Asymptotic capacity analysis under different adaptive transmission for TAS/MRC MIMO scheme subject to Weibull fading channels

Abstract

This paper investigates a unified average channel capacity (ACC) analysis under different adaptive transmission of multiple-input multiple-output (MIMO) system, combining single transmit antenna selection (TAS) and maximal-ratio combining (MRC) receiver operating under independent flat Weibull multipath fading channels (WFCs). Based on a tight approximate probability density function expression of the signal-to-noise ratio (SNR) at the considered receiver output, we derive new accurate closed-form expressions of ACC for TAS/MRC system employing several rate adaptation techniques, namely optimal rate adaptation, optimal simultaneous power and rate adaptation, channel inversion with fixed rate (CIFR), truncated CIFR (TCIFR), and continuous power TCIFR (CTCIFR). Further, asymptotic expressions, in both high and low-SNR regimes, are investigated. The results show high accuracy for significant values of $L_t\times L_r$ TAS/MRC system. Indeed, the smaller $L_t\times L_r$, the better the PDF’s accuracy, therefore, the better are the capacities. Numerical outcomes have been assessed by utilizing Mathematica and Matlab Software and corroborated via Monte-Carlo simulations.

Introduction

Multiple-input multiple-output (MIMO) regimes have been produced in order to alleviate the performance degradation of multipath fading in wireless frameworks and to fulfill the high information rate requests for future wireless communication through expanded spectral efficiency [1]. The performance of wireless transmission can be essentially enhanced by combining all the signals received by the individual antennas utilizing different combining methods, namely equal gain combining (EGC), selective combining (SC), and maximal-ratio combining (MRC), known to outperform all diversity techniques [1], [2].

The energy/spectral efficiency is becoming a key factor in the design of wireless communication systems. This has stimulated the enormous efforts that have been made so far within the communication theory communities to understand better the performance behavior of wireless communications in low/high power regimes and to develop new methods aiming to adapt/optimize the transmission rate in order to achieve a high capacity system.

Although the signal strength in large-scale communications is typically very low, one can take advantage of extremely important bandwidth to achieve high capacity [3], [4]. As the lifetime of nodes’ batteries in wireless sensor networks (WSNs) are limited, the signal is most often transmitted with a very low power to save as much as possible the batteries’ energy consumption [5], [6]. Also transmitting with high power would lead to higher interference. Therefore, reducing and adapting the transmission power of sensor nodes become the most important keys in designing WSNs [7], [8].

The Weibull distribution is used as a flexible statistical model for describing wireless indoor and outdoor channels [9]. It is worth mentioning that Weibull fading is more general than Rayleigh and negative exponential models, associated with $\beta =2$ and $\beta =1$, respectively, where $\beta$ denotes the shape parameter that controls the severity of fading. It is worthwhile that the closed-form PDF of the summation of Weibull RVs is not known in the literature. On the other hand, the Weibull distribution is a special case of both Generalized Gamma convolutions (GGC) and infinitely divisible (ID) distributions. The authors in [3], [10] investigate the ASER metric of the MRC/EGC diversity techniques under ID fading models. Interestingly, the summation of GGCs, IDs are GGC and ID distribution, respectively. Owing to this fact, seeing a Weibull RV as a GGC distribution might be useful while studying the performance of MRC diversity instead.

Among the MIMO technologies, transmit antenna selection (TAS) scheme at the transmitter with maximal-ratio combining at the receiver, referred to as (TAS/MRC) system, can be considered as a low complexity power efficient and accomplishes diversity gain at both the transmit and receive diversity branches [11], [12]. Indeed, in TAS/MRC system, a single transmit antenna, maximizing the total received signal power at the MRC receiver, is continuously selected for transmission. The antenna selection technique has been proposed in the literature, whereby the best performing method is chosen [12], within the number of radio frequency chains to decode at the receiver will be reduced.

The performance analysis of TAS/MRC under different fading scenarios and modulation schemes have been studied extensively in the literature. Indeed, the outage probability (OP) as well as the average bit/symbol error rate (ABER/ASER) have been investigated for Rayleigh and Nakagami-$m$ models in [12] and [13], respectively.

Also, several papers have been dealt with TAS/MRC MIMO relay networks undergoing WFCs based on the probability density function (PDF) expression of the sum of Weibull random variables (RVs) in terms of the well-known Meijer’s G-function. For instance, in [14], [15], [16], closed-form expressions for the asymptotic expression of the cumulative distribution function (CDF) and ASER of M-ary phase-shift keying (M-PSK) and M-ary quadrature amplitude modulation (M-QAM) have been derived for MIMO relay networks. Approximate closed-form of the post-processing SNR PDF has been investigated in [17], based on which, approximate analytical expressions such as outage probability, average channel capacity (ACC), and the ASER for several M-ary modulation techniques were derived, and their convergence has been proven using the truncated expression of Meijer’s G-function.

On the other hand, the authors in [18], [19] proved that the capacity of fading channels depends on the availability of channel state data either at the receiver (CSIR) or at the transmitter (CSIT). Also, they examined that the adaptive modulation achieves an average data rate within 7–10 dB of the capacity derived herein, while nonadaptive modulation exhibits a severe rate penalty. The CSIT techniques are also known as adaptive transmission (AT) policies, whereas the CSIR concerns only post-processing SNR level. That is, AT can be operated on a frame by frame to allow distinct adjustments among the achievable ACC. In [20], authors have derived the ACC of Rayleigh fading channels under different AT policies and diversity combining techniques as well. The ACC of a MIMO system without antenna selection over Weibull fading under different adaptive techniques has been derived in [21]. The ACC of Log-normal fading channels has been analytically derived in [22] for various adaptive policies. In [23], asymptotic capacity analysis at both high and low SNRs under AT schemes over distinct fading distributions (i.e., Rayleigh and Nakagami-$m$). A very tight approximate expression of ACC under various ATs of MRC receivers subject to independent WFCs have been evaluated in [24].

In recent years, previous works on ACC of TAS/MRC system employing different AT policies have been analyzed. The ACC of MRC with antenna selection under various ATs over Nakagami-$m$ fading model was studied in  [25]. The closed-form expressions of ACC for TAS/MRC system with channel estimation errors under different ATs over Rayleigh fading channels was evaluated in [26].

Motivated by the aforementioned discussion, our main contributions can be summarized as follows:

• We propose novel closed-form expressions for both the tight approximate and the asymptotic capacity under several ATs of TAS/MRC systems subject to independent and identically distributed (i.i.d.) Weibull RVs.

• The tightness of the approximate and the asymptotic capacity are proved and validated by Monte Carlo simulations.

• Moreover, relying on the output SNR PDF, we derive, in terms of univariate/bivariate Meijer’s G-functions, the OP, ASER for various M-ary modulation techniques, and ACC of TAS/MRC system under several variable power and variable rate ATs over i.i.d. Weibull fading channels. These results are valid for general operating scenarios to whatever diversity order on both branches of the transmit antenna and MRC receiver, and Weibull parameter $\beta$.

• We consider five adaptation transmission policies: (i) constant power with optimal rate adaptation (ORA), (ii) optimal simultaneous power and rate adaptation (OPRA), (iii) channel inversion with fixed rate (CIFR), (iv) truncated CIFR (TCIFR), and (v) continuous-power TCIFR (CTCIFR). Furthermore, their asymptotic behaviors at both low/high SNR regimes, are obtained using the residue theory. Based on our results, it is proved that the OPRA policy was the best in the performance compared to other adaptation policies. To the best of the authors’ knowledge, this result has not been investigated previously.

The remainder of this paper is structured as follows. In Section 2, we outline the channel and communication system model. The outage probability is evaluated in Section 3, while the channel capacities under five distinct AT policies are examined in Section 4. The ASER for several M-ary modulation techniques of TAS/MRC system is presented in Section 5. In Section 6, the numerical results are depicted and discussed. Lastly, a brief conclusion and some future works are pointed out in Section 7.