Frame Quantization Theory and Equiangular Tight Frames

Abstract

In this thesis, we first consider the finite frame quantization. We make a signalwise comparison of PCM and first order Sigma-Delta quantization. We show that Sigma-Delta quantization achieves smaller signal-wise quantization error bounds for a class of low amplitude signals. Then, we propose two new quantization methods for finite frames. First method is a variable bit-rate quantization algorithm. Given a finite signal and a predetermined error margin, this method calculates the number of bits necessary to quantize this signal within the pre-specified error margin. Second method is a 1-bit quantization technique that uses functional minimization methods. We first translate the combinatorial quantization problem into an analytic one. Then, we show that the solutions of this this analytic problem correspond to 1-bit quantized estimates of a given finite signal.

Second, we focus on finite equiangular tight frames. We show that equiangular tight frames are minimizers of certain functionals. We also give a characterization of equiangular tight frames with maximum possible redundancy.