57,391 research outputs found
Unit circle MVDR beamformer
The array polynomial is the z-transform of the array weights for a narrowband
planewave beamformer using a uniform linear array (ULA). Evaluating the array
polynomial on the unit circle in the complex plane yields the beampattern. The
locations of the polynomial zeros on the unit circle indicate the nulls of the
beampattern. For planewave signals measured with a ULA, the locations of the
ensemble MVDR polynomial zeros are constrained on the unit circle. However,
sample matrix inversion (SMI) MVDR polynomial zeros generally do not fall on
the unit circle. The proposed unit circle MVDR (UC MVDR) projects the zeros of
the SMI MVDR polynomial radially on the unit circle. This satisfies the
constraint on the zeros of ensemble MVDR polynomial. Numerical simulations show
that the UC MVDR beamformer suppresses interferers better than the SMI MVDR and
the diagonal loaded MVDR beamformer and also improves the white noise gain
(WNG).Comment: Accepted to ICASSP 201
Unit circle elliptic beta integrals
We present some elliptic beta integrals with a base parameter on the unit
circle, together with their basic degenerations.Comment: 15 pages; minor corrections, references updated, to appear in
Ramanujan
Polynomials with symmetric zeros
Polynomials whose zeros are symmetric either to the real line or to the unit
circle are very important in mathematics and physics. We can classify them into
three main classes: the self-conjugate polynomials, whose zeros are symmetric
to the real line; the self-inversive polynomials, whose zeros are symmetric to
the unit circle; and the self-reciprocal polynomials, whose zeros are symmetric
by an inversion with respect to the unit circle followed by a reflection in the
real line. Real self-reciprocal polynomials are simultaneously self-conjugate
and self-inversive so that their zeros are symmetric to both the real line and
the unit circle. In this survey, we present a short review of these
polynomials, focusing on the distribution of their zeros.Comment: Keywords: Self-inversive polynomials, self-reciprocal polynomials,
Pisot and Salem polynomials, M\"obius transformations, knot theory, Bethe
equation
Chebyshev constants for the unit circle
It is proven that for any system of n points z_1, ..., z_n on the (complex)
unit circle, there exists another point z of norm 1, such that
Equality holds iff the point system is a
rotated copy of the nth unit roots.
Two proofs are presented: one uses a characterisation of equioscillating
rational functions, while the other is based on Bernstein's inequality.Comment: 11 page
Boolean convolution of probability measures on the unit circle
We introduce the boolean convolution for probability measures on the unit
circle. Roughly speaking, it describes the distribution of the product of two
boolean independent unitary random variables. We find an analogue of the
characteristic function and determine all infinitely divisible probability
measures on the unit circle for the boolean convolution.Comment: 13 pages, to appear in volume 15 of Seminaires et Congre
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