Positive Effect of Severe Nakagami-m Fading on the Performance of Multiuser TAS/MRC Systems with High Selection Gain

Abstract

This paper investigates the positive effect of severe Nakagami-m fading on the performance of multiuser transmit antenna selection/maximal-ratio combining (TAS/MRC) systems with high selection gain. Both amount of fading (AF) and symbol error rate (SER) of M-QAM are derived as closed-form expressions for integer m. For arbitrary m, the AF and the SER are expressible as a single infinite series of Gamma function and Gauss hypergeometric function, respectively. The analytical results lead to the following observations. First, the SER performance can demonstrate the positive effect of severe Nakagami-m fading on multiuser TAS/MRC systems with high selection gain. Second, the AF performance only exhibits the negative impact of severe fading regardless of high selection gain. Last, the benefit of severe fading to the system performance diminishes at high signal-to-noise ratio (SNR).

1. Introduction

Multiple-input multiple-output (MIMO) communications have been considered as suitable ways to improve the performance of wireless communications. Various MIMO transmission schemes have been developed to obtain a high reliability through promised diversity gain and/or high rate transmission via spatial multiplexing (SM) [1–6]. Rectangular quadrature amplitude modulation (QAM) is a general modulation technique which includes important modulation schemes as particular cases, such as binary phase-shift keying (BPSK), pulse amplitude modulation (PAM), or square QAM. Although MIMO maximum-ratio combining (MRC) systems under different conditions have been investigated in the last few years, no results seem to be available for the average symbol error rate (SER) of rectangular QAM in noise-limited environments. In [7], an expression for the bit error rate (BER) of rectangular QAM is derived, yet the result is only valid for Gray code mapping, and no expression for the average SER is provided.

More specifically, the diverse techniques of coherent signal combining were developed depending on the specific treatment of the combined signals’ phase and amplitude, which are selection combining (SC), threshold combining, equal-gain combining (EGC) and MRC, as summarized in [8, 9]. Choosing a single transmit antenna, the one that maximizes the total received signal power at the receiver, can substantially reduce the transmitter’s hardware and software complexity. Thus, the transmit antenna selection (TAS) has occupied a considerable part of today’s communication researches [10]. On the other hand, MRC is an optimal diversity technique with a maximum SNR criterion. To retain the advantages of both TAS and MRC, an integrated TAS scheme with MRC at the receiver, labeled TAS/MRC, was proposed [11]. Besides the antenna diversity, multiuser diversity (MUD) is also utilized to improve performance in point-to-multipoint communication [12]. A proper scheduling algorithm is the key to gain the MUD in the multiuser systems [12]. Moreover, multiuser MIMO systems have recently attracted much attention as the technology enhances the total system capacity by generating a virtual large MIMO channel between a base station and multiple terminal stations [13]. In [14], tight closed-form expressions of outage performance were derived for the multiuser MIMO systems in Rayleigh fading channels. The analytical results demonstrate that users can be viewed as equivalent “virtual” transmit antennas [15]. The published papers are mainly focused on flat Rayleigh channels [11, 14, 15]. The closed-form expressions for the average symbol error rate of general rectangular QAM in MIMO MRC systems over Rayleigh fading channels are derived [16].

Via the m parameter, Nakagami model is able to cover both severe and weak fadings, which includes the Rayleigh fading (m = 1) as a special case. The Nakagami-m distribution also can closely approximate the Hoyt distribution and the Rice distribution [17]. Therefore, the Nakagami-m channels are usually used to model land-mobile, indoor mobile multipath propagation, as well as scintillating ionospheric radio links [17]. The severity of fading can be quantified by the amount of fading (AF) [17]. In [18], a closed-form expression for the AF was derived in a MIMO diversity system with space-time block coding (STBC) in Nakagami-m fading channels. In this paper, we derive the AF of multiuser TAS/MRC systems as a closed-form expression in independent and identically distributed (i.i.d.) Nakagami-m channels with a positive integer m. For arbitrary m, the AF is expressible as a single infinite series of Gamma function. The analytical results show that the AF of multiuser TAS/MRC systems decreases as m increases as expected. The positive effect of severe fading, however, cannot be observed from the standpoint of AF.

Performance analyses, including outage probability, capacity, and symbol error rate (SER), were investigated for the TAS/MRC systems in Nakagami-m fading channels in [19–21]. The performance is improved with the increase of m, that is, in weak fading channels [19–21]. However, the tendency becomes totally different because the paper [22] presented that the higher m causes a negative impact on the capacity of multiuser MIMO systems. In this paper, the performance of TAS/MRC system with MUD in Nakagami-m fading channels is analyzed. We derive a simple closed-form expression for the outage probability of multiuser TAS/MRC systems. Moreover, a closed-form SER of M-ary quadrature amplitude modulation (M-QAM) is also derived when m is an integer. For arbitrary m, the SER of M-QAM is expressible as a single infinite series of a Gauss hypergeometric function. The analytical results suggest that the effect of severe fading on the performance becomes beneficial at high selection gain, since a more scattering fading environment enables the selection scheduler to arrange transmissions at higher peaks of channel fading. Thus, the systems with high selection gain perform better in severe fading than in weak fading. However, the high signal-to-noise ratio (SNR) may dilute the function of selection scheduler in transmitters. All the derived expressions are verified by Monte Carlo simulations.

2. Amount of Fading for Nakagami-m Models

We consider a TAS/MRC system with a base station serving K users in the downlink. The best transmit antenna out of all 𝐿_𝑇 transmit candidates, which maximizes the postprocessing SNR at the MRC output of 𝐿_𝑅 received antennas, is selected to transmit data for the corresponding user. The channels between the transmit antennas and the users experience fading paths which are modeled as i.i.d. Nakagami-m fading channels. The probability density function (PDF) of the instantaneous postprocessing SNR of MRC combiner output for the kth user with respect to the ith transmit antenna is given by 𝑓𝛾_𝑖,𝑘=𝑚𝛾^𝑚𝐿𝑅𝛾^{𝑚𝐿𝑅−1}_𝑖,𝑘Γ𝑚𝐿_𝑅exp−𝑚𝛾_𝑖,𝑘𝛾, (1) where Γ(⋅) denotes the Gamma function and 𝛾 is the average SNR per symbol per branch. Then, the corresponding cumulative distribution function (CDF) denoted by 𝐹(𝛾_𝑖,𝑘) leads to 𝐹𝛾_𝑖,𝑘=^{𝛾𝑖,𝑘}₀𝑓(𝑥)𝛾𝑑𝑥=𝑚𝐿_𝑅,𝑚𝛾_𝑖,𝑘/𝛾Γ𝑚𝐿_𝑅, (2) where 𝛾(𝑦,𝑧) is the incomplete Gamma function [23, equation (8.350.1)]. For MUD in the TAS/MRC scheme, which is denoted as a (𝐿_𝑇,𝐿_𝑅,𝐾) multiuser TAS/MRC system, the scheduler at BS collects the MRC output SNR of all users and selects the target user according to the criterion 𝛾=max_{𝑖=1,2,…,𝐿𝑇𝑘=1,2,…,𝐾}𝛾_𝑖,𝑘. (3) Then, the selected PDF of effective postprocessing SNR in (3) can be expressed as [19, equation (4)] 𝑓_𝐼𝛾=𝐾𝐿_𝑇𝐹𝛾^{𝐾𝐿𝑇−1}𝑓𝛾=𝐾𝐿_𝑇Γ𝑚𝐿_𝑅^𝐾𝐿𝑇𝑚𝛾^𝑚𝐿𝑅𝛾^{𝑚𝐿𝑅−1}×exp−𝑚𝛾𝛾𝛾𝑚𝐿_𝑅,𝑚𝛾𝛾^{𝐾𝐿𝑇−1}. (4) Integrating (4), we can obtain the CDF as 𝐹_𝐼𝛾=𝐹𝛾^𝐾𝐿𝑇=𝛾𝑚𝐿_𝑅,𝑚𝛾/𝛾Γ𝑚𝐿_𝑅^𝐾𝐿𝑇. (5) From (3)–(5), one can see that the effective antenna selection gain is 𝐾𝐿_𝑇, since the users can be viewed as equivalent virtual transmit antennas [15]. We propose an expansion of the incomplete Gamma function in (4) by a series development for 𝛾(𝑚𝐿_𝑅,𝑚𝛾/𝛾̄), namely: 𝛾𝑚𝐿_𝑅,𝑚𝛾𝛾=exp−𝑚𝛾𝛾^∞_𝑛∑₌₀Γ𝑚𝐿_𝑅(𝑚𝛾/𝛾)^{𝑛+𝑚𝐿𝑅}Γ𝑚𝐿_𝑅,1+𝑛+1real𝑚≥2,Γ𝑚𝐿_𝑅1−exp−𝑚𝛾𝛾^{𝑚𝐿𝑅−1}_𝑛∑₌₀(𝑚𝛾/𝛾)^𝑛,𝑛!positiveinteger𝑚. (6) Next, using (6), we may infer the expansion of [𝛾(𝑚𝐿_𝑅,𝑚𝛾/𝛾)]^{𝐾𝐿𝑇−1} by the power series raised to powers [23, equation (0.314)] as follows: 𝛾𝑚𝐿_𝑅,𝑚𝛾𝛾^{𝐾𝐿𝑇−1}=^∞_𝑛∑₌₀𝛼_𝑛𝑚𝛾𝛾^{𝑚𝐿𝑅(𝐾𝐿𝑇−1)+𝑛}×exp−𝑚𝛾𝐾𝐿_𝑇−1𝛾,1real𝑚≥2,Γ𝑚𝐿∑_𝑅^{𝐾𝐿𝑇−1𝐾𝐿𝑇−1}_𝑗=0𝐾𝐿_𝑇𝑗()−1−1^𝑗×exp−𝑗𝑚𝛾𝛾^{𝑗(𝑚𝐿𝑅−1)}_𝑛∑₌₀𝛽_𝑛𝑚𝛾𝛾^𝑛,positiveinteger𝑚, (7) where 𝛼_𝑛 is the coefficient of (𝑚𝛾/𝛾)^{𝑚𝐿𝑅(𝐾𝐿𝑇−1)+𝑛} in the expansion of ^∞_𝑛=0Γ𝑚𝐿_𝑅𝑚𝛾/𝛾^{𝑚𝐿𝑅+𝑛}Γ𝑚𝐿_𝑅+1+𝑛^{𝐾𝐿𝑇−1}, (8) and [19, equation (6)] 𝛼₀=1𝑚𝐿_𝑅^{𝐾𝐿𝑇−1},𝛼_𝑛=Γ𝑚𝐿_𝑅+1𝑛^𝑛_𝑧=1𝑧𝐾𝐿_𝑇−𝑛Γ𝑚𝐿_𝑅𝛼+1+𝑧^𝑛−𝑧,for𝑛≥1. (9)𝛽_𝑛 in (7) is the coefficient of (𝑚𝛾/𝛾)^𝑛 in the expansion of {∑^{𝑚𝐿𝑅−1}_𝑛=01/𝑛!(𝑚𝛾/𝛾)^𝑛}^𝑗, and [19, equation (9)] 𝛽₀=1,𝛽_𝑛=1𝑛^{min(𝑛,𝑚𝐿𝑅−1)}_𝑘=1𝑘𝑗+1−𝑛𝛽𝑘!^𝑛−𝑘,for𝑛≥1. (10) Thus, substituting (7) into (4), we can rewrite the PDF of the multiuser TAS/MRC systems as 𝑓_𝐼𝛾=𝐾𝐿_𝑇Γ𝑚𝐿_𝑅^{𝐾𝐿𝑇∞}_𝑛∑₌₀𝛼_𝑛𝑚𝛾^{𝑚𝐾𝐿𝑇𝐿𝑅+𝑛}×𝛾^{𝑚𝐾𝐿𝑇𝐿𝑅+𝑛−1}exp−𝐾𝐿_𝑇𝑚𝛾𝛾,1real𝑚≥2𝐾𝐿_𝑇Γ𝑚𝐿_𝑅^{𝐾𝐿𝑇−1}_𝑗∑₌₀𝐾𝐿_𝑇𝑗()−1−1^𝑗×∑^{𝑗(𝑚𝐿𝑅−1)}_𝑛=0𝛽_𝑛𝑚𝛾^{𝑚𝐿𝑅+𝑛}×𝛾^{𝑚𝐿𝑅+𝑛−1}exp−𝑗+1𝑚𝛾𝛾,positiveinteger𝑚. (11) Similar to (11), the CDF of (5) can be expanded as 𝐹_𝐼𝛾=^∞_𝑛∑₌₀𝜂_𝑛𝑚𝛾𝛾^{𝑚𝐾𝐿𝑇𝐿𝑅+𝑛}exp−𝐾𝐿_𝑇𝑚𝛾𝛾,1real𝑚≥_𝐾𝐿𝑇2_𝑗∑₌₀𝐾𝐿_𝑇𝑗(−1)^𝑗×∑^{𝑗(𝑚𝐿𝑅−1)}_𝑛=0𝛽_𝑛𝑚𝛾𝛾^𝑛exp−𝑗𝑚𝛾𝛾,positiveinteger𝑚, (12) where 𝜂₀=1Γ𝑚𝐿_𝑅+1^𝐾𝐿𝑇,𝜂_𝑛=Γ𝑚𝐿_𝑅+1𝑛^𝑛_𝑧=1𝑧𝐾𝐿_𝑇+1−𝑛Γ𝑚𝐿_𝑅𝜂+1+𝑧^𝑛−𝑧,for𝑛≥1. (13) From (11), we can obtain the expectation and the second moment of effective SNR 𝛾 by the following expressions [20, equations (7), (8)]:𝜇_𝛾=𝛾/𝑚Γ𝑚𝐿_𝑅^𝐾𝐿𝑇×^∞_𝑛=0𝛼_𝑛Γ𝑚𝐾𝐿_𝑇𝐿_𝑅+𝑛+1𝐿_𝑇^{𝑚𝐾𝐿𝑇𝐿𝑅+𝑛}1,real𝑚≥2,𝜇_𝛾=𝐾𝐿_𝑇𝛾/𝑚Γ𝑚𝐿_𝑅^{𝐾𝐿𝑇−1}_𝑗=0𝐾𝐿_𝑇𝑗()−1−1^𝑗×^{𝑗(𝑚𝐿𝑅−1)}_𝑛=0𝛽_𝑛Γ𝑚𝐿_𝑅+𝑛+1𝑗+1^{𝑚𝐿𝑅+𝑛+1},positiveinteger𝑚, (14a)(14b)𝐸𝛾²=𝛾/𝑚²Γ𝑚𝐿_𝑅^𝐾𝐿𝑇×^∞_𝑛=0𝛼_𝑛Γ𝑚𝐾𝐿_𝑇𝐿_𝑅+𝑛+2𝐿_𝑇^{𝑚𝐾𝐿𝑇𝐿𝑅+𝑛+1}1,real𝑚≥2,𝐸𝛾²=𝐾𝐿_𝑇𝛾/𝑚²Γ𝑚𝐿_𝑅^{𝐾𝐿𝑇−1}_𝑗=0𝐾𝐿_𝑇𝑗)−1(−1^𝑗×^{𝑗(𝑚𝐿𝑅−1)}_𝑛=0𝛽_𝑛Γ𝑚𝐿_𝑅+𝑛+2𝑗+1^{𝑚𝐿𝑅+𝑛+2},positiveinteger𝑚. (15a)(15b)Using (14a), (14b), (15a), and (15b) yields the variance of the effective system SNR: 𝜎²_𝛾=𝐸𝛾²−𝜇²_𝛾. (16) Accordingly, from (14a)–(16), the quantity of AF for the multiuser TAS/MRC systems can be computed by 𝜎AF=^2𝛾𝜇²_𝛾. (17)

3. Performance Analysis: Outage Probability and SER

The outage probability is defined as the probability that the instantaneous capacity is less than a given capacity C [19], that is: 𝑃_out(𝐶,SNR)=Prlog₂𝛾21+SNR<𝐶=PrSNR<^𝐶−1. (18) Then, in light of the CDF in (5), we can express the outage probability for the multiuser TAS/MRC systems as 𝑃_out(𝐶,SNR)=𝐹_𝐼2^𝐶=𝛾−1𝑚𝐿_𝑅,2^𝐶−1𝑚/𝛾Γ𝑚𝐿_𝑅^𝐾𝐿𝑇. (19) On the other hand, the SER for M-QAM is given by [24, equation (31)] 𝑃_QAM𝛾=𝑎𝑄𝑏𝛾−𝑐𝑄²𝑏𝛾, (20) where 𝑄(𝑥)=(√2𝜋)⁻¹∫^∞_𝑥exp(−𝑧²/2)𝑑𝑧 is the Gaussian Q-function, (𝑎,𝑏,𝑐) = (4(√𝑀−1)/√𝑀,  3/(𝑀−1),  4(√𝑀−1)²/𝑀). Then, the derivative of (20) is [24, equation (32)] 𝑃^′_QAM𝛾=(𝑐−𝑎)𝑏𝑒8𝜋𝛾^{−𝑏𝛾/2}−𝑐𝑏𝑒2𝜋^1−𝑏𝛾𝐹₁31;2;𝑏𝛾2, (21) where ₁𝐹₁(⋅;⋅;⋅) is the confluent hypergeometric function [23, equation (9.210.2)].

Using the integral-by-parts method, the average SER in fading channels is given by 𝑃_SER=^∞₀𝑓_𝐼𝛾𝑃_QAM𝛾𝑑𝛾=−^∞₀𝐹_𝐼𝛾𝑃^′_QAM𝛾𝑑𝛾. (22) Substituting (12) and (21) into (22), we obtain the SER for M-QAM for real 𝑚≥1/2 as 𝑃_SER=12^∞_𝑛=0𝜂_𝑛⋅𝑚𝛾^{𝑚𝐾𝐿𝑇𝐿𝑅+𝑛}×𝑏()Γ2𝜋𝑎−𝑐𝑚𝐾𝐿_𝑇𝐿_𝑅+𝑛+0.5𝑚𝐾𝐿_𝑇/𝛾+𝑏/2^{𝑚𝐾𝐿𝑇𝐿𝑅+𝑛+0.5}+𝑏𝑐𝜋Γ𝑚𝐾𝐿_𝑇𝐿_𝑅+𝑛+1𝑚𝐾𝐿_𝑇/𝛾+𝑏^{𝑚𝐾𝐿𝑇𝐿𝑅+𝑛+1}×₂𝐹₁𝑚𝐾𝐿_𝑇𝐿_𝑅3+𝑛+1,1;2;𝑏2𝑚𝐾𝐿_𝑇/,1𝛾+𝑏real𝑚≥2, (23) where we have used the relation [23, equation (7.621.4)] exp(^∞₀−𝑠𝑡)𝑡₁^𝑑−1𝐹₁=Γ(𝑑)𝑓;𝑔;𝑘𝑡𝑑𝑡𝑠^𝑑2𝐹₁𝑘𝑓,𝑑;𝑔;𝑠 (24) and ₂𝐹₁(⋅,⋅;⋅) is the Gauss hypergeometric function [23, equation (9.14.1)]. Similarly, while the fading parameter m is a positive integer, we obtain a closed form as 𝑃_SER=12^𝐾𝐿𝑇_𝑗=0𝐾𝐿_𝑇𝑗(−1)^{𝑗𝑗(𝑚𝐿𝑅−1)}_𝑛=0𝛽_𝑛⋅𝑚𝛾^𝑛×𝑏()Γ()2𝜋𝑎−𝑐𝑛+0.5𝑚𝑗/𝛾+𝑏/2^𝑛+0.5+𝑏𝑐𝜋Γ(𝑛+1)𝑚𝑗/𝛾+𝑏^𝑛+1×₂𝐹₁31,𝑛+1;2;𝑏2𝑚𝑗/,𝛾+𝑏positiveinteger𝑚. (25)

4. Numerical and Simulation Results

In this section, we use the MATLAB software to present the numerical and simulation results of performance. The Nakagami-m samples are generated by the square root of Gamma distributed samples (i.e., the gamrnd(⋅) MATLAB function) [25]. The outage probability and the SER were simulated by the ratios when 10⁴ outages and 10⁴ symbol errors occurred, respectively. All the simulations, marked by the “×” symbols, agree closely with the analytic curves, validating the theoretical derivation. The number of received antenna 𝐿_𝑅 is chosen as 2 in all scenarios.

Figure 1 depicts the AF of the multiuser TAS/MRC systems against the selection gain 𝐾𝐿_𝑇 in various Nakagami-m fading channels. The curves with 𝛾=0dB are plotted, using (14a) and (15a) for 𝑚=0.5 by truncating the infinite series to 50 terms. It can be seen that the AF decreases as m increases; even the selection gain increases from 4 to 16. In other words, the fading index AF can quantify the severity of channel fading. For the high selection gain, however, the benefit of severe fading cannot be exhibited from the tendency of AF. Figure 2 shows the outage probability with 𝛾=0dB against the capacity for both low and high selection gains. The outage probability with the low selection gain (𝐾𝐿_𝑇=2) mostly performs better for the higher values of m, except that the outage probability already exceeds about 50%. On the other hand, the positive effect of severe fading on the performance is clearly demonstrated at the high selection gain (𝐾𝐿_𝑇=16). It is worth noting that the advantage of severe fading occurs in the outage range of practical interest. For example, the outage probability exceeds about 3×10⁻³ when the capacity is required larger than about 1.7 bps/Hz. The positive effect is further illustrated by the PDF values using (11) when 𝛾=0dB in Figure 3. As the selection gain increases from 2 to 16, the PDF values for 𝑚=0.5 shift to right much more than those for 𝑚=3. In the severe fading channel, indeed, the selection gain boosts the probability to distribute the PDF values over a higher instantaneous SNR region and hence demonstrates the positive effect.

Figure 1 

The amount of fading (AF) of multiuser TAS/MRC systems against the selection gain 𝐾𝐿𝑇 in various Nakagami-m fading channels when 𝐿𝑅=2 and 𝛾=0dB.

Figure 2 

Outage probability against the capacity for both low and high selection gains when 𝐿𝑅=2 and 𝛾=0dB.

Figure 3 

PDF of multiuser TAS/MRC systems with both low and high selection gains when 𝐿𝑅=2 and 𝛾=0dB.

In Figure 4, the SERs are plotted for the considered systems with 16-QAM in various fading channels. The SER for 𝑚=0.5 is evaluated by (23) with truncation (50 terms) for the infinite series. As expected, the SER performs better in the weak fading channels at low selection gain. However, the tendency becomes contrary for the SERs at high selection gain. It can be seen that the SER for 𝑚=0.5 performs best until 𝛾 is higher than about 15 dB, and the SER is smaller than the order of 10⁻⁵. Obviously, the signal with high SNR will dilute the character of severe fading even at high selection gain.

Figure 4 

SER of 16-QAM against average SNR per symbol per branch when 𝐿𝑅=2.

5. Conclusions

In general, system performance behaves better in weak fading channels than in severe fading channels. This paper presents a positive impact of severe Nakagami-m fading on the performance of multiuser TAS/MRC systems with high selection gain. The scheduler with high selection gain is the key to arrange the transmission at higher peaks in a more scattering fading channel. Two performance indexes, AF and SER of M-QAM, are derived as closed-form expressions for integer m. For arbitrary m, the AF and the SER are expressible as a single infinite series of Gamma function and Gauss hypergeometric function, respectively. Although the AF is unable to illustrate the benefit of severe fading, from the analytical and numerical results, we validate the favorable effect from the results of PDF, outage probability and SER, occurring in the range of practical interest.

Acknowledgment

The authors thank the anonymous referees for helpful comments. This research was supported by the Instrument Technology Research Center, National Applied Research Laboratories, and NSC 100-2221-E-251-004.