Performance evaluation of demodulation with diversity—a combinatorial approach II: bijective methods

Krob, Daniel; Vassilieva, Ekaterina A.

doi:10.1016/j.dam.2004.05.005

Cited by 2 publications

(6 citation statements)

References 13 publications

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“…As this algorithm is based on the Knuth correspondence (cf. [7,12,13]), it can be easily seen that it is reversible, thus, denoting by M (N) m the image of the mapping defined by this algorithm, we obtain a bijection between M (N) m and T (N) m . • Step 2.…”

Section: Description Of the Bijectionmentioning

confidence: 99%

“…Often, when appropriate channel decoding mechanism is used, the actual value of log-likelihood (called in this case a soft bit) represents the reliability of the decoder's input. One obtains equation 1by applying the parallelogram identity to (13). The situation undesirable for both demodulation (increased chances of taking incorrect decision) and soft decoding algorithms (unreliable input) is when the log-likelihood is close to zero, i.e.…”

Section: Signal Processing Backgroundmentioning

confidence: 99%

“…This expression for the probability P (U < V ) has been obtained in [6], while in [13] it has been given a combinatorial interpretation. Indeed, as λ ⊂ (N N −1 ) we have λ i ≤ N for all i, and thus (λ, N) is also a partition.…”

Section: Combinatorial Interpretation 51 a Special Casementioning

confidence: 99%

“…• Step 3. Apply the same procedure as in [13] to obtain a tableau Q of a shape conjugated to λ ∪r (Figure 3-c; see Example 5.2 below, or [13] for a formal description).…”

Section: Description Of the Bijectionmentioning

confidence: 99%

“…• Step 2. The fact that M belongs to M (N) m implies that both P and Q fit into (N N ), and thus we can again apply to Q the same procedure as in [13] to obtain a new Young tableau t µ of the shape complementary to that of P . and we transform it into a two-row array representing the positions of 1's: in each column of the array the element in the first row is the row number, and the one in the second row -the column number of a position containing 1.…”

Section: Algorithm 52 (Construction Of a Square Tabloid From A {0 1mentioning

confidence: 99%

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

Bliudze¹,

Krob²

2006

Electron. J. Combin.

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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.

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Section: Description Of the Bijectionmentioning

confidence: 99%

Section: Signal Processing Backgroundmentioning

confidence: 99%

Section: Combinatorial Interpretation 51 a Special Casementioning

confidence: 99%

“…• Step 3. Apply the same procedure as in [13] to obtain a tableau Q of a shape conjugated to λ ∪r (Figure 3-c; see Example 5.2 below, or [13] for a formal description).…”

Section: Description Of the Bijectionmentioning

confidence: 99%

Section: Algorithm 52 (Construction Of a Square Tabloid From A {0 1mentioning

confidence: 99%

See 3 more Smart Citations

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

Bliudze¹,

Krob²

2006

Electron. J. Combin.

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Towards a Functional Formalism for Modelling Complex Industrial Systems

Krob¹,

Bliudze²

2004

ComPlexUs

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This paper is devoted to the presentation of the main lines of an unifi ed functional formalism for modelling complex industrial systems, that is to say systems that typically mix a big number of different software and physical devices. Our approach is based on a discrete non-standard representation of time. It captures both the hierarchical architecture and the temporal and data multi-scale structures of a complex industrial system. We show in particular in this paper how our formalism allows to recover in an totally unifi ed way various sorts of simple systems such as Turing machines, elementary conservative physical systems and low level software/physical interfaces (sampler, modulator).

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Performance evaluation of demodulation with diversity—a combinatorial approach II: bijective methods

Cited by 2 publications

References 13 publications

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

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

Towards a Functional Formalism for Modelling Complex Industrial Systems

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