Energy-Efficiency Analysis of Per-Subcarrier Antenna Selection with Peak-Power Reduction in MIMO-OFDM Wireless Systems

Abstract

The use of per-subcarrier antenna subset selection in OFDM wireless systems offers higher system capacity and/or improved link reliability. However, the implementation of the conventional per-subcarrier selection scheme may result in significant fluctuations of the average power and peak power across antennas, which affects the potential benefits of the system. In this paper, power efficiency of high-power amplifiers and energy efficiency in per-subcarrier antenna selection MIMO-OFDM systems are investigated. To deliver the maximum overall power efficiency, we propose a two-step strategy for data-subcarrier allocation. This strategy consists of an equal allocation of data subcarriers based on linear optimization and peak-power reduction via cross-antenna permutations. For analysis, we derive the CCDF (complementary cumulative distribution function) of the power efficiency as well as the analytical expressions of the average power efficiency. It is proved from the power-efficiency perspective that the proposed allocation scheme outperforms the conventional scheme. We also show that the improvement in the power efficiency translates into an improved capacity and, in turn, increases energy efficiency of the proposed system. Simulation results are provided to validate our analyses.

1. Introduction

The last few years have seen an increasing demand for very fast data speeds in wireless multimedia applications. Meanwhile, reducing energy consumption in wireless networks has become a problem of concern among academic and industrial communities. As a result, high-speed systems with high-energy efficiency has emerged as a main stream for designing future wireless networks [1, 2]. The potential for improving energy efficiency in wireless systems could be in component level, link level, or network level [1]. From a physical layer viewpoint, it is well known that high-power amplifier (HPAs) is a major source of RF (radio frequency) power consumption. For example, in mobile networks, HPAs consume up to 50%–80% of overall power at a base station [1, 3]. Thus, increasing power efficiency of HPAs is of importance to achieve high energy-efficient wireless networks.

To date, MIMO-OFDM (multi-input multi-output orthogonal frequency division multiplexing) has been considered as a key technique for high-speed wireless transmission [4]. This is mainly because OFDM could offer high spectral efficiency and robustness against intersymbol interference in multipath fading channels. Also, an increase in capacity and/or diversity gains could be achieved with MIMO [5, 6]. In fact, MIMO-OFDM has been adopted in current and future standards, such as WiMAX, WLAN 802.11n, or LTE-Advanced. Among various MIMO schemes, antenna selection appears to be promising for OFDM systems. This MIMO scheme requires a low-cost implementation and small amount of feedback information, compared to other precoding or beamforming methods [7].

Many research works have considered the application of antenna selection in OFDM systems; for example, see [8–12]. To achieve a large coding gain resulting from the frequency-selective nature of the channels, a per-subcarrier antenna selection (i.e., selecting antennas on each subcarrier basis) is applied [9]. However, as this method selects antennas independently on each subcarrier, a large number of data symbols may be allocated to some antennas, depending on the channel condition. As a result, the peak power and average power of the signals on these antennas might be very large, whereas those on the other antennas might be small. The fluctuation of the powers clearly affects the power efficiency of HPAs or distorts signals, which in turn reduces the potential benefits of the antenna selection OFDM systems [13].

One possible approach to deal with the problem of imbalance allocation of data subcarriers is selecting antennas under a constraint that the number of data subcarriers allocated to each antenna is equal. Some research works have studied such a constrained selection approach in the literature; for example, see [10–12]. In [10], an ad hoc algorithm was developed to realize the constrained selection scheme. Meanwhile, the authors in [11] considered linear optimization to devise the constrained selection scheme. It was shown that the selection scheme based on optimization could offer a better performance than the suboptimal solution in [10]. In [12], we generalized the approach in [11] to the system with an arbitrary number of multiplexed data streams and analyzed the performance directly in nonlinear fading channels. However, all of these works only study the efficacy of the constrained selection scheme from error-performance perspective. Moreover, even though the same number of data subcarriers is allocated to each transmit antenna (i.e., all antennas have an equal average power), an occurrence of high peak-to-average power ratio (PAPR) still affects the system. Despite an increasing concern about energy consumption in wireless networks, to the best of our knowledge, a research on energy efficiency in the context of antenna selection MIMO-OFDM systems is still missing.

It is also essential to emphasize the need of research on improving energy efficiency in antenna selection MIMO-OFDM systems. First, per-subcarrier antenna selection MIMO configurations need multiple active RF (radio frequency) chains, which immediately raises a concern about energy consumption compared to single RF-chain OFDM systems. Second, OFDM inherently suffers from a high PAPR problem, which leads to a poor power efficiency of HPAs [13, 14]. In per-subcarrier antenna selection OFDM systems, this effect is intuitively more serious due to a combination of the high PAPR and power fluctuation resulting from an imbalance allocation of data subcarriers. This problem also becomes crucial in the OFDM systems where linear scaling (i.e., scale the peak power of the time-domain OFDM signals to the saturation level of the HPAs [15–17]) is implemented to realize OFDM transmissions with no nonlinear distortions.

In this paper, per-subcarrier antenna selection MIMO-OFDM systems with linear scaling are investigated from an energy perspective for the first time. The important contributions of this work include (a) the analysis of power efficiency of HPAs and energy efficiency and (b) the proposed strategy to improve these useful performance metrics. The main results are summarized as follows.(1)A two-step data allocation strategy is proposed to deliver a maximum overall power efficiency of HPAs. This strategy consists of an equal allocation of data subcarriers among transmit antennas based on linear optimization and a peak-power reduction algorithm via cross-antenna permutations.(2)Analytical expressions characterizing the achieved power efficiency of HPAs, including the CCDFs (complementary cumulative distribution function) and the average power efficiency, are derived. It is proved that, from the power-efficiency viewpoint, the proposed allocation scheme outperforms the conventional scheme.(3)The improvements in capacity and energy efficiency resulting from the improved power efficiency of HPAs are analyzed.

Numerical results are provided to verify the analyses as well as demonstrate the benefits in terms of the power efficiency of HPAs and capacity as well as energy efficiency.

The remainder of the paper is organized as follows. In Section 2, a per-subcarrier antenna subset selection MIMO-OFDM system with linear scaling is described. In Section 3, a data allocation strategy that could allocate evenly data subcarriers across antennas with a low peak power is proposed. Analysis of power efficiency is carried out in Section 4. The achievable capacity and energy efficiency are considered in Section 5. Numerical results are provided in Section 6. Finally, Section 7 concludes the paper.

Notation. A bold letter denotes a vector or a matrix, whereas an italic letter denotes a variable. (⋅)^𝑇, (⋅)^𝐻, 𝐸{⋅}, and det(⋅) denote transpose, Hermitian transpose, expectation, and determinant of a matrix, respectively. ⊗ denotes the Kronecker product. 𝐈_𝑛 indicates the 𝑛×𝑛 identity matrix, and 𝟏_𝐾 is a 𝐾×1 vector of ones. ℜ indicates the set of real numbers.

2. Antenna Selection for MIMO-OFDM Systems with Linear Scaling

2.1. System Model

We consider a MIMO-OFDM system with 𝐾 subcarriers, 𝑛_𝑇 transmit antennas, and 𝑛_𝑅 receive antennas as shown in Figure 1. At the transmitter, the input data are demultiplexed into 𝑛_𝐷 independent data streams. Each data stream is then mapped onto 𝑀-QAM (𝑀-ary quadrature amplitude modulation) constellations. For the 𝑘th subcarrier, we denote 𝑢^𝑘_𝑙 and 𝑥^𝑘_𝑖, 1≤𝑙≤𝑛_𝐷, 1≤𝑖≤𝑛_𝑇, 0≤𝑘≤𝐾−1, to be the symbols that the subcarrier block takes at its 𝑙th input and outputs at its 𝑖th output, respectively. The allocation block assigns the elements of 𝐮_𝑘=[𝑢^𝑘₁,𝑢^𝑘₂,…,𝑢^𝑘_𝑛𝐷]^𝑇to 𝑛_𝐷 selected antennas at the 𝑘th subcarrier based on feedback information. As a result, only 𝑛_𝐷 elements in a vector 𝐱_𝑘=[𝑥^𝑘₁,𝑥^𝑘₂,…,𝑥^𝑘_𝑛𝑇]^𝑇 are assigned values from 𝐮_𝑘, whereas the others are zeros. It is assumed that 𝐸{𝐮_𝑘𝐮^𝐻_𝑘}=𝜎²𝐈_𝑛𝐷. 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 𝑠_𝑖(𝑛)=1√𝐾^𝐾−1_𝑘=0𝑥^𝑘_𝑖𝑒^{𝑗2𝜋𝑛𝑘/𝐾},0≤𝑛≤𝐾−1,1≤𝑖≤𝑛_𝑇. (1)

(a)

(b)

Figure 1 

A simplified block diagram of a MIMO-OFDM system with per-subcarrier transmit antenna selection.

For simplicity, we consider ideal predistortion HPAs (i.e., soft envelope limiters) with a unity gain and class-A operation. To deliver the maximum power efficiency with no nonlinear distortions in the system with nonlinear HPAs, the peak power across transmit antennas is linearly scaled to the saturation level 𝑃_sat of the HPAs. In addition, as feedback information (i.e., the selected antenna indices which are calculated based on the channel state information) is deployed by the transmitter, all transmit branches are scaled with the same scaling factor [16]. Thus, the signal after linear scaling can be expressed as 𝑠_𝑖̃(𝑛)=√𝛼𝑠_𝑖(𝑛), (2) where the scaling factor 𝛼=𝑃_sat/𝑃, and the peak power across antennas 𝑃=max{𝑠_𝑖(𝑛)|𝑛=0,…,𝐾−1;𝑖=1,…,𝑛_𝑇}. Each time-domain OFDM signal is then amplified by the HPA before being transmitted via its corresponding transmit antenna.

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 𝑘th subcarrier can be expressed as 𝐲_𝑘=√𝛼𝐇_𝑘𝐱_𝑘+𝐧_𝑘=√𝛼𝐇_𝑘𝐮_𝑘+𝐧_𝑘, (3) where 𝐱_𝑘=𝑥^𝑘₁𝑥^𝑘₂⋯𝑥^𝑘_𝑛𝑇^𝑇,𝐇_𝑘=ℎ^𝑘_1,1ℎ^𝑘_1,2⋯ℎ^𝑘_1,𝑛𝑇ℎ^𝑘_2,1ℎ^𝑘_2,2⋯ℎ^𝑘_2,𝑛𝑇ℎ⋯⋯⋯⋯_{𝑘𝑛𝑅,1}ℎ^𝑘_𝑛𝑅,2⋯ℎ^𝑘_{𝑛𝑅,𝑛𝑇},𝐲_𝑘=𝑦^𝑘₁𝑦^𝑘₂⋯𝑦^𝑘_𝑛𝑅^𝑇,𝐧_𝑘=𝑛^𝑘₁𝑛^𝑘₂⋯𝑛^𝑘_𝑛𝑅^𝑇. (4)

In the above equations, ℎ^𝑘_𝑗,𝑖 indicates the channel coefficient between the 𝑖th transmit antenna and the 𝑗th receive antenna. The effective channel matrix 𝐇_𝑘 is obtained by eliminating the columns of 𝐇_𝑘 corresponding to the unselected transmit antennas. 𝑦^𝑘_𝑗 and 𝑛^𝑘_𝑗 denote the received signal and the noise at the 𝑗th receive antenna, respectively. Here, the noise is modeled as a Gaussian random variable with zero mean and 𝐸{𝐧_𝑘𝐧^𝐻_𝑘}=𝜎²_𝑛𝐈_𝑛𝑅. We assume that the receiver can perfectly estimate the channel coefficients, for example, using a block-type pilot arrangement for channel estimation [18].

2.2. Per-Subcarrier Antenna Subset Selection

In a MIMO-OFDM system with the conventional per-subcarrier antenna subset selection, antenna subsets are selected independently for each subcarrier [9]. On each subcarrier, only 𝑛_𝐷 antennas out of 𝑛_𝑇 available transmit antennas are active. Denote Γ_𝛾,𝛾=1,2,…,Γ, to be the 𝛾th subset consisting of 𝑛_𝐷 selected transmit antennas, where Γ=^𝑛𝐷_𝑛𝑇=𝑛_𝑇!/𝑛_𝐷!(𝑛_𝑇−𝑛_𝐷)! is the number of all possible 𝑛_𝐷-element subsets. Each subset consists of 𝑛_𝐷 transmit antenna indices that are chosen based on the feedback information from the receiver. For example, when 𝑛_𝑇=4 and 𝑛_𝐷=2, then Γ=6, and all possible subsets Γ_𝛾, 𝛾=1,2,…,6, are defined in Table 1. The choice of the best antenna subset depends on a particular selection criterion.

Table 1 

Antenna subsets (𝑛_𝑇=4, 𝑛_𝐷=2, and Γ=6).

Several antenna selection criteria, such as maximum capacity or maximum SNR (signal-to-noise ratio) [19] can be extended to this system. In this paper, we consider the capacity criterion. Accordingly, the optimal subset at the 𝑘th subcarrier is determined by maximizing the mutual information of the 𝑘th subchannel; that is, Γ_𝛾∗(𝑘)=argmax_{Γ𝛾,𝛾=1,…,Γ}𝐼^𝑘_𝛾, (5) where 𝐼^𝑘_𝛾=log₂det𝐈_𝑛𝑅+𝜌𝑛_𝐷𝐇_𝑘𝐇^𝐻_𝑘 (6) is the instantaneous mutual information associated with the 𝑘th subchannel [5]. Here, 𝜌=𝑃_𝑡/𝜎²_𝑛=𝛼𝑛_𝐷𝜎²/𝜎²_𝑛, and 𝑃_𝑡=𝛼𝑛_𝐷𝜎² is the total transmit power per subchannel. Also, we have assumed in (6) that the transmit power is allocated uniformly across antennas. This is due to the fact that the feedback information in our system is only the selected antenna indices (i.e., not sufficient enough to perform power allocation algorithms across subcarriers as well as antennas). The average mutual information across subcarriers can now be expressed as 𝐼𝜌,𝐇=1𝐾^𝐾−1_𝑘=0𝐼^𝑘_𝛾∗. (7)

3. A Proposed Strategy for Peak-Power Reduction

In Section 2, we have described the MIMO-OFDM system with the conventional selection scheme. It can be noted that the number of data subcarriers assigned to each transmit antenna might be significantly different depending on the channel condition. In the system with identical linear scaling, to achieve the maximal overall power efficiency of HPAs, the peak power across antennas should be as small as possible. We note that the peak power of the signal on each branch depends on both the transmitted constellation symbols and the number of data subcarriers allocated in each OFDM symbol (cf. (1)). Thus, it is not sufficient to reduce the peak power by solely implementing PAPR reduction techniques. More specifically, PAPR reduction techniques themselves cannot solve the problem of imbalance allocation of data subcarriers across antennas. To reduce the peak power across antennas, we propose a two-step strategy consisting of the following.

Step 1. Allocate the same number of data subcarriers to all transmit antennas (i.e., selecting antennas under a constraint that all antennas have the same number of data subcarriers as illustrated in Figure 2(b)). Once this is achieved, the time-domain signals on all transmit branches have the same average power. Moreover, as we will mathematically prove in Section 4.1, the peak power across antennas is reduced.

(a) Unbalanced-power selection

(b) Balanced-power selection

Figure 2 

Illustration of per-subcarrier antenna subset selection (𝑛𝑇=4, 𝑛𝐷=2, and 𝐾=12).

Step 2. Reallocate data symbols across antennas. This process will alter the statistical distribution of signals and thus further reduce the peak power.
We note that this paper does not focus on developing original techniques for either equal allocation of data subcarriers or peak-power reduction. Instead, we are interested in analyzing power efficiency of HPAs and energy efficiency in per-subcarrier antenna selection OFDM systems, which has not been considered so far. The two steps in our proposed strategy, described in Sections 3.1 and 3.2, are accomplished by extending the suitable approaches available in the literature to the context of the considered system.

3.1. Optimal Equal Allocation of Data Subcarriers

The optimal constrained antenna selection scheme based on linear optimization was considered in [11, 12] to improve error performance of OFDM systems suffering nonlinear distortions due to HPAs. We now consider this method for the first step of our strategy to achieve a better power delivery in a linearly scaled MIMO-OFDM system. Specifically, we define a variable 𝑧^𝑘_𝛾, where 𝑧^𝑘_𝛾=1, if Γ_𝛾 is chosen for the 𝑘th subcarrier, and 𝑧^𝑘_𝛾=0 otherwise. Also, denote 𝑐^𝑘_𝛾 to be the cost associated with the chosen subset Γ_𝛾. Here, 𝑐^𝑘_𝛾=𝐼^𝑘_𝛾 as the maximum capacity criterion is considered. By denoting vectors 𝐳=(𝑧⁰₁,…, 𝑧⁰_Γ,𝑧¹₁,…,𝑧¹_Γ,…,𝑧₁^𝐾−1,…,𝑧_Γ^𝐾−1)^𝑇∈{0,1}^𝐾Γ×1, and 𝐜=(𝑐⁰₁,…,𝑐⁰_Γ,𝑐¹₁,…,𝑐¹_Γ,…,𝑐₁^𝐾−1,…,𝑐_Γ^𝐾−1)^𝑇∈ℜ^𝐾Γ×1, an optimal solution for an equal allocation of data subcarriers is obtained by solving the following linear optimization problem [11, 12]: max_{𝐳∈{0,1}𝐾Γ×1}𝐜^𝑇𝐳,subjectto𝐀𝐳=𝐚, (8) where 𝐀₁=𝐈_𝐾⊗𝟏^𝑇_Γ∈{0,1}^𝐾×𝐾Γ, 𝐀₂=𝟏^𝑇_𝐾⊗𝐈_Γ∈{0,1}^Γ×𝐾Γ, 𝐀=(𝐀^𝑇₁,𝐀^𝑇₂)^𝑇∈{0,1}^{(𝐾+Γ)×𝐾Γ}, 𝐚=(𝟏^𝑇_𝐾,𝝀^𝑇)^𝑇, and 𝝀=(𝜆₁,𝜆₂,…,𝜆_Γ)^𝑇, where 𝜆_𝛾 is the number of times that the subset Γ_𝛾 is selected. The constraint in (8) means that only 𝑛_𝐷 antennas are allowed to transmit data symbols on each subcarrier and all transmit antennas have the same number of allocated data subcarriers. It is important to note that this binary linear optimization problem can be relaxed to linear programming with integral solutions [12]. Hence, (8) can be solved efficiently by the known linear programming methods, such as the simplex methods or interior point methods [20].

3.2. Data Allocation with Peak-Power Reduction

To further reduce the peak power of the whole system, various available PAPR reduction techniques (e.g., see [14] and the references therein) can be now adopted. In this paper, we are interested in a selected mapping (SLM) technique [21] as SLM is a distortionless PAPR technique that could achieve a good PAPR reduction [14, 21]. One of the most important steps in SLM is creating a set of candidates that represents the same data information. To exploit the available degrees of freedom in multiple-antenna systems for peak-power reduction, we propose to create a set of candidates using cross-antenna permutations. In the literature, a SLM-based scheme that could exploit the available degrees of freedom was first developed in [22]. The scheme in [22] creates candidates by performing cross-antenna rotation and inversion (CARI) based on a defined random matrix. However, that scheme is proposed for an Alamouti code based MIMO-OFDM system only. In a per-subcarrier antenna selection OFDM system, CARI cannot be implemented directly as only 𝑛_𝐷 out of 𝑛_𝑇 antennas are active on each subcarrier. To create candidates in our scheme, we perform cross-antenna permutations instead of CARI. In addition, we utilize an antenna allocation pattern that is already known by the transmitter and receiver, instead of storing a defined random matrix at both transmitter and receiver as in [22]. The proposed algorithm is described as follows.(1)Create 𝑊 candidates by performing cross-antennas permutations. An illustration of this process in the system with 𝑛_𝐷=2, 𝑛_𝑇=4, and 𝐾=4 is shown in Figure 3. Accordingly, the first candidate is the original data allocation. The second candidate is obtained by permuting all symbols on the first antennas with their associated symbols on the other antennas. The third and fourth candidates are created in a similar manner. To obtain a larger number of candidates, all symbols on the antenna that are going be permutated need first have their phase rotated (i.e., being multiplied with an element of a phase set, e.g., a 4-phase set is {0,𝜋/2,𝜋,3𝜋/2}).(2)Calculate the peak powers of all available candidates.(3)Select the candidate with the minimum peak power for transmission.

(a)

(b)

(c)

(d)

Figure 3 

Illustration of cross-antenna permutations (𝑛𝑇=4,𝑛𝐷=2, and 𝐾=4).

To recover the transmitted data, the transmitter needs to inform the receiver which candidates have been selected. Thus, the number of side information bits in this scheme is log₂𝑊, which is similar to that in [22].

3.3. Complexity Considerations

In this subsection, the complexity of the proposed allocation scheme is compared to that of the conventional (imbalance) allocation scheme. In the first step of the proposed scheme, to realize an equal allocation of data subcarriers, the optimization problem in (8) needs to be solved at the receiver. We note that this linear optimization problem can be solved in polynomial time. More specifically, the complexity to solve this problem using interior point methods can be reduced to 𝑂([(𝐾Γ)³/ln(𝐾Γ)]𝜉), where 𝑂(⋅) denotes an order of complexity and 𝜉 is the bit size of the optimization problem [23]. In addition, it is noted that this step is transparent to the transmitter (i.e., no additional complexity is required at the transmitter). In the second step, a major additional complexity lies in the required IFFT operations due to additional 𝑊 candidates. As an 𝐾 point-IFFT requires 𝐾log₂𝐾 complex additions and (𝐾/2)log₂𝐾 complex multiplications, the numbers of complex additions and complex multiplications in the conventional scheme are 𝑛_𝑇𝐾log₂𝐾 and 𝑛_𝑇(𝐾/2)log₂𝐾, respectively. Meanwhile, in the proposed scheme with 𝑊 candidates, 𝑊𝑛_𝑇𝐾log₂𝐾  complex additions and 𝑊𝑛_𝑇(𝐾/2)log₂𝐾complex multiplications are required. However, as we will show analytically in Section 4 and numerically in Section 6, an improvement in the peak-power reduction reduces when 𝑊 becomes large. Thus, a small value of 𝑊 is generally chosen, which does not incur much additional complexity. Finally, the amount of feedback information in the proposed scheme is similar to that in the conventional scheme.

4. Analysis of Power Efficiency

4.1. Statistical Distribution of Peak Power

Before proceeding to analyze the power efficiency of HPAs, we need to investigate the distribution of the peak power of the MIMO-OFDM signals (i.e., the peak power across transmit antennas). We consider the complementary cumulative distribution function (CCDF) of the peak power, defined as the probability that the peak power 𝑃 exceeds a given threshold 𝑃₀; that is, CCDF^𝑃𝑃₀=Pr𝑃>𝑃₀. (9) Note that although a procedure for calculating CCDF of PAPR in OFDM systems is known, to the best of our knowledge, all the CCDF expressions with respect to MIMO-OFDM signals available in the literature assume that all data subcarriers are active, which can be considered as a special case in the considered system when all the transmit antennas have the same number of allocated data symbols. In the following, we calculate the CCDF of the peak power in our system.

Let us begin with the discrete-time OFDM signal 𝑠_𝑖(𝑛), 𝑛=0,1,…,𝐾−1, corresponding to the 𝑖th transmit antenna. The peak power of this signal is defined as 𝑃_𝑖=max_{0≤𝑛≤𝐾−1}|𝑠_𝑖(𝑛)|². (10) For analytical tractability, we assume that both the real part and imaginary part of the signal 𝑠_𝑖(𝑛) are asymptotically independent and identically distributed Gaussian random variables. Note that this assumption, which is based on the central limit theorem [24], only holds when the number of assigned data subcarriers on the 𝑖th antenna, denoted as 𝐾_𝑖, is large enough. As a result, |𝑠_𝑖(𝑛)| follows the Rayleigh distribution, and |𝑠_𝑖(𝑛)|² has a chi-square distribution with two degrees of freedom. The probability density function of the signal |𝑠_𝑖(𝑛)|² can be expressed as [24] 𝑝_|𝑠|²𝑠_𝑖²=1𝜎²_𝐾𝑖𝑒^{−|𝑠𝑖|2/𝜎2𝐾𝑖}, (11) where 𝜎²_𝐾𝑖=𝜎²𝐾_𝑖/𝐾 is the variance of the signal |𝑠_𝑖(𝑛)|. Note that ∑^𝑛𝑇_𝑖=1𝐾_𝑖=𝑛_𝐷𝐾; thus, we have ∑^𝑛𝑇_𝑖=1𝜎²_𝐾𝑖=𝑛_𝐷𝜎². The CDF (cumulative distribution function) of the signal |𝑠_𝑖(𝑛)|² is given as Pr𝑠_𝑖²≤𝜃=1−𝑒^{−𝜃/𝜎2𝐾𝑖},𝜃≥0. (12) Suppose that 𝐾 samples of |𝑠_𝑖(𝑛)|, 𝑛=0,1,…,𝐾−1, are independent; the CDF of the peak power 𝑃_𝑖 can be expressed as CDF^𝑃𝑖=Pr𝑃_𝑖≤𝑃₀=Pr𝑠_𝑖(0)²≤𝑃₀Pr𝑠_𝑖(1)²≤𝑃₀⋯Pr𝑠_𝑖(𝐾−1)²≤𝑃₀=1−𝑒^{−𝑃0/𝜎2𝐾𝑖𝐾}. (13)

In MIMO-OFDM systems with linear scaling, the peak power across transmit antennas 𝑃 can be defined as 𝑃=max_{1≤𝑖≤𝑛𝑇}𝑃_𝑖. (14) Given the statistical independence of data among transmit antennas, which is the case in the considered spatial multiplexed OFDM system, the CDF of the peak power 𝑃 is calculated as CDF^𝑃=Pr𝑃≤𝑃₀=Pr𝑃₁≤𝑃₀Pr𝑃₂≤𝑃₀⋯Pr𝑃_𝑛𝑇≤𝑃₀=^𝑛𝑇_𝑖=11−𝑒^{−𝑃0/𝜎2𝐾𝑖𝐾}. (15) Therefore, the CCDF of the peak power of the antenna selection MIMO-OFDM signals can be expressed as CCDF^𝑃_imbalance𝑃₀=1−CDF^𝑃=1−^𝑛𝑇_𝑖=11−𝑒^{−𝑃0/𝜎2𝐾𝑖𝐾}. (16)

In the MIMO-OFDM system with a power balancing constraint, the number of allocated data subcarriers per transmit antenna is equal to one another (i.e., 𝐾_𝑖=𝑛_𝐷𝐾/𝑛_𝑇∶=𝐾, ∀𝑖=1,2,…,𝑛_𝑇). Thus, the variances of the signals are 𝜎²_𝐾𝑖=𝑛_𝐷𝜎²/𝑛_𝑇∶=𝜎²_𝐾, ∀𝑖=1,2,…,𝑛_𝑇. As a result, the CCDF expression could be simplified to as CCDF^𝑃_balance𝑃₀=1−1−𝑒^{−𝑃0/𝜎2𝐾𝑛𝑇𝐾}. (17) A comparison of the CCDF of the peak powers in the two systems is presented in the following theorem.

Theorem 1. In MIMO-OFDM transmission schemes that consist of inactive data subcarriers (e.g., per-subcarrier antenna selection), the probability of occurrences of high peak power is the smallest when the same number of data symbols is allocated to all transmit antennas; that is, 𝐶𝐶𝐷𝐹^𝑃_{𝑏𝑎𝑙𝑎𝑛𝑐𝑒}𝑃₀≤𝐶𝐶𝐷𝐹^𝑃_{𝑖𝑚𝑏𝑎𝑙𝑎𝑛𝑐𝑒}𝑃₀. (18)

Proof. The proof is given in Appendix A.
When the peak-power reduction algorithm proposed in Section 3.2 is implemented in the MIMO-OFDM system with a power-balancing constraint, the CCDF of the peak power can be expressed as CCDF^𝑃_{balance+reduced}𝑃₀=CCDF^𝑃_balance𝑃₀^𝑊=1−(1−𝑒^{−𝑃0/𝜎2𝐾})^{𝑛𝑇𝐾𝑊}, (19) where 𝑊 is the number of candidates that are assumed to be independent. Recall that, by definition, the CCDF value is always smaller than one (cf. (9)). Therefore, the CCDF value in (19) is smaller than that in (17); that is, CCDF^𝑃_{balance+reduced}𝑃₀≤CCDF^𝑃_balance𝑃₀. (20)

4.2. Power Efficiency of HPAs

We now analyze the power efficiency (PE) of high-power amplifiers (HPAs). The drain efficiency of HPAs, which is defined as a ratio between the power drawn from the DC source 𝑃_dc and the average output power 𝑃_out, is considered in this paper. Denote 𝑃^𝑖_in and 𝑃^𝑖_out to be the average input and output powers of the HPA for the 𝑖th antenna, respectively. Recall that all HPAs are assumed to have a unity gain; that is, 𝑃^𝑖_out=𝑃^𝑖_in, ∀𝑖=1,2,…,𝑛_𝑇. Hence, the instantaneous overall power efficiency of HPAs in the MIMO-OFDM system can be expressed as [16] 𝜂^PE=∑^𝑛𝑇_𝑖=1𝑃^𝑖_out𝑛_𝑇𝑃_dc=∑^𝑛𝑇_𝑖=1𝑃^𝑖_in𝑛_𝑇𝑃_dc𝑛=𝛼^𝐷𝜎²𝑛_𝑇𝑃_dc=𝑛_𝐷𝜎²𝑃_sat𝑛_𝑇𝑃_dc1𝑃=𝑛_𝐷𝜎²2𝑛_𝑇1𝑃. (21) In the above manipulations, we have used the fact that ∑^𝑛𝑇_𝑖=1𝑃^𝑖_in=𝛼𝑛_𝐷𝜎² and 𝑃_dc=2𝑃_sat for class-A HPAs, regardless of the average powers of the input time-domain signals. Denoting CCDF^PE(𝜂^PE₀)=Pr(𝜂^PE>𝜂^PE₀) to be the CCDF of the power efficiency, we obtain the following result with respect to 𝜂^PE.

Theorem 2. In per-subcarrier antenna selection MIMO-OFDM systems with linear scaling, the probability of achieving high instantaneous overall power efficiency of HPAs is the largest when all transmit antennas have the same number of allocated data symbols; that is, 𝐶𝐶𝐷𝐹^𝑃𝐸_{𝑏𝑎𝑙𝑎𝑛𝑐𝑒}𝜂₀^𝑃𝐸≥𝐶𝐶𝐷𝐹^𝑃𝐸_{𝑖𝑚𝑏𝑎𝑙𝑎𝑛𝑐𝑒}𝜂₀^𝑃𝐸, (22) where 𝐶𝐶𝐷𝐹^𝑃𝐸_{𝑏𝑎𝑙𝑎𝑛𝑐𝑒}𝜂₀^𝑃𝐸=1−𝑒^{−1/2𝜂0𝑃𝐸𝑛𝑇𝐾},𝐶𝐶𝐷𝐹^𝑃𝐸_{𝑖𝑚𝑏𝑎𝑙𝑎𝑛𝑐𝑒}𝜂₀^𝑃𝐸=^𝑛𝑇_𝑖=11−𝑒^{−𝑛𝐷𝐾/2𝑛𝑇𝐾𝑖𝜂0𝑃𝐸𝐾}. (23)

Proof. From (18) and (21), it is readily to obtain (22) and (23).
With respect to the average value of the overall power efficiency, from (21), we can express 𝜂^PE=𝐸𝜂^PE=𝑛_𝐷𝜎²2𝑛_𝑇𝐸1𝑃=𝑛_𝐷𝜎²2𝑛_𝑇1𝑥𝑝(𝑥)𝑑𝑥, (24) where 𝑝(𝑥) is the pdf (probability distribution function) of the peak power. In the system with a balance constraint (i.e., only use Step 1), the pdf of the peak power can be calculated as 𝑝_balance(𝑥)=𝑑𝑑𝑥CDF^𝑃_balance(𝑥)=𝑑𝑑𝑥1−𝑒^{−𝑥/𝜎2𝐾𝑛𝑇𝐾}=𝑛_𝑇𝐾𝜎²_𝐾𝑒^{−𝑥/𝜎2𝐾}1−𝑒^{−𝑥/𝜎2𝐾𝑛𝑇𝐾−1}. (25) Hence, 𝜂^PE_balance=𝑛_𝐷𝜎²2𝑛_𝑇1𝑥𝑝_balance(𝑥)=𝑛𝑑𝑥_𝑇𝐾2^{𝐾𝜎2𝐾}_𝜎²_𝐾1𝑥𝑒^{−𝑥/𝜎2𝐾}1−𝑒^{−𝑥/𝜎2𝐾𝑛𝑇𝐾−1}=𝑛𝑑𝑥_𝑇𝐾2^𝐾₁1𝑥𝑒^−𝑥1−𝑒^{−𝑥𝑛𝑇𝐾−1}𝑑𝑥. (26) Similarly, the power efficiency of HPAs in the conventional system is 𝜂^PE_imbalance=𝑛_𝐷𝜎²2𝑛_𝑇1𝑥𝑝_imbalance(𝑥)=𝑛𝑑𝑥_𝐷𝐾𝜎²2𝑛_𝑇^{𝐾max𝜎2max}_𝜎²_max1𝑥^𝑛𝑇_𝑗=1𝑒^{−𝑥/𝜎2𝐾𝑗}1−𝑒^{−𝑥/𝜎2𝐾𝑗}𝜎²_𝐾𝑗×^𝑛𝑇_𝑖=11−𝑒^{−𝑥/𝜎2𝐾𝑖𝐾}𝑑𝑥, (27) where 𝜎²_max=max{𝜎²_𝐾1,…,𝜎²_𝐾𝑛𝑇}, and 𝐾_max=max{𝐾₁,…,𝐾_𝑛𝑇}.
When the peak-power reduction algorithm is also implemented (i.e., the system employs both Steps 1 and 2), it is readily from (20) and (21) that CCDF^PE_{balance+reduced}𝜂^PE₀≥CCDF^PE_balance𝜂^PE₀, (28) where CCDF^PE_{balance+reduced}𝜂^PE₀=1−1−1−𝑒^{−1/2𝜂PE0𝑛𝑇𝐾𝑊}. (29) Also, the average power efficiency can now be calculated as 𝜂^PE_{balance+reduced}=𝑛_𝐷𝜎²2𝑛_𝑇1𝑥𝑝_{balance+reduced}(𝑥)=𝑛𝑑𝑥_𝑇𝐾𝑊2^𝐾₁1𝑥𝑒^−𝑥1−𝑒^{−𝑥𝑛𝑇𝐾−1}×1−1−𝑒^{−𝑥𝑛𝑇𝐾𝑊−1}𝑑𝑥. (30) It can be seen from (30) that the smaller the peak power is reduced (i.e., the larger the number of candidates 𝑊 is used), the higher the power efficiency could be achieved. Note that although the integrals in (26), (27), and (30) have no closed-form solutions, they can be evaluated numerically. Also, the average power efficiencies in (26), (27), and (30) are taken with respect to the input data. In other words, they are considered as instantaneous power efficiencies with respect to the channel distribution. Consequently, the power efficiency in the systems is obtained by averaging these values over the fading channel distribution.

5. Analyses of Capacity and Energy Efficiency

It has been shown in (22) and (28) that the proposed system could achieve a better power efficiency of HPAs than its counterpart. Thus, when the power 𝑃_dc is fixed, it is intuitive that an increased average power efficiency results in an increased average transmit power and, in turn, leads to an increase in the achievable rate. Moreover, an increase in the data rate under a constant consumption power will translate into an improvement in energy efficiency. The achieved capacity and energy efficiency are now investigated in this section.

5.1. System Capacity

We begin by rewriting the mutual information in (7) with respect to the average SNR value of 𝜌=𝑃_𝑡/𝜎²_𝑛, where 𝑃_𝑡=𝛼𝑛_𝐷𝜎²=𝜂^PE𝑛_𝑇𝑃_dc (cf. (21)). The ergodic capacity is then calculated by averaging the mutual information over the fading channel distribution; that is, 𝐶(𝜌)=𝐸_𝐇{𝐼(𝜌,𝐇)}. From (6) and (7), the mutual information in the proposed and conventional systems can be expressed, respectively, as 𝐼𝜌_proposed,𝐇=1𝐾^𝐾−1_𝑘=0log₂det𝐈_𝑛𝑅+𝜌_proposed𝑛_𝐷𝐇_𝑘𝐇^𝐻_𝑘,𝐼𝜌_imbalance,𝐇=1𝐾^𝐾−1_𝑘=0log₂det𝐈_𝑛𝑅+𝜌_imbalance𝑛_𝐷𝐇_𝑘𝐇^𝐻_𝑘, (31)(32) where 𝜌_proposed=𝜂^PE_proposed𝑛_𝑇𝑃_dc/𝜎²_𝑛 and 𝜌_imbalance=𝜂^PE_imbalance𝑛_𝑇𝑃_dc/𝜎²_𝑛. Here, 𝜂^PE_proposed = 𝜂^PE_{balance+reduced} if the peak-power reduction algorithm is implemented; otherwise, 𝜂^PE_proposed=𝜂^PE_balance. Also, 𝐇_𝑘 in (31) denotes the effective channel matrix on the 𝑘th subcarrier in the proposed system, which is obtained when solving the problem in (8). This channel matrix is generally different from the effective channel matrix 𝐇_𝑘 in the conventional system because the selected antenna subset may be different. The difference in the mutual information between the two systems can be now calculated as Δ𝐼=𝐼𝜌_proposed,𝐇−𝐼𝜌_imbalance,𝐇=1𝐾^𝐾−1_𝑘=0Δ𝐼_𝑘, (33) where Δ𝐼_𝑘=log₂det𝐈_𝑛𝑅+𝜌_proposed𝑛_𝐷𝐇_𝑘𝐇^𝐻_𝑘−log₂det𝐈_𝑛𝑅+𝜌_imbalance𝑛_𝐷𝐇_𝑘𝐇^𝐻_𝑘. (34)

For analytical simplicity, we focus on the high-SNR regime. At the high SNR, the mutual information at the 𝑘th subcarrier can be approximated as [25] 𝐼_𝑘=log₂det𝜌𝑛_𝐷Ω_𝑘, (35) where Ω_𝑘=Ω𝐇_𝑘=𝐇_𝑘𝐇^𝐻_𝑘,𝑛_𝑅≤𝑛_𝐷𝐇^𝐻_𝑘𝐇_𝑘,𝑛_𝑅>𝑛_𝐷. (36) Thus, the difference in the mutual information can be rewritten as 1Δ𝐼=𝐾^𝐾−1_𝑘=0log₂det𝜌_proposed𝑛_𝐷Ω_𝑘−log₂det𝜌_imbalance𝑛_𝐷Ω_𝑘=𝑝log₂𝜂^PE_proposed𝜂^PE_imbalance−1𝐾^𝐾−1_𝑘=0Δ_𝑘∶=𝑇₁+𝑇₂, (37) where 𝑝=min(𝑛_𝐷,𝑛_𝑅), Ω_𝑘=Ω(𝐇_𝑘), and Δ_𝑘=log₂detΩ_𝑘−log₂detΩ_𝑘 (38) is the loss in the mutual information associated with the 𝑘th subcarrier due to the constrained allocation. Note that if both systems have the same selected antenna subset at the 𝑘th subcarrier, then Δ_𝑘=0; otherwise, Δ_𝑘>0. Thus, the total loss in the mutual information Δ=(1/𝐾)∑^𝐾−1_𝑘=0Δ_𝑘>0.

We have some important observations with respect to the value of Δ𝐼 in (37). (i) The change in mutual information Δ𝐼 comes from two sources. The first source 𝑇₁ is a benefit in mutual information due to the improvement in the power efficiency of HPAs. The second source 𝑇₂ is a penalty that incurs because the chosen effective channel matrices in the proposed system are different from the ones in the conventional system. (ii) For each channel realization, the matrix 𝐇_𝑘 is fixed and the first term 𝑇₁ in (37) is a constant. Thus, the value of Δ𝐼 depends on how the effective channel matrix 𝐇_𝑘 is selected in the constrained selection scheme. From this observation, it is clear that, to make the value Δ𝐼 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 (8) could achieve the minimum possible value of the total cost. Hence, it is expected that the constrained selection scheme based on linear optimization will guarantee the maximum achievable value of Δ𝐼. In addition, to have an insight into the cost penalty, we derive the upper bound of the expected value of the cost penalty in 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 with an increasing value of 𝑞=max(𝑛_𝐷,𝑛_𝑅). (iii) As Δ > 0, the upper bound of the capacity improvement can be given as Δ𝐶=𝐸_𝐇{Δ𝐼}≤𝑝log₂𝐸_𝐇𝜂^PE_proposed𝜂^PE_imbalance. (39) In (39), we have used Jensen’s inequality of 𝐸{log(𝑥)}≤log(𝐸{𝑥}) as log(𝑥) is a concave function. Based on this bound, we could estimate the maximum improvement in capacity that could be realized in the proposed system compared to its counterpart.

It is now necessary to evaluate the change in capacity; that is, Δ𝐶. We note that although the distribution of the mutual information at high SNRs can be well approximated by a Gaussian distribution [25], it is still challenging to perform a mathematical evaluation of Δ𝐶 from a statistical viewpoint. This is mainly due to the fact that the two terms in (37) are complicated, dependent random variables. Thus, we perform a numerical evaluation of Δ𝐶 instead. Figure 4 plots the empirical CCDF of 𝑇₁, CDF of 𝑇₂, and CCDF of Δ𝐼. In the figure, “𝑊=1” stands for the case in which only Step 1 in Section 3.1 is implemented. The results are obtained in the systems with 𝑛_𝑇=4, 𝑛_𝐷=2, 𝑛_𝑅=2, 𝐾=128, and are averaged over 10³ channel realizations. Details about other simulation parameters are described in Section 6. The numerical results confirm that 𝑇₁>0 and 𝑇₂<0. Moreover, as shown in Figure 4(c), Δ𝐼 is always positive when the peak-power reduction algorithm is implemented (i.e., 𝑊=4 and 𝑊=8). For the case of 𝑊=1, the probability of ΔI being positive is significant. Therefore, the proposed system attains a better ergodic capacity than that in the conventional system. The achieved capacities in the considered systems will be provided in Section 6.

(a)

(b)

(c)

Figure 4 

Statistical distributions (note: 𝑇2 is independent of 𝑊).

5.2. Energy Efficiency

In this subsection, we further examine the efficacy of the proposed system from an energy-efficiency (EE) perspective. Normalized energy efficiency (bits/Hz/Joule) in MIMO-OFDM systems can be defined as [3, 26] 𝜂^EE=𝐶𝜌𝑃_total, (40) where 𝐶(𝜌) is the achievable rate and the total power consumption per-subchannel is 𝑃_total=𝑛_𝑇𝑃_dc+𝑛_𝑇𝑃_RF+𝑃_sp, where 𝑃_RF is the RF power consumption in each transmit branch excluding the associated HPA, and 𝑃_sp is the baseband processing power consumption. It can be seen from (40) that given a fixed value of 𝑃_total, a comparison of energy efficiency achieved in the two systems is based on the capacity comparison that has been analyzed in Section 5.1. We are now interested in evaluating a useful metric of energy efficiency-spectral efficiency performance. Recall that the average SNR 𝜌 is given as 𝜌=𝑃_𝑡/𝜎²_𝑛=𝜂^PE𝑛_𝑇𝑃_dc/𝜎²_𝑛. Thus, we can rewrite the energy efficiency in (40) as a function of 𝑃_dc as 𝜂^EE𝑃_dc=𝐶𝜂^PE𝑛_𝑇𝑃_dc/𝜎²_𝑛𝑛_𝑇𝑃_dc+𝑛_𝑇𝑃_RF+𝑃_sp. (41) The energy efficiency of the proposed and conventional systems can now be, respectively, expressed as 𝜂^EE_proposed𝑃_dc=𝐶_proposed𝜂^PE_proposed𝑛_𝑇𝑃_dc/𝜎²_𝑛𝑛_𝑇𝑃_dc+𝑛_𝑇𝑃_RF+𝑃_sp=𝐸_𝐇𝐼𝜂^PE_proposed𝑛_𝑇𝑃_dc/𝜎²_𝑛,𝐇𝑛_𝑇𝑃_dc+𝑛_𝑇𝑃_RF+𝑃_sp,𝜂^EE_imbalance𝑃_dc=𝐶_imbalance𝜂^PE_imbalance𝑛_𝑇𝑃_dc/𝜎²_𝑛𝑛_𝑇𝑃_dc+𝑛_𝑇𝑃_RF+𝑃_sp=𝐸_𝐇𝐼𝜂^PE_imbalance𝑛_𝑇𝑃_dc/𝜎²_𝑛,𝐇𝑛_𝑇𝑃_dc+𝑛_𝑇𝑃_RF+𝑃_sp. (42)(43) Similarly to the case of capacity, we compare the energy efficiency achieved in the two systems by means of numerical results in the next section. Note that the calculation of energy efficiency in the proposed system (i.e., (42)) has assumed that a reduction in spectral efficiency due to the side information as well as additional processing power required for the peak-power reduction algorithm is negligible. In fact, a reduction in spectral efficiency is very small. For example, in a system with 16-QAM, FFT size of 128, 𝑛_𝐷 = 2, and 𝑊= 4 (i.e., 2 bits are needed for side information), a spectral efficiency loss is 2 bits/(128 × 4 × 2 + 2) bits = 0.19%. Also, it was shown in [27] that the additional power cost when implementing SLM schemes is minuscule. Thus, the proposed peak-power reduction algorithm in Section 3.2, which is a SLM-based scheme, requires a small additional power cost.

6. Numerical Results

In this section, we provide numerical results to validate the analyses mentioned in the previous sections as well as demonstrate the effectiveness of the proposed allocation scheme over its counterpart. A MIMO-OFDM system with 𝑛_𝑇=4, 𝑛_𝐷=2, and 𝑛_𝑅=2 is considered in our simulations. The system parameters are listed in Table 2. These parameters are chosen based on the legacy WiMedia MB-OFDM UWB (Multiband-OFDM UWB) standard [28]. Also, the channel CM1, defined in the IEEE 802.15.3a channel model [29], 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. Moreover, the multipath gains are modeled as independent log normally distributed random variables. We assume that perfect channel state information is available at the receiver. Also, the feedback link has no delay and is error-free.

Table 2 

Simulation parameters.

6.1. Evaluations of Peak-Power Distribution

In Figure 5, we plot the CCDFs of the peak power of time-domain signals. The analytical curves are based on (16), (17), and (19). Meanwhile, the simulation curves are empirical CCDF values. The simulation result confirms that a system with the proposed allocation scheme offers a better CCDF performance than its counterpart. As expected, the occurrence of high peak power is significantly reduced when the peak-power reduction algorithm is implemented. Also, it can be seen that the improvement associated with this algorithm is reduced with increasing 𝑊. In other words, a very large value of 𝑊, while requiring higher complexity in terms on the number of IFFT operations, results in a marginal improvement. Thus, it is reasonable to choose a relatively small value for 𝑊 (e.g., 𝑊 = 4). It is also worth noting that the analytical curves are relatively close to the simulation curves. The small gaps exist due to the fact that the assumption of independent samples |𝑠_𝑖(𝑛)| to obtain (13) does not strictly hold as we have ∑^𝐾−1_𝑛=0|𝑠_𝑖(𝑛)|²=𝜎²𝐾_𝑖 by Parseval’s relation [24].

Figure 5 

Comparison of CCDFs of the peak powers.

6.2. Evaluations of Power Efficiency of HPAs

Figure 6 compares the CCDFs of the power efficiency achieved in the proposed and conventional systems. It can be seen that the probability of power efficiency being large highly likely occurs in the proposed system, compared to its counterpart. Also, the simulation results agree well with the analytical results derived in (23) and (29). In Table 3, we compare the average power efficiencies. Here, the analytical values are obtained according to (26), (27), and (30). Meanwhile, the simulation values are empirical values based on the original definition of the drain efficiency in (21). Also, these values are averaged over the fading channel realizations. It can be seen that the derived expressions approximate well the achieved power efficiencies. Table 3 also provides relative improvements of the power efficiencies achieved in the proposed system over the conventional system. Here, only 𝜂^PE (Simulation) values are used for calculating these improvements. It is clear that the proposed system could achieve a significant improvement in terms of average power efficiency.

Table 3 

A comparison of average power efficiencies.

Figure 6 

Comparison of CCDFs of the power efficiencies.

6.3. Evaluations of Capacity and Energy Efficiency

Figure 7 shows the system capacity in Mbps (i.e., the normalized value in (6) is scaled up with the system bandwidth) versus the SNR value of 𝜎²/𝜎²_𝑛. It is clear that a system with the proposed allocation scheme achieves a higher capacity than its counterpart. This agrees with the analysis in Section 5.1 that the change in capacity Δ𝐶 is positive. In Figure 8, we plot the energy efficiency (Mbits/Joule) versus spectral efficiency (Mbps/Hz). This figure is obtained based on (42) and (43) when varying 𝑃_dc. Other parameters are 𝑃_RF=100mW,𝑃_sp=10mW, and 𝜎²/𝜎²_𝑛=15dB. As expected, the improvement in the power efficiency of HPAs results in an improved energy efficiency. In addition, it can be observed that there exists an energy efficiency-spectral efficiency tradeoff in the systems. This tradeoff clearly needs to be taken into consideration when designing energy-efficient per-subcarrier antenna selection based OFDM wireless systems.

Figure 7 

Comparison of the ergodic capacities.

Figure 8 

Energy efficiency versus spectral efficiency.

7. Conclusions

In this paper, a per-subcarrier antenna subset selection MIMO-OFDM system with linear scaling has been investigated from an energy-efficiency perspective. We have shown that an imbalance allocation of data subcarriers associated with the conventional selection scheme affects the power efficiency of HPAs, as well as the energy-efficiency of the whole system. To deliver the maximum overall power efficiency, we have proposed the two-step strategy, consisting of equal allocation data subcarriers across antennas and peak-power reduction. It has been proved from the power-efficiency viewpoint that the proposed allocation scheme outperforms the conventional scheme. We have also derived the expressions for measuring the average power efficiency. Moreover, the improvements in terms of capacity and energy efficiency resulting from the improved power efficiency have been analyzed. The analytical results are validated by simulation results. The simulation result also shows that the system with the proposed allocation scheme could achieve a better efficiency efficiency-spectral energy tradeoff, compared to its counterpart.

Appendices

A. Proof of Theorem 1

Let us consider a function 𝑓(𝜐)=1−𝑒^{−𝑃0/𝜐}, 0<𝜐≤𝜎²_max, where 𝜎²_max=max{𝜎²_𝐾1,𝜎²_𝐾2,…,𝜎²_𝐾𝑛𝑇}. The second derivative of this function is 𝑓^′′(𝜐)=2𝑃₀𝜐−𝑃²₀𝜐⁴𝑒^{−𝑃0/𝜐}. (A.1) From a HPAs’ viewpoint, it is of interest to consider the situation when the peak power across antennas 𝑃₀ is large. Thus, we consider the scenarios of 𝑃₀>2𝜎²_max, where 𝜎²_max is the maximum average power across antennas. Under these situations, it is clear that 𝑓^′′(𝜐)<0,∀𝜐∈(0,𝜎²_max]. Hence, the function 𝑓(𝜐) is concave. By applying Jensen’s inequality, we obtain the following inequality: 1𝑛_𝑇^𝑛𝑇_𝑖=1𝑓𝜎²_𝐾𝑖1≤𝑓𝑛_𝑇^𝑛𝑇_𝑖=1𝜎²_𝐾𝑖1=𝑓𝑛_𝑇^𝑛𝑇_𝑖=1𝜎²_𝐾=𝑓𝜎²_𝐾, (A.2) where the equality comes from the fact that ∑^𝑛𝑇_𝑖=1𝜎²_𝐾𝑖=∑^𝑛𝑇_𝑖=1𝜎²_𝐾=𝑛_𝐷𝜎². Expression (A.2) can be rewritten as ^𝑛𝑇_𝑖=11−𝑒^{−𝑃0/𝜎2𝐾𝑖}≤𝑛_𝑇1−𝑒^{−𝑃0/𝜎2𝐾}, (A.3) with equality if and only if 𝜎²_𝐾𝑖=𝜎²_𝐾, ∀𝑖=1,…,𝑛_𝑇.

On the other hand, by applying the arithmetic-geometric mean inequality, and note that (1−𝑒^−𝑥)>0, ∀𝑥>0, we have ^𝑛𝑇_𝑖=11−𝑒^{−𝑃0/𝜎2𝐾𝑖}≤1𝑛_𝑇^𝑛𝑇_𝑖=11−𝑒^{−𝑃0/𝜎2𝐾𝑖𝑛𝑇}. (A.4) Note that the equality in (A.4) holds if and only if 𝜎²_𝐾𝑖=𝜎²_𝐾, ∀𝑖=1,…,𝑛_𝑇.

Combining (A.3) and (A.4) results in ^𝑛𝑇_𝑖=11−𝑒^{−𝑃0/𝜎2𝐾𝑖}≤1−𝑒^{−𝑃0/𝜎2𝐾𝑛𝑇}, (A.5) with equality if and only if 𝜎²_𝐾𝑖=𝜎²_𝐾, ∀𝑖=1,…,𝑛_𝑇. Thus, we get the following desired inequality: 1−^𝑛𝑇_𝑖=11−𝑒^{−𝑃0/𝜎2𝐾𝑖𝐾}≥1−1−𝑒^{−𝑃0/𝜎2𝐾𝑛𝑇𝐾}, (A.6) or CCDF^𝑃_imbalance𝑃₀≥CCDF^𝑃_balance𝑃₀. (A.7) This completes the proof.

B. Upper Bound of an Expected Value of Cost Penalty

In this appendix, we derive an upper bound of the expected value of Δ_𝑘. It can be seen from (5) that, among all possible matrices 𝐇_𝑘, the matrix 𝐇_𝑘 with the highest value of log₂(det(Ω_𝑘)),Ω_𝑘=Ω(𝐇_𝑘), will be selected as the effective channel matrix for the 𝑘th subcarrier in the conventional scheme. Meanwhile, in the proposed scheme, due to the balance constraint, the effective channel matrix associated with the kth subcarrier is not necessarily the one with the highest log₂(det(Ω_𝑘)); that is, log₂(det(Ω_𝑘))≤log₂(det(Ω_𝑘)). Thus, the expected value of Δ_𝑘 can be computed by using order statistics. In particular, an upper bound on the expected difference of two-order statistics, the Γth and 𝛾th, 1≤𝛾<Γ, is given by [30] 𝐸Δ_𝑘=𝐸log₂detΩ_𝑘−𝐸log₂detΩ_𝑘≤𝜎_𝐶Γ𝛾+1𝛾, (B.1) where 𝜎²_𝐶 is the variance of log₂(det(Ω_𝑘)) that is assumed to be the same for all possible matrices 𝐇_𝑘.

On the other hand, suppose that the entries of the 𝑛_𝑅×𝑛_𝑇 matrix 𝐇_𝑘 are i.i.d. complex Gaussian random variables with zero mean and unit variance; then for any effective channel matrix 𝐇_𝑘, Ω_𝑘 is a complex Wishart matrix. It follows from [25] that the variance of log₂(det(Ω_𝑘)) can be expressed as 𝜎²_𝐶=log₂(𝑒)^2𝑝_𝑚=1𝜓^′𝑞−𝑚+1, (B.2) where 𝑝=min(𝑛_𝐷,𝑛_𝑅), 𝑞=max(𝑛_𝐷,𝑛_𝑅), and 𝜓^′(𝑥)=∑^∞_𝜑=11/(𝜑+𝑥−1)² is the first derivative of the digamma function. By approximating 𝜓^′(𝑥)≈1/𝑥 [25], the simpler expression for 𝜎²_𝐶 in (B.2) is given as 𝜎²_𝐶=log₂(𝑒)^2𝑝_𝑚=11.𝑞−𝑚+1 (B.3) Substitute (B.3) into (B.1), we finally arrive at 𝐸Δ_𝑘≤log₂(𝑒)^𝑝_𝑚=11×Γ𝑞−𝑚+1𝛾+1𝛾. (B.4)

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.