Leveraging redundant information in wavelet transforms to measure plasma dispersion processes otherwise unresolved by Fourier analysis

On its inaugural uses, the Fourier transform yields profound insight into stationary wave data, but including transient evolution on an additional, larger, scale presents deficiencies. The usual practice is to window the data, leading to the Short-Term Fourier Transform. This amounts to convolving the signal with the product of complex exponentials and a static windowing function.

Rather than an ad-hoc repetition of a simple transform to study a more complicated scenario, one can gain more rigorous understanding by furthering the theory within the utilized transform. In wavelet analysis [1], the window and fluctuating part of the convolution filter are mixed: the window size corresponds to the wavelength -- together, a scale. This allows them to be moved in unison, creating a unique convolution at every point in the original signal, preserving its resolution, and adding a dimension of scale. This yields redundant information, part of which is imprinting the wavelet oscillations onto the original dimension of the decomposed signal. When one performs multiple (independent) e.g. spatial transformations at varying instances in time, one can clearly track the imprinted wave crests, measuring their k-dependant phase and group velocities. Further decomposing the (now complex) signal in time separates the motion of the analytic and anti-analytic portions of these waves, conditionally distinguishing waves travelling in opposing directions. Due to the ubiquity of disperse waves in plasma physics, the potential applications for this technique are profound.