The Lth and (L+1)th order asymptotic symbol error rate based SNR expressions as a function of modulation scheme, constellation size and diversity order in a Rayleigh fading channel

In this paper, the expression for symbol error rate (SER) or BER of linear modulation in Rayleigh fading as a function of signal-to-noise ratio (SNR), modulation scheme and diversity order, is inverted, thereby giving SNR as a function of SER. The key is to represent the PDF (or MGF) by a Maclaurin series and discard all but the first one or two terms. For large SNRs, the first term in the series is accurate and simple closed-form SNR expressions for an arbitrary diversity order are derived, in terms of asymptotic SER (ASER), in SNR. Here ASER only has the term of the inverse SNR raised to the system diversity order L, thereby called the Lth order ASER. When the SNR is not large enough, the first two terms in the series, the terms containing Lth and (L_1)th order, are used and proven to be accurate. Based on the derived ASERs, we then present closed-form SNR expressions for smaller SNRs. In this paper, closed-form expressions for BPSK, MQAM and MPSK signals are presented, respectively. Using the derived closed-form expressions, one can easily design a link budget for a given BER requirement without going through tedious and time consuming simulations (for example, in MATLAB). As an application, several examples are presented to illustrate the ease of usage of the presented SNR closed-form expressions in a wireless SISO and MIMO link budget design. Tables are given to show the SNR expressions for their SER ranges with different modulation schemes/constellation sizes for diversity order varying from one to four.