CAZAC Sequences and Haagerup’s Characterization of Cyclic N-roots

Benedetto, John J.; Cordwell, Katherine; Magsino, Mark

doi:10.1007/978-3-030-32353-0_1

Cited by 9 publications

(3 citation statements)

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“…Cyclic n-roots can be used to show that the number of prime length CAZACs is finite. This was proved by Haagerup [23], but a summary, along with more results on CAZACs, can be found in [5]. Note that for both the Chu and P4 sequence, the DPAF is only nonzero on the diagonal m = n. We leverage this fact for several examples in Section 5, and the examples use the Chu and P4 sequences interchangably.…”

Section: Cazac Sequencesmentioning

confidence: 88%

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Constructing Tight Gabor Frames Using CAZAC Sequences

Magsino

2017

STSIP

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The construction of finite tight Gabor frames plays an important role in many applications. These applications include significant ones in signal and image processing. We explore when constant amplitude zero autocorrelation (CAZAC) sequences can be used to generate tight Gabor frames. The main theorem uses Janssen's representation and the zeros of the discrete periodic ambiguity function to give necessary and sufficient conditions for determining whether any Gabor frame is tight. The relevance of the theorem depends significantly on the construction of examples. These examples are necessarily intricate, and to a large extent, depend on CAZAC sequences. Finally, we present an alternative method for determining whether a Gabor system yields a tight frame. This alternative method does not prove tightness using the main theorem, but instead uses the Gram matrix of the Gabor system.

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Section: Cazac Sequencesmentioning

confidence: 88%

“…We give a brief exposition of the known results on CAZAC sequences, but a more detailed exposition on CAZAC sequences can be found in [5]. Although CAZAC sequences are defined in the introduction, for the sake of a self-contained exposition, we provide the definition once again.…”

Section: Cazac Sequencesmentioning

confidence: 99%

Constructing Tight Gabor Frames Using CAZAC Sequences

Magsino

2017

STSIP

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“…The discrete periodic ambiguity function A(u) of u is defined by the formula, A recent survey on the theory and applicability of CAZAC sequences is [9]. The construction of all CAZAC sequences remains a tantalizing and applicable venture.…”

Section: An Application Of Gleason Functionsmentioning

confidence: 99%

A Generalization of Gleason's Frame Function for Quantum Measurement

Benedetto¹,

Koprowski²,

Nolan³

2020

Preprint

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The goal is to extend Gleason's notion of a frame function, which is essential in his fundamental theorem in quantum measurement, to a more general function acting on 1-tight, socalled, Parseval frames. We refer to these functions as Gleason functions for Parseval frames. The reason for our generalization is that positive operator valued measures (POVMs) are essentially equivalent to Parseval frames, and that POVMs arise naturally in quantum measurement theory. We prove that under the proper assumptions, Gleason functions for Parseval frames are quadratic forms, as well as other results analogous to Gleason's original theorem. Further, we solve an intrinsic problem relating Gleason functions for Parseval frames of different lengths. We use this solution to weaken the hypotheses in the finite dimensional version of Busch's theorem, that itself is an analog of Gleason's mathematical characterization of quantum states.

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