![gaussian window functiin in inmr gaussian window functiin in inmr](https://paco-cpu.github.io/paco-cpu/images/gauss_filter_grid.png)
The Gabor representation suffers from another inherent difficulty when applied to wideband field representations. A reasonable solution is found at an oversampling of order 4/3 or larger for 1D problems ((4/3) 2 for 2D). This poses a tradeoff between the oversampling ratio and the stability of the representation. The overcomplete nature of this representation smoothes out and localizes the dual function, ending up with stable and local coefficients at the expense of having to calculate more coefficients and trace more beam propagators. This difficulty has been circumvented recently by using a frame-based beam summation representation, which is considered in section 3. A major difficulty in these formulations is the nonlocality and instability of the expansion coefficients that follow from the highly irregular and distributed form of the “analysis function” (the Gabor dual or bi-orthogonal function) which is used to calculate the coefficients (see section 2.3). One example is the well known Gabor-based beam algorithms that have been utilized in various applications involving radiation and scattering in complex environments or for local inverse scattering. An important feature of these formulations is that they may be a priori discretized on a phase-space lattice. The beam representations are based on windowed configuration-spectrum transforms of the source distributions, e.g., the local Fourier transform in the frequency domain and the local-slant-stack-Radon transform in the time domain. Thus the beam representations combine the algorithmic ease of geometrical optics with the uniform features of spectral representations and, therefore, have been used recently in various applications. The advantages of the beam formulations over the more traditional distributed (Maslov-based) phase-space formulations are (1) the phase-space data is a priori localized in the vicinity of the Lagrange manifold that forms the phase-space skeleton of geometrical optics, (2) further localization is due to the fact that only those beam propagators that pass near the observation point actually contribute there, and (3) the beam propagators can be tracked locally in inhomogeneous media or through interactions with interfaces and, unlike ray or plane wave propagators, they are insensitive to transition zones. Of particular interest are beam-based phase-space formulations, where the source field is expanded as a spectrum of beam propagators.
![gaussian window functiin in inmr gaussian window functiin in inmr](http://4.bp.blogspot.com/-7y7G_hLj0sE/U1rfjl_JM1I/AAAAAAAABEU/IYt3iRKrM-A/s1600/Gaussian+Kernel.jpg)
Phase-space representations are a fundamental tool in wave theory as they provide a framework for the construction of spectrally uniform solutions in configurations of increasing complexity.