Redundancy of finite frames with a discussion on Gabor frames = /

Frames represent an important theoretical concept for signal processing. A frame is a collection of vectors satisfying the frame inequality, i.e. a relaxed form of Parseval’s identity for which the vectors involved do not need to form an orthonormal basis. Frames are used in order to provide a redundant representation of a vector (signal) in terms of coefficients associated to the frame’s elements. The goal of this thesis is to quantify redundancy for frames. Therefore we present a combinatorial- and an analytical redundancy measure. Both yield information on the maximal number of spanning sets and the minimal number of linearly independent sets one can partition the frame into, though in general the two measures are not equivalent. For the combinatorial measure we show a proof of the Rado-Horn theorem and two results basing upon it. We then examine a well-known example: the Fourier frame. Instead the analytical measure is characterized in terms of properties of the redundancy function. It is known that these two measures coincide for an equal norm Parseval frame, but when relaxing equal norm, this does not hold anymore. Exploiting the properties of the redundancy function, we show that the two redundancy measures still coincide if we consider Parseval frames with some additional property. Finally, we examine Gabor frames. The frame elements are modulated translates of a window function. We characterize the frame both in general terms and with respect to its redundancy.