384 research outputs found
Noncommutative Uncertainty Principles
The classical uncertainty principles deal with functions on abelian groups.
In this paper, we discuss the uncertainty principles for finite index
subfactors which include the cases for finite groups and finite dimensional Kac
algebras. We prove the Hausdorff-Young inequality, Young's inequality, the
Hirschman-Beckner uncertainty principle, the Donoho-Stark uncertainty
principle. We characterize the minimizers of the uncertainty principles. We
also prove that the minimizer is uniquely determined by the supports of itself
and its Fourier transform. The proofs take the advantage of the analytic and
the categorial perspectives of subfactor planar algebras. Our method to prove
the uncertainty principles also works for more general cases, such as Popa's
-lattices, modular tensor categories etc.Comment: 41 pages, 71 figure
Fusion Bialgebras and Fourier Analysis
We introduce fusion bialgebras and their duals and systematically study their
Fourier analysis. As an application, we discover new efficient analytic
obstructions on the unitary categorification of fusion rings. We prove the
Hausdorff-Young inequality, uncertainty principles for fusion bialgebras and
their duals. We show that the Schur product property, Young's inequality and
the sum-set estimate hold for fusion bialgebras, but not always on their duals.
If the fusion ring is the Grothendieck ring of a unitary fusion category, then
these inequalities hold on the duals. Therefore, these inequalities are
analytic obstructions of categorification. We classify simple integral fusion
rings of Frobenius type up to rank 8 and of Frobenius-Perron dimension less
than 4080. We find 34 ones, 4 of which are group-like and 28 of which can be
eliminated by applying the Schur product property on the dual. In general,
these inequalities are obstructions to subfactorize fusion bialgebras.Comment: 39 pages; 8 figures; addition of a classification in Subsection 9.2;
the long lists in Subsection 9.3 are now more pleasant to read; addition of
Section 7 providing a categorical proof of Schur Product Theore
Algorithms for Blind Equalization Based on Relative Gradient and Toeplitz Constraints
Blind Equalization (BE) refers to the problem of recovering the source symbol sequence from a signal received through a channel in the presence of additive noise and channel distortion, when the channel response is unknown and a training sequence is not accessible. To achieve BE, statistical or constellation properties of the source symbols are exploited. In BE algorithms, two main concerns are convergence speed and computational complexity.
In this dissertation, we explore the application of relative gradient for equalizer adaptation with a structure constraint on the equalizer matrix, for fast convergence without excessive computational complexity. We model blind equalization with symbol-rate sampling as a blind source separation (BSS) problem and study two single-carrier transmission schemes, specifically block transmission with guard intervals and continuous transmission. Under either scheme, blind equalization can be achieved using independent component analysis (ICA) algorithms with a Toeplitz or circulant constraint on the structure of the separating matrix. We also develop relative gradient versions of the widely used Bussgang-type algorithms. Processing the equalizer outputs in sliding blocks, we are able to use the relative gradient for adaptation of the Toeplitz constrained equalizer matrix. The use of relative gradient makes the Bussgang condition appear explicitly in the matrix adaptation and speeds up convergence.
For the ICA-based and Bussgang-type algorithms with relative gradient and matrix structure constraints, we simplify the matrix adaptations to obtain equivalent equalizer vector adaptations for reduced computational cost. Efficient implementations with fast Fourier transform, and approximation schemes for the cross-correlation terms used in the adaptation, are shown to further reduce computational cost.
We also consider the use of a relative gradient algorithm for channel shortening in orthogonal frequency division multiplexing (OFDM) systems. The redundancy of the cyclic prefix symbols is used to shorten a channel with a long impulse response. We show interesting preliminary results for a shortening algorithm based on relative gradient
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