## 3 Explicit Calculation of Phase Space Representations of Arbitrary Digital Holograms
### Introduction to Spectrograms  
Let \(g\in\mathbb{C}^{N}\) be a discrete 1D complex-valued signal of length \(N\) and \(w\in\mathbb{C}^{M}\) be another function, called (spatial) window \(w\) with \(M\leq N\), e.g., Figure 5a. We will refer to \(M:=2M^{\prime}+1,M^{\prime}\in\mathbb{N}_{0}\) as the window length. Contrary to analyzing the signal in its native (space) domain or its Fourier domain, we strive to understand when which frequency components are active. Due to the perfect delocalization of sinusoids, the elementary components of a Fourier transform, we can only do so by taking the Fourier transform of our signal over many small intervals. This is where the window function comes into play. The element-wise (Hadamard) product \(g(\xi^{px}+m)w(m),\xi^{px}\in\{0,\ldots,N-1\},f_{g}^{px}\in\{0,\ldots,M-1\}\) will have only at most \(M\) non-zeros (in case \(\xi^{px}+m\) exceeds \([0,N-1]\) the signal \(g\) is continued periodically). A discrete Fourier transform of \(gw\) will provide a frequency analysis localized on the at most \(M\) coefficients of the signal selected by the window. This local frequency analysis will be quasi-stationary in that it assumes a periodic continuation of the signal beyond the \(M\) non-zeros of the window and constant frequencies within, see Figure 5b. Shifting the window with respect to the signal and re-evaluating the Fourier transform each time, will provide a (quasi-stationary) representation of excited frequencies over space. Mathematically this concept of a windowed Fourier transform is known as the short-term Fourier transform (STFT), which is given at a location \(\xi^{px}\in\{0,\ldots,N-1\}\) (around the window center) and a frequency \(f_{\xi}^{px}\in\{0,\ldots,M-1\}\) (excited within this window) as  
\[\text{STFT}_{w}(g;\xi^{px},f_{\xi}^{px}):=\sum_{m=0}^{M-1}g\left(m-M^{\prime} \right)w\left(\xi^{px}+m-M^{\prime}\right)e^{-2\pi if_{\xi}^{px}m/M} \tag{10}\]  
and the spectrogram \(SP_{w}(g;\xi^{px},f_{\xi}^{px})\) is then defined as  
\[SP_{w}(g;\xi^{px},f_{\xi}^{px}):=\left|\text{STFT}_{w}(g;\xi^{px},f_{\xi}^{px} )\right|^{2}\,. \tag{11}\]  
Figure 4: Comparison of phase space representations evaluated on two examples of wavefronts expanding along \(z\) illustrate the arising interference terms—\(2\times\) parabolic (top); \(1\times\) spherical (bottom). (a) Wigner–Ville; (b) spectrogram; (c) S-method; (d) Wigner–Ville; (e) spectrogram; (f) S-method. While general window functions are possible, we shall content ourselves here with the simplest case of a symmetric, non-negative (real-valued), Hann window ([11], p. 532) of length \(M\geq 1\), defined as  
\[w(m):=\frac{1}{2}\left(1-\cos\left(\frac{2\pi m}{N}\right)\right)=\sin\left( \frac{\pi m}{M}\right)^{2},\quad m\in\left[0,M-1\right]. \tag{12}\]  
and normalized in the \(\ell_{2}\)-norm via  
\[\sum_{m=0}^{M-1}|w(m)|^{2}=1\,, \tag{13}\]  
to preserve the \(\ell_{2}\)-energy of the signal. A window of length \(M\) allows for \(M\) distinct frequencies to be analyzed. Thus if \(M\) is decreased, the spatial resolution is increased (as more distinct windows exist) but the frequency resolution (per window) decreases. If the converse is true, the frequency resolution is increased and traded for spatial resolution. A formal result of this is Lieb's uncertainty principle for the short-term Fourier transform [39]. Selecting \(M=\lfloor\sqrt{N}\rfloor\), where \(\lfloor\cdot\rfloor\) is the nearest lower integer, will provide the most balanced space-frequency resolution. In practice, overlapping windows are used to enhance the resolution through an increase of the redundancy in the representation. Good results are achieved by choosing maximal overlap \(M-1\) between long windows, e.g., setting \(M=4\lfloor\sqrt{N}\rfloor\) as was done in Figure 4. [42] describes how window size affects the confidence of instantaneous frequency estimates. Despite these measures, the joint resolution of the spectrogram remains low. Further enhancements are possible by abolishing the invertibility of the underlying STFT.