Phase Space Approach to Signal Processing and Quantum Mechanical Calculations: Thinking Inside the Box
In 1946, Gabor proposed using a lattice of Gaussians in the time-frequency phase space to provide an intuitive and compact representation of arbitrary signals. The same idea had actually been discovered fifteen years earlier by his countryman von Neumann, in searching for a way to represent arbitrary quantum mechanical wavefunctions in the position-momentum phase space. Despite the great interest in this approach in both the signal processing and quantum mechanical communities, the method has never succeeded as hoped due to severe convergence problems. We have recently discovered a simple and surprising solution to the convergence problem, based on introducing periodic boundary conditions into the Gabor/von Neumann lattice. The resulting method provides a simple and compact representation of arbitrary signals and images, and opens the door to unprecedentedly large quantum calculations based on exploiting the underlying classical phase space structure. In the classical limit the method reaches the remarkable efficiency of 1 basis function per 1 eigenstate. We illustrate the method by calculating the vibrational eigenstates of a polyatomic with 104 bound states and by simulating attosecond electron dynamics in the presence of combined strong XUV and NIR laser fields. We also present examples of audio and image compression where we show that the method is competitive with or superior to state-of-the-art wavelet methods.
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