90 research outputs found
Numerically erasure-robust frames
Given a channel with additive noise and adversarial erasures, the task is to
design a frame that allows for stable signal reconstruction from transmitted
frame coefficients. To meet these specifications, we introduce numerically
erasure-robust frames. We first consider a variety of constructions, including
random frames, equiangular tight frames and group frames. Later, we show that
arbitrarily large erasure rates necessarily induce numerical instability in
signal reconstruction. We conclude with a few observations, including some
implications for maximal equiangular tight frames and sparse frames.Comment: 15 page
Harmonic equiangular tight frames comprised of regular simplices
An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a
Euclidean space whose coherence achieves equality in the Welch bound, and thus
yields an optimal packing in a projective space. A regular simplex is a simple
type of ETF in which the number of vectors is one more than the dimension of
the underlying space. More sophisticated examples include harmonic ETFs which
equate to difference sets in finite abelian groups. Recently, it was shown that
some harmonic ETFs are comprised of regular simplices. In this paper, we
continue the investigation into these special harmonic ETFs. We begin by
characterizing when the subspaces that are spanned by the ETF's regular
simplices form an equi-isoclinic tight fusion frame (EITFF), which is a type of
optimal packing in a Grassmannian space. We shall see that every difference set
that produces an EITFF in this way also yields a complex circulant conference
matrix. Next, we consider a subclass of these difference sets that can be
factored in terms of a smaller difference set and a relative difference set. It
turns out that these relative difference sets lend themselves to a second,
related and yet distinct, construction of complex circulant conference
matrices. Finally, we provide explicit infinite families of ETFs to which this
theory applies
Polyphase equiangular tight frames
An equiangular tight frame (ETF) is a type of optimal packing of lines in a finite-dimensional Hilbert space. ETFs arise in various applications, such as waveform design for wireless communication, compressed sensing, quantum information theory and algebraic coding theory. In a recent paper, signature matrices of ETFs were constructed from abelian distance regular covers of complete graphs. We extend this work, constructing a new infinite family of complex ETFs. Our approach involves designing matrices whose entries are polynomials over a finite abelian group, namely polyphase matrices of finite filter banks
Filter Bank Fusion Frames
In this paper we characterize and construct novel oversampled filter banks
implementing fusion frames. A fusion frame is a sequence of orthogonal
projection operators whose sum can be inverted in a numerically stable way.
When properly designed, fusion frames can provide redundant encodings of
signals which are optimally robust against certain types of noise and erasures.
However, up to this point, few implementable constructions of such frames were
known; we show how to construct them using oversampled filter banks. In this
work, we first provide polyphase domain characterizations of filter bank fusion
frames. We then use these characterizations to construct filter bank fusion
frame versions of discrete wavelet and Gabor transforms, emphasizing those
specific finite impulse response filters whose frequency responses are
well-behaved.Comment: keywords: filter banks, frames, tight, fusion, erasures, polyphas
Frame completions for optimally robust reconstruction
In information fusion, one is often confronted with the following problem:
given a preexisting set of measurements about an unknown quantity, what new
measurements should one collect in order to accomplish a given fusion task with
optimal accuracy and efficiency. We illustrate just how difficult this problem
can become by considering one of its more simple forms: when the unknown
quantity is a vector in a Hilbert space, the task itself is vector
reconstruction, and the measurements are linear functionals, that is, inner
products of the unknown vector with given measurement vectors. Such
reconstruction problems are the subject of frame theory. Here, we can measure
the quality of a given frame by the average reconstruction error induced by
noisy measurements; the mean square error is known to be the trace of the
inverse of the frame operator. We discuss preliminary results which help
indicate how to add new vectors to a given frame in order to reduce this mean
square error as much as possible
Equiangular tight frames from group divisible designs
An equiangular tight frame (ETF) is a type of optimal packing of lines in a
real or complex Hilbert space. In the complex case, the existence of an ETF of
a given size remains an open problem in many cases. In this paper, we observe
that many of the known constructions of ETFs are of one of two types. We
further provide a new method for combining a given ETF of one of these two
types with an appropriate group divisible design (GDD) in order to produce a
larger ETF of the same type. By applying this method to known families of ETFs
and GDDs, we obtain several new infinite families of ETFs. The real instances
of these ETFs correspond to several new infinite families of strongly regular
graphs. Our approach was inspired by a seminal paper of Davis and Jedwab which
both unified and generalized McFarland and Spence difference sets. We provide
combinatorial analogs of their algebraic results, unifying Steiner ETFs with
hyperoval ETFs and Tremain ETFs
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