The construction of some Riesz basis families and their application to coefficient quantization, sampling theory, and wavelet analysis

We develop construction techniques in the context of Riesz bases which facilitate the conception of a new family of multi-channel, multi-sampling rate theorems. This family takes advantage of the dilation property of the Fourier transform. In each channel a simple transform is taken which merely involves the addition and subtraction of subsets of the given samples. Each channel is then multiplied by a scalar and added together to construct the Fourier transform. This process can be characterised as the parallel decomposition of the Fourier transform into arrays of step, or binary, transforms. We then generalize this construction to include realizations of other Riesz basis families. Each family is seen to quantize uniquely an orthonormal basis by using different constructions. The dynamic range, or resolution, required to implement the basis coefficients depends upon the relative divisibility of the range of the basis. A measure which evaluates the required resolution is introduced. This leads to the construction of the uniform step transform which minimizes the dynamic range of the basis coefficients. This results in a coefficient quantization strategy which offers an optimal balance between programme memory, speed, and efficacy. We also show how the constructions are used to develop a novel method which refines wavelet bases. The refinement method modifies an existing wavelet such that the efficacy of the wavelet filter characteristic is increased. The ratios of main lobe and side lobe power and of maximal main lobe and side lobe magnitude of a wavelet filter characteristic are used as measures of efficacy. We extend these measures and propose the localized ratio of main lobe and side lobe power and the globalized ratio of maxima main lobe and side lobe magnitude. When these generalized measures are taken it is again