## 3 Explicit Calculation of Phase Space Representations of Arbitrary Digital Holograms
### Selecting the Right Phase Space Representation  
There exist many PSRs, which is explained by the fact that it is impossible to craft one representation that can precisely analyze instantaneous frequencies at every point in space due to the Heisenberg-Pauli-Weyl uncertainty principle [39]. Henceforth, there are various trade-offs to be made with respect to properties that shall be retained, such as: visual interpretability, positivity, energy conservation, covariance with space and frequency, invertibility, degree of adaptivity, and many more. We will restrict ourselves to [PERSON]'s class [40], which is the general set of bilinear PSRs that are invariant under space-frequency translations and encloses all off the most frequently used PSRs.  
Most of the references cited earlier on, chose the Wigner-Ville (WV) PSR, because it offers a variety of nice mathematical properties such as invertibility, compatibility with linear filtering and modulations, conservation of scalar product and instantaneous frequency (in terms of marginals), and invertibility. Put differently, it lends itself naturally to many mathematical derivations and offers perfect localization (\"best resolution\") for any signal described by a single, straight line in phase space; such as: pure sinusoids, impulses, or linear frequency chirps. However, visual interpretation becomes unfeasible for signals described in phase space by either multiple lines and/or arbitrary curves because of the introduction of interference terms and their specific placement.  
This is illustrated by two examples shown in Figure 4 where several PSRs are evaluated on the central horizontal cross-section of 2D holograms. The holograms consists of PSFs from point-sources placed at different positions and recorded in the Fresnel approximation using 2 parabolic wavefronts, Equation (6), on top and in the close-proximity near field using 1 spherical wave, Equation (3), below. In Figure 4a the WV PSR is shown to exhibit an interference pattern stronger in intensity than the signals themselves and located in between them. Self-interference in the vicinity of non-linear sections of non-linear signals can be observed well in Figure 4d by the broadening along the curved sections of the PSF.  
Another common PSR is the spectrogram. As can be seen in the middle column of Figure 4, it does not introduce new false signals; however, it has lower resolution. Although its resolution can be increased by means of high-resolution STFTs [11], it will always remain inferior to the comparatively recent S-method [12] depicted in the right column of Figure 4. A similar argument was already made for the superior resolution of the smoothed pseudo Wigner-Ville representation over the spectrogram in [41]. The smoothed pseudo Wigner-Ville representation is a special case of the S-method.  
In the following, we will introduce first the spectrogram and then the S-method, because the efficient computation of the more general S-method is based on the spectrogram. Section 3.4 and Section 3.5 thereafter will explain how to apply the PSRs to DHs and list the advantages and disadvantages of both methods essential for a visual interpretation of holograms.