## 1 Introduction  
The phase space of a signal is the joint space of the signals native domain and its Fourier domain. It can consist of time and frequency for temporal signals or spatial positions and spatial frequencies for spatial signals. Phase space representations (PSRs; in signal processing literature the term time-frequency representations is more common) are an essential tool initially proposed to aid phase space analysis, which tries to understand systems whose frequency spectrum changes over the course of the signal by visualizing quasi-instantaneous frequencies as they change. They quickly proved themselves useful in many applications of science and engineering. Examples are: incoherent optics [1], biomedical engineering [2; 3], radar [4; 5]/sonar/seismic wave-processing [6], telecommunications [7], power generation [7], combustion analysis [7], machine-condition monitoring [7], geophysics [7], or quantum mechanics [7]. When PSRs are applied to temporal signals and are equipped with a logarithmic decrease in frequency resolution,favoring frequency over time resolution at low-frequencies, they are used in auditory applications such as instrument or musical score identification [8], or even reconstruction of long lost antiquities as for example Brahms Hungarian Rhapsody [9]. Even in the recent past, PSRs continue to provide vital, new insights in research, such as the S-method applied to gravitational waves [10]. In digital holography (DH) the Fourier conjugated variables are usually spatial position \(\xi,\eta\) and spatial frequency \(f_{\xi},f_{\eta}\), loosely speaking the number of fringe oscillations per unit of space. To avoid confusion, we shall continue to refer to the joint space-frequency domain as phase space, and representations thereof as PSRs. Several authors have recognized the potential of phase space discussions in DH, since point-spread functions \(P\) (PSFs) are a prime example of signals whose frequency distribution varies with space. Figure 1a shows the real parts of two exemplary PSFs, recorded at some distance \(z\) along the optical axis in an off-axis position denoted by \(\Delta x\). Note, the oscillations in the spatial domain are more rapid the further off-axis the recording is considered. We will elaborate in more depth on this later on. In Figure 1b we provide an overview of how to use PSRs to perform phase space analysis on DH and over the involved dimensions. Although 2D holograms span a 4D phase space, it will be often sufficient for analysis to consider only 2D cross-sections of phase space obtained from 1D cross-sections of the hologram. Most of this paper will use horizontal or vertical cross-sections centered on the hologram. The majority of previously published work about the application of PSRs to DH is based on the Wigner-Ville (WV) distribution (Section 9.6 in [11]) and the short-term Fourier transform (STFT) (Section 9.2.6 in [11]). These are vital for performing calculations, as they are invertible; however, upon visualizing phase space they suffer from substantial artifacts as we shall see later on. Henceforth, most of the time only qualitative WV charts are used for visualization, which are not as useful to practitioners confronted with imperfect, partially unknown, or processed data. In science, the intuition for finding a solution to a problem is often guided by appropriate visualizations, therefore we propose the use of the S-method [12] for visual exploration of phase space in DH and performing a diagnostic of processing artifacts that are not easily visible otherwise. For this, it is sufficient to evaluate it on individual 1D slices of 2D holograms even for deep 3D scenes (see Figure 1b). We show that this representation stands out from many other PSRs by being fast to calculate, generic, and allowing for a good visual interpretation. Figure 1: (**a**) Real part of a 1D point-spread function of a spherical (dashed, black; Rigorous form) and a quadratic wavefront (continuous, red; Fresnel approx.).