Frame Multiplication Theory and a Vector-Valued DFT and Ambiguity Function

Abstract

Vector-valued discrete Fourier transforms (DFTs) and ambiguity functions are defined. The motivation for the definitions is to provide realistic modeling of multi-sensor environments in which a useful time-frequency analysis is essential. The definition of the DFT requires associated uncertainty principle inequalities. The definition of the ambiguity function requires a component that leads to formulating a mathematical theory in which two essential algebraic operations can be made compatible in a natural way. The theory is referred to as frame multiplication theory. These definitions, inequalities, and theory are interdependent, and they are the content of the paper with the centerpiece being frame multiplication theory.The technology underlying frame multiplication theory is the theory of frames, short time Fourier transforms (STFTs), and the representation theory of finite groups. The main results have the following form: frame multiplication exists if and only if the finite frames that arise in the theory are of a certain type, e.g., harmonic frames, or, more generally, group frames.In light of the complexities and the importance of the modeling of time-varying and dynamical systems in the context of effectively analyzing vector-valued multi-sensor environments, the theory of vector-valued DFTs and ambiguity functions must not only be mathematically meaningful, but it must have constructive implementable algorithms, and be computationally viable. This paper presents our vision for resolving these issues, in terms of a significant mathematical theory, and based on the goal of formulating and developing a useful vector-valued theory. FRAME MULTIPLICATION THEORY AND A VECTOR-VALUED DFT AND AMBIGUITY FUNCTION 3 d. The short-time Fourier transform (STFT) of v with respect to a window function wfor a definitive mathematical treatment. Thus, we think of the window w as centered at t, and we havee. A(v, w) and V w (v) can clearly be defined for functions v, w on R d and for other function spaces besides L 2 (R d ). The quantity |V w (v)| is the spectrogram of v, that is so important in power spectrum analysis, see, e.g., [89], [17], [68], [70], [24], [65], [83].Our goals are the following.• Ultimately, we shall establish the theory of vector-valued ambiguity functions of vector-valued functions v on R d in terms of their discrete periodic counterparts on Z/N Z. • To this end, in this paper, we define vector-valued DFTs and discrete periodic vectorvalued ambiguity functions on Z/N Z in a natural way. The STFT is the guide and the theory of frames, especially the theory of DFT, harmonic, and group frames, is the framework (sic) to formulate these goals. The underlying technology that allows us to obtain these goals is frame multiplication theory.1.3. Outline. We begin with an extended exposition on the theory of frames (Section 2). The reason is that frames are essential for our results, and our results are sometimes not conceived in terms of frames. As such, it made sense to add sufficient background material.The vect...

A Generalization of Gleason's Frame Function for Quantum Measurement

“…c. The fact that there are equi-normed Parseval frames for H having N > d elements fits into the theory of harmonic frames [83], [4], which itself is part of the group frame theory mentioned in Definition 3.1. For an explicit calculation to show the existence of equi-normed Parseval frames for C d , consider the N × N DFT matrix, let c > 0, and let s : {1, .…”

“…On the other hand, the inequality (24) gives perspective with regard to Zauner's conjecture. Further, these CAZAC sequences are an essential component of the background and goals dealing with phase-coded waveforms that were the driving force leading to the role of group frames in the vector-valued theory of [4].…”

“…4.2 (Generalizations of Theorem 4.1). a. Theorem 4.1 is true for normal operators over C, since the spectral theorem is true in that setting, e.g., see[42], page 377.…”

“…On the other hand, we can formulate the definition of a Gleason function to consider the class of all equiangular Parseval frames, thereby interleaving the power of Gleason's theorem with fundamental problems of equiangularity as they relate to the Welch bound and optimal ambiguity function behavior, see Definition 3.1 c and Appendix A. This is inextricably related to the construction of constant amplitude finite sequences with 0-autocorrelation, whose narrow-band ambiguity function is comparable to the Welch bound, e.g., see [8] and [4]. 1.3.…”

Reactive Sensing and Multiplicative Frame Super-resolution

“…As our simulations have shown, this is a challenging task. • Develop multiplicative frames as a full-fledged mathematical theory; and especially analyze its relation with frame multiplication theory and group frames, see [2].…”