# On foundational discretization barriers in STFT phase retrieval

Philipp Grohs*,†,‡ *Faculty of Mathematics, University of Vienna, Oskar-
Morgenstern-Platz 1, 1090 Vienna, Austria  †Research Network
DataScience@UniVie, University of Vienna, Kolingasse 14-16, 1090 Vienna,
Austria  ‡Johann Radon Institute of Applied and Computational Mathematics,
Austrian Academy of Sciences, Altenbergstrasse 69, 4040 Linz, Austria
<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Lukas Liehr*
<EMAIL_ADDRESS>
###### Abstract.

We prove that there exists _no_ window function $g\in{L^{2}({\mathbb{R}})}$
and _no_ lattice $\mathcal{L}\subset{\mathbb{R}}^{2}$ such that every
$f\in{L^{2}({\mathbb{R}})}$ is determined up to a global phase by spectrogram
samples $|V_{g}f(\mathcal{L})|$ where $V_{g}f$ denotes the short-time Fourier
transform of $f$ with respect to $g$. Consequently, the forward operator

$f\mapsto|V_{g}f(\mathcal{L})|$

mapping a square-integrable function to its spectrogram samples on a lattice
is never injective on the quotient space
${L^{2}({\mathbb{R}})}\mathclose{\hbox{$\left/\vbox
to8.5pt{}\right.\n@space$}}\mathopen{}\sim$ with $f\sim h$ identifying two
functions which agree up to a multiplicative constant of modulus one. We will
further elaborate this result and point out that under mild conditions on the
lattice $\mathcal{L}$, functions which produce identical spectrogram samples
but do not agree up to a unimodular constant can be chosen to be real-valued.
The derived results highlight that in the discretization of the STFT phase
retrieval problem from lattice measurements, a prior restriction of the
underlying signal space to a proper subspace of ${L^{2}({\mathbb{R}})}$ is
inevitable.

Keywords. phase retrieval, time-frequency analysis, short-time Fourier
transform, lattices, sampling, signal reconstruction

AMS subject classifications. 42A38, 44A15, 94A12, 94A20

##  1\. Introduction

The problem of recovering a function $f\in{L^{2}({\mathbb{R}})}$ from the
absolute value of its short-time Fourier transform (STFT) has attracted a
great deal of attention in recent years. This so-called _STFT phase retrieval
problem_ has seen tremendous development of the corresponding theory and
applications. It arises, for instance, in ptychography, which provides an
attractive setting for the detailed understanding of the structure of
materials [20, 22]. Starting from a window function
$g\in{L^{2}({\mathbb{R}})}$ and a set $\mathcal{L}\subseteq{\mathbb{R}}^{2}$
of the time-frequency plane, one aims at recovering
$f\in{L^{2}({\mathbb{R}})}$ from phaseless measurements of the form

$|V_{g}f(\mathcal{L})|=\left\\{|V_{g}f(x,\omega)|:(x,\omega)\in\mathcal{L}\right\\}$

with $V_{g}f$ denoting the short-time Fourier transform of $f$ with respect to
the window function $g$, defined by

(1) $V_{g}f(x,\omega)=\int_{\mathbb{R}}f(t)\overline{g(t-x)}e^{-2\pi i\omega
t}\,dt.$

The modulus of the STFT, $|V_{g}f|$, is called the spectrogram and it measures
the distribution of the time-frequency content of $f$. Note that two functions
$f,h$ which agree up to a global phase, i.e. there exists a constant
$\nu\in{\mathbb{T}}\coloneqq\\{z\in{\mathbb{C}}:|z|=1\\}$ of modulus one such
that $f=\nu h$, produce identical spectrograms. It follows that a
reconstruction of $f$ from $|V_{g}f(\mathcal{L})|$ is only possible up to the
ambiguity of a global phase factor. While the classical Fourier phase
retrieval problem and Pauli problem suffer from non-trivial ambiguities or
non-uniqueness, the usage of the STFT constitutes an attractive transform
since suitable assumptions on the window function $g$ implies unique recovery
(up to a global phase) from $|V_{g}f(\mathcal{L})|$, provided that
$\mathcal{L}$ is a continuous domain such as $\mathcal{L}={\mathbb{R}}^{2}$
[11, Section 4]. In applications, however, the spectrogram is only accessible
at a discrete set, most notably at samples on a lattice. In this article, we
address the natural question of the uniqueness of the STFT phase retrieval
problem in the situation where $\mathcal{L}$ is a lattice in the time-
frequency plane. We prove that __

_no window function_ $g\in{L^{2}({\mathbb{R}})}$ and no lattice
$\mathcal{L}\subset{\mathbb{R}}^{2}$

_achieves unique recovery (up to a global phase) of signals in_
${L^{2}({\mathbb{R}})}$.

This statement reveals a fundamental barrier in the discretization of the STFT
phase retrieval problem from lattice measurements: the prior restriction to a
proper subspace of ${L^{2}({\mathbb{R}})}$ in the discretization of the STFT
phase retrieval problem from lattice samples is inevitable.

###  1.1. Main results

We now state the main results of the paper in a mathematically precise form.
To that end, we introduce the equivalence relation $f\sim h$ which indicates
that $f$ and $h$ equal up to a global phase, i.e.

$f\sim h\iff\exists\nu\in{\mathbb{T}}:f=\nu h.$

######  Definition 1.1.

Let $g\in{L^{2}({\mathbb{R}})}$ and $\mathcal{L}\subseteq{\mathbb{R}}^{2}$. We
say that $(g,\mathcal{L})$ is a uniqueness pair of the STFT phase retrieval
problem if every $f\in{L^{2}({\mathbb{R}})}$ is determined up to a global
phase by $|V_{g}f(\mathcal{L})|$, i.e. the implication

$|V_{g}f(\mathcal{L})|=|V_{g}h(\mathcal{L})|\implies f\sim h$

holds true for every $f,h\in{L^{2}({\mathbb{R}})}$.

In a more abstract language, the property of a uniqueness pair may be replaced
by the demand on the forward operator $f\mapsto|V_{g}f(\mathcal{L})|$ being
injective on the the quotient space
${L^{2}({\mathbb{R}})}\mathclose{\hbox{$\left/\vbox
<EMAIL_ADDRESS>Recall that a lattice is a set
$\mathcal{L}=L{\mathbb{Z}}^{2}$ with $L\in\mathrm{GL}_{2}({\mathbb{R}})$ being
an invertible matrix, the generating matrix of $\mathcal{L}$. A shifted
lattice is a set $\mathcal{S}\subset{\mathbb{R}}^{2}$ of the form
$\mathcal{S}=z+\mathcal{L}$ where $\mathcal{L}$ is a lattice and
$z\in{\mathbb{R}}^{2}$ is a vector. In [1, Theorem 1 and Remark 4] the authors
show that $(\varphi,\mathcal{L})$ is not a uniqueness pair provided that
$\varphi(t)=e^{-\pi t^{2}}$ is a centered Gaussian and $\mathcal{L}$ is a
shifted lattice. The following theorem significantly generalizes this result
(see Section 3).

######  Theorem 1.2.

Suppose that $g\in{L^{2}({\mathbb{R}})}$ is an arbitrary window function and
let $\mathcal{S}\subset{\mathbb{R}}^{2}$ be a subset of the time-frequency
plane. Then $(g,\mathcal{S})$ is never a uniqueness pair, provided that
$\mathcal{S}$ is a shifted lattice, i.e. $\mathcal{S}=z+L{\mathbb{Z}}^{2}$ for
some $L\in\mathrm{GL}_{2}({\mathbb{R}})$ and some $z\in{\mathbb{R}}^{2}$.

The previous statement is in stark contrast to the case where phase
information is available: it is known that for every lattice
$\mathcal{L}=L{\mathbb{Z}}^{2},L\in\mathrm{GL}_{2}({\mathbb{R}})$, which
satisfies $|\det(L)|\leq 1$ there exists a window function
$g\in{L^{2}({\mathbb{R}})}$ such that every $f\in{L^{2}({\mathbb{R}})}$ is
uniquely determined by $V_{g}f(\mathcal{L})$ [14, Theorem 11]. In addition,
mild conditions on a window function $g$ imply that the Gabor system
$\\{e^{2\pi i\ell\cdot}g(\cdot-k):(k,\ell)\in a{\mathbb{Z}}\times
b{\mathbb{Z}}\\}$ forms a frame for ${L^{2}({\mathbb{R}})}$ (which is a much
stronger statement than being complete) provided that $a,b>0$ are sufficiently
small, see [14, Section 3.5] and the references therein. In fact, Theorem 1.2
above follows from a more general observation that shifted parallel lines do
not guarantee uniqueness. We say that $\mathcal{P}\subset{\mathbb{R}}^{2}$ is
a set of shifted parallel lines if $\mathcal{P}$ arises from a set of the form
${\mathbb{R}}\times h{\mathbb{Z}},h>0,$ via a rotation around the origin,
followed by a translation.

######  Theorem 1.3.

Suppose that $g\in{L^{2}({\mathbb{R}})}$ is an arbitrary window function and
let $\mathcal{P}\subset{\mathbb{R}}^{2}$. Then $(g,\mathcal{P})$ is never a
uniqueness pair, provided that $\mathcal{P}$ is a set of shifted parallel
lines in the time-frequency plane.

Figure 1. Plot of the pointwise distance $Q$ of the square of two
spectrograms, $Q\coloneqq||V_{g}f_{1}|^{2}-|V_{g}f_{2}|^{2}|$, on the
rectangle $[-2,2]\times[-1.5,1.5]$ with window function $g(t)=e^{-|t|}$. The
functions $f_{1}$ and $f_{2}$ are chosen in such a way that $Q$ vanishes on
parallel lines in the time-frequency plane, one of which is highlighted by the
white dotted line. Such functions $f_{1}$ and $f_{2}$ can be obtained for
every window function $g$ and they have the additional property that they do
not agree up to a global phase. Their precise construction is the content of
Section 3.

The conclusion of Theorem 1.3 is visualized in Figure 1. Prominent choices of
window functions in Gabor analysis and its applications have the property that
they are real-valued (Gaussians, Hermite functions, Airy disc functions,
Hanning windows, rectangular windows). One could inquire about the question of
whether a real-valuedness assumption on the underlying signal space achieves
uniqueness from lattice samples, i.e. one specializes the considered input
functions to belong to the space

$L^{2}({\mathbb{R}},{\mathbb{R}})=\left\\{f\in{L^{2}({\mathbb{R}})}:f\
\text{is real-valued}\right\\}.$

Under mild conditions on a lattice $\mathcal{L}$, the STFT phase retrieval
problem still fails to be unique.

######  Theorem 1.4.

Let $g\in L^{2}({\mathbb{R}},{\mathbb{R}})$ be a real-valued window function
and $\mathcal{L}=L{\mathbb{Z}}^{2}$ a lattice generated by

$L=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\mathrm{GL}_{2}({\mathbb{R}}).$

If $a$ and $b$ as well as $c$ and $d$ are linearly dependent over
${\mathbb{Q}}$ then there exist two real-valued functions $f_{1},f_{2}\in
L^{2}({\mathbb{R}},{\mathbb{R}})$ which do not agree up to a global phase,
$f_{1}\nsim f_{1}$, but their spectrograms agree on $\mathcal{L}$,

$|V_{g}f_{1}(\mathcal{L})|=|V_{g}f_{2}(\mathcal{L})|.$

In particular, the conclusion of the previous theorem holds true if the
generating matrix $L$ has rational entries,
$L\in\mathrm{GL}_{2}({\mathbb{Q}}),$ or if the lattice is rectangular, i.e.
$\mathcal{L}=\alpha{\mathbb{Z}}\times\beta{\mathbb{Z}}$ for some
$\alpha,\beta\in{\mathbb{R}}\setminus\\{0\\}$.

###  1.2. Outline

We quickly outline the structure of the paper. In Section 2 we define
frequently used operators and their interrelations. Besides, we present the
necessary background on the fractional Fourier transform and shift-invariant
spaces. Based on this discussion we proceed in proving the main results of the
article in Section 3. We conclude the article with Section 4, where we compare
our results to previous work and outline a list of open problems suggested by
the reported results.

##  2\. Preliminaries on fractional Fourier transform and associated shift-
invariant spaces

In this section, we collect several preliminary definitions and results about
operators arising in time-frequency analysis, the fractional Fourier
transform, and shift-invariant spaces.

###  2.1. Basic operations

The short-time Fourier transform of a function $f\in{L^{2}({\mathbb{R}})}$
with respect to a window function $g\in{L^{2}({\mathbb{R}})}$ is the map

$V_{g}f(x,\omega)=\int_{\mathbb{R}}f(t)\overline{g(t-x)}e^{-2\pi i\omega
t}\,dt,$

which is defined on the time-frequency plane ${\mathbb{R}}^{2}$. This is a
uniformly continuous map which measures the contribution to $f(t)$ of the
frequency $\omega$ near $t=x$. In particular, it is pointwise defined. Hence,
the STFT phase retrieval problem from lattice samples is well-defined. The so-
called spectrogram of $f$ with respect to $g$ is the modulus of the STFT, i.e.
the map

${\mathbb{R}}^{2}\to{\mathbb{R}}_{\geq 0},\ \
(x,\omega)\mapsto|V_{g}f(x,\omega)|.$

Note that $V_{g}f(x,\omega)$ is the Fourier transform of
$f\overline{g(\cdot-x)}$ evaluated at $\omega$. The Fourier transform of a map
$u\in L^{1}({\mathbb{R}})$ is given by

${\mathcal{F}}u(\omega)\coloneqq\hat{u}(\omega)\coloneqq\int_{\mathbb{R}}u(t)e^{-2\pi
i\omega t}\,dt.$

Recall that ${\mathcal{F}}u$ is an element of $C_{0}({\mathbb{R}})$, the
Banach space of all continuous functions on the real line vanishing at
infinity. The inverse Fourier transform is the map
${\mathcal{F}^{-1}}u(t)={\mathcal{F}}u(-t)$. The operator ${\mathcal{F}}$
extends from $L^{1}({\mathbb{R}})\cap{L^{2}({\mathbb{R}})}$ to a unitary
operator mapping ${L^{2}({\mathbb{R}})}$ bijectively onto
${L^{2}({\mathbb{R}})}$. In addition to the Fourier operator, we introduce
some key-operations in time-frequency analysis. For $\tau,\nu\in{\mathbb{R}}$,
the translations operator
$T_{\tau}:{L^{2}({\mathbb{R}})}\to{L^{2}({\mathbb{R}})}$, the modulation
operator $M_{\nu}:{L^{2}({\mathbb{R}})}\to{L^{2}({\mathbb{R}})}$ and the
reflection operator
$\mathcal{R}:{L^{2}({\mathbb{R}})}\to{L^{2}({\mathbb{R}})}$ are defined via

$\begin{split}T_{\tau}f(t)&=f(t-\tau),\\\ M_{\nu}f(t)&=e^{2\pi i\nu t}f(t),\\\
\mathcal{R}f(t)&=f(-t).\end{split}$

Using translation and modulation, the short-time Fourier transform of $f$ with
respect to $g$ simplifies to the expression $V_{g}f(x,\omega)=\langle
f,M_{\omega}T_{x}g\rangle$ where $\langle
a,b\rangle=\int_{\mathbb{R}}a(t)\overline{b(t)}\,dt$ denotes the $L^{2}$-inner
product of two functions $a,b\in{L^{2}({\mathbb{R}})}$. The next Lemma
summarizes elementary relations between the operators
${\mathcal{F}},V_{g},T_{\tau},M_{\nu}$ and $\mathcal{R}$ which will be used
throughout the article. The simple proofs can be found, for instance, in [9,
Chapter 1–3].

######  Lemma 2.1.

For every $\tau,\nu,x,\omega\in{\mathbb{R}}$ and every
$f,g\in{L^{2}({\mathbb{R}})}$, the operators defined above satisfy the
relations

  1. (1)

$T_{\tau}M_{\nu}=e^{-2\pi i\tau\nu}M_{\nu}T_{\tau}$

  2. (2)

${\mathcal{F}}T_{\tau}=M_{-\tau}{\mathcal{F}}$

  3. (3)

${\mathcal{F}^{-1}}T_{\tau}=M_{\tau}{\mathcal{F}^{-1}}$

  4. (4)

$T_{\tau}\mathcal{R}=\mathcal{R}T_{-\tau}$

  5. (5)

$\overline{{\mathcal{F}}f}=\mathcal{R}{\mathcal{F}}\overline{f}$.

Moreover, the STFT satisfies the covariance property

  1. (6)

$V_{g}(T_{\tau}M_{\nu}f)(x,\omega)=e^{-2\pi
i\tau\omega}V_{g}f(x-\tau,\omega-\nu)$.

###  2.2. Fractional Fourier transform

Denote by $\\{H_{n}\\}_{n\in{\mathbb{N}}_{0}}$ the Hermite basis functions on
${\mathbb{R}}$, defined by

$H_{n}(t)=\frac{2^{1/4}}{\sqrt{n!}}\left(-\frac{1}{\sqrt{2\pi}}\right)^{n}e^{\pi
t^{2}}\left(\frac{d}{dt}\right)^{n}e^{-2\pi t^{2}}.$

The system $\\{H_{n}\\}_{n\in{\mathbb{N}}_{0}}$ constitutes and orthonormal
basis for ${L^{2}({\mathbb{R}})}$. For $\theta\in{\mathbb{R}}$, the unitary
operator
${\mathcal{F}}_{\theta}:{L^{2}({\mathbb{R}})}\to{L^{2}({\mathbb{R}})}$,

${\mathcal{F}}_{\theta}f=\sum_{n=0}^{\infty}e^{-i\theta n}\langle
f,H_{n}\rangle H_{n},$

is called the fractional Fourier transform of order $\theta$. A detailed
investigation of this transform and its appearance in quantum mechanics can be
found in the early paper by Namias [19]. Besides, Namias’ article provides an
alternative and frequently used definition of the operator
${\mathcal{F}}_{\theta}$ as an integral transform invoking chirp modulations
[19, Section 3]. The following properties of the fractional Fourier transform
are used throughout the present article [19, Section 4].

######  Lemma 2.2.

For every $\theta\in{\mathbb{R}}$ and every $\xi\in{\mathbb{R}}$ the
fractional Fourier transform ${\mathcal{F}}_{\theta}$ has the following
properties:

  1. (1)

${\mathcal{F}}_{0}=\mathrm{Id},{\mathcal{F}}_{\frac{\pi}{2}}={\mathcal{F}},{\mathcal{F}}_{-\frac{\pi}{2}}={\mathcal{F}}^{-1}$
and ${\mathcal{F}}_{\pi}=\mathcal{R}$

  2. (2)

${\mathcal{F}}_{\theta+\xi}={\mathcal{F}}_{\theta}{\mathcal{F}}_{\xi}$

  3. (3)

${\mathcal{F}}_{\theta}$ commutes with $\mathcal{R}$

  4. (4)

If $f\in{L^{2}({\mathbb{R}})}$ then
$\overline{{\mathcal{F}}_{\theta}f}={\mathcal{F}}_{-\theta}\overline{f}$

In the context of time-frequency analysis, the perhaps most significant
property of the operator ${\mathcal{F}}_{\theta}$ is its relation to rotations
in the time-frequency plane [16, 18]. To present the precise mathematical
formulation of this fact, we define for $\theta\in{\mathbb{R}}$ the rotation
matrix $R_{\theta}\in{\mathbb{R}}^{2\times 2}$ via

(2) $R_{\theta}=\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}.$

The rotation property of the fractional Fourier transform reads as follows.

######  Theorem 2.3.

If $R_{\theta}$ denotes the rotation matrix as given in equation (2) then for
every $f,g\in{L^{2}({\mathbb{R}})}$ and every $(x,\omega)\in{\mathbb{R}}^{2}$
one has

$V_{g}f(R_{\theta}(x,\omega))=e^{-\pi iq(\theta)}e^{\pi
ix\omega}V_{{\mathcal{F}}_{\theta}g}({\mathcal{F}}_{\theta}f)(x,\omega)$

with $q(\theta)=(x\cos\theta-\omega\sin\theta)(x\sin\theta+\omega\cos\theta)$.

A proof of Theorem 2.3 can be found in [3]. Moreover, the reader may consult
[16, Section 3.4] for a discussion of the rotation property. The phase
retrievability of functions $f\in{L^{2}({\mathbb{R}})}$ from measurements of
the form $\\{|{\mathcal{F}}_{\theta}f|:\theta\in\Theta\\}$, with
$\Theta\subset{\mathbb{R}}$ constituting a set of orders, was studied by
Jaming [16] and Carmeli et al. [6].

###  2.3. Shift-invariant spaces

In this section, we shall discuss shift-invariant spaces, both in the
classical sense as well as generalizations in the context of fractional
Fourier transform. We start by introducing classical shift-invariant spaces.
Consider a generating function $u\in{L^{2}({\mathbb{R}})}$ and a constant
${\mathfrak{s}}>0$. The subspace
$\mathcal{V}_{\mathfrak{s}}(u)\subset{L^{2}({\mathbb{R}})}$, defined as the
$L^{2}$-closure of the ${\mathfrak{s}}{\mathbb{Z}}$-shifts of $u$,

(3)
$\mathcal{V}_{\mathfrak{s}}(u)\coloneqq\overline{{\mathrm{span}}\left\\{T_{{\mathfrak{s}}k}u:k\in{\mathbb{Z}}\right\\}},$

is called the (principle) shift-invariant space generated by $u$ and step-size
${\mathfrak{s}}>0$. Shift-invariance means that whenever
$h\in\mathcal{V}_{\mathfrak{s}}(u)$, so is $T_{{\mathfrak{s}}n}h$ for every
$n\in{\mathbb{Z}}$. A characterization of functions belonging to
$\mathcal{V}_{\mathfrak{s}}(u)$ was given in [8]. If
$h\in\mathcal{V}_{\mathfrak{s}}(u)$ is such that $h$ has a representation of
the form $h=\sum_{k\in{\mathbb{Z}}}c_{k}T_{{\mathfrak{s}}k}u$ for some
sequence $\\{c_{k}\\}\subset{\mathbb{C}}$ then we call $\\{c_{k}\\}$ a
defining sequence of $h$. If $\\{c_{k}\\}$ belongs to the space
$c_{00}({\mathbb{Z}})$ of sequences with only finitely many nonzero components
then clearly $\sum_{k\in{\mathbb{Z}}}c_{k}T_{{\mathfrak{s}}k}u$ is a well-
defined function in $\mathcal{V}_{\mathfrak{s}}(u)$ and no convergence issues
appear. Care should be taken if $\\{c_{k}\\}\notin c_{00}({\mathbb{Z}})$:
since we did not make any additional assumptions on the generating function
$u\in{L^{2}({\mathbb{R}})}$ (in particular, we are not assuming that
$\\{T_{{\mathfrak{s}}k}u:k\in{\mathbb{Z}}\\}$ is a Riesz basis or frame for
$\mathcal{V}_{\mathfrak{s}}(u)$), we may not be able to represent every
function $f\in\mathcal{V}_{\mathfrak{s}}(u)$ in the form
$\sum_{k\in{\mathbb{Z}}}c_{k}T_{{\mathfrak{s}}k}u$. An assumption which
guarantees unconditional convergence for every square-summable sequence
$\\{c_{k}\\}\in\ell^{2}({\mathbb{Z}})$ is the requirement that
$\\{T_{{\mathfrak{s}}k}u:k\in{\mathbb{Z}}\\}$ is a Bessel sequence [7,
Corollary 3.2.5]. Recall that a sequence $\\{x_{k}\\}$ in a Hilbert space $H$
is a Bessel sequence if

$\forall x\in H:\sum_{k\in{\mathbb{Z}}}|\langle x,x_{k}\rangle|^{2}<\infty.$

######  Theorem 2.4.

If $\\{x_{k}:k\in{\mathbb{Z}}\\}$ is a Bessel sequence in a Hilbert space $H$
then for every $\\{c_{k}\\}\in\ell^{2}({\mathbb{Z}})$ the series
$\sum_{k\in{\mathbb{Z}}}c_{k}x_{k}$ converges unconditionally in $H$.

If one would apply the Fourier transform to $\mathcal{V}_{\mathfrak{s}}(u)$
then the resulting space

$\mathcal{M}_{\mathfrak{s}}(u)=\left\\{{\mathcal{F}}h:h\in\mathcal{V}_{\mathfrak{s}}(u)\right\\}$

is modulation invariant (or: invariant under a Fourier shift), i.e.
$M_{{\mathfrak{s}}n}h\in\mathcal{M}_{\mathfrak{s}}(u)$ for every
$n\in{\mathbb{Z}}$ provided that $h\in\mathcal{M}_{\mathfrak{s}}(u)$. Now
observe that

$T_{{\mathfrak{s}}k}=\mathrm{Id}\,T_{{\mathfrak{s}}k}\,\mathrm{Id},\ \
M_{{\mathfrak{s}}k}={\mathcal{F}}^{-1}T_{{\mathfrak{s}}k}{\mathcal{F}},$

or, in the language of fractional Fourier transform,

$T_{{\mathfrak{s}}k}={\mathcal{F}}_{0}T_{{\mathfrak{s}}k}{\mathcal{F}}_{0},\ \
M_{{\mathfrak{s}}k}={\mathcal{F}}_{-\frac{\pi}{2}}T_{{\mathfrak{s}}k}{\mathcal{F}}_{\frac{\pi}{2}}.$

The previous identities motivate the definition of a fractional Fourier shift.

######  Definition 2.5.

Let $\tau,\theta\in{\mathbb{R}}$. The operator $T_{\tau}^{\theta}$, defined by

$T_{\tau}^{\theta}\coloneqq{\mathcal{F}}_{-\theta}T_{\tau}{\mathcal{F}}_{\theta}$

is called the fractional Fourier shift by $\tau$ of order $\theta$.

We have $T_{\tau}^{0}=T_{\tau}$ and $T_{\tau}^{\frac{\pi}{2}}=M_{\tau}$.
Hence, the operator $T_{\tau}^{\theta}$ interpolates between the ordinary
shift and the modulation, i.e. a Fourier shift. In a natural way, the
fractional Fourier shift extends the concept of classical shift-invariant
spaces.

######  Definition 2.6.

Let $u\in{L^{2}({\mathbb{R}})}$ be a generating function and
${\mathfrak{s}}>0$. The space

$\mathcal{V}_{\mathfrak{s}}^{\theta}(u)\coloneqq\overline{{\mathrm{span}}\left\\{T_{{\mathfrak{s}}k}^{\theta}u:k\in{\mathbb{Z}}\right\\}}$

is called the shift-invariant space associated to the fractional Fourier shift
generated by $u$ and step-size ${\mathfrak{s}}>0$. In the case $\theta=0$ we
have $\mathcal{V}_{\mathfrak{s}}^{0}(u)=\mathcal{V}_{\mathfrak{s}}(u)$ with
$\mathcal{V}_{\mathfrak{s}}(u)$ being defined as in (3).

For details and properties on shift-invariant spaces associated to the
fractional Fourier shift, the reader may consult the article [4] and the
references therein. A direct application of Theorem 2.4 gives the following
statement.

######  Corollary 2.7.

Let $u\in{L^{2}({\mathbb{R}})}$ be a generating function, ${\mathfrak{s}}>0$ a
step-size and $\theta\in{\mathbb{R}}$ an order. If the system of fractional
Fourier shifts $\\{T_{{\mathfrak{s}}k}^{\theta}u:k\in{\mathbb{Z}}\\}$
constitutes a Bessel sequence in ${L^{2}({\mathbb{R}})}$ then for every
$\\{c_{k}\\}\in\ell^{2}({\mathbb{Z}})$ the series
$\sum_{k\in{\mathbb{Z}}}c_{k}T_{{\mathfrak{s}}k}^{\theta}u$ converges
unconditionally to an element in $\mathcal{V}_{\mathfrak{s}}^{\theta}(u)$.

##  3\. Main results

In this section, we present and prove the main results of the paper. We derive
functions $f_{1},f_{2}\in{L^{2}({\mathbb{R}})}$ which have the property that
their spectrograms agree on certain selected subsets of the time-frequency
plane. Moreover, we discuss under which assumptions such functions do not
agree up to a global phase and which conditions imply that they can be chosen
to be real-valued.

###  3.1. Identical spectrograms and non-equivalence conditions

Let $u\in{L^{2}({\mathbb{R}})},{\mathfrak{s}}>0$ and $\theta\in{\mathbb{R}}$.
In the sequel we shall fix the following notation: whenever
$f\in\mathcal{V}_{\mathfrak{s}}^{\theta}(u)$ has a representation of the form
$f=\sum_{k\in{\mathbb{Z}}}c_{k}T_{{\mathfrak{s}}k}^{\theta}u$ with a defining
sequence $\\{c_{k}\\}\subset{\mathbb{C}}$ then the map $f_{\times}$ is defined
by

$f_{\times}\coloneqq\sum_{k\in{\mathbb{Z}}}\overline{c_{k}}T_{{\mathfrak{s}}k}^{\theta}u,$

i.e. $f_{\times}$ arises from $f$ via complex conjugation of the defining
sequence $\\{c_{k}\\}$. We now show that a suitable choice of the generator
$u$ implies equality of the spectrogram of $f$ and $f_{\times}$ on parallel
lines of the form

(4) $R_{\theta}({\mathbb{R}}\times
h{\mathbb{Z}})\coloneqq\left\\{\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}\begin{pmatrix}x\\\
\omega\end{pmatrix}:x\in{\mathbb{R}},\,\omega\in h{\mathbb{Z}}\right\\},\ \
h>0.$

######  Theorem 3.1.

Let $g\in{L^{2}({\mathbb{R}})}$ be a window function, ${\mathfrak{s}}>0$ a
step-size and $\theta\in{\mathbb{R}}$ an order. If
$f\in\mathcal{V}_{\mathfrak{s}}^{\theta}(\mathcal{R}g)$ has defining sequence
$\\{c_{k}\\}\in c_{00}({\mathbb{Z}})$ then

$|V_{g}f(R_{\theta}({\mathbb{R}}\times\tfrac{1}{{\mathfrak{s}}}{\mathbb{Z}}))|=|V_{g}(f_{\times})(R_{\theta}({\mathbb{R}}\times\tfrac{1}{{\mathfrak{s}}}{\mathbb{Z}}))|.$

If, in addition, the system
$\\{T_{{\mathfrak{s}}k}^{\theta}(\mathcal{R}g):k\in{\mathbb{Z}}\\}$ is a
Bessel sequence then the same conclusion holds true, provided that $f$ has
defining sequence $\\{c_{k}\\}\in\ell^{2}({\mathbb{Z}})$.

###### Proof.

Let $(x,\omega)\in{\mathbb{R}}^{2}$ and let $R_{\theta}$ be the rotation
matrix as defined in equation (2). Suppose that
$f\in\mathcal{V}_{\mathfrak{s}}^{\theta}(\mathcal{R}g)$ has defining sequence
$\\{c_{k}\\}\in c_{00}({\mathbb{Z}})$. Invoking the definition of the
fractional Fourier shift, $f$ can be written as

$f={\mathcal{F}}_{-\theta}\sum_{k\in{\mathbb{Z}}}c_{k}T_{{\mathfrak{s}}k}{\mathcal{F}}_{\theta}\mathcal{R}g.$

We now evaluate the spectrogram of $f$ with respect to the window function $g$
at a $\theta$-rotation of $(x,\omega)$. Consulting the properties of the
fractional Fourier transform as presented in Lemma 2.2 yields

$\displaystyle|V_{g}f(R_{\theta}(x,\omega))|$
$\displaystyle\mathrel{\overset{\scriptsize{\eqmakebox[0]{Theorem\ref{thm:frac_rotation}}}}{=}}\left|V_{{\mathcal{F}}_{\theta}g}\left(\sum_{k\in{\mathbb{Z}}}c_{k}T_{{\mathfrak{s}}k}{\mathcal{F}}_{\theta}\mathcal{R}g\right)(x,\omega)\right|$
$\displaystyle\mathrel{\overset{\scriptsize{\eqmakebox[0]{linearity}}}{=}}\left|\sum_{k\in{\mathbb{Z}}}c_{k}V_{{\mathcal{F}}_{\theta}g}(T_{{\mathfrak{s}}k}{\mathcal{F}}_{\theta}\mathcal{R}g)(x,\omega)\right|$
$\displaystyle\mathrel{\overset{\scriptsize{\eqmakebox[0]{Lemma\ref{lemma:comm_relations}(6)}}}{=}}\left|\sum_{k\in{\mathbb{Z}}}c_{k}e^{-2\pi
i{\mathfrak{s}}k\omega}V_{{\mathcal{F}}_{\theta}g}({\mathcal{F}}_{\theta}\mathcal{R}g)(x-{\mathfrak{s}}k,\omega)\right|$
$\displaystyle\mathrel{\overset{\scriptsize{\eqmakebox[0]{}}}{=}}\left|\sum_{k\in{\mathbb{Z}}}c_{k}e^{-2\pi
i{\mathfrak{s}}k\omega}{\mathcal{F}}\left(({\mathcal{F}}_{\theta}\mathcal{R}g)(\overline{T_{x-{\mathfrak{s}}k}{\mathcal{F}}_{\theta}g})\right)(\omega)\right|$
(5)
$\displaystyle\mathrel{\overset{\scriptsize{\eqmakebox[0]{Lemma\ref{lemma:fracft_properties}(4)}}}{=}}\left|\sum_{k\in{\mathbb{Z}}}c_{k}e^{-2\pi
i{\mathfrak{s}}k\omega}{\mathcal{F}}\left(({\mathcal{F}}_{\theta}\mathcal{R}g)(T_{x-{\mathfrak{s}}k}{\mathcal{F}}_{-\theta}\overline{g})\right)(\omega)\right|.$

We continue by inspecting the spectrogram of $f_{\times}$. Replacing $c_{k}$
with $\overline{c_{k}}$ in equation (3.1) shows that the spectrogram of
$f_{\times}$ satisfies

$\displaystyle|V_{g}(f_{\times})(R_{\theta}(x,\omega))|$
$\displaystyle\mathrel{\overset{\scriptsize{\eqmakebox[0]{}}}{=}}\left|\sum_{k\in{\mathbb{Z}}}\overline{c_{k}}e^{-2\pi
i{\mathfrak{s}}k\omega}{\mathcal{F}}\left(({\mathcal{F}}_{\theta}\mathcal{R}g)(T_{x-{\mathfrak{s}}k}{\mathcal{F}}_{-\theta}\overline{g})\right)(\omega)\right|$
$\displaystyle\mathrel{\overset{\scriptsize{\eqmakebox[0]{}}}{=}}\left|\sum_{k\in{\mathbb{Z}}}c_{k}e^{2\pi
i{\mathfrak{s}}k\omega}\overline{{\mathcal{F}}\left(({\mathcal{F}}_{\theta}\mathcal{R}g)(T_{x-{\mathfrak{s}}k}{\mathcal{F}}_{-\theta}\overline{g})\right)(\omega)}\right|$
$\displaystyle\mathrel{\overset{\scriptsize{\eqmakebox[0]{Lemma\ref{lemma:comm_relations}(5)}}}{=}}\left|\sum_{k\in{\mathbb{Z}}}c_{k}e^{2\pi
i{\mathfrak{s}}k\omega}\mathcal{R}{\mathcal{F}}\left(\overline{({\mathcal{F}}_{\theta}\mathcal{R}g)(T_{x-{\mathfrak{s}}k}{\mathcal{F}}_{-\theta}\overline{g})}\right)(\omega)\right|$
$\displaystyle\mathrel{\overset{\scriptsize{\eqmakebox[0]{Lemma\ref{lemma:fracft_properties}(4)}}}{=}}\left|\sum_{k\in{\mathbb{Z}}}c_{k}e^{2\pi
i{\mathfrak{s}}k\omega}\mathcal{R}{\mathcal{F}}\left(({\mathcal{F}}_{-\theta}\mathcal{R}\overline{g})(T_{x-{\mathfrak{s}}k}{\mathcal{F}}_{\theta}g)\right)(\omega)\right|$
$\displaystyle\mathrel{\overset{\scriptsize{\eqmakebox[0]{Lemma\ref{lemma:fracft_properties}(3)}}}{=}}\left|\sum_{k\in{\mathbb{Z}}}c_{k}e^{2\pi
i{\mathfrak{s}}k\omega}{\mathcal{F}}\left(\mathcal{R}[({\mathcal{F}}_{-\theta}\mathcal{R}\overline{g})(T_{x-{\mathfrak{s}}k}{\mathcal{F}}_{\theta}g)]\right)(\omega)\right|$
$\displaystyle\mathrel{\overset{\scriptsize{\eqmakebox[0]{Lemma\ref{lemma:fracft_properties}(3)}}}{=}}\left|\sum_{k\in{\mathbb{Z}}}c_{k}e^{2\pi
i{\mathfrak{s}}k\omega}{\mathcal{F}}\left(({\mathcal{F}}_{-\theta}\overline{g})(T_{{\mathfrak{s}}k-x}{\mathcal{F}}_{\theta}\mathcal{R}g)\right)(\omega)\right|$
$\displaystyle\mathrel{\overset{\scriptsize{\eqmakebox[0]{Lemma\ref{lemma:comm_relations}(2)}}}{=}}\left|\sum_{k\in{\mathbb{Z}}}c_{k}e^{2\pi
i{\mathfrak{s}}k\omega}e^{-2\pi
i\omega({\mathfrak{s}}k-x)}{\mathcal{F}}\left((T_{x-{\mathfrak{s}}k}{\mathcal{F}}_{-\theta}\overline{g})({\mathcal{F}}_{\theta}\mathcal{R}g)\right)(\omega)\right|$
$\displaystyle\mathrel{\overset{\scriptsize{\eqmakebox[0]{}}}{=}}\left|\sum_{k\in{\mathbb{Z}}}c_{k}{\mathcal{F}}\left((T_{x-{\mathfrak{s}}k}{\mathcal{F}}_{-\theta}\overline{g})({\mathcal{F}}_{\theta}\mathcal{R}g)\right)(\omega)\right|.$

Now let $\omega=\frac{n}{{\mathfrak{s}}}$ for some $n\in{\mathbb{Z}}$. In this
case, the term $e^{-2\pi i{\mathfrak{s}}k\omega}$ appearing in the equation
(3.1) simplifies to $e^{-2\pi i{\mathfrak{s}}k\omega}=e^{-2\pi ikn}=1$. This
shows that
$|V_{g}f(R_{\theta}(x,\tfrac{n}{{\mathfrak{s}}}))|=|V_{g}(f_{\times})(R_{\theta}(x,\tfrac{n}{{\mathfrak{s}}}))|$
for every $n\in{\mathbb{Z}}$. Since $x\in{\mathbb{R}}$ was arbitrary, we
obtain the desired equality of the two spectrograms on the parallel lines
$R_{\theta}({\mathbb{R}}\times\tfrac{1}{{\mathfrak{s}}}{\mathbb{Z}})$. Under
the additional assumption that the system
$\\{T_{{\mathfrak{s}}k}^{\theta}(\mathcal{R}g):k\in{\mathbb{Z}}\\}$ forms a
Bessel sequence, Corollary 2.7 implies that for every
$\\{c_{k}\\}\in\ell^{2}({\mathbb{Z}})$ the series
$\sum_{k\in{\mathbb{Z}}}c_{k}T_{{\mathfrak{s}}k}\mathcal{R}g$ converges
unconditionally in ${L^{2}({\mathbb{R}})}$. Since
$\\{\overline{c_{k}}\\}\in\ell^{2}({\mathbb{Z}})$ if and only if
$\\{c_{k}\\}\in\ell^{2}({\mathbb{Z}})$, the same holds for the series
$\sum_{k\in{\mathbb{Z}}}\overline{c_{k}}T_{{\mathfrak{s}}k}\mathcal{R}g$. By
continuity and linearity of the fractional Fourier transform, the exact same
argument as above implies equality of the spectrogram of $f$ and $f_{\times}$
on $R_{\theta}({\mathbb{R}}\times\tfrac{1}{{\mathfrak{s}}}{\mathbb{Z}})$. ∎

The statement obtained in Theorem 3.1 is visualized in Figure 2 and Figure 3.
The functions $f$ and $f_{\times}$ defined in Theorem 3.1 constitute
candidates of functions for which their spectrograms agree on parallel lines
whereas the functions themselves might not agree up to a global phase. We now
examine criteria on the defining sequence $\\{c_{k}\\}$ of $f$ so that $f$ and
$f_{\times}$ do not agree up to a global phase. Before doing so, we recall the
concept of $\omega$-independence [24, p. 34].

######  Definition 3.2.

A sequence $\\{x_{n}\\}_{n\in{\mathbb{Z}}}$ of elements in a Banach space $X$
is said to be $\omega$-independent if the equality

(6) $\sum_{n\in{\mathbb{Z}}}c_{n}x_{n}=0$

is possible only for $c_{n}=0,n\in{\mathbb{Z}}$.

Recall that in the definition of a linearly independent system, the norm-
convergent series in (6) is replaced by a finite sum. Systems of translates of
a function which constitute an $\omega$-independent sequence were studied in
[23]. We continue by introducing a class of sequences which obeys a geometric
property.

######  Definition 3.3.

The subset $\ell^{2}_{\mathcal{O}}({\mathbb{Z}})\subset\ell^{2}({\mathbb{Z}})$
of square-summable sequences whose elements do not lie on a line in the
complex plane passing through the origin is defined as

$\ell^{2}_{\mathcal{O}}({\mathbb{Z}})\coloneqq\left\\{\\{c_{k}\\}\in\ell^{2}({\mathbb{Z}}):\nexists\,\alpha\in{\mathbb{R}}\
\mathrm{s.t.}\ \\{c_{k}\\}\subset e^{i\alpha}{\mathbb{R}}\right\\}.$

Note that $\ell^{2}_{\mathcal{O}}({\mathbb{Z}})$ is invariant under complex
conjugation, i.e. $\\{c_{k}\\}\in\ell^{2}_{\mathcal{O}}({\mathbb{Z}})$ if and
only if $\\{\overline{c_{k}}\\}\in\ell^{2}_{\mathcal{O}}({\mathbb{Z}})$. In
order to show that sequences in $\ell^{2}_{\mathcal{O}}({\mathbb{Z}})$ produce
functions $f$ and $f_{\times}$ which do not agree up to a global phase, we
make use of the following simple observation.

######  Lemma 3.4.

Let $Y$ be an arbitrary set and $f:Y\to{\mathbb{C}}$ a complex-valued map.
Then $f\sim\overline{f}$ if and only if there exists an
$\alpha\in{\mathbb{R}}$ such that $f(y)\in e^{i\alpha}{\mathbb{R}}$ for every
$y\in Y$.

###### Proof.

We first prove the sufficiency of the property $f\sim\overline{f}$. To this
end, let $\nu\in{\mathbb{R}}$ such that $f(y)=e^{i\nu}\overline{f(y)}$ for
every $y\in Y$. Let $f(y)=e^{i\phi(y)}|f(y)|$ and
$\overline{f(y)}=e^{-i\phi(y)}|f(y)|$ with $\phi$ being a phase function
corresponding to $f$. Let $\Lambda=\\{y\in Y:f(y)\neq 0\\}$. It follows that

$e^{i(2\phi(y)-\nu)}=1$

for every $y\in\Lambda$ which implies that there exists an
$n(y)\in{\mathbb{Z}}$ so that $\phi(y)=\pi n(y)+\tfrac{\nu}{2}$. Thus,

$f(y)=e^{i\pi
n(y)}e^{i\frac{\nu}{2}}|f(y)|=e^{i\frac{\nu}{2}}\left((-1)^{n(y)}|f(y)|\right)$

which shows that $f$ takes values on the line
$e^{i\frac{\nu}{2}}{\mathbb{R}}$. Conversely, suppose that $f$ takes values on
a line $e^{i\alpha}{\mathbb{R}}$. Then
$f=e^{i\alpha}|f|=e^{2i\alpha}\overline{f}$ which implies that $f$ and
$\overline{f}$ agree up to a global phase. ∎

With the aid of the previous Lemma we can now show that the assumption on a
defining sequence belonging to $\ell^{2}_{\mathcal{O}}({\mathbb{Z}})$ gives
rise to non-equivalent function pairs. This is the main content of

######  Theorem 3.5.

Let $0\neq u\in{L^{2}({\mathbb{R}})},{\mathfrak{s}}>0$ and suppose that
$f\in\mathcal{V}_{\mathfrak{s}}^{\theta}(u)$ has defining sequence
$\\{c_{k}\\}\subset{\mathbb{C}}$. Then the following holds:

  1. (1)

If $\\{c_{k}\\}\in\ell^{2}_{\mathcal{O}}({\mathbb{Z}})\cap
c_{00}({\mathbb{Z}})$ then $f\nsim f_{\times}$.

  2. (2)

If the generating function $u$ has the property that the system of fractional
Fourier translates $\\{T_{{\mathfrak{s}}k}^{\theta}u:k\in{\mathbb{Z}}\\}$
forms an $\omega$-independent Bessel sequence then

$\\{c_{k}\\}\in\ell^{2}_{\mathcal{O}}({\mathbb{Z}})\implies f\nsim
f_{\times}.$

(a)

(b)

Figure 2. Visualisation of the conclusion of Theorem 3.1: the window $g$ is
chosen to be the first Hermite basis function, $g=H_{1}$. The function
$f\in\mathcal{V}_{\mathfrak{s}}^{\theta}(\mathcal{R}g)$ is chosen to have
defining sequence $\\{c_{k}\\}\in\ell^{2}_{\mathcal{O}}({\mathbb{Z}})$ with
$c_{-1}=1,c_{1}=i$ and $c_{k}=0$ for $k\in{\mathbb{Z}}\setminus\\{-1,1\\}$.
The constant ${\mathfrak{s}}$ is set to one. The above plots are contour plots
of the function $Q\coloneqq||V_{g}f|^{2}-|V_{g}(f_{\times})|^{2}|$ on the
rectangle $[-2,2]\times[-2,2]$. We observe, that $Q$ vanishes on parallel
lines of the form
$R_{\theta}({\mathbb{R}}\times\tfrac{1}{2}{\mathbb{Z}})\supset
R_{\theta}({\mathbb{R}}\times{\mathbb{Z}})$. In plot (A) we set $\theta=0$ and
in plot (B) we set $\theta=1$. Note that plot (B) is simply a rotation of (A)
since $H_{1}$ is an eigenfunction of the fractional Fourier transform.

(a)

(b)

Figure 3. Visualisation of the conclusion of Theorem 3.1 with window function
$g(t)\coloneqq e^{-|t|}+\frac{1}{1+t^{2}}$. The function
$f\in\mathcal{V}_{\mathfrak{s}}^{\theta}(\mathcal{R}g)$ is chosen to have
defining sequence $\\{c_{k}\\}\in\ell^{2}_{\mathcal{O}}({\mathbb{Z}})$ with
$c_{-1}=1,c_{1}=i$ and $c_{k}=0$ for $k\in{\mathbb{Z}}\setminus\\{-1,1\\}$.
The constant ${\mathfrak{s}}$ is set to one. The above plots are contour plots
of the function $Q\coloneqq||V_{g}f|^{2}-|V_{g}(f_{\times})|^{2}|$ on the
rectangle $[-2,2]\times[2,5]$ resp. $[-2,2]\times[-2,2]$. We observe, that $Q$
vanishes on parallel lines of the form
$R_{\theta}({\mathbb{R}}\times\tfrac{1}{2}{\mathbb{Z}})\supset
R_{\theta}({\mathbb{R}}\times{\mathbb{Z}})$. In plot (A) we set $\theta=0$ and
in plot (B) we set $\theta=-1$.

###### Proof.

(1) Assume by contradiction that $f\sim f_{\times}$ and let
$\nu\in{\mathbb{T}}$ such that $f=\nu f_{\times}$. Further, let
$J\coloneqq\\{k\in{\mathbb{Z}}:c_{k}\neq 0\\}$. Since $\\{c_{k}\\}\in
c_{00}({\mathbb{Z}})$, the set $J$ is finite and we have

$\sum_{k\in
J}(c_{k}-\nu\overline{c_{k}})T_{{\mathfrak{s}}k}^{\theta}u=\sum_{k\in
J}(c_{k}-\nu\overline{c_{k}}){\mathcal{F}}_{-\theta}T_{{\mathfrak{s}}k}{\mathcal{F}}_{\theta}u=0.$

The invertibility of the fractional Fourier transform implies that

$\sum_{k\in
J}(c_{k}-\nu\overline{c_{k}})T_{{\mathfrak{s}}k}({\mathcal{F}}_{\theta}u)=0.$

Since $0\neq u\in{L^{2}({\mathbb{R}})}$ it holds that
${\mathcal{F}}_{\theta}u\neq 0$. Therefore, the system of translates
$\\{T_{{\mathfrak{s}}k}({\mathcal{F}}_{\theta}u):k\in{\mathbb{Z}}\\}$ is a
linearly independent system in the vector space ${L^{2}({\mathbb{R}})}$ [7,
Proposition 9.6.2]. From the additional property that $|J|<\infty$, it follows
that

(7) $c_{k}-\nu\overline{c_{k}}=0\ \ \ \forall k\in J.$

Equation (7) implies that the map $C:J\to{\mathbb{C}}$, defined by
$C(k)=c_{k}$, is equivalent to its complex conjugate, $C\sim\overline{C}$. By
Lemma 3.4, the values $\\{c_{k}\\}$ lie on a line in the complex plane passing
through the origin, contradicting the assumption that
$\\{c_{k}\\}\in\ell^{2}_{\mathcal{O}}({\mathbb{Z}})$.

(2) Since $\\{T_{{\mathfrak{s}}k}^{\theta}u:k\in{\mathbb{Z}}\\}$ is a Bessel
sequence and
$\ell_{\mathcal{O}}^{2}({\mathbb{Z}})\subset\ell^{2}({\mathbb{Z}})$, the
series $f=\sum_{k\in{\mathbb{Z}}}c_{k}T_{{\mathfrak{s}}k}^{\theta}u$ and
$f_{\times}=\sum_{k\in{\mathbb{Z}}}\overline{c_{k}}T_{{\mathfrak{s}}k}^{\theta}u$
converge unconditionally by Corollary 2.7. In a similar fashion as in the
first part of the present proof, we assume by contradiction that $f\sim
f_{\times}$. This implies that there exists a $\nu\in{\mathbb{T}}$ such that

$\sum_{k\in{\mathbb{Z}}}(c_{k}-\nu\overline{c_{k}})T_{{\mathfrak{s}}k}^{\theta}u=0.$

By assumption, the system
$\\{T_{{\mathfrak{s}}k}^{\theta}u:k\in{\mathbb{Z}}\\}$ is
$\omega$-independent. Hence, $c_{k}=\nu\overline{c_{k}}$ for every
$k\in{\mathbb{Z}}$ which shows that the points $\\{c_{k}\\}$ lie on a line in
the complex plane passing through the origin. This is a contradiction to the
assumption that $\\{c_{k}\\}\in\ell^{2}_{\mathcal{O}}({\mathbb{Z}})$. ∎

######  Remark 3.6 ($\omega$-independent Bessel sequences: necessary and
sufficient conditions).

Let $u\in{L^{2}({\mathbb{R}})}$. Since ${\mathcal{F}}_{\theta}$ is a unitary
operator, it follows from the definition of the fractional Fourier shift that
the system $\\{T_{{\mathfrak{s}}k}^{\theta}u:k\in{\mathbb{Z}}\\}$ is an
$\omega$-independent Bessel sequence if and only if the system
$\\{T_{{\mathfrak{s}}k}h:k\in{\mathbb{Z}}\\}$ is an $\omega$-independent
Bessel sequence where $h\coloneqq{\mathcal{F}}_{\theta}u$. Assume for the sake
of simplicity that ${\mathfrak{s}}=1$. Then $\\{T_{k}h:k\in{\mathbb{Z}}\\}$ is
a Bessel sequence if and only if the _periodization_ $\Phi_{h}$ of
$|{\mathcal{F}}h|^{2}$, defined by

$\Phi_{h}(t)=\sum_{k\in{\mathbb{Z}}}|{\mathcal{F}}h(t+k)|^{2},$

is an element of $L^{\infty}[0,1]$ [15, Theorem 10.19]. Assume, in addition,
that $h\in W_{0}$ where $W_{0}$ denotes the Wiener amalgam space,

$W_{0}=\left\\{h\in
C({\mathbb{R}}):\sum_{k\in{\mathbb{Z}}}\max_{t\in[k,k+1]}|h(t)|<\infty\right\\}.$

Then $\\{T_{k}h:k\in{\mathbb{Z}}\\}$ is $\ell^{\infty}$-independent, i.e.
$\sum_{k\in{\mathbb{Z}}}c_{k}T_{k}h\neq 0$ for every
$\\{c_{k}\\}\in\ell^{\infty}({\mathbb{Z}})\setminus\\{0\\}$, if and only if
the periodization $\Phi_{h}$ does not vanish on $[0,1]$ [10, Theorem 2.1].
This holds, in particular, under the assumptions of the previous theorem where
$\\{c_{k}\\}\in\ell^{2}({\mathbb{Z}})\subset\ell^{\infty}({\mathbb{Z}})$.

###  3.2. Consequences for STFT phase retrieval

At this juncture, we are prepared to transfer the foregoing considerations to
the uniqueness problem arising in STFT phase retrieval. Recall that
$(g,\mathcal{L})$ is said to be a uniqueness pair of the STFT phase retrieval
problem with window function $g\in{L^{2}({\mathbb{R}})}$ and sampling set
$\mathcal{L}\subseteq{\mathbb{R}}^{2}$ if every $f\in{L^{2}({\mathbb{R}})}$ is
determined up to a global phase by $|V_{g}f(\mathcal{L})|$ (see Definition
1.1). Call a set $\mathcal{P}\subset{\mathbb{R}}^{2}$ a set of shifted
parallel lines if

$\mathcal{P}=z+R_{\theta}({\mathbb{R}}\times h{\mathbb{Z}})$

for some $z\in{\mathbb{R}}^{2},h>0$ and some $\theta\in{\mathbb{R}}$
($R_{\theta}$ denotes the rotation matrix defined in equation (2)). Combining
the statements derived in Section 3.1 yields the following result.

######  Theorem 3.7.

Let $g\in{L^{2}({\mathbb{R}})}$ be a window function and
$\mathcal{P}\subset{\mathbb{R}}^{2}$. Then $(g,\mathcal{P})$ is never a
uniqueness pair of the STFT phase retrieval problem, provided that
$\mathcal{P}$ is a set of shifted parallel lines.

###### Proof.

The case $g=0$ is trivial. Therefore, assume that $g$ does not vanish
identically. Since $\mathcal{P}$ is a set of shifted parallel lines, there
exist constants $\theta\in{\mathbb{R}}$ and ${\mathfrak{s}}>0$ as well as a
vector $z=(a,b)\in{\mathbb{R}}^{2}$ such that

$\mathcal{P}=z+R_{\theta}({\mathbb{R}}\times\tfrac{1}{{\mathfrak{s}}}{\mathbb{Z}}).$

Case 1: $z=0$. Choose a function
$f\in\mathcal{V}_{\mathfrak{s}}^{\theta}(\mathcal{R}g)$ with defining sequence
belonging to the (non-empty) intersection
$\ell^{2}_{\mathcal{O}}({\mathbb{Z}})\cap c_{00}({\mathbb{Z}})$. By Theorem
3.1, we have
$|V_{g}f(R_{\theta}({\mathbb{R}}\times\tfrac{1}{{\mathfrak{s}}}{\mathbb{Z}}))|=|V_{g}(f_{\times})(R_{\theta}({\mathbb{R}}\times\tfrac{1}{{\mathfrak{s}}}{\mathbb{Z}}))|$
whereas $f\nsim f_{\times}$ by Theorem 3.5. This yields the assertion for
$z=0$.

Case 2: $z\neq 0$. If $p=(x,\omega)\in{\mathbb{R}}^{2}$ then the covariance
property of the STFT (Lemma 2.1(6)) shows that for every
$f\in{L^{2}({\mathbb{R}})}$ one has

$V_{g}f(z+p)=V_{g}f(a+x,b+\omega)=e^{-2\pi ia\omega}V_{g}(T_{-a}M_{-b}f)(p).$

Taking absolute values, it follows that if
$h_{1},h_{2}\in{L^{2}({\mathbb{R}})}$ are two functions which produce
identical spectrogram values on the parallel lines
$R_{\theta}({\mathbb{R}}\times\tfrac{1}{{\mathfrak{s}}}{\mathbb{Z}})$ then
$f_{1}\coloneqq M_{b}T_{a}h_{1},f_{2}\coloneqq M_{b}T_{a}h_{2}$ produce
identical spectrogram values on the shifted parallel lines
$\mathcal{P}=z+R_{\theta}({\mathbb{R}}\times\tfrac{1}{{\mathfrak{s}}}{\mathbb{Z}})$.
Moreover, one has $h_{1}\sim h_{2}$ if and only if $f_{1}\sim f_{2}$ (the
equivalence relation $\sim$ is invariant under time-frequency shifts).
Therefore, if $h\in\mathcal{V}_{\mathfrak{s}}^{\theta}(\mathcal{R}g)$ has
defining sequence in the intersection
$\ell^{2}_{\mathcal{O}}({\mathbb{Z}})\cap c_{00}({\mathbb{Z}})$, then the
spectrograms of $f_{1}=M_{b}T_{a}h$ and $f_{2}=M_{b}T_{a}(h_{\times})$ agree
on $\mathcal{P}$ whereas $f_{1}\nsim f_{2}$. ∎

Note that if the window function $g$ is assumed to be the centered Gaussian
$g(t)=e^{-\pi t^{2}}$ then it was shown in [1] that $(g,\mathcal{P})$ is not a
uniqueness pair provided that $\mathcal{P}$ is a set of shifted parallel
lines. In comparison, Theorem 3.7 above makes no assumption on the window
function $g$. In analogy to the definition of a set of shifted parallel lines
we say that $\mathcal{S}\subset{\mathbb{R}}^{2}$ is a shifted lattice if
$\mathcal{S}$ arises from an ordinary lattice
$\mathcal{L}=L{\mathbb{Z}}^{2},L\in\mathrm{GL}_{2}({\mathbb{R}}),$ via a
translation by a vector $z\in{\mathbb{R}}^{2}$, i.e.

$\mathcal{S}=z+\mathcal{L}.$

As a consequence of Theorem 3.7 we obtain the statement that lattices never
achieve uniqueness no matter how the window function $g$ is chosen.

######  Theorem 3.8.

Let $g\in{L^{2}({\mathbb{R}})}$ be a window function and
$\mathcal{S}\subset{\mathbb{R}}^{2}$. Then $(g,\mathcal{S})$ is never a
uniqueness pair of the STFT phase retrieval problem, provided that
$\mathcal{S}$ is a shifted lattice.

###### Proof.

Since $\mathcal{S}$ is a shifted lattice, there exist constants
$\theta\in{\mathbb{R}}$ and ${\mathfrak{s}}>0$ as well as a vector
$z\in{\mathbb{R}}^{2}$ such that

$\mathcal{S}\subset
z+R_{\theta}({\mathbb{R}}\times\tfrac{1}{{\mathfrak{s}}}{\mathbb{Z}}).$

Thus, the assertion follows from Theorem 3.7. ∎

######  Remark 3.9.

If $\mathcal{S}=z+L{\mathbb{Z}}^{2},z\in{\mathbb{R}}^{2},$ with a non-
invertible matrix $L\in{\mathbb{R}}^{2\times 2}$ then the column vectors of
$L$ are linearly dependent which implies that $\mathcal{S}$ is contained in a
single line. Clearly, the conclusions made in Theorem 3.8 are still valid in
this setting.

######  Remark 3.10.

A subset $\mathcal{C}\subseteq X$ of a (real or complex) vector space $X$ is
said to be a cone if $\kappa\mathcal{C}\subseteq\mathcal{C}$ for every
$\kappa>0$. A cone is called infinite-dimensional if it is not contained in
any finite-dimensional subspace of $X$. Observe that the intersection
$\ell^{2}_{\mathcal{O}}({\mathbb{Z}})\cap c_{00}({\mathbb{Z}})$ is a cone in
the vector space of complex sequences and this cone is infinite-dimensional.
Thus, for every ${\mathfrak{s}}>0,\theta\in{\mathbb{R}}$ and
$z=(a,b)\in{\mathbb{R}}^{2}$ the set

$\mathcal{K}_{\mathfrak{s}}^{\theta}(z,g)\coloneqq\left\\{M_{b}T_{a}\sum_{k\in{\mathbb{Z}}}c_{k}T_{{\mathfrak{s}}k}^{\theta}g:\\{c_{k}\\}\in\ell^{2}_{\mathcal{O}}({\mathbb{Z}})\cap
c_{00}({\mathbb{Z}})\right\\}$

is an infinite-dimensional cone in ${L^{2}({\mathbb{R}})}$. In an abstract
language, Theorem 3.7 and Theorem 3.8 imply that for every window function
$g\in{L^{2}({\mathbb{R}})}$ and every shifted lattice $\mathcal{S}$ (resp. set
of shifted parallel lines $\mathcal{P}$), one can associate an infinite-
dimensional cone of the form $\mathcal{K}_{\mathfrak{s}}^{\theta}(z,g)$ with
the following property: if $f_{1}\in\mathcal{K}_{\mathfrak{s}}^{\theta}(z,g)$
then there is an $f_{2}\in\mathcal{K}_{\mathfrak{s}}^{\theta}(z,g)$ such that

  1. (1)

$|V_{g}f_{1}(\mathcal{S})|=|V_{g}f_{2}(\mathcal{S})|$ (resp.
$|V_{g}f_{1}(\mathcal{P})|=|V_{g}f_{2}(\mathcal{P})|$)

  2. (2)

$f_{1}\nsim f_{2}$.

###  3.3. Real-valuedness

As outlined in the introduction, real-valued window functions are prominent
choices in many applications. Note that the previously derived function pairs
$f,f_{\times}$ which implied the non-uniqueness of the STFT phase retrieval
problem from lattice samples were, in general, complex-valued (even if the
window function is real-valued). We now examine the question of whether a
prior real-valuedness assumption on the underlying signal space achieves
uniqueness from lattice samples, i.e. one specializes the considered input
functions to belong to the space
$L^{2}({\mathbb{R}},{\mathbb{R}})=\left\\{f\in{L^{2}({\mathbb{R}})}:f\
\text{is real-valued}\right\\}.$ To that end, we first introduce the subspace
of Hermitian sequences.

######  Definition 3.11.

The subspace $\ell^{2}_{\mathcal{H}}({\mathbb{Z}})$ of Hermitian sequences is
defined by

$\ell^{2}_{\mathcal{H}}({\mathbb{Z}})\coloneqq\left\\{\\{c_{k}\\}\in\ell^{2}({\mathbb{Z}}):c_{k}=\overline{c_{-k}}\right\\}.$

As for the set $\ell^{2}_{\mathcal{O}}({\mathbb{Z}})$, the space
$\ell^{2}_{\mathcal{H}}({\mathbb{Z}})$ is invariant under complex conjugation.
Functions
$f\in\mathcal{V}_{\mathfrak{s}}(u)=\mathcal{V}_{\mathfrak{s}}^{0}(u)$, with
defining sequence belonging to $\ell^{2}_{\mathcal{H}}({\mathbb{Z}})$, have
the following property.

######  Proposition 3.12.

Let $u\in{L^{2}({\mathbb{R}})}$ and suppose that
$h\in\mathcal{V}_{\mathfrak{s}}(u)$. If $h$ has defining sequence
$\\{c_{k}\\}\in\ell^{2}_{\mathcal{H}}({\mathbb{Z}})\cap c_{00}({\mathbb{Z}})$
then ${\mathcal{F}}h$ is real-valued provided that ${\mathcal{F}}u$ is real-
valued. If, in addition, the system of translates
$\\{T_{{\mathfrak{s}}k}u:k\in{\mathbb{Z}}\\}$ forms a Bessel sequence, then
the same conclusion holds true, under the weaker assumption that
$\\{c_{k}\\}\in\ell^{2}_{\mathcal{H}}({\mathbb{Z}})$.

###### Proof.

Suppose that $h=\sum_{k\in{\mathbb{Z}}}c_{k}T_{{\mathfrak{s}}k}u$ is such that
$\\{c_{k}\\}\in\ell^{2}_{\mathcal{H}}({\mathbb{Z}})\cap c_{00}({\mathbb{Z}})$.
Then

$\begin{split}{\mathcal{F}}h=\sum_{k\in{\mathbb{Z}}}c_{k}M_{-{\mathfrak{s}}k}{\mathcal{F}}u=c_{0}{\mathcal{F}}u+\sum_{k=1}^{\infty}c_{k}M_{-{\mathfrak{s}}k}{\mathcal{F}}u+\sum_{k=1}^{\infty}c_{-k}M_{{\mathfrak{s}}k}{\mathcal{F}}u\end{split}$

and since ${\mathcal{F}}u$ is real-valued and
$\\{c_{k}\\}\in\ell^{2}_{\mathcal{H}}({\mathbb{Z}})$ we have
$c_{0}=\overline{c_{0}}$ which implies that $c_{0}\in{\mathbb{R}}$. Moreover,

$\sum_{k=1}^{\infty}c_{-k}M_{{\mathfrak{s}}k}{\mathcal{F}}u=\overline{\sum_{k=1}^{\infty}c_{k}M_{-{\mathfrak{s}}k}{\mathcal{F}}u},$

thereby proving the first part of the statement. To prove the second part,
assume that $\\{T_{{\mathfrak{s}}k}u:k\in{\mathbb{Z}}\\}$ forms a Bessel
sequence. By Corollary 2.7 the series
$\sum_{k\in{\mathbb{Z}}}c_{k}T_{{\mathfrak{s}}k}u$ converges unconditionally.
Invoking the continuity and linearity of the Fourier transform yields the
statement with the same argument as above. ∎

As an application of Proposition 3.12, we construct non-equivalent, real-
valued functions for which their spectrograms agree on certain lattices in the
time-frequency plane.

######  Theorem 3.13.

Let $g\in L^{2}({\mathbb{R}},{\mathbb{R}})$ be a real-valued window function
and let $\mathcal{L}=L{\mathbb{Z}}^{2}\subset{\mathbb{R}}^{2}$ be a lattice
with generating matrix

$L=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\mathrm{GL}_{2}({\mathbb{R}}).$

If $a,b$ are linearly dependent over ${\mathbb{Q}}$ and $c,d$ are linearly
dependent over ${\mathbb{Q}}$ then there exist two real-valued functions
$f_{1},f_{2}\in L^{2}({\mathbb{R}},{\mathbb{R}})$ with the property
$|V_{g}f_{1}(\mathcal{L})|=|V_{g}f_{2}(\mathcal{L})|$ but $f_{1}\nsim f_{2}$.

###### Proof.

The case $g=0$ is trivial. Therefore, assume in the following that $g\neq 0$
and let $\theta\in{\mathbb{R}}$.

Case 1: rectangular lattices. First assume that the lattice $\mathcal{L}$ is
rectangular, i.e. $\mathcal{L}=\alpha{\mathbb{Z}}\times\beta{\mathbb{Z}}$ for
some $\alpha,\beta\in{\mathbb{R}}\setminus\\{0\\}$. In this case,
$\mathcal{L}$ is generated by the diagonal matrix
$L=\mathrm{diag}(\alpha,\beta)$ which clearly satisfies the assumptions of the
Theorem. Noting that the intersection
$\ell^{2}_{\mathcal{O}}({\mathbb{Z}})\cap\ell^{2}_{\mathcal{H}}({\mathbb{Z}})\cap
c_{00}({\mathbb{Z}})$ is non-empty, we select a function
$h\in\mathcal{V}_{\mathfrak{s}}^{\theta}(\mathcal{R}g)$ with defining sequence
$\\{c_{k}\\}\in\ell^{2}_{\mathcal{O}}({\mathbb{Z}})\cap\ell^{2}_{\mathcal{H}}({\mathbb{Z}})\cap
c_{00}({\mathbb{Z}})$ and set

$f_{1}=h,\ \ f_{2}=h_{\times}.$

For $\theta=\frac{\pi}{2}$ we have

(8)
$f_{1}=\mathcal{R}\left({\mathcal{F}}\sum_{k\in{\mathbb{Z}}}c_{k}T_{{\mathfrak{s}}k}{\mathcal{F}}\mathcal{R}g\right),$

and $u\coloneqq{\mathcal{F}}\mathcal{R}g$ has a real-valued Fourier transform.
Identity (8) together with Proposition 3.12 shows that $f_{1}$ is real-valued.
Since
$\ell^{2}_{\mathcal{O}}({\mathbb{Z}}),\ell^{2}_{\mathcal{H}}({\mathbb{Z}})$
and $c_{00}({\mathbb{Z}})$ are all invariant under complex conjugation we
immediately conclude that $f_{2}$ is real-valued as well. Theorem 3.5 shows
that $f_{1}$ and $f_{2}$ do not agree up to a global phase. But for
${\mathfrak{s}}=\frac{1}{\alpha}$ we have

$R_{\theta}({\mathbb{R}}\times\tfrac{1}{{\mathfrak{s}}}{\mathbb{Z}})=R_{\frac{\pi}{2}}({\mathbb{R}}\times\tfrac{1}{{\mathfrak{s}}}{\mathbb{Z}})=\tfrac{1}{{\mathfrak{s}}}{\mathbb{Z}}\times{\mathbb{R}}\supset\alpha{\mathbb{Z}}\times\beta{\mathbb{Z}},$

whence $|V_{g}f_{1}(\mathcal{L})|=|V_{g}f_{2}(\mathcal{L})|$ by Theorem 3.1,
thereby proving the statement for rectangular lattices.

Case 2: generalization. Suppose that $\mathcal{L}=L{\mathbb{Z}}^{2}$ satisfies
the assumptions above, i.e. the row elements of $L$ are linearly dependent
over ${\mathbb{Q}}$. Hence, there exist $p,q\in{\mathbb{Q}}$ such that $a=pb$
and $c=qd$. Write $p\coloneqq\frac{p_{1}}{p_{2}}\in{\mathbb{Q}}$ with
$p_{1}\in{\mathbb{Z}},p_{2}\in{\mathbb{Z}}\setminus\\{0\\}$ and
$q\coloneqq\frac{q_{1}}{q_{2}}\in{\mathbb{Q}}$ with
$q_{1}\in{\mathbb{Z}},q_{2}\in{\mathbb{Z}}\setminus\\{0\\}$. Then for
$v=(v_{1},v_{2})\in{\mathbb{Z}}^{2}$ we have

$Lv=\begin{pmatrix}pbv_{1}+bv_{2}\\\
qdv_{1}+dv_{2}\end{pmatrix}=\underbrace{\begin{pmatrix}\frac{b}{p_{2}}&0\\\
0&\frac{d}{q_{2}}\end{pmatrix}}_{\coloneqq L^{\prime}}\begin{pmatrix}w_{1}\\\
w_{2}\end{pmatrix}$

with $w_{1}=p_{1}v_{1}+p_{2}v_{2}\in{\mathbb{Z}}$ and
$w_{2}=q_{1}v_{1}+q_{2}v_{2}\in{\mathbb{Z}}$. This shows that $\mathcal{L}$ is
contained in a rectangular lattice generated by $L^{\prime}$. Therefore, Case
1 applies and the existence of two real-valued, non-equivalent functions
$f_{1},f_{2}\in L^{2}({\mathbb{R}},{\mathbb{R}})$, for which their
spectrograms agree on $\mathcal{L}$, follows. ∎

##  4\. Conclusions and open problems

In the concluding section of the paper, we shall compare our results to
previous work and address open problems suggested by the statements made in
Section 3.

###  4.1. Gaussian windows

The authors of [1] show that in the case of the Gaussian window
$g(t)=\varphi(t)=e^{-\pi t^{2}}$, there exists no lattice of the form
$\mathcal{L}=\alpha{\mathbb{Z}}\times\beta{\mathbb{Z}}$ such that every
$f\in{L^{2}({\mathbb{R}})}$ is determined up to a unimodular constant by
$|V_{g}f(\mathcal{L})|$. In their article, the provided functions
$f_{1},f_{2}\in{L^{2}({\mathbb{R}})}$ which produce the same spectrogram
values on the parallel lines ${\mathbb{R}}\times\beta{\mathbb{Z}}$ but are not
equivalent, $f_{1}\nsim f_{2}$, are of the form

$\begin{split}f_{1}(t)&=e^{-\pi t^{2}}\left(\cosh(\tfrac{\pi
t}{\beta})+i\sinh(\tfrac{\pi t}{\beta})\right)\\\ f_{2}(t)&=e^{-\pi
t^{2}}\left(\cosh(\tfrac{\pi t}{\beta})-i\sinh(\tfrac{\pi
t}{\beta})\right).\end{split}$

A simple calculation shows that
$f_{1},f_{2}\in\mathcal{V}_{\frac{1}{2\beta}}(\varphi)$. Moreover, $f_{1}$ has
defining sequence $\\{c_{k}\\}\in c_{00}({\mathbb{Z}})$ given by

$c_{k}=\begin{cases}\frac{1-i}{2}e^{\frac{\pi}{4b^{2}}},&k=-1\\\
\frac{1+i}{2}e^{\frac{\pi}{4b^{2}}},&k=1\\\ 0,&\text{else}\end{cases}$

and the defining sequence of $f_{2}$ is given by $\\{\overline{c_{k}}\\}$. In
particular, one has
$\\{c_{k}\\}\in\ell^{2}_{\mathcal{O}}({\mathbb{Z}})\cap\ell^{2}_{\mathcal{H}}({\mathbb{Z}})\cap
c_{00}({\mathbb{Z}})$ and $f_{2}=(f_{1})_{\times}$. This shows that $f_{1}$
and $f_{2}$ are special cases of the functions derived in Section 3.

###  4.2. Restriction to proper subspaces and comparison to previous work

Theorem 3.8 states that no matter how the window function is chosen, a
discretization of the STFT phase retrieval problem from lattice samples or
parallel lines forces one to restrict the input space of functions to a proper
subspace or subset $S\subsetneq{L^{2}({\mathbb{R}})}$. This was successfully
done in previous work: in [12] it was shown that if the window function is a
Gaussian and the underlying signal space satisfies a mild support condition,
then sampling on a lattice implies uniqueness. Additionally, in the same
article, it was shown that if the signal space is a Gaussian shift-invariant
space with an irrational step-size, sampling on a lattice again suffices to
achieve uniqueness. In [13] it was further proved that without an assumption
on the step-size of the Gaussian shift-invariant space, the uniqueness
property holds, assuming knowledge of the spectrogram on parallel lines in the
time-frequency plane. Finally, the article [2] establishes uniqueness results
for the class of real-valued, band-limited functions from samples in a
1-dimensional lattice and window functions which satisfy a non-vanishing
condition in Fourier space.

###  4.3. Open problems

In this article, we have established foundational barriers in the
discretization of the STFT phase retrieval problem from lattice samples.
Theorem 3.8 showed that uniqueness from (shifted) lattices is never guaranteed
for the signal space ${L^{2}({\mathbb{R}})}$. A natural question arising at
this point is the inquiry if Theorem 3.13 also holds in this generality. For
instance, a hexagonal lattice is not covered by the assumptions of Theorem
3.13.

######  Problem 4.1.

Suppose that $g\in L^{2}({\mathbb{R}},{\mathbb{R}})$ is a real-valued window
function. Does there exist a lattice $\mathcal{L}\subset{\mathbb{R}}^{2}$ such
that every real-valued map $f\in L^{2}({\mathbb{R}},{\mathbb{R}})$ is
determined up to a sign by $|V_{g}f(\mathcal{L})|$?

A lattice is a special case of a separated (or: uniformly discrete) set.
Recall that this is a set $\mathcal{L}\subset{\mathbb{R}}^{2}$ with the
property

$\inf_{\begin{subarray}{c}\ell,\ell^{\prime}\in\mathcal{L}\\\
\ell\neq\ell^{\prime}\end{subarray}}|\ell-\ell^{\prime}|>0,$

where $|\ell-\ell^{\prime}|$ denotes the Euclidean distance of $\ell$ and
$\ell^{\prime}$. A fruitful question for future research suggested by the
results reported here is the following.

######  Problem 4.2.

Does there exist a window function $g\in{L^{2}({\mathbb{R}})}$ and a separated
set $\mathcal{L}\subset{\mathbb{R}}^{2}$ such that $(g,\mathcal{L})$ is a
uniqueness pair for the STFT phase retrieval problem?

A more narrow form of Problem 4.2 is the question of whether or not adjoining
a finite set of points $\\{p_{1},\dots,p_{N}\\}\subset{\mathbb{R}}^{2}$ to a
lattice $\mathcal{L}$ produces a sampling set with the uniqueness property.
Such questions arise in the completeness problem of complex exponentials [5,
17, 21] which is equivalent to the uniqueness problem in Paley-Wiener spaces.
Recall that a set $\Lambda\subset{\mathbb{R}}^{d},d\in{\mathbb{N}}$, is called
a uniqueness set for a subset $Q\subset C({\mathbb{R}}^{d})$ of the space of
continuous functions on ${\mathbb{R}}^{d}$ if the implication

$f(\lambda)=h(\lambda)\ \forall\lambda\in\Lambda\implies f(t)=h(t)\ \forall\in
t\in{\mathbb{R}}^{d}$

holds true for all $f,h\in Q$. The notion of exactness, excess, and deficiency
concerns economy [24, Chapter 3.1].

######  Definition 4.3.

Let $Q\subset C({\mathbb{R}}^{d}),d\in{\mathbb{N}}$, and
$\Lambda\subset{\mathbb{R}}^{d}$. Then $\Lambda$ is said to be exact in $Q$ if
it is a uniqueness set for $Q$ but fails to be a uniqueness set for $Q$ on the
removal of any one term of $\Lambda$. If $\Lambda$ becomes exact when $N$
terms are removed, then we say that it has excess $N$; if it becomes exact
when $N$ terms $\\{p_{1},\dots,p_{N}\\}\subset{\mathbb{R}}^{d}$ are adjoint,
then $\Lambda$ is said to have deficiency $N$. If it never becomes exact, then
we say that $\Lambda$ has deficiency $N=\infty$.

Now let $g\in{L^{2}({\mathbb{R}})}$ be a window function. The notions provided
in Definition 4.3 directly transfer to phase retrieval: we say that
$\Lambda\subset{\mathbb{R}}^{2}$ is a uniqueness set for the STFT phase
retrieval problem with deficiency $N\in{\mathbb{N}}_{0}\cup\\{\infty\\}$ if
there exists a set $\\{p_{1},\dots,p_{N}\\}\subset{\mathbb{R}}^{2}$ (the case
$N=\infty$ is included) such that every $f\in{L^{2}({\mathbb{R}})}$ is
determined up to a global phase by
$|V_{g}f(\Lambda\cup\\{p_{1},\dots,p_{N}\\})|$. According to Theorem 3.8,
every lattice has deficiency $N>0$.

######  Problem 4.4.

Does there exists a window functions $g\in{L^{2}({\mathbb{R}})}$ and a lattice
$\mathcal{L}\subset{\mathbb{R}}^{2}$ with deficiency $N<\infty$?

Note that a positive answer to Problem 4.4 provides a positive answer to
Problem 4.2.

Acknowledgements. The authors highly appreciate the valuable and helpful
comments made by the reviewers.

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