Topic
Biorthogonal system
About: Biorthogonal system is a research topic. Over the lifetime, 2190 publications have been published within this topic receiving 32209 citations.
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TL;DR: The paper investigates the properties of certain biorthogonal polynomials appearing in a specific simultaneous Hermite-Pade approximation scheme and defines and proves that their zeros are simple and positive, and shows how to characterize them in terms of a Riemann-Hilbert problem.
Abstract: The paper investigates the properties of certain biorthogonal polynomials appearing in a specific simultaneous Hermite-Pade' approximation scheme. Associated to any totally positive kernel and a pair of positive measures on the positive axis we define biorthogonal polynomials and prove that their zeroes are simple and positive. We then specialize the kernel to the Cauchy kernel 1/{x+y} and show that the ensuing biorthogonal polynomials solve a four-term recurrence relation, have relevant Christoffel-Darboux generalized formulae, and their zeroes are interlaced. In addition, these polynomial solve a combination of Hermite-Pade' approximation problems to a Nikishin system of order 2. The motivation arises from two distant areas; on one side, in the study of the inverse spectral problem for the peakon solution of the Degasperis-Procesi equation; on the other side, from a random matrix model involving two positive definite random Hermitian matrices. Finally, we show how to characterize these polynomials in term of a Riemann-Hilbert problem.
28 citations
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TL;DR: In this article, it was shown that the existence of a pair of biorthogonal multiple vector-valued scaling functions guarantees the presence of pair of BVM wavelet functions.
Abstract: In this paper, we introduce biorthogonal multiple vector-valued wavelets which are wavelets for vector fields. We proved that, like in the scalar and multiwavelet case, the existence of a pair of biorthogonal multiple vector-valued scaling functions guarantees the existence of a pair of biorthogonal multiple vector-valued wavelet functions. Finally, we investigate the construction of a class of compactly supported biorthogonal multiple vector-valued wavelets.
28 citations
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TL;DR: This paper develops conditions for the existence of MIMO biorthogonal partners and conditions under which FIR solutions are possible and exploits the nonuniqueness of the solution, which will lead to the design of flexible fractionally spaced MIMo zero-forcing equalizers.
Abstract: Multiple input multiple output (MIMO) biorthogonal partners arise in many different contexts, one of them being multiwavelet theory. They also play a central role in the theory of MIMO channel equalization, especially with fractionally spaced equalizers. In this paper, we first derive some theoretical properties of MIMO biorthogonal partners. We develop conditions for the existence of MIMO biorthogonal partners and conditions under which FIR solutions are possible. In the process of constructing FIR MIMO biorthogonal partners, we exploit the nonuniqueness of the solution. This will lead to the design of flexible fractionally spaced MIMO zero-forcing equalizers. The additional flexibility in design makes these equalizers more robust to channel noise. Finally, other situations where MIMO biorthogonal partners occur are also considered, such as prefiltering in multiwavelet theory and deriving the vector version of the least squares signal projection problem.
27 citations
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04 Oct 1998
TL;DR: This work designs the entire class of antisymmetric biorthogonal coiflet systems, whose filterbanks have even lengths and are linear phase, and shows that one of the novel filterbanks achieves noticeably better rate-distortion performance than several state-of-the-art filterbanks in image coding.
Abstract: Wavelet techniques have achieved a tremendous success in image data compression. In designing wavelet coding algorithms, the choice of wavelet systems is of great importance for compression performance. We design the entire class of antisymmetric biorthogonal coiflet systems, whose filterbanks have even lengths and are linear phase. We show that one of the novel filterbanks achieves noticeably better rate-distortion performance than several state-of-the-art filterbanks in image coding.
27 citations
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TL;DR: In this article, the authors show that imposing a certain number of vanishing moments on a scaling function (e.g., coiflets) leads to fairly small phase distortion on its associated filter bank in the neighborhood of DC.
Abstract: We show that imposing a certain number of vanishing moments on a scaling function (e.g., coiflets) leads to fairly small phase distortion on its associated filter bank in the neighborhood of DC. However, the phase distortion at the other frequencies can be much larger. We design a new class of real-valued, compactly supported, orthonormal, and nearly symmetric wavelets (we call them generalized coiflets) with a number of nonzero-centered vanishing moments equally distributed on scaling function and wavelet. Such a generalization of the original coiflets offers one more free parameter, the mean of the scaling function, in designing filter banks. Since this parameter uniquely characterizes the first several moments of the scaling function, it is related to the phase response of the lowpass filter at low frequencies. We search for the optimal parameter to minimize the maximum phase distortion of the filter bank over the lowpass half-band. Also, we are able to construct nearly odd-symmetric generalized coiflets, whose associated lowpass filters are surprisingly similar to those of some biorthogonal spline wavelets. These new wavelets can be useful in a broad range of signal and image processing applications because they provide a better tradeoff between the two desirable but conflicting properties of the compactly supported and real-valued wavelets, i.e., orthonormality versus symmetry, than the original coiflets.
27 citations