all the subcarriers on the i th antenna experience the same attenuation α i and the variance σ 2 d i [15]. Thus, the factors of α and α, the variance of d, denoted as , are the same for all subcarriers. Here, the indices k associated with α i and σ 2 d i are dropped for simplicity. The thermal noise is modeled as a Gaussian random variable with zero mean and E{n k n H k } = σ 2 n I n R . Also, it is assumed that per-subcarrier power loading is not an option due to the complexity of power loading and the strict regulation of a power spectral mask, such as in UWB systems.
Several MIMO detection techniques can be employed in this system to detect signals. For simplicity, we only consider a ZF (zero-forcing) receiver. The equalized signal at the k th subcarrier is computed as [19]q where G k = H k α and G † k = (G H k G k ) −1 G H k denotes the Moore-Penrose pseudo-inverse 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 α −1 d k , and the thermal noise after equalization G † k n k .
Note that the attenuations introduced by nonlinear HPAs need to be compensated to obtain a proper decision variable for detection. Also, the receiver can estimate the attenuations and the variances of the distortion noises, supposing that it knows the characteristics of HPAs.
III Several antenna selection criteria that originally derived in linear channels, such as MMSE (minimum mean-squared error) [9], maximum capacity [20], or maximum SNR (signal-to-noise ratio) [20] can be extended to this system. For brevity, only the MMSE criterion is investigated in this paper. The MMSE criterion selects the best antenna subset from the viewpoint of minimum mean-squared error (i.e., minimizing the Euclidean distance between the estimated symbols and the transmit symbols). Therefore, it also aims to reduce the error rate. When a ZF receiver is used, the error covariance matrix corresponding to the k th subcarrier and the subset Γ γ is computed as where . Note that the third equality comes from the fact that the distortion noise and the thermal noise are independent. Recall that the mean-squared error (MSE) between the estimated symbols and the transmitted symbols is the trace of an error covariance matrix. Hence, the selected subset at the k th subcarrier is determined by minimizing the trace of the above matrix, i.e., From (15) 5)). Therefore, (15) can be simplified to as which is similar to that in the systems with ideal HPAs.
ii) On the other hand, if the above condition is not satisfied, per-subcarrier antenna selection criteria, e.g., MMSE criterion in (15), cannot be realized due to a non-causal problem.
The non-causality arises because the selection of antenna subset for each subcarrier, i.e., calculating a metric MSE k γ , requires the values α and σ 2 d . Meanwhile, the calculations of these two values require the total number of data subcarriers assigned on each antenna to be known. To realize per-subcarrier 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 non-causal problem occurs with all per-subcarrier antenna selection criteria in OFDM systems suffering nonlinear distortions.

B. Feedback Considerations
With respect to a feedback mechanism used in this system, the selected antenna indices could be directly transmitted through reverse links in a TDD (time-division duplex) mode. In addition, it is typical in indoor wireless applications that the channel might not be changed during the transmission of several consecutive frames. In that scenario, the transmitter will reallocate data subcarriers according to the updated feedback information. Finally, in MIMO-OFDM systems with large values of Γ and/or K, the number of feedback bits might be high. Reduced feedback could be realized by combining subcarriers into a cluster and using only one antenna subset for all subcarriers in the cluster. This is due to the fact that neighboring subcarriers within each OFDM symbol are correlated. Therefore, it is likely that an optimal antenna subset for a particular subcarrier remains optimal for its neighbor subcarriers. If the cluster size is L, the number of feedback bits is reduced by 1/L. Note that the choice of value L is a matter of tradeoff between feedback overhead and error performance. We propose the following criterion for choosing a proper subset for the m th cluster,1 ≤ m ≤ M, M = K/L, input powers result in severe distortion of signal. In this case, power back-off is required. However, the back-off will degrade the system performance. In addition, the imbalance allocation of data subcarriers on antennas leads to the non-causality as discussed in Section III. It is intuitive that these problems can be avoided if the same number of data subcarriers is allocated to all transmit antennas, as illustrated in Fig. 3. When a balance selection of data subcarriers is required, the designed selection scheme should retain the benefits in terms of capacity or error performance as large as possible. To this end, we formulate a linear optimization problem to realize such a scheme.
As mentioned in the Introduction, the linear optimization approach was considered for an OFDM system with n D = 1 in [13]. Before proceeding to formulating a generalized optimization problem for systems with n D ≥ 1, we make some evaluations with respect to the formulated problem in [13]: i) 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.
ii) 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 the complexity to solve the optimization problem also becomes increased. 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.

A. Optimization Formulation
We define a variable z k γ , where z k γ = 1 if Γ γ is chosen for the k th subcarrier, and z k γ = 0 otherwise. Also, denote c k γ to be the cost associated with the chosen subset Γ γ . The type of the cost depends on antenna selection criteria, e.g., c k γ = tr{MSE k γ } if the MMSE selection criterion is used. The total cost function can be expressed as As mentioned in Section III, only n D antennas in this system are allowed to transmit data symbols on each subcarrier. This is equivalent to choosing only one subset of n D elements among Γ subsets Γ γ , γ = 1, 2, ..., Γ per subcarrier. Thus, the first constraint can be expressed as Γ γ=1 z k γ = 1, ∀k = 0, 1, ..., K − 1.
The second constraint is that all transmit antennas have the same number of allocated data subcarriers. In the case that Kn D is not divisible by n T , some antennas will be allowed to have one more subcarrier than others. This will guarantee that the transmit power will be evenly distributed over the transmit antennas as much as it could. This constraint can be expressed as where the parameter λ γ is the number of times that the subset Γ γ is selected. The values λ γ are chosen to satisfy where Ψ i denotes a set of n D −1 n T −1 subsets Γ γ that contains the i th antenna, and ⌈a⌉ indicates the smallest integer that is larger than or equal to a. For example, from Table I, we (21) can be simplified to as For instance, if n T = 4, n D = 2, and K = 12, then λ γ = 12 6 = 2, ∀γ = 1, 2, ..., 6. As all subsets are chosen twice, from Table I, we know that each antenna has six data symbols (cf. Fig. 3b).
The optimization problem is now a minimization of the cost function (18) subject to two constraints (19) and (20). Note that, in the system without power balancing, a problem of subcarrier allocation is equivalent to minimizing (18) subject to the constraint (19) only.
In what follows, we will represent the above optimization problem in a matrix form. Let us define a vector z = (z 0 Then, (18) can be rewritten as f = c T z.
Also, the first and the second constraints can now be expressed as where A 1 = I K ⊗ 1 T Γ ∈ {0, 1} K×KΓ , and where A 2 = 1 T K ⊗ I Γ ∈ {0, 1} Γ×KΓ and λ = (λ 1 λ 2 ... λ Γ ) T ∈ ℜ Γ×1 . These constraints could be combined in a concise form as It is obvious that (26) 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 A). As a result, the optimization problem in (26) can be solved efficiently by well-known linear programming methods, such as simplex methods or interior point method [21]. When n D = 1, the formulated problem in (26) is identical to the one in [13]. In addition, it is worth noting that, as the optimization problem in (26) 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.

B. Optimization in the Systems with Reduced Feedback
In the system with feedback reduction, an efficient approach to formulate the optimization problem is based on a cluster basis rather than on a subcarrier basis. Let us define z m γ and c m γ = mL k=(m−1)L+1 tr{MSE k γ } to be the variable and the cost associated with the m th cluster and the subset Γ γ that is applied to all subcarriers within the m th cluster. By doing similar steps as in Section IV.A, we arrive at an optimization formula similar to (26), excepting that: i) The number of variables is KΓ/L, i.e., z ∈ {0, 1} (KΓ/L)×1 , ii) A cost vector is c = ℜ KΓ/L×1 and its elements are c m γ , iii) 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 where O(.) denotes an order of complexity, and ξ is the bit size of the optimization problem [22]. 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.

V. PERFORMANCE ANALYSIS
In Section IV, a linear optimization problem has been formulated to realize an optimal (constrained) selection scheme from a viewpoint of minimum MSE (mean-squared error). 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 average SNDR (signal-to-noise-plus-distortion ratio), in the constrained system is better than that in its counterpart. Without loss of generality, it is assumed that all HPAs have the input saturation level of P i,sat and operate with an input back-off of IBO = P i,sat /σ 2 K . In the unconstrained system, the power back-off is required on the antennas where the numbers of allocated data subcarriers are larger than K, i.e., K i > 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 β i = σ 2 K /σ 2 K i < 1. Meanwhile, the powers of the signals on the other antennas, i.e., K i ≤ K, are not scaled up due to an EIRP restriction as well as the complexity of power loading.
Let us first rewrite the received signal y k in (6) when the back-off operation is included as where β = diag([β 1 β 2 . . . β n T ]), and β = diag([β 1 β 2 . . . β n D ]) is obtained by eliminating the rows of β that are corresponding to the unselected transmit antennas. Note that β i = 1 if no back-off is required on the i th antenna. The error covariance matrix can now be expressed as (cf. (14)) From (28), we can express the MSE corresponding to the data symbol transmitted at the u th selected antenna on the k th subcarrier as where [A] u,u denotes the (u, u) th entry of the matrix A. Thus, the average MSE across subcarriers and transmit antennas can be calculated as For notational simplicity, we denote where Ω k is a mapping from the u th selected antenna index to the i th real antenna index at the k th subcarrier, i.e., i = Ω k (u), 1 ≤ u ≤ n D , 1 ≤ i ≤ n T , which depends on the selected subset.
Note that β 2 u and β 2 Ω k (u) are the same. We can rewrite (30) as As mentioned above, in the unconstrained system, the powers of signals on the antennas that have a large number of data subcarriers will be scaled by a factor β 2 Ω k (u) < 1. Therefore, the average MSE in this system can now be expressed as where V denotes a set of antennas that the number of allocated data subcarriers on these antennas are smaller than or equal to K, and V is a set of the remain antennas.
In the constrained system, the same number of data subcarriers K is allocated to all antennas.
Thus, all subcarriers will be scaled by the same factor α, and distorted by the distortion noises with the same variance σ 2 d . Recall that, for a given K, the values α and σ 2 d can be calculated using (4) and (5), respectively. In addition, it is important to note that the effective channel matrix on the k th subcarrier in the constrained system, denoted as H k , is generally different from the channel matrix H k obtained in the unconstrained system because the selected antenna subsets may be different. From (30), we can express the average MSE in this system as On the other hand, let us define ∆ to be the difference in the total cost between the constrained and unconstrained scheme 2 , i.e., (cf. (16), (18)) Note that the value ∆ is positive due to the fact that the total cost in the constrained optimization (i.e., minimization problem) is always larger than that in its unconstrained counterpart. Substitute (35) into (34), we arrive at The difference in the average MSE between the unconstrained system and the constrained one can now be computed as where It can be seen from (37) that the change in the average MSE when implementing balanced allocation compared to the case of imbalanced allocation comes from I V , I V , and I ∆ , where: • I V is a kind of MSE penalty that is associated with data subcarriers on the antennas where K i < K. It can be seen from (4) and (5) that when K i increases, α i decreases and σ 2 η i increases. Thus, the value of the function F (u, H k , K i ), defined in (31), increases when K i increases. Consequently, the value of I V in (38) is always negative (i.e., I V < 0).
• I V is a MSE benefit that is associated with data subcarriers on the antennas where K i > K, i = Ω k (u). As the scale factor β 2 i < 1, it is clear that I V > 0. The more data subcarriers are allocated to some particular antennas, the smaller the value β 2 i = σ 2 K /σ 2 K i = K/K i is required, and, thus, I V becomes larger.
• I ∆ 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 ∆ < 0 because ∆ > 0 as mentioned before.
It is important to note that, for a given system with defined HPAs in terms of nonlinear characteristics, only I ∆ among the three components depends on the effective channel matrices H k , k = 0, 1, ..., K − 1. Therefore, while different balanced selection schemes introduce different changes in the average MSE, the difference in the average MSEs indeed comes from the difference in I ∆ . 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 ∆ as small as possible. We note that the formulated optimization in (26) could achieve the minimum possible value of the total cost. Hence, with the definition of ∆ as shown in (35), it is expected that the proposed constrained selection scheme based on linear optimization will guarantee the minimum achievable value of ∆.
In addition, an upper bound of the expected value of the cost penalty is derived in the Appendix B. 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.
Finally, it is necessary to evaluate the value of Θ. As it is too challenging to mathematically evaluate Θ from a statistical viewpoint due to the fact that all components I V , I V , and I ∆ are complicated and dependent random variables, we perform a numerical evaluation of (37) instead. The numerical results confirm that I V < 0, I V > 0, and I ∆ < 0. Moreover, as shown in Fig.   4.d, the probability of Θ being possitive is very significant. Therefore, the proposed system could achieve a smaller average MSE (i.e., a better MSE performance) than that in the unconstrained system. In case that the receiver first estimates the value of Θ, and then apply the constrained method when Θ > 0, then the value of Θ is always positive. In addition, in an spatial multiplexing MIMO system with a ZF receiver, we have SN DR k,u = σ 2 M SE k,u , where SN DR k,u is the SNDR corresponding to the data symbol transmitted at the u th selected antenna on the k th subcarrier [19]. Thus, it can be shown that the proposed system could achieve a better average SNDR than the unconstrained system. Error-rate performance comparison will be provided and discussed in the next section. It is also worth mentioning that the analysis in this section holds for both full feedback and reduced feedback systems.

VI. PERFORMANCE EVALUATIONS
In this section, the error performance of the proposed system is evaluated via simulation results.
The simulation parameters are listed in Table II. These parameters are based on a data-rate mode of 960 Mbps. Thus, the data rate in the proposed system when n D = 2 is 1920 Mbps. The system performance is measured in terms of packet-error rate (PER) over the channel model of CM1 defined in the IEEE 802.15.3a channel model [24]. This channel is based on a measurement of a line-of-sight scenario where the distance between the transmitter and the receiver is up to 4 m.
Additionally, the multipath gains are modeled as independent log-normally distributed random variables. Perfect channel state information is assumed to be available at the receiver. Also, the feedback link is assumed to have no delay and is error-free. The average energy of transmitted data symbols is normalized to unity, i.e., σ 2 = 1. It can be seen that there is a significant improvement in terms of PER performance in the proposed system. This agrees with the analysis in Section V that an imbalance allocation of data subcarriers in the unconstrained system results in a reduced average MSE, as well as average SNDR, compared to that in the proposed system. Similar observations can be made in the systems equipped with n R = 3 and n R = 4 receive antennas as shown in Fig. 6. Fig. 7 shows the PER performance of the proposed system with the reduced-complexity approach. Here, the feedback reduction of L = 8 is used. As predicted, there is some loss in performance when applying feedback reduction compared to full feedback. However, we note that the system with feedback reduction requires only 12.5% of the number of feedback bits and has lower computational effort for solving the optimization problem. In addition, the proposed system with power balancing still outperforms its counterpart under reduced feedback. These results illustrate the efficacy of the proposed system with power balancing for practical MIMO-OFDM wireless systems.

VII. CONCLUSIONS
In this paper, per-subcarrier antenna subset selection for MIMO-OFDM systems in the presence of nonlinear HPAs has been investigated. The optimal antenna selection scheme that can equally allocate data subcarriers among transmit antennas has been devised by means of linear optimization. The system deploying this selection method could avoid performance degradation due to large power back-off as well as the non-causal issue associated with the per-subcarrier antenna selection. The formulated optimization problem 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. The efficacy of the proposed system over the system without power-balancing has been analyzed from the MSE viewpoint. Simulation results have confirmed the benefit in the average MSE when the powerbalancing selection is employed. The results have also shown that a significant improvement in terms of error performance could be achieved in the system with power balancing compared to its counterpart. The optimization problem in (26) can be relaxed to linear programming (LP) relaxation using a similar approach as in [13], even though the constraint matrices in the two formulated problems are defined differently. Specifically, the feasible set of the LP relaxation of (26) can be expressed as where As the matrix A, defined in (25), is totally unimodular (i.e., every square submatrix of A has determinant +1, -1, or 0), it follows from [25] (also in [13], Proposition 1] that B is also a totally unimodular matrix. On the other hand, the vector b, defined in (44), is an integer vector.
Therefore, the solution obtained by solving the LP relaxation using known programming methods is integral [25]. In other words, the optimal solution of the LP relaxation is also optimal for the original problem in (26 [27]. When n R > n D + 1 it is shown in [27,Lemma 6] that and Thus, the variance of tr{(H H k H k ) −1 } can be computed as [16] Substitute (50) into (47), we finally arrive at