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34
Life Beyond Bases: The Advent of Frames (Part I)
, 2007
"... Redundancy is a common tool in our daily lives. Before we leave the house, we double and triplecheck that we turned off gas and lights, took our keys, and have money (at least those worrywarts among us do). When an important date is coming up, we drive our loved ones crazy by confirming “just onc ..."
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Cited by 72 (8 self)
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Redundancy is a common tool in our daily lives. Before we leave the house, we double and triplecheck that we turned off gas and lights, took our keys, and have money (at least those worrywarts among us do). When an important date is coming up, we drive our loved ones crazy by confirming “just once more” they are on top of it. Of course, the reason we are doing that is to avoid a disaster by missing or forgetting something, not to drive our loved ones crazy. The same idea of removing doubt is present in signal representations. Given a signal, we represent it in another system, typically a basis, where its characteristics are more readily apparent in the transform coefficients. However, these representations are typically nonredundant, and thus corruption or loss of transform coefficients can be serious. In comes redundancy; we build a safety net into our representation so that we can avoid those disasters. The redundant counterpart of a basis is called a frame [no one seems to know why they are called frames, perhaps because of the bounds in (25)?]. It is generally acknowledged (at least in the signal processing and harmonic analysis communities) that frames were born in 1952 in the paper by Duffin and Schaeffer [32]. Despite being over half a century old, frames gained popularity only in the last decade, due mostly to the work of the three wavelet pioneers—Daubechies, Grossman, and Meyer [29]. Framelike ideas, that is, building redundancy into a signal expansion, can be found in pyramid
Mutually Unbiased Bases are Complex Projective 2Designs
 PROC. 2005 IEEE INTERNATIONAL SYMPOSIUM ON INFORMATION THEORY
, 2005
"... Mutually unbiased bases (MUBs) are a primitive used in quantum information processing to capture the principle of complementarity. While constructions of maximal sets of d+1 such bases are known for system of prime power dimension d, it is unknown whether this bound can be achieved for any nonpri ..."
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Cited by 31 (1 self)
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Mutually unbiased bases (MUBs) are a primitive used in quantum information processing to capture the principle of complementarity. While constructions of maximal sets of d+1 such bases are known for system of prime power dimension d, it is unknown whether this bound can be achieved for any nonprime power dimension. In this paper we demonstrate that maximal sets of MUBs come with a rich combinatorial structure by showing that they actually are the same objects as the complex projective 2designs with angle set {0, 1/d}. We also give a new and simple proof that symmetric informationally complete POVMs are complex projective 2designs with angle set {1/(d+1)}.
A physical interpretation of tight frames
 In: Harmonic Analysis and Applications, Applied and Numerical Harmonic Analysis
, 2006
"... Summary. We find finite tight frames when the lengths of the frame elements are predetermined. In particular, we derive a “fundamental inequality ” which completely characterizes those sequences which arise as the lengths of a tight frame’s elements. Furthermore, using concepts from classical physic ..."
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Cited by 18 (6 self)
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Summary. We find finite tight frames when the lengths of the frame elements are predetermined. In particular, we derive a “fundamental inequality ” which completely characterizes those sequences which arise as the lengths of a tight frame’s elements. Furthermore, using concepts from classical physics, we show that this characterization has an intuitive physical interpretation. 1
Minimization of convex functionals over frame operators
 Adv. Comput. Math
"... Abstract. We present results about minimization of convex functionals defined over a finite set of vectors in a finite dimensional Hilbert space, that extend several known results for the BenedettoFickus frame potential. Our approach depends on majorization techniques. We also consider some perturb ..."
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Cited by 15 (9 self)
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Abstract. We present results about minimization of convex functionals defined over a finite set of vectors in a finite dimensional Hilbert space, that extend several known results for the BenedettoFickus frame potential. Our approach depends on majorization techniques. We also consider some perturbation problems, where a positive perturbation of the frame operator of a set of vectors is realized as the frame operator of a set of vectors which is close to the original one. 1.
Life Beyond Bases: The Advent of Frames
"... Redundancy is a common tool in our daily lives. We doubleand triplecheck that we turned off gas and lights, took our keys, money, etc. (at least those worrywarts among us do). When an important date is coming up, we drive our loved ones crazy by confirming âjust once more â they are on top of i ..."
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Cited by 15 (1 self)
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Redundancy is a common tool in our daily lives. We doubleand triplecheck that we turned off gas and lights, took our keys, money, etc. (at least those worrywarts among us do). When an important date is coming up, we drive our loved ones crazy by confirming âjust once more â they are on top of it. Of course, the reason we are doing that is to avoid a disaster by missing or forgetting something, not to drive our loved ones crazy. The same idea of removing doubt is present in signal representations. Given a signal, we represent it in another system, typically a basis, where its characteristics are more readily apparent in the transform coefficients (for example, waveletbased compression). However, these representations are typically nonredundant, and thus corruption or loss of transform coefficients can be fatal. In comes redundancy; we build a safety net into our representation so that we can avoid those fatal disasters. The redundant counterpart of a basis is called a frame (no one seems to know why they are called frames, perhaps because they are bounded from two sides as in (15)?). It is generally acknowledged 1 that frames were born in 1952 in the paper by Duffin and Schaeffer [57]. Despite being over half a century old, frames gained popularity only in the last decade, due mostly to the work of the three wavelet pioneersâ
Second order sigmadelta (Σ∆) quantization of finite frame expansions
 Applied and Computational Harmonic Analysis
, 2004
"... The second order SigmaDelta (Σ∆) scheme with linear quantization rule is analyzed for quantizing finite unitnorm tight frame expansions for R d. Approximation error estimates are derived, and it is shown that for certain choices of frames the quantization error is of order 1/N 2, where N is the fr ..."
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Cited by 7 (5 self)
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The second order SigmaDelta (Σ∆) scheme with linear quantization rule is analyzed for quantizing finite unitnorm tight frame expansions for R d. Approximation error estimates are derived, and it is shown that for certain choices of frames the quantization error is of order 1/N 2, where N is the frame size. However, in contrast to the setting of bandlimited functions there are many situations where the second order scheme only gives approximation error of order 1/N. For example, this is the case when quantizing harmonic frames of odd length in even dimensions. An important component of the error analysis involves extending existing stability results to yield smaller invariant sets for the linear second order Σ ∆ scheme.
A Physical Interpretation for Finite Tight Frames
, 2003
"... Though finite tight frames arise in many applications, they have often proved difficult to understand and construct. We investigate the nonlinear problem of finding a tight frame for which the lengths of the frame elements have been prescribed in advance. Borrowing several ideas from Classical Mecha ..."
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Cited by 6 (0 self)
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Though finite tight frames arise in many applications, they have often proved difficult to understand and construct. We investigate the nonlinear problem of finding a tight frame for which the lengths of the frame elements have been prescribed in advance. Borrowing several ideas from Classical Mechanics, we show that this problem has a natural, intuitive interpretation. In particular, we show that such frames may be characterized as the minimizers of a potential energy function, and justify their interpretation as “maximally orthogonal” sequences. By exploiting this idea, we are able toshowthat such frames always exist, provided the requisite lengths satisfy a “fundamental inequality.” In so doing, we characterize those sequences of nonnegative numbers which arise as the lengths of a tight frame’s elements.
Prime tight frames
 Adv. Comput. Math
"... Abstract: We introduce a class of finite tight frames called prime tight frames and prove some of their elementary properties. We show that any finite tight frame can be written as a union of prime tight frames. We then characterize all prime harmonic tight frames as well as all prime frames constr ..."
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Cited by 5 (1 self)
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Abstract: We introduce a class of finite tight frames called prime tight frames and prove some of their elementary properties. We show that any finite tight frame can be written as a union of prime tight frames. We then characterize all prime harmonic tight frames as well as all prime frames constructed from the spectral tetris method. As a byproduct of this last result, we obtain a characterization of when the spectral tetris construction works for redundancies below two.
Some remarks on Heisenberg frames and sets of equiangular lines
, 2006
"... We consider the long standing problem of constructing d 2 equiangular lines in C d, i.e., finding a set of d 2 unit vectors (φj) in C d with 〈φj, φk〉  = 1 √ d + 1, j � = k. Such ‘equally spaced configurations ’ have appeared in various guises, e.g., as complex spherical 2–designs, equiangular tigh ..."
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Cited by 5 (0 self)
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We consider the long standing problem of constructing d 2 equiangular lines in C d, i.e., finding a set of d 2 unit vectors (φj) in C d with 〈φj, φk〉  = 1 √ d + 1, j � = k. Such ‘equally spaced configurations ’ have appeared in various guises, e.g., as complex spherical 2–designs, equiangular tight frames, isometric embeddings ℓ2(d) → ℓ4(d 2), and most recently as SICPOVMs in quantum measurement theory. Analytic solutions are known only for d = 2, 3, 4, 8. Recently, numerical solutions which are the orbit of a discrete Heisenberg group H have been constructed for d ≤ 45. We call these Heisenberg frames. In this paper we study the normaliser of H, which we view as a group of symmetries of the equations that determine a Heisenberg frame. This allows us to simplify the equations for a Heisenberg frame. From these simplified equations we are able construct analytic solutions for d = 5, 7, and make conjectures about the form of a solution when d is odd. Most notably, it appears that solutions for d odd are eigenvectors of some element in the normaliser which has (scalar) order 3. Key Words: complex spherical 2–design, equiangular lines, equiangular tight frame, Grassmannian frame, Heisenberg frame, isometric embeddings, discrete Heisenberg group
The size of optimal sequence sets for synchronous CDMA systems
 IEEE TRANSACTIONS ON INFORM. THEORY
, 2006
"... The sum capacity on a symbolsynchronous CDMA system having processing gain N and supporting K power constrained users is achieved by employing at most 2N − 1 sequences. Analogously, the minimum received power (energyperchip) on the symbolsynchronous CDMA system supporting K users that demand spe ..."
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Cited by 2 (2 self)
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The sum capacity on a symbolsynchronous CDMA system having processing gain N and supporting K power constrained users is achieved by employing at most 2N − 1 sequences. Analogously, the minimum received power (energyperchip) on the symbolsynchronous CDMA system supporting K users that demand specified data rates is attained by employing at most 2N −1 sequences. If there are L oversized users in the system, at most 2N −L−1 sequences are needed. 2N −1 is the minimum number of sequences needed to guarantee optimal allocation for single dimensional signaling. N orthogonal sequences are sufficient if a few users (at most N − 1) are allowed to signal in multiple dimensions. If there are no oversized users, these split users need to signal only in two dimensions each. The above results are shown by proving a converse to a wellknown result of Weyl on the interlacing eigenvalues of the sum of two Hermitian matrices, one of which is of rank 1. The converse is analogous to Mirsky’s converse to the interlacing eigenvalues theorem for bordering matrices.