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{\bf S. Bliudze and D. Krob}
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{\bf A Combinatorial Approach to Evaluation of Reliability of the Receiver Output for BPSK Modulation with Spatial Diversity}
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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.
\bye