An accurate frame error rate approximation of coded diversity systems with non-identical diversity branches

This paper presents an accurate approximation of the frame error rate (FER) of coded wireless communication systems with receiver diversity, such as single-input multiple- output (SIMO) systems with maximum ratio combining (MRC) or hybrid automatic repeat request (HARQ) systems with Chase combining. The signals at different diversity branches experience independent but non-identically distributed Rayleigh fading. The FER approximation is obtained with a threshold-based method. Specifically, the threshold value, which is critical to the FER approximation accuracy, is modeled as a linear function of the frame length in the log-domain, with the slope and intercept of the linear function determined by the underlying modulation and channel coding schemes. The analytical FER approximation is expressed as an explicit function of parameters related to modulation, coding, frame length, number of diversity branches, and the power distribution across the diversity branches. Such an FER approximation summarizes the complex physical layer operations into a few parameters, and it provides the para- metric flexibility that is not available in most existing FER approximations. Simulation results show that the proposed FER approximation can accurately predict the FER performance of a wide range of receiver diversity systems. I. INTRODUCTION Bit error rate (BER) and frame error rate (FER) are two important metrics used to measure the quality of service (QoS) of communication systems. BER provides a measurement of the end-to-end communication quality in the physical (PHY) layer. FER is also a PHY layer metric, but it can be used to facilitate the design across multiple protocol layers in a wireless network. For example, a frame error after PHY layer channel decoding can be detected with cyclic redundancy check (CRC) or parity check in the media access control layer (MAC), and frames with unrecoverable errors will be retransmitted if automatic repeat request (ARQ) is employed. The knowledge of the FER at the upper layers can thus greatly facilitate the design of wireless networks, especially for cross-layer designs that aim to improve the system performance by jointly optimizing the system parameters across two or more protocol layers (1) and (13). Recently there are growing interests in FER analysis for wireless communication systems due to the increasing im- portance of cross-layer designs. It is extremely difficult, if not impossible, to obtain the exact analytical FER expression, especially for coded systems. Therefore almost all the works in the literature focus on the analytical approximations of the actual FER. In (3), an FER approximation is provided for systems with low-density parity-check (LDPC) codes over bi- nary symmetric channel (BSC) by enumerating a large number of error patterns. A threshold-based FER approximation for a system with quasi-static Rayleigh fading is proposed in (4). The threshold-based method is developed by using an assumption that the FER in an additive white Gaussian noise (AWGN) channel can be approximated by either 1 or 0 depending on whether the signal-to-noise ratio (SNR) is less than or greater than a waterfall threshold. The results in (4) are extended to diversity systems with independent and identically distributed (i.i.d.) Rayleigh fading channels in (5) and to 2-hop relay networks with incremental hybrid relay selection in (6). The FER approximations in (4) - (6) are semi-analytical in that the threshold values of coded systems need to be evaluated through FER simulations in AWGN channels. None of the above FER approximations can explicitly quantify many important design parameters, such as frame length, modulation, coding, and the diversity orders, etc. In this paper, a new FER approximation is proposed for wireless communication systems with receiver diversity and independent but non-identically distributed (i.n.d.) diversity branches. Receiver diversity exists in many practical systems, such as single-input multiple-output (SIMO) systems (7), multi- node relay systems (8), and hybrid ARQ systems (9), where the receiver can coherently combine multiple copies of the transmitted message. Signals at different diversity branches usually experience non-identically distributed fading due to the effects of pathloss and differences in transmission power. The statistical properties of the SNR for receiver diversity systems undergoing i.n.d. quasi-static Rayleigh fading are derived with the help of the moment generating function (MGF). The results are used to develop the FER approximation with the threshold- based method. Different from existing threshold-based methods that need to calculate a new threshold value for each frame length, the proposed FER analysis models the threshold value as a linear function of the frame length in the log-domain, with the slope and intercept of the linear function determined by the modulation and coding schemes. It is discovered that the threshold value is independent of the actual fading distributions or the number of diversity branches. Enabled by the log-domain linear threshold modelling, the new FER is expressed as an explicit function of various parameters related to modulation,