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 \(V_gf({\mathcal {L}})\) where \(V_gf\) denotes the shorttime Fourier transform of f with respect to g. Consequently, the forward operator
mapping a squareintegrable function to its spectrogram samples on a lattice is never injective on the quotient space 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 realvalued. 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.
1 Introduction
The problem of recovering a function \(f \in {L^2(\mathbb {R})}\) from the absolute value of its shorttime Fourier transform (STFT) has attracted a great deal of attention in recent years. This socalled 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
with \(V_gf\) denoting the shorttime Fourier transform of f with respect to the window function g, defined by
The modulus of the STFT, \(V_gf\), 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 \(\nu \in {{\mathbb {T}}}:=\{ z \in {{\mathbb {C}}}: z=1 \}\) of modulus one such that \(f = \nu h\), produce identical spectrograms. It follows that a reconstruction of f from \(V_gf(\mathcal {L})\) is only possible up to the ambiguity of a global phase factor. While the classical Fourier phase retrieval problem and Pauli problem suffer from nontrivial ambiguities or nonuniqueness, 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 \(V_gf(\mathcal {L})\), provided that \(\mathcal {L}\) is a continuous domain such as \(\mathcal {L}=\mathbb {R}^2\) [10, Sect. 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
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.
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 \(V_gf(\mathcal {L})\), i.e. the implication
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 \(f \mapsto V_gf(\mathcal {L})\) being injective on the the quotient space 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 Sect. 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 \(\det (L)\le 1\) there exists a window function \(g \in {L^2(\mathbb {R})}\) such that every \(f \in {L^2(\mathbb {R})}\) is uniquely determined by \(V_gf(\mathcal {L})\) [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, Sect. 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.
The conclusion of Theorem 1.3 is visualized in Fig. 1. Prominent choices of window functions in Gabor analysis and its applications have the property that they are realvalued (Gaussians, Hermite functions, Airy disc functions, Hanning windows, rectangular windows). One could inquire about the question of whether a realvaluedness assumption on the underlying signal space achieves uniqueness from lattice samples, i.e. one specializes the considered input functions to belong to the space
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 realvalued window function and \(\mathcal {L}=L{{\mathbb {Z}}}^2\) a lattice generated by
If a and b as well as c and d are linearly dependent over \({{\mathbb {Q}}}\) then there exist two realvalued functions \(f_1,f_2 \in L^2(\mathbb {R},\mathbb {R})\) which do not agree up to a global phase, \(f_1 \not \sim f_1\), but their spectrograms agree on \(\mathcal {L}\),
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 Sect. 2 we define frequently used operators and their interrelations. Besides, we present the necessary background on the fractional Fourier transform and shiftinvariant spaces. Based on this discussion we proceed in proving the main results of the article in Sect. 3. We conclude the article with Sect. 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 ShiftInvariant Spaces
In this section, we collect several preliminary definitions and results about operators arising in time–frequency analysis, the fractional Fourier transform, and shiftinvariant spaces.
2.1 Basic Operations
The shorttime 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
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 welldefined. The socalled spectrogram of f with respect to g is the modulus of the STFT, i.e. the map
Note that \(V_gf(x,\omega )\) 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
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 keyoperations in time–frequency analysis. For \(\tau ,\nu \in \mathbb {R}\), the translations operator \(T_\tau : {L^2(\mathbb {R})}\rightarrow {L^2(\mathbb {R})}\), the modulation operator \(M_\nu : {L^2(\mathbb {R})}\rightarrow {L^2(\mathbb {R})}\) and the reflection operator \(\mathcal {R} : {L^2(\mathbb {R})}\rightarrow {L^2(\mathbb {R})}\) are defined via
Using translation and modulation, the shorttime Fourier transform of f with respect to g simplifies to the expression \(V_gf(x,\omega ) = \langle f, M_\omega T_x g \rangle \) 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 [13, 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)
\(T_\tau M_\nu = e^{2\pi i \tau \nu } M_\nu T_\tau \)

(2)
\({{\mathcal {F}}}T_\tau = M_{\tau } {{\mathcal {F}}}\)

(3)
\({{\mathcal {F}}^{1}}T_\tau = M_\tau {{\mathcal {F}}^{1}}\)

(4)
\(T_\tau \mathcal {R} = \mathcal {R} T_{\tau }\)

(5)
\(\overline{{{\mathcal {F}}}f} = \mathcal {R} {{\mathcal {F}}}\overline{f}\).
Moreover, the STFT satisfies the covariance property

(6)
\(V_g(T_\tau M_\nu f)(x,\omega ) = e^{2\pi i \tau \omega } V_gf(x\tau , \omega \nu )\).
2.2 Fractional Fourier Transform
Denote by \(\{ H_n \}_{n \in {{\mathbb {N}}}_0}\) the Hermite basis functions on \(\mathbb {R}\), defined by
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 \({{\mathcal {F}}}_\theta : {L^2(\mathbb {R})}\rightarrow {L^2(\mathbb {R})}\),
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, Sect. 3]. The following properties of the fractional Fourier transform are used throughout the present article [19, Sect. 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)
\({{\mathcal {F}}}_0 = \mathrm {Id}, {{\mathcal {F}}}_\frac{\pi }{2} = {{\mathcal {F}}}, {{\mathcal {F}}}_{\frac{\pi }{2}} = {{\mathcal {F}}}^{1}\) and \({{\mathcal {F}}}_\pi = \mathcal {R}\)

(2)
\({{\mathcal {F}}}_{\theta +\xi } = {{\mathcal {F}}}_\theta {{\mathcal {F}}}_\xi \)

(3)
\({{\mathcal {F}}}_\theta \) commutes with \(\mathcal {R}\)

(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
The rotation property of the fractional Fourier transform reads as follows.
Theorem 2.3
If \(R_\theta \) denotes the rotation matrix as given in Eq. (2) then for every \(f,g\in {L^2(\mathbb {R})}\) and every \((x,\omega ) \in \mathbb {R}^2\) one has
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, Sect. 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 ShiftInvariant Spaces
In this section, we shall discuss shiftinvariant spaces, both in the classical sense as well as generalizations in the context of fractional Fourier transform. We start by introducing classical shiftinvariant 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,
is called the (principle) shiftinvariant space generated by u and stepsize \({\mathfrak {s}}>0\). Shiftinvariance 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 welldefined 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 squaresummable 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
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
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
or, in the language of fractional Fourier transform,
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
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 shiftinvariant spaces.
Definition 2.6
Let \(u \in {L^2(\mathbb {R})}\) be a generating function and \({\mathfrak {s}}>0\). The space
is called the shiftinvariant space associated to the fractional Fourier shift generated by u and stepsize \({\mathfrak {s}}>0\). 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 shiftinvariant 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 stepsize 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 realvalued.
3.1 Identical Spectrograms and Nonequivalence 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
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
Theorem 3.1
Let \(g \in {L^2(\mathbb {R})}\) be a window function, \({\mathfrak {s}}>0\) a stepsize 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
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 Eq. (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
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
We continue by inspecting the spectrogram of \(f_\times \). Replacing \(c_k\) with \(\overline{c_k}\) in Eq. (5) shows that the spectrogram of \(f_\times \) satisfies
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 Eq. (5) simplifies to \(e^{2 \pi i {\mathfrak {s}}k \omega } = e^{2 \pi i kn}=1\). This shows that \(V_gf(R_\theta (x,\tfrac{n}{{\mathfrak {s}}})) = V_g(f_\times )(R_\theta (x,\tfrac{n}{{\mathfrak {s}}}))\) 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 \(R_\theta (\mathbb {R}\times \tfrac{1}{{\mathfrak {s}}} {{\mathbb {Z}}})\). \(\square \)
The statement obtained in Theorem 3.1 is visualized in Figs. 2 and 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
is possible only for \(c_n = 0, n \in {{\mathbb {Z}}}\).
Recall that in the definition of a linearly independent system, the normconvergent 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 squaresummable sequences whose elements do not lie on a line in the complex plane passing through the origin is defined as
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 \(f : Y \rightarrow {{\mathbb {C}}}\) a complexvalued 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 \(\Lambda = \{ y \in Y : f(y) \ne 0 \}\). It follows that
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,
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. \(\square \)
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 nonequivalent function pairs. This is the main content of
Theorem 3.5
Let \(0 \ne u \in {L^2(\mathbb {R})}, {\mathfrak {s}}>0\) and suppose that \(f \in \mathcal {V}_{\mathfrak {s}}^\theta (u)\) has defining sequence \(\{ c_k \} \subset {{\mathbb {C}}}\). Then the following holds:

(1)
If \(\{ c_k \} \in \ell ^2_\mathcal {O}({{\mathbb {Z}}}) \cap c_{00}({{\mathbb {Z}}})\) then \(f \not \sim f_\times \).

(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
$$\begin{aligned} \{ c_k \} \in \ell ^2_\mathcal {O}({{\mathbb {Z}}}) \implies f \not \sim f_\times . \end{aligned}$$
Proof
(1) Assume by contradiction that \(f \sim f_\times \) and let \(\nu \in {{\mathbb {T}}}\) such that \(f = \nu f_\times \). Further, let \(J :=\{ k \in {{\mathbb {Z}}}: c_k \ne 0 \}\). Since \(\{ c_k \} \in c_{00}({{\mathbb {Z}}})\), the set J is finite and we have
The invertibility of the fractional Fourier transform implies that
Since \(0 \ne u \in {L^2(\mathbb {R})}\) it holds that \({{\mathcal {F}}}_\theta u \ne 0\). 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
Equation (7) implies that the map \(C : J \rightarrow {{\mathbb {C}}}\), 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
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 \(\{ c_k \} \in \ell ^2_\mathcal {O}({{\mathbb {Z}}})\). \(\square \)
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 \(h :={{\mathcal {F}}}_\theta u\). 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
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,
Then \(\{ T_k h : k \in {{\mathbb {Z}}}\}\) is \(\ell ^\infty \)independent, i.e. \(\sum _{k \in {{\mathbb {Z}}}} c_k T_k h \ne 0\) 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] [9, 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 \(V_gf(\mathcal {L})\) (see Definition 1.1). Call a set \(\mathcal {P} \subset \mathbb {R}^2\) a set of shifted parallel lines if
for some \(z \in \mathbb {R}^2, h>0\) and some \(\theta \in \mathbb {R}\) (\(R_\theta \) denotes the rotation matrix defined in Eq. (2)). Combining the statements derived in Sect. 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
Case 1: \(z=0\). Choose a function \(f \in \mathcal {V}_{\mathfrak {s}}^\theta (\mathcal {R}g)\) with defining sequence belonging to the (nonempty) intersection \(\ell ^2_\mathcal {O}({{\mathbb {Z}}}) \cap c_{00}({{\mathbb {Z}}})\). By Theorem 3.1, we have \(V_gf(R_\theta (\mathbb {R}\times \tfrac{1}{{\mathfrak {s}}}{{\mathbb {Z}}})) = V_g(f_\times )(R_\theta (\mathbb {R}\times \tfrac{1}{{\mathfrak {s}}}{{\mathbb {Z}}}))\) whereas \(f \not \sim f_\times \) by Theorem 3.5. This yields the assertion for \(z=0\).
Case 2: \(z \ne 0\). 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
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 \(f_1 :=M_bT_a h_1, f_2 :=M_b T_a h_2\) 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 \(f_1 = M_bT_a h\) and \(f_2 = M_bT_a(h_\times )\) agree on \(\mathcal {P}\) whereas \(f_1 \not \sim f_2\). \(\square \)
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.
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
Thus, the assertion follows from Theorem 3.7. \(\square \)
Remark 3.9
If \(\mathcal {S} = z + L{{\mathbb {Z}}}^2, z \in \mathbb {R}^2,\) with a noninvertible 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 infinitedimensional if it is not contained in any finitedimensional 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 infinitedimensional. Thus, for every \({\mathfrak {s}}>0, \theta \in \mathbb {R}\) and \(z=(a,b) \in \mathbb {R}^2\) the set
is an infinitedimensional cone in \({L^2(\mathbb {R})}\). In an abstract language, Theorems 3.7 and 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 infinitedimensional 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)
\(V_gf_1(\mathcal {S}) = V_gf_2(\mathcal {S})\) (resp. \(V_gf_1(\mathcal {P}) = V_gf_2(\mathcal {P})\))

(2)
\(f_1 \not \sim f_2\).
3.3 RealValuedness
As outlined in the introduction, realvalued window functions are prominent choices in many applications. Note that the previously derived function pairs \(f,f_\times \) which implied the nonuniqueness of the STFT phase retrieval problem from lattice samples were, in general, complexvalued (even if the window function is realvalued). We now examine the question of whether a prior realvaluedness assumption on the underlying signal space achieves uniqueness from lattice samples, i.e. one specializes the considered input functions to belong to the space \( L^2(\mathbb {R},\mathbb {R}) = \left\{ f \in {L^2(\mathbb {R})}: f \ \text {is realvalued} \right\} . \) 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
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 realvalued provided that \({{\mathcal {F}}}u\) is realvalued. 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
and since \({{\mathcal {F}}}u\) is realvalued 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,
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. \(\square \)
As an application of Proposition 3.12, we construct nonequivalent, realvalued 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 realvalued window function and let \(\mathcal {L}=L{{\mathbb {Z}}}^2 \subset \mathbb {R}^2\) be a lattice with generating matrix
If a, b are linearly dependent over \({{\mathbb {Q}}}\) and c, d are linearly dependent over \({{\mathbb {Q}}}\) then there exist two realvalued functions \(f_1,f_2 \in L^2(\mathbb {R},\mathbb {R})\) with the property \(V_gf_1(\mathcal {L}) = V_gf_2(\mathcal {L})\) but \(f_1 \not \sim f_2\).
Proof
The case \(g = 0\) is trivial. Therefore, assume in the following that \(g \ne 0\) 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 nonempty, 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
For \(\theta = \frac{\pi }{2}\) we have
and \(u :={{\mathcal {F}}}\mathcal {R}g\) has a realvalued Fourier transform. Identity (8) together with Proposition 3.12 shows that \(f_1\) is realvalued. 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 realvalued 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
whence \(V_gf_1(\mathcal {L})=V_gf_2(\mathcal {L})\) 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 \(p :=\frac{p_1}{p_2} \in {{\mathbb {Q}}}\) with \(p_1 \in {{\mathbb {Z}}}, p_2 \in {{\mathbb {Z}}}\setminus \{ 0 \}\) and \(q :=\frac{q_1}{q_2} \in {{\mathbb {Q}}}\) 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
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 \(L'\). Therefore, Case 1 applies and the existence of two realvalued, nonequivalent functions \(f_1,f_2 \in L^2(\mathbb {R},\mathbb {R})\), for which their spectrograms agree on \(\mathcal {L}\), follows. \(\square \)
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 Sect. 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 \(V_gf(\mathcal {L})\). 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, \(f_1 \not \sim f_2\), are of the form
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
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 Sect. 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 [11] 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 shiftinvariant space with an irrational stepsize, sampling on a lattice again suffices to achieve uniqueness. In [12] it was further proved that without an assumption on the stepsize of the Gaussian shiftinvariant 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 realvalued, bandlimited functions from samples in a 1dimensional lattice and window functions which satisfy a nonvanishing 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 realvalued window function. Does there exist a lattice \(\mathcal {L} \subset \mathbb {R}^2\) such that every realvalued map \(f \in L^2(\mathbb {R}, \mathbb {R})\) is determined up to a sign by \(V_gf(\mathcal {L})\)?
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
where \(\ell  \ell '\) denotes the Euclidean distance of \(\ell \) and \(\ell '\). 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 PaleyWiener 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
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 \(V_gf(\Lambda \cup \{ p_1, \dots , p_N \})\). 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.
References
Alaifari, R., Wellershoff, M.: Phase retrieval from sampled Gabor transform magnitudes: counterexamples. J. Fourier Anal. Appl. 28(1), 9 (2021). https://doi.org/10.1016/10.1007/s00041021099017
Alaifari, R., Wellershoff, M.: Uniqueness of STFT phase retrieval for bandlimited functions. Appl. Comput. Harmon. Anal. 50, 34–48 (2021). https://doi.org/10.1016/j.acha.2020.08.003
Almeida, L.: The fractional Fourier transform and timefrequency representations. IEEE Trans. Signal Process. 42(11), 3084–3091 (1994). https://doi.org/10.1109/78.330368
Bhandari, A., Zayed, A.I.: Shiftinvariant and sampling spaces associated with the fractional Fourier transform domain. IEEE Trans. Signal Process. 60(4), 1627–1637 (2012). https://doi.org/10.1109/TSP.2011.2177260
Boivin, A., Zhong, H.: Completeness of systems of complex exponentials and the Lambert W functions. Trans. Am. Math. Soc. 359(4), 1829–1849 (2007). https://doi.org/10.2307/20161653
Carmeli, C., Heinosaari, T., Schultz, J., Toigo, A.: Nonuniqueness of phase retrieval for three fractional Fourier transforms. Appl. Comput. Harmon. Anal. 39(2), 339–346 (2015). https://doi.org/10.1016/j.acha.2014.11.001
Christensen, O.: An Introduction to Frames and Riesz Bases. Basel, Birkhäuser (2016)
Deboor, C., Devore, R., Ron, A.: The structure of finitely generated shiftinvariant spaces in \(L^2(\mathbb{R}^d)\). J. Funct. Anal. 119(1), 37–78 (1994). https://doi.org/10.1006/jfan.1994.1003
Gröchenig, K., Romero, J.L., Stöckler, J.: Sampling theorems for shiftinvariant spaces, Gabor frames, and totally positive functions. Invent. Math. 211(3), 1119–1148 (2018). https://doi.org/10.1007/s0022201707602
Grohs, P., Koppensteiner, S., Rathmair, M.: Phase retrieval: uniqueness and stability. SIAM Rev. 62(2), 301–350 (2020). https://doi.org/10.1137/19M1256865
Grohs, P., Liehr, L.: Injectivity of Gabor phase retrieval from lattice measurements. arXiv:2008.07238 (2020)
Grohs, P., Liehr, L.: Stable Gabor phase retrieval in Gaussian shiftinvariant spaces via biorthogonality. arXiv:2109.02494 (2021)
Gröchenig, K.: Foundations of TimeFrequency Analysis. Basel, Birkhäuser (2001)
Heil, C.: History and evolution of the density theorem for Gabor frames. J. Fourier Anal. Appl. 13(2), 113–166 (2007). https://doi.org/10.1007/s0004100660732
Heil, C.: A Basis Theory Primer. Birkhäuser, Boston (2011)
Jaming, P.: Uniqueness results in an extension of Pauli’s phase retrieval problem. Appl. Comput. Harmon. Anal. 37(3), 413–441 (2014). https://doi.org/10.1016/j.acha.2014.01.003
Levinson, N.: Gap and Density Theorems, volume 26. American Mathematical Society Colloquium Publications (1940). https://doi.org/10.1090/coll/026
Lohmann, A.W.: Image rotation, Wigner rotation, and the fractional Fourier transform. J. Opt. Soc. Am. A 10(10), 2181–2186 (1993). https://doi.org/10.1364/JOSAA.10.002181
Namias, V.: The fractional order Fourier transform and its application to quantum mechanics. IMA J. Appl. Math. 25(3), 241–265 (1980). https://doi.org/10.1093/imamat/25.3.241
Pfeiffer, F.: Xray ptychography. Nat. Photonics 12(1), 9–17 (2018). https://doi.org/10.1038/s4156601700725
Redheffer, R.M.: Completeness of sets of complex exponentials. Adv. Math. 24(1), 1–62 (1977). https://doi.org/10.1016/S00018708(77)800029
Rodenburg, J.: Ptychography and related diffractive imaging methods. volume 150 of Adv. Imaging Electron Phys., pp. 87–184. Elsevier (2008). https://doi.org/10.1016/S10765670(07)000031
Ron, A.: Introduction to shiftinvariant spaces. linear independence. In: Dyn, N., Leviatan, D., Levin, D., Pinkus, A. (eds.) Multivariate Approximation and Applications, pp. 152–211. Cambridge University Press, Cambridge (2001)
Young, R.: An Introduction to NonHarmonic Fourier Series. Academic Press, revised edition (2001)
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Grohs, P., Liehr, L. On Foundational Discretization Barriers in STFT Phase Retrieval. J Fourier Anal Appl 28, 39 (2022). https://doi.org/10.1007/s00041022099355
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DOI: https://doi.org/10.1007/s00041022099355
Keywords
 Phase retrieval
 Time–frequency analysis
 Shorttime Fourier transform
 Lattices
 Sampling
 Signal reconstruction
Mathematics Subject Classification
 42A38
 44A15
 94A12
 94A20