OptFROG – Analytic signal spectrograms with optimized time-frequency resolution
A Python package for the calculation of spectrograms with optimized time and frequency resolution for application in the analysis of numerical simulations on ultrashort pulse propagation is presented. Gabor’s uncertainty principle prevents both resolutions from being optimal simultaneously for a given window function employed in the underlying short-time Fourier analysis. Our aim is to yield a time-frequency representation of the input signal with marginals that represent the original intensities per unit time and frequency similarly well. As use-case we demonstrate the implemented functionality for the analysis of simulations on ultrashort pulse propagation in a nonlinear waveguide.
keywords:Spectrogram, Short-time Fourier analysis, Analytic signal, Optics, Ultrashort pulse propagation
1 Motivation and significance
The spectrogram, providing a particular time-frequency representation of signals that vary in time Cohen:1989 (), represents an inevitable tool in the analysis of the characteristics of ultrashort optical pulses. E.g., allowing to monitor the change in frequency of pulse features as function of time permits to determine quantities that cannot be obtained from either the time or frequency domain representation of the optical pulse alone. The applicability of the spectrogram to both, data retrieved from experiments Trebino:1993 (); Linden:2000 (); Trebino:2000 (); Efimov:2005 () where it is referred to as frequency resolved optical gating (FROG) analysis, as well as from numerical simulations Dudley:2002 (); Skryabin:2005 (); Guo:2013 (), carried out to complement experiments and to provide a basis for the interpretation of the observed effects, highlights the relevance of signal processing in the field of nonlinear optics and demonstrates the need to be able to compute such spectrograms in the first place. Here, we consider the issue of obtaining optimal time-frequency representations of signals for the interpretation of numerical experiments on ultrashort pulse propagation in nonlinear waveguides.
In principle, a spectrogram measures the properties of the signal under scrutiny as well as those of a user-specified window function for localizing parts of the signal during analysis. Exhibiting features of both, the interpretation of the spectrogram is strongly affected by the particular function used for windowing. Different window functions estimate different signal properties, e.g., if a given function achieves a good approximation of the intensities per unit time of the underlying signal, its approximation of the intensities per unit frequency might be bad. Consequently, the spectrogram might suffer from distortion yielding an unreasonable characterization of the time-frequency features of the signal under scrutiny. The usual approach for deciding on a particular window function is by trial-and-error and guided by the liking and experience of the individual.
Here we present a software tool that aims at minimizing the mismatch between the intensities per unit time and frequency and their corresponding estimates based on the spectrogram itself, obtained for a user-supplied parameterized window function. The resulting spectrograms are “optimal” in the sense that their visual inspection exhibits a minimal amount of distortion and thus allow for a reliable interpretation of the time-frequency composition of the input signal. Such an approach was previously shown to result in a reasonable characterization of the underlying time-frequency features Cohen:2003 (). It is further independent of the experience of the individual user and thus yields reproducible results.
2 Software description
The presented package facilitates the construction of spectrograms for the analytic signal (AS) Gabor:1947 () of the real field . In the Fourier domain, the angular frequency components of both are related via Marple:1999 (). Due to its one-sided spectral definition the time-domain representation of the AS is complex, further satisfying . The construction of an AS spectrogram relies on the repeated calculation of the spectrum of the modified signal at different delay times in terms of the short-time Fourier transform
wherein specifies a narrow window function centered at and decaying to zero for increasing . The latter allows to selectively filter parts of the AS and to estimate its local frequency content. Scanning over a range of delay times then yields the spectrogram as , providing a joint time-frequency distribution of both, the AS and the window function WignerNote (). For assessing the approximation quality of we utilize its time and frequency marginals
Note that in the limit where approaches a delta function, the time marginal will approach the intensity per unit time but the frequency marginal will represent the intensity per unit frequency only poorly. As a result, time resolution will be good and frequency resolution will be bad, see the discussion in section 3 below. The time-frequency uncertainty principle prevents both resolutions from being optimal simultaneously Cohen:1989 ().
The aim of the presented package is to obtain a time-frequency representation of the input signal for which the integrated absolute error (IAE) between its normalized marginals and the original intensities per unit time and frequency are minimal. We consider a single parameter window function , e.g. a Gaussian function with mean and root-mean-square (rms) width , and solve for
Above, the underlying spectrogram is computed via , indicated by the superscript on the marginals, and we assume normalization to and a total signal energy in terms of the spectrogram. For a good agreement of the marginals and the original intensities, the objective function assumes a small value. The additional parameter might be adjusted to give more weight to frequency resolution () or time resolution () if appropriate. The particular choice yields a balanced time-frequency representation, see the example provided in section 3. The optimized spectrogram is then computed by using for windowing.
2.1 Software Architecture
2.2 Software Functionalities
The current version of optfrog comprises five software units having the subsequent responsibilities:
Compute a standard spectrogram for the normalized time-domain analytic signal for a particular window function .
Compute a time-frequency resolution optimized spectrogram for the normalized time-domain analytic signal using the window function that minimizes the total IAE of both marginals. Note: for the minimization of the scalar function in the variable , the scipy native function scipy.optimize.minimize_scalar is employed in bounded mode.
Compute the marginal distribution in time based on the spectrogram.
Compute the marginal distribution in frequency based on the spectrogram.
Compute the total energy provided by the spectrogram approximation of the time-frequency characteristics of the signal.
For a more detailed description of function parameters and return values we refer to the documentation provided within the code optfrog_GitHub:2018 ().
2.3 Sample code snippet
In our research work we use optfrog mainly in script mode. An exemplary data postprocessing script, reproducing Fig. 1(b) discussed in section 3 below, is shown in listing 1. Therein, after importing the functionality of numpy, optfrog, and a custom figure generating routine in lines 1–3, the location of the input data (line 5) and filter options for the spectrogram output-data (lines 6p) are specified. Note that the user defined window function (lines 9p) does not need to be normalized. After loading the input data (lines 12p) the routine optFrog is used to compute an optimized spectrogram in line 15. Finally, a visual account of the latter is prepared by the routine spectrogramFigure in line 17.
3 Illustrative Examples
So as to demonstrate the functionality of optfrog we consider the numerical propagation of a short and intense few-cycle optical pulse in presence of the refractive index profile of an “endlessly single mode” () photonic crystal fiber Birks:1997 (); Stone:2007 (). The underlying unidirectional propagation model includes the Kerr effect and a delayed Raman response of Hollenbeck-Cantrell type Hollenbeck:2002 (). For the preparation of the initial condition we considered a single soliton with duration , i.e. approximately cycles, and soliton order , prepared at a center frequency . See Refs. Amiranashvili:2010b (); Amiranashvili:2011 () for a detailed account of the propagation model and Ref. Melchert:2018 () for a more thorough discussion of the particular problem setup. In Fig. 1 we illustrate the time-frequency characteristics of the pulse at propagation distance by using a Gaussian window function centered at and having rms-width . Note that the delay time has to be interpreted as being relative to the origin of a co-moving frame of reference in which the soliton is initially at rest.
In Figs. 1(a,c) we demonstrate an inevitable drawback of a trial-and-error choice of a window function used for calculating a spectrogram. As discussed earlier, the properties of the window implies a trade-off in resolution that might be achieved. I.e., if the user opts for a window function that is either too wide or too narrow in comparison to the signal features in the time domain, only one marginal will approximate its underlying original intensity well and, as a result, the spectrogram will appear distorted. This is shown in Fig. 1(a), where a vanillaFrog trace using demonstrates a good frequency resolution and a bad time resolution. Conversely, as evident from Fig. 1(c), a vanillaFrog trace using yields a good time resolution and a bad frequency resolution. In contrast, if the IAEs of both marginals are minimized simultaneously by aid of a numerical algorithm, both marginals of the optimized spectrogram are found to approximate the original intensities per unit time and frequency similarly well. Consequently, the resulting spectrogram provides a most reasonable time-frequency representation of the underlying signal. To demonstrate this, the balanced () optFrog trace for the optimized window function, obtained for with , is shown in Fig. 1(b).
Computing reliable spectrograms represents an integral part in the analysis of the characteristics of ultrashort optical pulses. The publicly available and free Python package optfrog performs the nontrivial task of computing such spectrograms with optimized time-frequency resolution. It is based on a computational approach to parameter optimization in opposition to common trial-and-error approaches, helping to save time and effort and yielding reproducible results independent of the skill of the individual user. It addresses researchers in the field of ultrashort pulse propagation and related disciplines where signal analysis in terms of short-time Fourier transforms is of relevance. As independent software postprocessing tool it is ideally suited for the analysis of output data obtained by existing pulse propagation codes, as, e.g., the open source LaserFOAM (Python) Amorim:2009 () and gnlse (Matlab) Travers:2010 () solver for the generalized nonlinear Schrödinger equation.
The optfrog Python package provides easy-to-use tools that yield a time-frequency representation of a real valued input signal and allow to quantify how well the resulting spectrogram approximates the signal under scrutiny for a user supplied window function.
We have shown how optfrog can be used to calculate analytic signal based spectrograms that are optimal in the sense that their visual inspection exhibits a minimal amount of distortion, allowing for a reliable interpretation of the time-frequency composition of the input signal.
We acknowledge support from the VolkswagenStiftung within the Niedersächsisches Vorab program (HYMNOS; Grant ZN 3061) and Deutsche Forschungsgemeinschaft (Grand MO 850/20-1).
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