SEP of rectangular QAM in composite fading channels

Abstract

In this paper, we analytically evaluate the average symbol error probability (SEP) of rectangular quadrature amplitude modulation (QAM) signalling in composite fading channels modelled by the generalized-K (K_G) distribution. The analysis is based on a fast converging infinite series representation of the average of the product of two Gaussian-Q functions over K_G fading that has been extracted. Considering integer and a half values for the distribution's shaping parameters, exact closed-form expressions have been also derived. Numerical evaluated results, complemented by equivalent computer simulated ones, are presented to verify the accuracy of the proposed analysis.

Introduction

Contemporary cellular communication systems should support the continuously increasing data rate demands of a higher number of subscribers. Towards this objective, a quite promising approach that is found to considerable increase the spectral efficiency represent the non-constant envelope modulation schemes. In this context, quadrature amplitude modulation (QAM) constellation is a class of non-constant modulation scheme that can achieve higher data rates as compared to constant envelope schemes, e.g., phase shift keying (PSK). Such a characteristic is very important for numerous of communication systems, where bandwidth efficiency is more important than power efficiency. In addition to the bandwidth efficiency offered, rectangular (or square) QAM can be easily generated as two independent pulse amplitude modulation (PAM) signals, while its demodulation is also easy [1], [2]. This is the main reason why QAM schemes have been employed in many practical communication systems, e.g., Digital Video Broadcasting (DVB) digital cable transmission [3], DVB-satellite to handheld [4] and DVB-digital terrestrial television [5].

Among the various QAM constellations that have been proposed and studied in the past, rectangular QAM has gained an increased interest as it is proved by the various contributions that have been reported on this topic. In particular, rectangular QAM is a generic modulation technique which includes various modulation schemes as special cases, namely square QAM, binary PSK (BPSK), orthogonal binary frequency-shift keying, quadrature PSK and multilevel amplitude shift-keying modulation techniques [6]. Thus, its performance has been evaluated in various communication scenarios, including single channel reception, e.g., [1], [7], [8], multichannel reception, e.g., [9], [10], [11], multiple-input-multiple-output (MIMO) systems, e.g., [12], [13], [14] and cooperative communications, e.g., [15], [16], [17], [18]. More specifically, in [11], assuming a L-branch selection combiner operating over Nakagami-m fading channels, a novel closed-form expression was derived for the average symbol error rate (ASER) of general order rectangular QAM. In [13], the ASER of general rectangular QAM in Rayleigh fading was studied for MIMO maximal ratio combining (MRC) systems, with arbitrary number of antennas on both sides. Finally in [18], a lower bound of the ASER for amplify and forward cooperative systems with best-relay selection over Rayleigh fading channels is derived for general order rectangular QAM. A common observation in all these works is that the proposed analysis takes into consideration only the small scale fading (multipath) that indeed represents a major impairment factor in mobile communication links.

However, in many practical communication scenarios large scale fading effects (shadowing) become dominant, originating what is known as composite multipath/shadowing environment. This scenario arises in situations of slow moving or stationary users, for example congested downtown areas with slow moving pedestrians and vehicles, where the receiver is unable to average over the effects of fading and a composite distribution is necessary for evaluating link performance and other quantities [19], [20]. Depending upon the statistical description of the shadowing effects, several families of composite fading distributions have been proposed, such as the lognormal-based ones, for example Rayleigh, Nakagami-lognormal [19], and gamma based ones, for example K, generalized-K (K_G) and Weibull-gamma [21], [19], [22]. Employing gamma distribution as an alternative to the log-normal one leads to simpler composite distributions and mathematical more tractable analytical approach. In the past, many researchers have used the K and K_G distributions for investigating various phenomena and parameters of the composite fading propagation environments, e.g., [21], [23], [24], [25], [26], [27], and this is the case that is also going to be investigated in the current paper.

In particular the scope of this paper is to analyze the performance of rectangular QAM in composite fading environments. To this aim, novel exact expressions are derived for the average symbol error probability (ASEP) of rectangular QAM constellation in composite fading channels, modelled by the K_G distribution. The proposed approach is based on fast converging infinite series representation of the average of the product of two Gaussian-Q functions over K_G fading that has been derived. In addition our analysis also includes simplified closed-form expressions for the case of special values of the distribution's parameters.

The remainder of this paper is organized as follows. The system and channel model are described in Section 2. In Section 3, exact solutions for an integral representing the average of the product of two Gaussian Q-functions over K_G fading are presented. These solutions are employed in Section 4 to analyze the ASEP for both general as well as special cases. In Section 5, the numerically evaluated performance results are provided, while the concluding remarks are given in Section 6.