## 3 Explicit Calculation of Phase Space Representations of Arbitrary Digital Holograms
### Important Properties for Phase Space Visualization of Digital Holograms  
The spectrogram and the S-method are both visually faithful representations in that they share two mathematical properties which are essential for an accurate visual interpretation. First, as members of [PERSON]'s class they transform linear under space, frequency shifts of the signal and, second, they largely avoid introducing artificial signal components. Thus, if a signal component is visible in either PSR, it was actually present in the signal at these phase space coordinates. The first property is captured by the notion of covariance under space and frequency translations. That is, if a signal \(g\) is shifted in space or frequency, the PSR \(\rho\) shifts by the same amount:  
\[ \begin{split}\forall\Delta\xi,\Delta f_{\xi}\in\mathbb{R}:& h(\xi)=g(\xi-\Delta\xi)e^{2\pi i\Delta f_{\xi}}\to\\ &\rho(h;\xi,f_{\xi})=\rho(g;\xi-\Delta\xi,f_{\xi}-\Delta f_{\xi} )\,.\end{split}  \tag{15}\]Covariance is essential to locate a holographic signal in phase space, see for example Figure 2. For near-field, non-Fourier hologram lateral translations of the object are reflected by lateral translations in the phase space of the object wave. Frequency shifts of the recorded hologram occur, for example when tilting a plane wave illumination. We will show explicit examples in Section 4 and Section 5.5. Artificial signal components introduced through \"cross-talk\" of multiple signal components in the original signal are a mathematical necessity and their position and shape has been thoroughly studied in the past. It is understood for the entire Cohen class [40], and can render PSRs useless for the visual interpretation of many component signals such as DHs. Thus, it is worth understanding how they are handled in the spectrogram and the S-method, as to still allow for a clear visual interpretation. The spectrogram hides cross terms by locating the \"cross-talk\" on top of the actual signal. One speaks of inner interference terms in this case. Inner interferences lead to a broadening of the signal in phase space, perceivable as low resolution. The S-method follows a different strategy. Implicitly it introduces outer interferences, which are located next to the original signal, but they are filtered out via the frequency filter window. This guarantees a high-definition of the signal components itself, which is comparable to that of the WF PSR. We quote [11], p. 647: \"The S-method can produce a representation of a multi-component signal such that the distribution of each component is its [pseudo] Wigner distribution, avoiding cross-terms, if the STFTs of the components do not overlap in time-frequency plane.\" (Note: time-frequency plane = phase space). As the frequency filtering is imperfect, weak cross-terms can occur when signals overlap in phase space or are in very close proximity. In either case they do not impair visual interpretation and the visual degradation is trumped by far by the increased phase space resolution compared to the spectrogram. In Figure 6 both PSRs were computed with equal parameters (\(N=8192\), \(M=1024\), overlap length \(M-1\), \(L=5\)) and the spatial separation of the dices is better visible with the S-method. Note, because the Fourier transform of a Hann window is given as a sum of 3 delta distributions, for \(M>3\) its frequency resolution is better than its spatial resolution. Thus, a long spatial Hann window will act as a stronger frequency filter than a spatial filter. This is why the inner interferences in the spectrogram appear to be smeared out in the spatial domain in Figure 5(a). Conversely, the additional Hann frequency filter window of the S-method, acts as a stronger spatial than frequency filter and recovers the spatial phase space resolution. Both the spectrogram and the S-method can be computed much faster than the WV PSR, due to the reliance on small filter windows. Figure 6: A hologram of 2 dices in close proximity is used to contrast the phase space resolution of the spectrogram and the S-method.