Asymptotic analysis of the MMSE multiuser detector fornonorthogonal multipulse modulation

We develop the minimum mean-squared-error (MMSE) multiuser detector for nonorthogonal multipulse modulation over the noncoherent additive white Gaussian noise channel. We analyze the asymptotic performance of the detector and show that, unlike the case of linear modulation, the MMSE detector does not generally approach the generalized maximum-likelihood (GML) detection rule as the noise power vanishes. It does, however, approach a detector which nulls out the multiaccess interference. This detector is termed the multipulse decorrelating detector due to its similarity to the linear decorrelating detector. The probability of error for this detector is derived and used to find the asymptotic multiuser efficiencies of both the multipulse decorrelating detector and the MMSE detector. It is shown that for noncoherent binary signaling, in which the multipulse modulation is two-dimensional, the multipulse decorrelating detector is superior to the GML detector asymptotically. This result does not generalize to larger dimensionality signal sets     

Alkaline aqueous solution promoted debromination of 1,2 dibromo-fluorocarbons – A convenient method for electron deficient perfluorovinyl ethers

A facile and efficient base-mediated protocol for debromination of vic-dibromides in perfluoroalkyl(aryl) compounds in aqueous medium has been demonstrated. With mild reaction conditions, the developed strategy has a good substrate scope and electron-deficient olefin products were obtained in good yields. A mechanistic explanation of the debromination is offered with three key experimental observations: (1) the reactions are accelerated by the more electron-rich nucleophiles, (2) the reactions are promoted by the more electron poor vic-dibromides in perfluoroalkyl compounds, and (3) the nucleophilic side reaction is preventable. It is evident that the electronic factors strongly dictate vic-dibromides elimination to the perfluorovinyl ethers, which are the precursors for various perfluorinated polymers. The different reaction conditions were tested in implicit solvent (water) conditions, which helped to confirm the E2-like mechanism.     

A connection between Schur multiplication and Fourier interpolation. II

Given m × n matrices A = [a_jk ] and B = [b_jk ], their Schur product is the m × n matrix A ○ B = [a_jkb_jk ]. For any matrix T, define ‖T‖ _S = max_{X ≠O} ‖T ○ X ‖/‖X ‖ (where ‖·‖ denotes the usual matrix norm). For any complex (2n – 1)‐tuple μ = (μ _{–n +1}, μ _{–n +2}, …, μ _{n –1}), let T_μ be the Hankel matrix [μ –_{n +j +k –1}]_j,k and define ??_μ = {f ∈ L ₁[–π, π] : f? (2j ) = μ_j for –n + 1 ≤ j ≤ n – 1} . It is known that ‖T_μ‖ _S ≤ infequation/tex2gif-inf-18.gif ‖f ‖₁. When equality holds, we say T_μ is distinguished. Suppose now that μ _j ∈ ? for all j and hence that T_μ is hermitian. Then there is a real n × n hermitian unitary X and a real unit vector y such that 〈(T_μ ○ X )y, y 〉 = ‖T_μ ‖_S . We call such a pair a norming pair for T_μ . In this paper, we study norming pairs for real Hankel matrices. Specifically, we characterize the pairs that norm some distinguished Schur multiplier T_μ . We do this by giving necessary and suf.cient conditions for (X, y ) to be a norming pair in the n ‐dimensional case. We then consider the 2‐ and 3‐dimensional cases and obtain further results. These include a new and simpler proof that all real 2 × 2 Hankel matrices are distinguished, and the identi.cation of new classes of 3 × 3 distinguished matrices. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)