Time-Frequency-Scale Transforms

Abstract

In this thesis, we will recall some familiar definitions regarding the Continuous Wavelet Transform and the reconstruction of signals. We shall talk about exploring how the transform evolves as we go from continuous dilation and translation parameter to the semi-discrete scenario, and finally to the fully discrete world of frames and frame operators.

From here, we then evaluate how we can construct frames generated by mother wavelets through modifying a result of Ole Christensen. We shall discover how to include families of modulated wavelets in the arguments to come up with new ways of constructing frames. Interestingly, we can also derive certain well-known results already in literature about Gabor Analysis by choosing the choice of our family of functions wisely.