857 research outputs found
Binary matrices of optimal autocorrelations as alignment marks
We define a new class of binary matrices by maximizing the peak-sidelobe
distances in the aperiodic autocorrelations. These matrices can be used as
robust position marks for in-plane spatial alignment. The optimal square
matrices of dimensions up to 7 by 7 and optimal diagonally-symmetric matrices
of 8 by 8 and 9 by 9 were found by exhaustive searches.Comment: 8 pages, 6 figures and 1 tabl
Polyomino convolutions and tiling problems
We define a convolution operation on the set of polyominoes and use it to
obtain a criterion for a given polyomino not to tile the plane (rotations and
translations allowed). We apply the criterion to several families of
polyominoes, and show that the criterion detects some cases that are not
detectable by generalized coloring arguments.Comment: 8 pages, 8 figures. To appear in \emph{J. of Combin. Theory Ser. A
Parity properties of Costas arrays defined via finite fields
A Costas array of order is an arrangement of dots and blanks into
rows and columns, with exactly one dot in each row and each column, the
arrangement satisfying certain specified conditions. A dot occurring in such an
array is even/even if it occurs in the -th row and -th column, where
and are both even integers, and there are similar definitions of odd/odd,
even/odd and odd/even dots. Two types of Costas arrays, known as Golomb-Costas
and Welch-Costas arrays, can be defined using finite fields. When is a
power of an odd prime, we enumerate the number of even/even odd/odd, even/odd
and odd/even dots in a Golomb-Costas array. We show that three of these numbers
are equal and they differ by from the fourth. For a Welch-Costas array
of order , where is an odd prime, the four numbers above are all equal
to when , but when , we show
that the four numbers are defined in terms of the class number of the imaginary
quadratic field , and thus behave in a much less
predictable manner.Comment: To appear in Advances in Mathematics of Communication
Compressive and Noncompressive Power Spectral Density Estimation from Periodic Nonuniform Samples
This paper presents a novel power spectral density estimation technique for
band-limited, wide-sense stationary signals from sub-Nyquist sampled data. The
technique employs multi-coset sampling and incorporates the advantages of
compressed sensing (CS) when the power spectrum is sparse, but applies to
sparse and nonsparse power spectra alike. The estimates are consistent
piecewise constant approximations whose resolutions (width of the piecewise
constant segments) are controlled by the periodicity of the multi-coset
sampling. We show that compressive estimates exhibit better tradeoffs among the
estimator's resolution, system complexity, and average sampling rate compared
to their noncompressive counterparts. For suitable sampling patterns,
noncompressive estimates are obtained as least squares solutions. Because of
the non-negativity of power spectra, compressive estimates can be computed by
seeking non-negative least squares solutions (provided appropriate sampling
patterns exist) instead of using standard CS recovery algorithms. This
flexibility suggests a reduction in computational overhead for systems
estimating both sparse and nonsparse power spectra because one algorithm can be
used to compute both compressive and noncompressive estimates.Comment: 26 pages, single spaced, 9 figure
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