Redundancy is a common tool in our daily lives. Before we leave the house, we double- and triple-check 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]. Frame-like ideas, that is, building redundancy into a signal expansion, can be found in pyramid

This paper shows what are called Welch bound equality (WBE) sequences by the signal processing community are precisely the isometric/equal norm/normalized/uniform tight frames which are currently being investigated for a number of applications, and in the real case are the spherical 2–designs of combinatorics. Recent applications include wavelet expansions, Grassmannian frames, frames robust to erasures, and quantum measurements. This is done by giving an elementary proof of a generalisation of Welch’s inequality to vectors which need not have equal energy, and then showing that equality occurs in this exactly when the vectors form a tight frame.

The aim of this paper is to investigate symmetry properties of tight frames, with a view to constructing tight frames of orthogonal polynomials in several variables which share the symmetries of the weight function, and other similar applications. This is achieved by using representation theory to give methods for constructing tight frames as orbits of groups of unitary transformations acting on a given finite-dimensional Hilbert space. Along the way, we show that a tight frame is determined by its Gram matrix and discuss how the symmetries of a tight frame are related to its Gram matrix. We also give a complete classification of those tight frames which arise as orbits of an abelian group of symmetries.

Baraniuk 1. Characterization of the scaling and wavelet functions Consider the vector m of N sensor measurements given by m = [m1, m2,..., mN] The sensors are irregularly spaced on a 2D grid. We assume that the sensors sample a slowly varying field. To reconstruct the original field, we interpolate the sensor values over R 2 in the following manner. The approximate value of the field at any arbitrary location is equal to the value recorded by the sensor closest to it. Thus, the function space F spanned by the reconstructed field is the space of piecewise constants whose discontinuities lie on voronoi cell boundaries. Our goal is to first find a suitable set of functions that span the function space F. Motivated by the Haar wavelet transform, we pick one function that captures the average value of the sensor field, and N funtions that capture the deviation of each sensor value from the average. Since we use (N + 1) functions that span the N dimensional function space F, we refer to the functions as frame functions. Example of such frame functions in 1 − D are shown in Figure 1. The scaling function is constant over the entire region. The wavelet functions have discontinuities only at the boundaries of one voronoi cell. The wavelet functions have zero mean. Thus, the dot product of a wavelet function and the scaling function is zero. We need to pick the parameters of the frame functions (namely the heights of the wavelet functions) appropriately. We choose these parameters such that the set of frame functions form a Parseval tight frame [1]. For Parseval tight frames, we can establish bounds on the energy of the error when wavelet coefficients are thresholded. Let the support of the voronoi regions be ∆i, for i = 1, 2,..., N. Let ∆tot = � N i=1 ∆i. Define the scaling function as The i th wavelet function, Wi, takes the value ki for all the voronoi cells except at the i th cell, where it takes value k ′ i. Since the average value of the wavelet functions are zero, we have k ′ i ∆i + ki(∆tot − ∆i) = 0. Let ∆ be the N × N diagonal matrix of the areas ∆i, i = 1...N. To satisfy the Parseval tight frame conditions, we look at the energy of the signal in the signal and wavelet domains. The energy of the signal is given by

For radio systems there are two resources, frequency and time. Division by frequency (or time), so that each pair of communicators is allocated part of the spectrum for all of the time (or all of the spectrum for part of the time), results in Frequency (or Time) Division Multiple Access (FDMA or TDMA). In Code Division Multiple Access (CDMA), every communicator will be allocated the entire spectrum all of the time (Fig. 1). CDMA has applications in wireless cellular communication as well as navigation (e.g., GPS) systems. CDMA uses codes (here referred to as signature sequences) to identify connections and is an interference limited multiple access system. Because all users transmit on the same frequency, internal interference generated by the system is the most significant factor in determining system capacity and communication quality (e.g., quality of voice in mobile phones). In what follows, we will review frame-theoretic results associated with the problem of designing optimal, i.e. minimum interference, signature sequences in CDMA communication systems. (a) (b) Fig. 1. (a) Demonstration of code-division multiplexing (b) Transmitter structure of a CDMA system.

Abstract not found