18 IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 1, NO. 1, FEBRUARY 2012 ASER of Rectangular MQAM in Noise-Limited and Interference-Limited MIMO MRC Systems Juan M. Romero-Jerez, Member, IEEE, and Juan P. Pe˜ na-Martin Abstract—Novel expressions for the average symbol error rate (ASER) of general rectangular quadrature amplitude modulation (QAM) in Rayleigh fading are derived in this work for MIMO MRC systems. The number of antennas is assumed to be arbitrary at both the transmit and receive ends and, unlike previous related work, expressions are given in terms of finite summations where all the terms are explicitly given. Both noise- limited and interference-limited systems are analyzed. Monte Carlo simulations have been carried out to validate the accuracy of the results. Index Terms—Symbol error rate, rectangular QAM, MIMO MRC. I. I NTRODUCTION M ULTIPLE-input/multiple-output (MIMO) maximal ra- tio combining (MRC) has been proposed to maximize the output signal-to-noise ratio (SNR) in wireless systems with multiple transmit and receive antennas [1], [2]. Rect- angular 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 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 (ASER) of rectangular QAM in noise-limited environments. In [3], 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 ASER is provided. Additionally, the analysis in [3] is actually semi-analytical, as the derived expression rely on coefficients not given in explicit form and which have to be calculated numerically. The use of these coefficients is avoided in [4] for the same scheme, but only for a system with two receive antennas, and again no ASER result is provided. In practice, wireless communication systems are usually limited by co-channel interference (CCI). An analysis of the ASER of MIMO MRC with rectangular QAM in the presence of CCI was presented in [5]. However, the analysis is based on an approximated expression of the instatantaneous error rate and the provided results are again based on coefficients which are not given explicitly. In this work, we present novel and general expressions for the ASER of general rectangular QAM in MIMO MRC Manuscript received October 22, 2011. The associate editor coordinating the review of this letter and approving it for publication was G. Vitetta. This work has been supported by FEDER-MCINN Project No.TEC2009- 13763-C02-01. The material in this paper was presented in part at the 2010 IEEE International Conference on Communications, Cape Town, South Africa, May 2010. The authors are with the Department of Electronic Technology, E.T.S.I. Telecomunicaci´ on, University of M´ alaga, 29071 M´ alaga, Spain (e-mail: romero@dte.uma.es, jppena@uma.es). Digital Object Identifier 10.1109/WCL.2012.120211.110111 with no restriction in the number of transmit and receive antennas. Unlike previous related work, all the terms in the derived expressions are explicitly given, as our results do not rely on coefficients to be computed numerically. For systems without CCI a closed-form expression of the ASER in terms of elementary functions and independent of the code mapping of the constellation symbols is derived for the first time. For systems with CCI the derived expression is given in terms of an integral which can be solved fast and as accurately as desired using the generalized Laguerre polynomials. This work also presents a novel expression for the probability of outage when CCI is present. The analysis presented here has been validated by means of Monte Carlo simulations. II. SER ANALYSIS WITHOUT CCI A. System model We assume a MIMO system with transmit and receive antennas. The desired signal undergoes flat Rayleigh fading that is uncorrelated across the antennas. We also assume that the signal at each of the receive antennas is corrupted by additive white Gaussian noise (AWGN). The received noise vector is assumed to be zero mean with covariance matrix 2 I , where I denotes the × identity matrix. Thus, after matched filtering and sampling at the symbol interval, the × 1 received vector r can be modeled as r = H w + n, (1) where H is an × matrix whose (, ) ℎ element represents the channel complex gain from transmit antenna to receive antenna ; n is the -dimensional received noise vector; is the transmitted symbol of the desired user and w is the -dimensional weight vector associated with the transmit antenna array. We assume that the channel matrix H is perfectly known at both the transmitter and receiver. For simplicity, we also assume ∣ ∣=1, although this assumption is easily relaxed. In MIMO MRC systems without CCI, the combiner SNR is maximized with appropriate weighting at the transmitter and receiver. These weights, derived in [1] and [2], are given by w = √ Ω 0 u, and w = H u, where √ Ω 0 is the norm of vector w , u is a unitary (∣∣u∣∣=1) eigenvector corresponding to the largest eigenvalue () of the matrix F = H H , and (⋅) is the Hermitian operator. Thus, the received SNR, which we denote as , is a random variable (RV) given by = Ω 0 2 = , (2) where =Ω 0 / 2 is the average SNR per receive antenna. 2162-2337/12$31.00 c ⃝ 2012 IEEE