A Combinatorial Approach to Evaluation of Reliability of the Receiver Output for BPSK Modulation with Spatial Diversity

In the context of soft demodulation of a digital signal modulated with Binary Phase Shift Keying (BPSK) technique and in presence of spatial diversity, we show how the theory of symmetric functions can be used to compute the probability that the log-likelihood of a recieved bit is less than a given threshold $\varepsilon$. We show how such computation can be reduced to computing the probability that $U-V < \varepsilon$ (denoted $P(U-V < \varepsilon)$) where $U$ and $V$ are two real random variables such that $U=\sum_{i=1}^N |u_i|^2$ and $V=\sum_{i=1}^N |v_i|^2$ where the $u_i$'s and $v_i$'s are independent centered complex Gaussian variables with variances ${\Bbb E}[\,|u_i|^2\,]=\chi_i$ and ${\Bbb E}[\,|v_i|^2\,]=\delta_i$. We give two expressions in terms of symmetric functions over the alphabets $\Delta=(\delta_1,\dots,\delta_N)$ and $X=(\chi_1,\dots,\chi_N)$ for the first $2N-1$ coefficients of the Taylor expansion of $P(U-V < \varepsilon)$ in terms of $\varepsilon$. The first one is a quotient of multi-Schur functions involving two alphabets derived from alphabets $\Delta$ and $X$, which allows us to give an efficient algorithm for the computation of these coefficients. The second expression involves a certain sum of pairs of Schur functions $s_\lambda(\Delta)$ and $s_\mu(X)$ where $\lambda$ and $\mu$ are complementary shapes inside a $N\times N$ rectangle. We show that such a sum has a natural combinatorial interpretation in terms of what we call square tabloids with ribbons and that there is a natural extension of the Knuth correspondence that associates a (0,1)-matrix to each square tabloid with ribbon. We then show that we can completely characterise the (0,1)-matrices that arise from square tabloids with ribbons under this correspondence.