On the Maximal Cross-Correlation of Algebraically Constructed Costas Arrays

Drakakis, Konstantinos; Gow, Rod; Rickard, Scott; Sheekey, John; Taylor, Ken

doi:10.1109/tit.2011.2145890

Cited by 19 publications

(10 citation statements)

References 10 publications

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“…The absence of periodicity to cancel wrapped crossproduct sums and the lack of asymmetric sign-pairing of elements both conspire to drastically diminish the relative auto-correlation peaks of all perfect arrays. Costas arrays [34], [35] have aperiodic off-peak auto-correlations of 0 or 1, but they comprise mostly zero terms with just N Costas elements being 1 in an otherwise zero N × N array. Hunt and Ackroyd [15] The Methods section (Sec.…”

Section: A Quasi-huffman Arrays In One Dimensionmentioning

confidence: 99%

Sharp images from diffuse beams: factorisation of the discrete delta function

Svalbe,

Paganin,

Petersen

2019

Preprint

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Discrete delta functions define the limits of attainable spatial resolution for all imaging systems. Here we construct broad, multi-dimensional discrete functions that replicate closely the action of a Dirac delta function under aperiodic convolution. These arrays spread the energy of a sharp probe beam to simultaneously sample multiple points across the volume of a large object, without losing image sharpness. A diffuse pointspread function applied in any imaging system can reveal the underlying structure of objects less intrusively and with equal or better signal-to-noise ratio. These multi-dimensional arrays are related to previously known, but relatively rarely employed, onedimensional integer Huffman sequences. Practical point-spread functions can now be made sufficiently large to span the size of the object under measure. Such large arrays can be applied to ghost imaging, which has demonstrated potential to greatly improve signal-to-noise ratios and reduce the total dose required for tomographic imaging. The discrete arrays built here parallel the continuum self-adjoint or Hermitian functions that underpin wave theory and quantum mechanics.

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Section: A Quasi-huffman Arrays In One Dimensionmentioning

confidence: 99%

Sharp images from diffuse beams: factorisation of the discrete delta function

Svalbe,

Paganin,

Petersen

2019

Preprint

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“…Studying the maximal cross-correlation of a family of Costas permutations is not only a very interesting mathematical problem, since families with small maximal cross-correlation are of high practical importance, see [2] and references therein.…”

Section: Introductionmentioning

confidence: 99%

“…

We build on the work of Drakakis et al (2011) on the maximal crosscorrelation of the families of Welch and Golomb Costas permutations. In particular, we settle some of their conjectures.

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mentioning

confidence: 99%

A Note on the Cross-Correlation of Costas Permutations

Gómez-Pérez¹,

Winterhof²

2020

Preprint

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We build on the work of Drakakis et al. (2011) on the maximal crosscorrelation of the families of Welch and Golomb Costas permutations. In particular, we settle some of their conjectures. More precisely, we prove two results.First, for a prime p ≥ 5, the maximal cross-correlation of the family of the ϕ(p − 1) different Welch Costas permutations of {1, . . . , p − 1} is (p − 1)/t, where t is the smallest prime divisor of (p − 1)/2 if p is not a safe prime and at most 1 + p 1/2 otherwise. Here ϕ denotes Euler's totient function and a prime p is a safe prime if (p − 1)/2 is also prime.Second, for a prime power q ≥ 4 the maximal cross-correlation of a subfamily of Golomb Costas permutations of {1, . . . , q − 2} is (q − 1)/t − 1 if t is the smallest prime divisor of (q − 1)/2 if q is odd and of q − 1 if q is even provided that (q − 1)/2 and q − 1 are not prime, and at most 1 + q 1/2 otherwise. Note that we consider a smaller family than Drakakis et al. Our family is of size ϕ(q − 1) whereas there are ϕ(q − 1) 2 different Golomb Costas permutations. The maximal cross-correlation of the larger family given in the tables of Drakakis et al. is larger than our bound (for the smaller family) for some q.

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“…This is unusual. Among sparse arrays, Costas arrays have optimal autocorrelation, but there are no large families with optimal, or even good crosscorrelation [3]. Among dense arrays, m-sequences have perfect autocorrelation, but only small families, called maximal connected sets have good cross-correlation [4].…”

Section: Introductionmentioning

confidence: 99%

Extended families of 2D arrays with near optimal auto and low cross-correlation

Svalbe

Tirkel²

2017

EURASIP J. Adv. Signal Process.

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Families of 2D arrays can be constructed where each array has perfect autocorrelation, and the cross-correlation between any pair of family members is optimally low. We exploit equivalent Hadamard matrices to construct many families of p p × p arrays, where p is any 4k-1 prime. From these families, we assemble extended families of arrays with members that exhibit perfect autocorrelation and next-to-optimally low cross-correlation. Pseudo-Hadamard matrices are used to construct extended families using p = 4k + 1 primes. An optimal family of 31 31 × 31 perfect arrays can provide copyright protection to uniquely stamp a robust, low-visibility watermark within every frame of each second of high-definition, 30 fps video. The extended families permit the embedding of many more perfect watermarks that have next-to-minimal cross-correlations.

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On the Maximal Cross-Correlation of Algebraically Constructed Costas Arrays

Cited by 19 publications

References 10 publications

Sharp images from diffuse beams: factorisation of the discrete delta function

Sharp images from diffuse beams: factorisation of the discrete delta function

A Note on the Cross-Correlation of Costas Permutations

Extended families of 2D arrays with near optimal auto and low cross-correlation

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