Biorthogonal system

About: Biorthogonal system is a research topic. Over the lifetime, 2190 publications have been published within this topic receiving 32209 citations.

Unbounded Control and Observation Systems and Their Duality

Abstract: The paper considers some questions arising from a general theory of observation and control by Russell and Dolecki.For a distributed parameter system the observation may be restricted to the boundary or some other subset, so the observation operator may be unbounded. If this operator satisfies certain conditions the dual control system operator can be defined. If also there is an appropriate Green’s formula, the dual system is interpreted as one of control on a subset. For systems associated with biorthogonal sequences it is shown that the attainable sets cannot be fully characterized in terms of the sequence unless it is basic.

Biorthogonal bases with local support and approximation properties

Abstract: We construct locally supported basis functions which are biorthogonal to conforming nodal finite element basis functions of degree p in one dimension. In contrast to earlier approaches, these basis functions have the same support as the nodal finite element basis functions and reproduce the conforming finite element space of degree p - 1. Working with Gaus-Lobatto nodes, we find an interesting connection between biorthogonality and quadrature formulas. One important application of these newly constructed biorthogonal basis functions are two-dimensional mortar finite elements. The weak continuity condition of the constrained mortar space is realized in terms of our new dual bases. As a result, local static condensation can be applied which is very attractive from the numerical point of view. Numerical results are presented for cubic mortar finite elements.

Perfect reconstruction conditions and design of oversampled DFT-modulated transmultiplexers

Abstract: This paper presents a theoretical analysis of oversampled complex modulated transmultiplexers. The perfect reconstruction (PR) conditions are established in the polyphase domain for a pair of biorthogonal prototype filters. A decomposition theorem is proposed that allows it to split the initial system of PR equations, that can be huge, into small independent subsystems of equations. In the orthogonal case, it is shown that these subsystems can be solved thanks to an appropriate angular parametrization. This parametrization is efficiently exploited afterwards, using the compact representation we recently introduced for critically decimated modulated filter banks. Two design criteria, the out-of-band energy minimization and the time-frequency localization maximization, are examined. It is shown, with various design examples, that this approach allows the design of oversampled modulated transmultiplexers, or filter banks with a thousand carriers, or subbands, for rational oversampling ratios corresponding to low redundancies. Some simulation results, obtained for a transmission over a flat fading channel, also show that, compared to the conventional OFDM, these designs may reduce the mean square error.

Biorthogonal cosine-modulated filter bank

Abstract: Cosine-modulated filter banks have been studied extensively because of their design ease and efficient implementation. These filter banks either have restricted lengths or assume the paraunitary property for the polyphase matrices. In this paper, the biorthogonal cosine-modulated filter bank with arbitrary length is considered and the perfect reconstruction (PR) conditions are derived. These conditions are the general form of the PR conditions reported in the literature. Examples of PR systems with variable overall delay are designed using the quadratic constrained least squares formulation.