# Phase-Based Signal Representations for Scattering
††thanks: This work was supported by the Austrian Science Fund (FWF) projects
Y 551-N13 (FLAME) and I 3067-N30 (MERLIN).
_This is the Author’s Accepted Manuscript version of work presented at
EUSIPCO21. It is licensed under the terms of theCreative Commons Attribution
4.0 International License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original author and source are
credited. The published version is available at:_
https://ieeexplore.ieee.org/abstract/document/9616285

1st Daniel Haider  Acoustics Research Institute
Austrian Academy of Sciences
Vienna, Austria
<EMAIL_ADDRESS>2nd Peter Balazs  Acoustics Research Institute
Austrian Academy of Sciences
Vienna, Austria
<EMAIL_ADDRESS>3rd Nicki Holighaus  Acoustics Research Institute
Austrian Academy of Sciences
Vienna, Austria
<EMAIL_ADDRESS>
###### Abstract

The scattering transform is a non-linear signal representation method based on
cascaded wavelet transform magnitudes. In this paper we introduce phase
scattering, a novel approach where we use phase derivatives in a scattering
procedure. We first revisit phase-related concepts for representing time-
frequency information of audio signals, in particular, the partial derivatives
of the phase in the time-frequency domain. By putting analytical and numerical
results in a new light, we set the basis to extend the phase-based
representations to higher orders by means of a scattering transform, which
leads to well localized signal representations of large-scale structures. All
the ideas are introduced in a general way and then applied using the STFT.

##  I Introduction

For most applications using time-frequency representations, only the magnitude
of the obtained complex coefficients is considered as, e.g., in the cases of
spectrogram and scalogram [1]. This is a standard method for analyzing the
time-frequency content of an audio signal, which is also heavily used in deep
learning for audio, where neural networks and time-frequency (magnitude)
representations often go hand-in-hand [2, 3, 4]. In this context, the
_scattering transform_ provides an important link, as it may be interpreted as
a deterministic neural network, computing a cascade of filter magnitude
transformations, i.e. a sequence of filtering and application of the modulus
as non-linearity. As signal representation, it is able to reveal large-scale
structures of the signal [5], which is partly caused by the non-linearity,
i.e. discarding the phase information. The phase is visually very unintuitive
but its derivative in the time, as well as in the frequency direction yield
very informative and intuitive phase-based signal representations. The
resulting quantities overcome some of the issues of the (pure) phase and are
directly linked to the concepts of _instantaneous frequency_ and _group
delay_. The interest of this work is to investigate the idea of scattering
with a complementary approach as before, namely considering the partial
derivatives of the phase instead of the modulus as non-linear step. As in the
magnitude case, larger-scale structures can be revealed with this phase
scattering procedure, additionally inheriting the precision of the
localization of time-frequency information that is provided by these phase-
based signal representations.
In Section II we revisit basic results around the concepts of instantaneous
frequency and group delay, in particular a result related to the short-time
Fourier transform (STFT). This revision shows the nature of the introduced
concepts from a different perspective, which is the key for the main part of
the paper. Analytical and numerical results are provided in order to
strengthen the intuition. In Section III we start by briefly presenting the
scattering transform in a time-frequency setting and adapt its principle
structure to define the _phase scattering coefficients_. Finally we show
promising experiments on proof-of-concept examples in a direct comparison to
(magnitude) scattering based on the STFT. This first study sets the basis and
intuition for further research on this idea.

The content of this paper was mainly developed in the Master’s thesis of the
first author [6]. More details, including proofs and Matlab scripts are
available under https://github.com/danedane-haider/Phase-Scattering-
Masterthesis. The scripts use the LTFAT toolbox [7, 8].

##  II Instantaneous Frequency and Group Delay

Instantaneous frequency (IF) and group delay (GD) are important concepts in
signal analysis [1] as they provide precise information about frequency, resp.
temporal positioning of signal components [9]. Local versions are defined by
the partial derivatives of the phase in the time-frequency plane. Let
$\Phi_{\Psi}^{x}(t,\omega)$ denote the phase function of a complex-valued
time-frequency representation, based on a family of localization functions
$\Psi$, then

$\widehat{\omega}_{\scriptscriptstyle\Psi}(x;t,\omega)=\frac{1}{2\pi}\frac{\partial}{\partial
t}\Phi_{\Psi}^{x}(t,\omega)$ (1)

is called _channelized instantaneous frequency_ (CIF) and

$\widehat{\tau}_{\scriptscriptstyle\Psi}(x;t,\omega)=-\frac{\partial}{\partial\omega}\Phi_{\Psi}^{x}(t,\omega)$
(2)

_local group-delay_ (LGD) of $x$ w.r.t. $\Psi$. These concepts have been used
already many times, e.g. for the phase vocoder [10] or reassignment methods
[11, 12] and other types of phase-based processing [13]. In this work we focus
on the STFT as underlying time-frequency representation. According to the
scaling conventions we use, its phase function $\Phi_{\scriptscriptstyle
g}^{x}(t,\omega)$ using a window $g$ is found in its magnitude-phase
representation in the following way,

$\mathcal{V}_{g}x(t,\omega)=|\mathcal{V}_{g}x(t,\omega)|\cdot e^{2\pi
i\Phi_{\scriptscriptstyle g}^{x}(t,\omega)}.$ (3)

Usually, when working with the STFT, a specific phase convention is assumed.
The resulting phase can then be converted into a time-invariant and a
frequency-invariant version. In contrast to previous work, we select specific
invariances in the STFT beforehand for computing CIF and LGD respectively. For
$x,g\in\textbf{L}^{2}(\mathbb{R})$ the _frequency-invariant_ STFT is defined
by

$\mathcal{V}_{g}x(t,\omega)=\int_{\mathbb{R}}x(\tau)\overline{g(\tau-t)}e^{-2\pi
i\omega\tau}d\tau,$ (4)

$t,\omega\in\mathbb{R}$. Here, the exponential term is independent of the time
index $t$. The _time-invariant_ STFT is defined by changing the order of
modulation and translation, giving

$\mathcal{V}_{g}^{t}x(t,\omega)=\int_{\mathbb{R}}x(\tau)\overline{g(\tau-t)}e^{-2\pi
i\omega(\tau-t)}d\tau.$ (5)

Our approach is to consider CIF in a frequency-invariant (4) and LGD in a
time-invariant setting (5). This results directly in modes of these quantities
that are necessary for the last chapter of the paper. Note that this setting
only agrees with the original definitions (1) and (2) up to the induced
invariances. Therefore, we will use the notation CIFf and LGDt further on. The
following proposition shows how CIFf and LGDt can be computed based on a
result by Auger and Flandrin [11]. We additionally emphasize the formal
requirements [14] and consider the introduced STFT-invariances (4) and (5).

######  Proposition 1.

Let $x\in L^{2}(\mathbb{R})$ and $g\in
L^{2}(\mathbb{R})\cap\mathcal{C}^{1}(\mathbb{R})$, such that $Tg\in
L^{2}(\mathbb{R})$, where the time-weighted window $Tg$ is given by
$Tg(\tau)=\tau g(\tau)$ for all $\tau\in\mathbb{R}$. Then
$\mathcal{V}_{g}x(t,\omega)$ and $\mathcal{V}_{g}^{t}x(t,\omega)$ are
infinitely partially differentiable in both variables, $t$ and $\omega$. Let
further $g^{\prime}$ denote the differentiated window
$g^{\prime}(\tau)=\frac{d}{d\tau}g(\tau)$. Then, for
$\mathcal{V}_{g}x(t,\omega)\neq 0$, CIFf and LGDt can be computed by

$\displaystyle\widehat{\omega}_{\scriptscriptstyle\text{STFT}}(x;t,\omega)$
$\displaystyle=-\frac{1}{2\pi}\operatorname{Im}\Biggl{\\{}\frac{\mathcal{V}_{g^{\prime}}x(t,\omega)}{\mathcal{V}_{g}x(t,\omega)}\Biggr{\\}}$
(6) $\displaystyle\widehat{\tau}_{\scriptscriptstyle\text{STFT}}(x;t,\omega)$
$\displaystyle=\operatorname{Re}\Biggl{\\{}\frac{\mathcal{V}^{t}_{Tg}x(t,\omega)}{\mathcal{V}^{t}_{g}x(t,\omega)}\Biggr{\\}},\
\ t,\omega\in\mathbb{R}.$ (7)

A compact proof of this can be found in [6]. The impact of the STFT
invariances gets clear in the following.

###  II-A Analytical Shapes of CIF and LGD - Revisited

Based on Proposition 1, one finds the CIFf of stationary sinusoids and the
LGDt of Dirac impulse to be affine linear functions. The results may be
considered as known, however, they show an alternative point of view via the
induced STFT-invariances which is crucial for the construction of the phase
scattering coefficients in the last section.

####  II-A1 CIFf Analytically

The CIFf representation of a stationary sinusoid (based on a frequency-
invariant STFT) has linear behaviour along the frequency axis, centered at the
IF of the sinusoid.

######  Lemma 1.

Let $x(\tau)=e^{2\pi i\xi_{0}\tau}$ be a complex sinusoid with frequency
$\xi_{0}$ and $g$ a window function with $g,Tg\in L^{2}(\mathbb{R})\cap
L^{1}(\mathbb{R})$ such that the Fourier transform of $g$ has no zeros and the
Fourier transform of $Tg$ only a single one at $0$. Then Equation (6) reduces
to

$\widehat{\omega}_{\scriptscriptstyle\text{STFT}}(x;t,\omega)=\xi_{0}-\omega$
(8)

$t,\omega\in\mathbb{R}$.

Note that using a time-invariant STFT results in
$\widehat{\omega}_{\scriptscriptstyle\text{STFT}}(x;t,\omega)=\xi_{0}$,
yielding the IF directly, whereas here a frequency shift function is computed,
measuring the distance to $\xi_{0}$ linearly.

####  II-A2 LGDt Analytically

The LGDt representation of a Dirac impulse (based on a time-invariant STFT)
has linear behaviour along the time axis, centered at the impulse.

######  Lemma 2.

Let $x(\tau)=\delta(\cdot-\tau_{0})$ where $\delta$ denotes a Dirac impulse
and $g\in L^{2}(\mathbb{R})\cap\mathcal{C}^{1}$ with $g^{\prime}\in
L^{2}(\mathbb{R})$ where $g$ has no zeros and $g^{\prime}$ only a single one
at $0$. Then Equation (7) reduces to

$\widehat{\tau}_{\scriptscriptstyle\text{STFT}}(x;t,\omega)=\tau_{0}-t$ (9)

$t,\omega\in\mathbb{R}$.

Also here, using the time-invariant version of the STFT is key to obtain this
affine linear function and is not obtained when using the frequency-invariant
STFT. The results of the Lemmata can be deduced directly from Proposition 1,
see [6].

Thus, by choosing the specific STFT versions in this discriminative way, CIFf
and LGDt are linear functions in frequency resp. time direction. We briefly
illustrate this also numerically to strengthen the intuition.

###  II-B Numerical Shapes of CIF and LGD - Revisited

Proposition 1 provides a convenient method to compute phase derivatives
numerically by point-wise operations of two STFTs. However, when working with
finitely supported windows in applications, the computations (6) and (7)
clearly have to be restricted to the support of the window, i.e. CIFf and LGDt
coefficients are considered to be zero outside the supports. Therefore, in an
actual numerical representation the linear shapes from (8) and (9) are
localized naturally to the range of the window support as well. Using the toy
examples from Lemma 1 and 2 with $\xi_{0}=1000$Hz and $\tau_{0}=0.5$s, one
finds the piece-wise linearly decreasing functions locally around $\xi_{0}$,
resp. $\tau_{0}$. Figure 1 (a),(b) illustrates well how CIFf and LGDt resemble
one another in terms of their shape in time, resp. frequency direction,
emphasizing the duality of the two quantities. Panels (c),(d) show the direct
comparisons to STFT magnitudes. Besides the intuition this brings for phase-
based signal representations in general, the chosen STFT conventions allow to
use CIFf and LGDt in a scattering procedure in a meaningful way.

For the calculation of the plots we used the function gabphasederiv from the
Large Time-Frequency Toolbox (LTFAT) [7] with the additional flags ’freqinv’
and ’timeinv’, which follow the STFT conventions used in Proposition 1, based
on the discrete Gabor transform dgt. Discrete versions of a dilated Gaussian
are used as window functions.

Figure 1: (a) Full CIFf of sinusoid and one column for fixed $t$ (right). (b)
full LGDt of impulse and one column for fixed $\omega$ (bottom). (c) full STFT
magnitude of sinusoid and one column for fixed $t$ (right). (d) full STFT
magnitude of impulse and one column for fixed $\omega$ (bottom).

##  III Phase Scattering

Indeed, the partial derivatives of the time-frequency domain phase are useful
quantities for describing time-frequency information since they provide
precise time and frequency localization of harmonic resp. transient events of
a signal. Using the analytical results and the intuition from the previous
section, we aim for expanding the representations to higher orders in the
manner of the _scattering transform_.

###  III-A Time-Frequency Scattering

The computational scheme of the scattering transform is a cascade of absolute
values of filter transformations [15, 16]. A network-like structure emerges
that results in layers of magnitude filter decompositions. In a time-frequency
setting this can be seen as taking one set of localization functions (band-
pass filters) per layer, applied via convolution (originally using wavelet
filters). Several papers show this transform as promising tool to represent
large-scale features of audio signals, such as amplitude or frequency
modulation, chord structures (intervals), tempo and rhythm, etc. [17, 18, 19].
Therefore, it is often used as external feature extraction step for deep
learning tasks and moreover, is celebrated as being an interpretable neural
network itself with fixed filters and a solid mathematical foundation,
allowing for rigorous analysis.

######  Definition 1 (Time-Frequency Scattering).

Let $x\in L^{2}(\mathbb{R})$ and
$\Psi_{k}=\\{\psi_{i}^{k}\\}_{i\in\Lambda_{k}}$ denote the set of filters for
the $k$-th layer. $\Lambda_{k}$ is the corresponding (frequency-)index set and
the operator $\mathcal{U}_{k}[p_{0}]=|\psi_{p{0}}\ast\cdot\ |$ computes the
magnitude filtering at index $p_{0}$. Then, the scattering transform is a
cascade of the operators $\mathcal{U}_{1},\dots,\mathcal{U}_{k}$ along a
frequency-index path
$p=(p_{1},\dots,p_{k})\in\Lambda_{1}\times\dots\times\Lambda_{k}$ and we call

$\mathcal{S}[p]x=\mathcal{U}_{k}[p_{k}]\dots\mathcal{U}_{2}[p_{2}]\mathcal{U}_{1}[p_{1}]x$
(10)

the $k$-th order (magnitude) scattering coefficients w.r.t. $p$.

The original idea of the scattering transform was to create a translation-
invariant representation on $L^{2}(\mathbb{R})$ by integrating over the
emerging scattering coefficients [15]. In applications, a final low-pass
filtering step is used to simulate this property locally. In this work we
preliminarily do not consider this step and focus merely on the nature of the
coefficients.

Figure 2: (a),(b) $1$st and $2$nd order CIFf and STFT scattering layers of
$x_{1}$ with propagation channel at $880$Hz. (c),(d) $1$st and $2$nd order
CIFf and STFT scattering layers of $x_{2}$.

###  III-B CIF and LGD Scattering

Instead of taking the modulus of the complex time-frequency coefficients, we
use the mappings to the partial derivatives of the phase as non-linearity in
the scattering cascade. This procedure can be written in terms of operators,
defined in the following way.

######  Definition 2 (Phase Scattering Coefficients).

Let $x\in L^{2}(\mathbb{R})$ and $\Psi$ a family of localization functions
with index set $\Lambda$, then the operators

$\displaystyle\big{(}\widehat{\Omega}_{\scriptscriptstyle\Psi}[p_{0}]x\big{)}(t):=\widehat{\omega}_{\scriptscriptstyle\Psi}(x;t,p_{0})$
(11)
$\displaystyle\big{(}\widehat{T}_{\scriptscriptstyle\Psi}[p_{0}]x\big{)}(t):=\widehat{\tau}_{\scriptscriptstyle\Psi}(x;t,p_{0})$
(12)

compute CIF and LGD of $x$ at the frequency index $p_{0}\in\Lambda$ as
function of time. Let
$p=(p_{1},\dots,p_{k})\in\Lambda_{1}\times\dots\times\Lambda_{k}$ denote a
frequency-index path associated to $k$ families of localization functions
$\Psi_{1},\dots,\Psi_{k}$, then cascading the operators (11) and (12) along
$p$ defines the $k$-th order CIF scattering coefficients

$\displaystyle\widehat{\Omega}[p]x=\widehat{\Omega}_{\scriptscriptstyle\Psi_{k}}[p_{k}]\dots\widehat{\Omega}_{\scriptscriptstyle\Psi_{1}}[p_{1}]x$
(13)

and the $k$-th order LGD scattering coefficients

$\displaystyle\widehat{T}[p]x=\widehat{T}_{\scriptscriptstyle\Psi_{k}}[p_{k}]\dots\widehat{T}_{\scriptscriptstyle\Psi_{1}}[p_{1}]x.$
(14)

of $x$ w.r.t. $p$. We shall call
$\widehat{\Omega}[(q_{1},...,q_{k-1},\lambda)]x$, considered for all
$\lambda\in\Lambda_{k}$, the $k$-th CIF scattering layer of $x$ w.r.t. the
path $q=(q_{1},...,q_{k-1})$. Analog for the LGD case.

Building on the results from the previous chapter, we use a phase scattering
procedure that use CIFf and LGDf computed by the STFT to provide numerical
examples.

######  Definition 3 (STFT-Phase Scattering Coefficients).

Under the conditions from Proposition 1 we can define the STFT-phase (or CIFf
resp. LGDf) scattering coefficients analogously to (13) and (14) by cascading
the operators

$\displaystyle\big{(}\widehat{\Omega}_{\scriptscriptstyle\text{STFT}}[p_{0}]x\big{)}(t)=-\frac{1}{2\pi}\operatorname{Im}\Biggl{\\{}\frac{\mathcal{V}_{g^{\prime}}x(t,p_{0})}{\mathcal{V}_{g}x(t,p_{0})}\Biggr{\\}},$
(15)
$\displaystyle\big{(}\widehat{T}_{\scriptscriptstyle\text{STFT}}[p_{0}]x\big{)}(t)=\operatorname{Re}\Biggl{\\{}\frac{\mathcal{V}^{t}_{Tg}x(t,p_{0})}{\mathcal{V}^{t}_{g}x(t,p_{0})}\Biggr{\\}},\quad
t\in\mathbb{R}$ (16)

along a frequency-index path $p=(p_{1},...,p_{k})\in\mathbb{R}^{k}$,
associated to $k$ STFTs using windows $g_{1},\dots,g_{k}$.

Inspired by the experiments from [18, 17, 19], we show that $2$nd order phase
scattering coefficients are capable of capturing wider scale information,
comparable to magnitude scattering.

####  III-B1 CIFf Scattering Application - Frequency Modulation

We first show that CIFf scattering of a frequency modulated sinusoid finds the
modulation frequency in the $2$nd layer.

######  Lemma 3.

Let $x(\tau)=e^{2\pi i(\xi_{0}\tau+\gamma(\tau))}$ with
$\gamma(\tau)=\sin(2\pi\xi_{1}\tau)$ denote a vibrato signal with center
frequency $\xi_{0}$ and modulation frequency $\xi_{1}$. Then the $2$nd order
CIFf scattering layer of $x$ w.r.t. $p_{1}=\xi_{0}$ is given by

$(\widehat{\Omega}_{\scriptscriptstyle\text{STFT}}[\xi_{0},\omega]x)(t)=\xi_{1}-\omega$
(17)

$t,\omega\in\mathbb{R}$.

This result shows the natural extension of the ($1$st order) CIFf
representation from Lemma 1. Let $\phi(\tau)$ denote the temporal phase of
$x$, given as above. We see that the instantaneous frequency of $x$ is
$\omega(x;\tau)=\frac{1}{2\pi}\frac{d}{d\tau}\phi(\tau)=\xi_{0}+2\pi\xi_{1}\cos(2\pi\xi_{1}\tau)$,
which when considered as a time-domain signal itself possesses the IF
$\omega(\omega(x;\tau);\tau)=\xi_{1}$, the modulation frequency of the
vibrato. Using $2$nd order CIFf scattering, we find $\xi_{1}$ as zero of an
affine linear function, coming from
$\big{(}\widehat{\Omega}_{\scriptscriptstyle\text{STFT}}[\omega]x\big{)}(t)=\omega(x;t)-\omega$.
A more detailed discussion can be found in [6].

Note that this result theoretically does not depend on the chosen propagation
index $p_{1}$, however, in the numerical case we have to care about staying
inside the support of the window, i.e.
$p_{1}\in\operatorname{supp}(\widehat{g})$.

As numerical examples we set up two sinusoidal signals $x_{1},x_{2}$ at
$880$Hz and modulate $x_{1}$ by $\gamma_{1}(\tau)=\sin(2\pi 20\tau)$, creating
a constant vibrato of $20$Hz and $x_{2}$ by
$\gamma_{2}(\tau)=\sin(2\pi\tau(20+e^{\tau}))$, creating a vibrato that
increases exponentially in frequency, beginning at $20$Hz. Figure 2 shows
$1$st and $2$nd order CIFf and STFT scattering layers of $x_{1}$ and $x_{2}$,
using the propagation frequency index corresponding to $880$Hz. Although the
modulation frequencies induced by $\gamma_{1}$ and $\gamma_{2}$ are extremely
low, they are captured precisely in the $2$nd CIFf layers respectively as
zeros of linear functions. This can be observed also for more general
frequency modulation functions. Magnitude scattering finds the modulation
frequencies as well, but here in terms of shifted window functions.

###  III-C Mixed Phase Scattering

To provide an intuitive application of LGD scattering, we refer to an earlier
work [19] where we have shown how STFT magnitude scattering can capture the
frequency of temporal patterns among transients in a signal. In this sense,
our approach is to use the LGDt representation as transient detector and use a
subsequent CIFf transformation to get the frequency information of the LGDt-
peak arrangement.

######  Definition 4 (Mixed Phase Scattering).

Let $x\in L^{2}(\mathbb{R})$ and $\Psi_{1},\Psi_{2}$ families of localization
functions, then we define

$\widehat{M}[p_{1},p_{2}]x=\widehat{\Omega}_{\scriptscriptstyle\Psi_{2}}[p_{2}]\widehat{T}_{\scriptscriptstyle\Psi_{1}}[p_{1}]x$
(18)

as $2$nd order mixed phase scattering coefficients of $x$ w.r.t. the path
$(p_{1},p_{2})$.

####  III-C1 Mixed Phase Scattering Application - Dirac Comb

We show that $2$nd order mixed scattering finds the fundamental frequency of a
Dirac comb signal.

######  Lemma 4.

Let $x(\tau)=\sum_{\ell\in\mathbb{Z}}\delta(\cdot-\frac{2\pi\ell}{\xi_{0}})$
be a Dirac comb with fundamental frequency $\xi_{0}$. Then the $2$nd order
mixed phase scattering layer of $x$ is given by

$\big{(}\widehat{M}[p_{1},\omega]x\big{)}(t)=\xi_{0}-\omega$ (19)

for any $p_{1}$ and $t,\omega\in\mathbb{R}$.

Crucial for this result is to use a window $g$ with
$\operatorname{supp}(g)\geq 2\pi/\xi_{0}$ to avoid regions of zero amplitude
between the impulses. Then
$\big{(}\widehat{T}_{\scriptscriptstyle\text{STFT}}[\omega]x\big{)}(t)$ is a
saw-tooth wave for every $\omega\in\mathbb{R}$ with a fundamental frequency of
$\xi_{0}$ and the same phase function as its sinusoidal equivalent, i.e. the
IF is found by the subsequent CIFf transformation analog to Lemma 3. A more
detailed discussion can be found in [6]. Figure 3 shows the mixed phase
scattering and STFT scattering layers of a Dirac comb signal with fundamental
frequency of $20$Hz.

Figure 3: (a) $1$st and $2$nd order mixed phase scattering layers of the Dirac
comb w.r.t. an arbitrary $p_{1}$. (b) $1$st and $2$nd order STFT scattering
layers.

##  IV Conclusion and Outlook

We revisited channelized instantaneous frequency and local group delay by
pointing out their linear characters when considering two different STFT-
invariance settings. Analytical and numerical arguments are provided in order
to emphasize this perspective. Based on that we introduced an extension of
these representations in the manner of the scattering transform by cascading
the computations, called _phase scattering_. It turned out that this procedure
is, like the original scattering, capable of extracting information that lives
in larger scales. We found modulation frequencies of vibrato signals and the
tempo of a Dirac comb emerging in the second layer of the transform. However,
other than the blurred representation of time-frequency information that
magnitude scattering provides, CIFf and LGDt scattering allows for precise
allocation of large-scale time-frequency information by means of zeros of
linear functions.

This first study provides the foundation and intuition for further
investigations on scattering phase information. The many existing results
related to the scattering transform may serve as source of inspiration. In
particular, to provide a mathematically well-defined setting one could
formulate phase scattering in terms of distribution theory. Then, the expected
properties regarding translation-invariance could be approached in theory. An
immediate next step for continuing numerical experiments should be the
development of computational algorithms that produce clean plots and further
on, how phase scattering performs on real world audio signals. Finally, given
the precise allocation properties, it seems promising to evaluate a set of
phase-based scattering features as pre-processing step for a classification
task.

## Acknowledgment

The work on this paper was partially supported by the Austrian Science Fund
(FWF) START-project FLAME (Frames and Linear Operators for Acoustical Modeling
and Parameter Estimation; Y 551-N13) and MERLIN (I 3067–N30).

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