Determining the nature of band gaps in transition-metal compounds remains
challenging. We present a first-principles study on electronic and optical
properties of CoO using hybrid functional pseudopotentials. We show that
optical absorption spectrum can provide a clear fingerprint to distinguish
between Mott-Hubbard, intermediate and charge-transfer insulators. This
discrimination is reflected by the qualitative difference in peak satellites
due to unique interplay between $d$-$d$ and $p$-$d$ excitations, thus allowing
identification from experimental data alone, unlike the existing methods that
require additional theoretical interpretation. We conclude that the CoO is an
intermediate, rather than a Mott-Hubbard insulator as is initially believed.

# Optical signature for distinguishing between Mott-Hubbard, intermediate and
charge-transfer insulators

Xiangtian Bu  Key Lab of advanced optoelectronic quantum architecture and
measurement (MOE), and Advanced Research Institute of Multidisciplinary
Science, Beijing Institute of Technology, Beijing 100081, China  Yuanchang Li
<EMAIL_ADDRESS>Key Lab of advanced optoelectronic quantum architecture and
measurement (MOE), and Advanced Research Institute of Multidisciplinary
Science, Beijing Institute of Technology, Beijing 100081, China

Transition-metal compounds form a rich and fruitful field of research, not
only for its scientific significance in high-$T_{c}$ superconductivity,
colossal magnetoresistance, phase transition and multiferroicityKhomskii ;
Imada ; Lee ; MARIANETTI2004 , but also for its commercial applications in
batteries and catalystsLiu2018 ; Ozawa1994 ; Westerhaus2013 ; Iravani2020 .
Explaining these phenomena and improving device performance require a
quantitative understanding of their fundamental electronic structures,
particularly the nature of the gap, which is associated with the competition
between on-site Coulomb repulsion $U$ of $d$-orbitals and charge-transfer
energy $\Delta$ from anion to cation. According to the Zaanen-Sawatzky-Allen
diagramZaanen1985 , if $U<\Delta$, the gap is of Mott-Hubbard type; If
$U>\Delta$, the gap is of charge-transfer type; If $U\approx\Delta$, the gap
is of intermediate type. As schematically illustrated in Fig. 1, different
gaps correspond to completely different lowest charge excitations, such as
$d$-$d$ and $p$-$d$ excitations for Mott-Hubbard and charge-transfer
insulators, respectively. This distinction is particularly important for
high-$T_{c}$ superconductivity in the case of hole dopingLee ; Zhang ; PNAS ,
because it determines whether the doped holes predominantly sit in the
metal-$d$ or anion-$p$ state.

However, unambiguously determining the nature of the gap remains
challengingVelasco2011 . There has been no experimental way to
straightforwardly derive $U$ and $\Delta$. Although they can be derived from
electronic structure calculations, the accuracy is not yet satisfactory when
dealing with partially filled $d$-electron systemsTanJCP2019 . Popular methods
such as density functional theory (DFT) often fail even to open energy gaps.
To remedy the problem, several post-DFT methods have been developed, which
include tunable parameters that need to be fitted experimentally. This, in
turn, requires quantitative agreement between experiment and theory, which is
almost impossible. Different manners whereby the parameters are fitted further
increase the uncertainty. As a result, different interpretations of
experimental data from scanning tunneling microscopy/photon-electron
spectroscopy give conflicting results in the literatureBrookes1989 ; Shen1990
; Nakata ; LiuLW ; LiuMK ; LiuZY . A qualitatively discriminating signature,
especially the kind that can be directly observed from experiments without
theoretical assistance, naturally becomes a key issue to be addressed in
theoretical studies, which will be the focus here.

The difference in the lowest excitation leads us to the optical scheme (see
Fig. 1). According to the Laporte selection ruleLaporte , the $d$-$d$
transition is optically-forbidden while the $p$-$d$ transition is optically-
allowed for a centrosymmetric complex. Although, in solids, this selection
rule is partially relaxed as a result of orbital hybridization, such
differences still substantially affect the optical propertiesJiang2019 ,
probably leaving fingerprints in the absorption spectrum or the exciton
structure. Herein, we will take cobalt monoxide (CoO) as an example for
illustration. CoO is theoretically classified as a Mott-Hubbard
insulatorBrandow , but an increasing number of experiments point to a pivotal
role of charge transferShen1990 ; Lee1991 ; Elp1991 ; Magnuson2002 . At
present, to determine the nature of the CoO gap, people first need its
experimental value to calibrate the tunable parameter in the calculations and
then derive $U$ and $\Delta$ from the corresponding band structure. However,
two facts limit the usefulness of this scheme: i) The reported gaps of CoO
span a wide range from 2.5 to 6.0 eVElp1991 ; Kang2007 ; Pratt1959 ; Shen1990
; Gvishi1972 . ii) Some of the adopted calculations contain more than one
parameter that cannot be determined using only one gap valueFuchs2009 ;
Lanata2019 . Therefore, the nature of the CoO gap remains elusive to today. On
the other hand, although optical absorption of CoO has been measured more than
60 years agoPratt1959 , its explicit interpretation is still absent.

Figure 1:  Schematics of Mott-Hubbard, intermediate and charge-transfer
insulators, as well as the different lowest excitations. The $d$-$d$
transition is optically-forbidden while the $p$-$d$ transition is optically-
allowed.

In this work, we seek to reveal qualitative differences in the optical spectra
of Mott-Hubbard, intermediate and charge-transfer insulators. To this end, we
employ the recently developed Co hybrid functional pseudopotentials which has
been shown to open for CoO a gap at the DFT levelTanJCP2019 . With the
increase of exact exchange in generating hybrid functional pseudopotentials,
we find that the resulting CoO band structure shows a continuous transition
from Mott-Hubbard to intermediate and then to charge-transfer insulating
phase. This allows us to directly compare the optical absorption features of
the three different insulators, which show significant differences in terms of
peak satellites. In particular, the absorption spectrum of the Mott-Hubard
phase contains only a two-peak skeleton while a third satellite peak appears
between the two peaks in the case of the intermediate phase. This satellite
peak in the charge-transfer phase further evolves into a plateau. Thus, a
simple light absorption experiment will fully reveal the fingerprints of Mott-
Hubard and charge-transfer physics in transition-metal compounds. Comparison
with available experimental data suggests that CoO is an intermediate rather
than a Mott-Hubbard insulator.

All DFT calculations were carried out using the Quantum Espresso
packageGiannozzi2017  with the Perdew-Burke-Ernzerhof (PBE) Perdew1996
exchange-correlation functional. Hybrid functional pseudopotentialsJing2018
generated from the OPIUMopium  code were used for Co while norm-conserving
Vanderbilt pseudopotentials Hamann ; SCHLIPF2015  were used for O. Note that
here the $\alpha$ which specifies the exact exchange weight of the exchange-
correlation functional used in hybrid pseudopotential construction is taken as
an adjustable parameterTanJCP2019 ; Jing2018 . An energy cutoff of 60 Ry was
set after a convergence test. A $k$-grid of $12\times 12\times 12$ was
employed to sample the Brillouin zone throughout this work. The optical
adsorption spectrum was obtained by solving the Bethe-Salpeter equation (BSE)
Rohlfing2000  as implemented in the YAMBO code marini2009yambo ; Sangalli2019
. A total of 300 empty bands were used to calculate the dielectric function
matrix, with 45 and 12 Ry cutoff respectively for exchange and correlation
parts. Top two valence bands and bottom two conduction bands were considered
to construct the BSE Hamiltonian. Test calculations involving more bands yield
the same conclusion. Here studied CoO has the rock salt structure with
antiferromagnetic alignment along [111] directionJauch2001 . The cubic lattice
parameter is fixed at the experimental value 4.26 Å.

We first calculate the electronic structures of CoO using different Co hybrid
functional pseudopotentials with the $\alpha$ ranging from 0.001 to 1, and
selectively plot the results in Fig. 2(a) for $\alpha$ = 0.001, 0.25, 0.50 and
0.75note . These bands look very similar to each other, except for the
parabolic O-2$p$ unoccupied band (highlighted in red) which continues to move
up as the $\alpha$ increases. It constitutes the conduction band bottom at
$\alpha$ = 0.001, analogous to previous studiesFuchs2009 ; Engel . Whilst, the
parabolic O-2$p$ band crosses the lowest Co 3$d$-band at $\alpha$ = 0.25 and
all three Co 3$d$-bands at $\alpha$ = 0.75.

Figure 2: (Color online) (a) Band structures (Left) and projected density of
states (Right) of CoO at typical $\alpha$. Here only one spin channel is
shown. For the antiferromagnetic case, the other spin is fully degenerate. The
valence band maximum is set to energy zero. (b) Dielectric constant, direct-
and indirect-gap, and (c) local magnetic moment of Co as a function of
$\alpha$.

All bands are semiconducting with a fundamental gap $>$ 1.4 eV. It is
noteworthy that only a very small amount of exact exchange, e.g., $\alpha$ =
0.001, is sufficient to yield a gap. This is very interesting because even
when applied to antiferromagnetic configuration, the PBE fails to produce a
gap in CoO, unlike the case in MnO and NiOTerakura1984 ; Wei1994 ; Gopal2017 ;
Fabien2006 ; LiuJJ2019 . While it is still debated whether orbital-dependent
potentials are necessaryTerakura1984 ; Engel , it is recognized that the gap
opening in CoO originates from the population imbalance among the Co minority
$t_{2\rm g}$ orbitals, as we obtain here. The difference between our
calculation and the standard DFT is all in the way the Co core-electrons are
treated. While the physics behind the gap opening is attractive to the DFT
community, it belongs to another work outside the scope of the current study
and will be discussed in the future.

As $\alpha$ increases, several interesting aspects are found. Firstly, there
is a counter-intuitive correlation between the tendency of the fundamental gap
and the dielectric constant (electronic contribution) to vary with $\alpha$.
Generally speaking, the gap affects the system screening: the larger the gap,
the weaker the screening and therefore the smaller the dielectric constant.
This should have led to a synchronous variation relationship between the
fundamental gap and the dielectric constant. However, this is not the case
here. As can be seen in Fig. 2(b), either the minimum indirect gap or the
direct gap shows a quadratic variation of increasing and then decreasing.
Around $\alpha$ = 0.75, the gap undergoes an indirect-to-direct transition.
The largest direct (indirect) gap reaches 2.62 (2.06) eV, very close to the
photoemission result of 2.5 $\pm$ 0.3 eVElp1991 . In contrast, the dielectric
constant increases monotonically from 5.45 to 8.18, always larger than the
experimental value of 5.35Powell1970 , as a result of the gap underestimation.
This counter-intuitive gap-dielectric constant correspondence suggests that
the electronic structure changes substantially during the $\alpha$ increase.

Secondly, the Co local magnetic moment decreases [see Fig. 2(c)]. Under the
local crystal field with $O_{h}$ symmetry, the 3$d$-orbitals of Co split into
a lower-lying $t_{2\rm g}$ triplet and a higher-lying $e_{\rm g}$ doublet. In
case of a Mott-Hubbard insulator, only the $d$-electron occupation is
considered. Seven Co $d$-electrons fully occupy five states of the majority
spin channel and the remaining two electrons partially fill the minority spin
$t_{2\rm g}$ triplet, giving rise to a local moment of 3 $\mu_{\rm B}$. In
this sense, the 3 $\mu_{\rm B}$ local moment can be regarded as a sign of the
Mott-Hubbard gap. Our calculations show that the moment decreases from 2.74
$\mu_{\rm B}$ at $\alpha$ = 0.001 to 2.36 $\mu_{\rm B}$ at $\alpha$ = 1. This
decrease comes from the hybridization between the Co-3$d$ and the O-2$p$
states, and the hybridization is enhanced with increasing $\alpha$. Obviously,
the charge-transfer role is becoming increasingly important. Note that the
moments obtained here are comparable to other calculationsLiuJJ2019 , but much
smaller than the experimental 3.35$\sim$3.98 $\mu_{\rm B}$ Jauch2001 ;
Khan1970 ; Ronzaud1978 , since the orbital contribution is neglectedFabien2006
.

Thirdly, the evolution of the projected density of states as $\alpha$
increases visualizes the enhancement of $p$-$d$ hybridization [see right
panels in Fig. 2(a)], manifested by the increasing contribution of O-2$p$ to
the valence band top. Its contribution increases from $\sim$10% at $\alpha$ =
0.001 to 24% at $\alpha$ = 0.25 and reaches $\sim$70% at $\alpha$ = 0.75. At
$\alpha$ = 0.5, the contributions of Co-3$d$ and O-2$p$ are about the same. On
the other hand, the bottom conduction band is always dominated by the Co
3$d$-states. Referring to Fig. 1, the band undergoes a continuous transition
from Mott-Hubbard-type to intermediate-type to charge-transfer-type as
$\alpha$ increases. This is interesting because it provides a basis for
exploring the optical properties related to the nature of the gap.

Next, we solve the BSE for the imaginary part of the dielectric function using
the bands at different $\alpha$ ranging from 0.001 to 1 as inputs, and typical
results are presented in Fig. 3(a). Experimental result extracted from Ref.
Pratt1959,  is also added for comparison. Note that here we focus on the
structural differences in the spectra given by different $\alpha$, so for
clarity we uniformly adjust the first absorption peak (labeled as $p_{1}$) to
energy zero (Its absolute energy corresponds to the optical gap as summarized
in Fig. 3(b)). This is physically reasonable, since the gap underestimation
can usually be handled by applying scissors during the solution of the BSE,
which results in a rigid translation of the optical absorption spectrum along
the energy axis without changing the spectral shape.

Figure 3:  (Color online) (a) Imaginary part of the dielectric function by
solving the BSE at typical $\alpha$ (colored solid lines), with the
experimental adsorption spectrum extracted from Ref. Pratt1959,  (black dash
line) for comparison. The first peak $p_{1}$, whose absolute energy
corresponds to the optical gap ($E_{\rm g}^{\rm opt}$), is aligned uniformly
to the energy zero. (b) Optical gap and exciton binding energy as a function
of $\alpha$. (c) Plots of the change of the real-space exciton wave-function
under different $\alpha$. For instance, $\alpha_{0.25}-\alpha_{0.001}$
represents the difference between the exciton wave-functions at $\alpha$ =
0.25 and $\alpha$ = 0.001 with the holes placed at the same central position
(Red balls). The electron accumulation and depletion regions are depicted in
yellow and blue, respectively

Each spectrum contains a basic skeleton consisting of two peaks (labeled as
$p_{1}$ and $p_{3}$). The intensity of $p_{1}$ does not change much as
$\alpha$ increases, but the intensity of $p_{3}$ increases rapidly. Meanwhile,
some satellite signals appear on the shoulder of $p_{3}$ and become more and
more prominent as $\alpha$ increases. All spectra can be divided into three
categories according to the different characteristics of such peak satellites:
1) When $\alpha\leq$ 0.25, the satellite is almost invisible; 2) When
$\alpha\in$ (0.25, 0.75), a new adsorption peak (labeled as $p_{2}$) appears;
3) When $\alpha\geq$ 0.75, the peak of $p_{2}$ develops into a plateau
connecting $p_{1}$ and $p_{3}$.

By comparing with the electronic structures shown in Fig. 2(a), we find that
the change in the optical spectral features is correlated with the transition
in the gap nature . When $\alpha\leq$ 0.25, the band is of Mott-Hubbard type,
which corresponds to the two-peak structure of the optical spectrum. When
$\alpha\in$ (0.25, 0.75), the band is of intermediate type, which corresponds
to the three-peak structure of the optical spectrum. When $\alpha\geq$ 0.75,
the band is of charge-transfer type, which corresponds to the structure of a
plateau connecting two peaks in the optical spectrum. We observe a three-peak
feature in the experimental spectrum of CoO. From above criterion we infer
that CoO is an intermediate insulator. Nevertheless, we should note that
quantitatively, we did not reproduce the experiment, including the energies of
the characteristic absorption peaks and the separation between them.

Although the spectra here are calculated using CoO as an example, the
differences in characteristic structures on the spectrum essentially stems
from different excitations across different types of gaps (see Fig. 1). In
other words, such differences depend on the properties of the band-edge states
rather than on the elements. Therefore, a simple light absorption measurement
would reveal these fingerprints and thus determine whether a transition-metal
compound is a Mott-Hubbard, an intermediate or a charge-transfer insulator.

Different $d$-$d$ and $p$-$d$ transitions may also lead to different excitons,
serving to distinguish different gap behaviors. Indeed, $p_{1}$ in Fig. 3(a)
corresponds to an exciton inside the fundamental gap. Its energy defines the
optical gap while its difference from the fundamental gap defines the exciton
binding energy. Figure 3(b) summarizes the situation of both at different
$\alpha$. As $\alpha$ increases, the optical gap decreases monotonically from
2.1 eV to 1.3 eV. This trend is consistent with a monotonic increase in the
dielectric constant [see Fig. 2(b)], in contrast to the non-monotonic
dependence of the fundamental gap. The exciton binding energy shows a complex
variation, first increasing and then decreasing.

Generally speaking, the larger the exciton size, the more it shows an electron
aggregation towards the outside away from the hole. By contrast, the smaller
the exciton size, the more it shows an electron aggregation towards the hole
in the center. In this sense, we fix the holes at the same central position
and plot the change of the exciton wave-function for two different $\alpha$ in
Fig. 3(c), whereby the exciton sizes are compared visually. One can see that
as compared to the cases of $\alpha$ = 0.001 and $\alpha$ = 0.5, the exciton
of $\alpha$ = 0.25 shows more electron aggregation toward the center, so it is
smaller in size. This is consistent with its larger binding energy [see Fig.
3(b)]. When $\alpha\geq$ 0.5, the exciton binding energy continues to decrease
and the corresponding exciton radius becomes larger and larger, as in the case
of $\alpha$ = 0.75. Despite these differences, it seems no direct
correspondence between the exciton feature and the nature of gap.

In summary, the electronic structure and optical properties of CoO has been
reexamined using hybrid-functional pseudopotentials. The results show that CoO
is an intermediate insulator rather than a Mott-Hubbard insulator as is
initially believed. This conclusion is drawn based on our new knowledge that
Mott-Hubbard, intermediate insulators and charge-transfer insulators have
different optical spectral characteristics. In contrast to traditionally used
methods that combine experimental gap as calibration with theoretical
calculations, our new scheme allows direct discrimination from experimental
data without relying on calculations, thus eliminating uncertainties caused by
differences in calculation methods. In addition, the introduction of the exact
exchange to core-electrons opens up the CoO gap within the PBE, which points
to some as yet unrevealed role of core-electrons in the DFT gap problem. Our
work provides new insights to understand the complex correlation effects and
charge-transfer interactions in transition-metal compounds, especially for the
interesting physical properties of cobalt oxides.

This work was supported by the Ministry of Science and Technology of China
(Grant No. 2020YFA0308800) and the National Natural Science Foundation of
China (Grant No. 12074034).

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# The HRT conjecture for two classes of special configurations

Kasso A. Okoudjou  Department of Mathematics, Tufts University, Medford MA
02131, USA<EMAIL_ADDRESS>and  Vignon Oussa  Department of
Mathematics, Bridgewater State University, Bridgewater, MA 0235, USA
<EMAIL_ADDRESS>
###### Abstract.

The HRT (Heil-Ramanathan-Topiwala) conjecture stipulates that the set of any
finitely many time-frequency shifts of a non-zero square Lebesgue integrable
function is linearly independent. The present work settles two special cases
of this conjecture, namely, the cases where the set of time-frequency shifts
has cardinality $N+1$ such that either $N$ of the points lie on some integer
lattice and the last point is arbitrary, or $N$ of the points are on a line,
while the last point does not belong this line. In both cases, we prove that
the HRT conjecture holds appealing mainly to various forms of the ergodic
theorem. We note that, in recent years, the latter case has been the subject
of many investigations–notably, the subcase where $N=3$– and our work
completely resolves it.

###### Key words and phrases:

HRT conjecture, Ergodic maps, Zak transform

###### 2000 Mathematics Subject Classification:

Primary 42C15 Secondary 42C40

##  1\. Introduction

For a nonzero function $f\in L^{2}\left(\mathbb{R}\right),$ a time-frequency
shift of $f$ is a function of the type

$M_{y}T_{x}f\left(t\right)=e^{-2\pi iyt}f\left(t-x\right)\text{ \ \
}\left(x,y\right)\in\mathbb{R}^{2}.$

The operator $M_{y}$ is called a modulation or frequency-shift operator, and
$T_{x}$ is an operator acting by shifting $f$ in time domain. Generally,
modulation and translation operators do not commute, and in fact, these two
families of unitary operators generate a non-commutative group called the
Heisenberg group. The ubiquitous nature of the Heisenberg group and its
relevance across numerous subjects within harmonic analysis is a fact that has
been extensively documented in the literature [17]. For instance, the
Heisenberg group is fundamental to the foundation of time-frequency or Gabor
analysis and is a source of important examples in frame theory [12, 15]. For a
historical development, as well as an in-depth treatment of Gabor analysis and
connection with topics such as frame theory, we refer the reader to [7, 8, 12,
14].

We recall that given a countable subset $\Lambda$ of $\mathbb{R}^{2},$ the
collection of functions

$\mathcal{G}\left(f,\Lambda\right)=\left\\{M_{y}T_{x}f:\left(x,y\right)\in\Lambda\right\\}$

is called a Gabor system. For comparison, the system obtained by substituting
modulation for dilation operators is often called a wavelet system. The
underlying groups for these systems: the Heisenberg group and the affine
group, share many striking similarities, as documented in [23, 22]. However,
there are also several instances in which these systems behave in drastically
different ways. For example, it is well-known that there exist finite systems
produced by time-dilation operators which are linearly dependent. This
property guarantees the existence of a scaling function as a solution to a
suitable functional equation–a central ingredient in the construction of
multiresolution orthonormal wavelets [15]. However, to this date, we do not
know the extent to which an analogous statement holds for finite systems of
time-frequency shifts. In fact, all progress made in this direction suggests
that any finite system of time-frequency shifted copies of a nonzero function
is linearly independent. This conjecture, first formulated by Heil,
Ramanathan, and Topiwala, is now known as the HRT Conjecture [16] and may be
stated as follows.

######  Conjecture 1.

[16] (HRT Conjecture) Given a fixed finite set $\Lambda\subset\mathbb{R}^{2},$
and a nonzero function $f\in L^{2}\left(\mathbb{R}\right),$ the associated
finite Gabor system, $\mathcal{G}\left(f,\Lambda\right)$ is linearly
independent.

For the specific case where all the points lie on the same line (which is
equivalent to stating that all modulation parameters are identical or all the
translation parameters are identical), a straightforward application of the
Fourier transform verifies the conjecture. Generally, however, the problem is
much more challenging than one might initially anticipate. To this date, the
most general result is attributed to Linnell [18], who was able to corroborate
the conjecture for the case where $\Lambda$ is a shift of a finite subset of a
discrete subgroup of $\mathbb{R}^{2d},d\in\mathbb{N}.$

In terms of the number of points, the smallest case not handled by Linnell’s
result can only occur for specific configurations of a set $\Lambda$
containing four points. To date, the status of the HRT conjecture for sets of
four points can be summarized as follows.

######  Proposition 1.

Let $\Lambda\subset\mathbb{R}^{2}$ be such that $\\#\Lambda=4$, and let $0\neq
f\in L^{2}(\mathbb{R})$. The finite Gabor system $\mathcal{G}(f,\Lambda)$ is
linearly independent in each of the following cases:

  1. (1)

$\Lambda$ is $(2,2)-$configuration, that is two of the points are on a line
and the other two on a parallel line, and $0\neq f\in L^{2}(\mathbb{R})$ is
arbitrary, [4, 5];

  2. (2)

$\Lambda$ is $13-$configuration, that is three of the points are on a line and
the fourth point off that line, and $0\neq f\in\mathcal{S}(\mathbb{R})$ [4];

  3. (3)

$\Lambda$ is an arbitrary set of four points, but extra restrictions are
imposed on $f$, e.g., [2, 3, 21].

Still, the general problem of the HRT conjecture for four points remains
unsettled, and the main aim of this paper is to present new results, including
the proof of the HRT conjecture for all $(1,3)$ configurations and all $L^{2}$
functions.

Our work is motivated by the two special cases of $4$-point HRT, namely the
$(1,3)$ configurations, as well as well the case in which three of the points
are on the $\mathbb{Z}^{2}$-lattice and the fourth is arbitrary.

(a) $(1,3)$-configuration

(b) $(1,\mathbb{Z}^{2})$-configuration.

Figure 1. Examples of four-point configurations considered in the sequel.

For example, the configuration
$\left\\{T_{x_{1}}f,T_{x_{2}}f,T_{x_{3}}f,Mf\right\\},$ for arbitrary
$x_{1},x_{2},x_{3}\in\mathbb{R}$ falls into the first category, while the
configuration $\left\\{f,Mf,Tf,M_{\alpha}T_{\beta}f\right\\}$ falls into the
second, see Figure 1.

Our results will deal with generalization of these two cases to the setting of
$N+1$ time-frequency shifts with $N\geq 3$. In particular, our results are
summarized as follows.

######  Theorem 2.

Let
$\\{(x_{k},y_{k})\\}_{k=1}^{N}\cup\\{(\alpha,\beta)\\}\subset\mathbb{R}^{2}$
be such that, either

  1. (i)

$\\{(x_{k},y_{k})\\}_{k=1}^{N}\subset\mathbb{Z}^{2}$, and
$(\alpha,\beta)\in\mathbb{R}^{2}\setminus\mathbb{Z}^{2}$, or

  2. (ii)

$\\{(x_{k},y_{k})\\}_{k=1}^{N}\subset\mathbb{R}^{2}$ lie on a line $L$ and
$(\alpha,\beta)\in\mathbb{R}^{2}\setminus L$.

If $f\in L^{2}(\mathbb{R})$ is such that for some nonzero complex
coefficients, $c_{1},\cdots,c_{N}$ the following linear dependency relation
holds:

${\displaystyle\sum\limits_{k=1}^{N}}c_{k}M_{y_{k}}T_{x_{k}}f=M_{\alpha}T_{\beta}f,$

then $f$ must be identically zero.

As will be seen in most of our proofs, Ergodic theory plays a critical role.
This is primarily due to the structure of the configurations of interest in
this work. For any time-frequency equations of the type
${\displaystyle\sum\limits_{k=1}^{N}}c_{k}M_{y_{k}}T_{x_{k}}f=M_{\alpha}T_{\beta}f,$
or ${\displaystyle\sum\limits_{k=1}^{N}}c_{k}T_{x_{k}}f=Mf,$ we prove that
there exist unitary operators that diagonalize the left hand side of each
equation. This allows us to reformulate each time-frequency equation in the
form $|p(z)F(z)|=|F(z+\gamma)|$ with a varying parameter $z,$ some polynomial
$p$ in the variable $z,$ and some fixed parameter $\gamma.$ The overarching
strategy consists of step-wise iterations of $|p(z)F(z)|=|F(z+\gamma)|$
combined with suitable applications of various versions of the Ergodic Theorem
and its relatives (such as the Poincaré Recurrence Theorem.) For each of these
applications, we refer the interested reader to the following relevant
textbooks [6, 24] for detailed treatment.

The rest of the paper is organized as follows. In Section 2, we consider the
case of $(1,\mathbb{Z}^{2})$ configurations, and state and prove our first
main result, namely Theorem 3. In Section 3, we prove our second main result,
Theorem 6, which settles the HRT conjecture for all $(1,N)$ configurations.

##  2\. The HRT conjecture for $(1,\mathbb{Z}^{2})$-configurations

A collection of $N+1$ points
$\Lambda=\\{(x_{k},y_{k})\\}_{k=1}^{N}\cup\\{(\alpha,\beta)\\}\subset\mathbb{R}^{2}$
is a $(1,\mathbb{Z}^{2})$-configuration if
$\\{(x_{k},y_{k})\\}_{k=1}^{N}\subset\mathbb{Z}^{2}$ and $(\alpha,\beta)$ is
arbitrary. We refer to Figure 1 for an illustration in the case $N=3$. Our
first result establishes the HRT conjecture for all such configurations.

######  Theorem 3.

Fix $\left\\{\left(x_{k},y_{k}\right):1\leq k\leq
N\right\\}\subset\mathbb{Z}^{2}$, and let
$(\alpha,\beta)\in\mathbb{R}^{2}\setminus\mathbb{Z}^{2}$. If there exists a
function $f\in L^{2}\left(\mathbb{R}\right)$ such that

(1)
${\displaystyle\sum\limits_{k=1}^{N}}c_{k}M_{y_{k}}T_{x_{k}}f=M_{\alpha}T_{\beta}f$

for some nonzero complex coefficients $c_{1},\cdots,c_{N},$ then $f$ must be
the zero function.

An immediate consequence of Theorem 3, is the follwing result which answers a
sub-conjecture of C. Heil, [15, Conjecture 11.39].

######  Corollary 4.

For any function $0\neq f\in L^{2}(\mathbb{R}),$ the finite collection of
time-frequency shifts $\\{f,M_{1}f,T_{1}f,M_{\alpha}T_{\beta}f\\}$ is linearly
independent in $L^{2}(\mathbb{R})$ for any point
$(\alpha,\beta)\in\mathbb{R}^{2}.$

Theorem 3 can also be extended to $(1,A\mathbb{Z}^{2})$-configurations where
$A$ is any $2\times 2$ symplectic matrix, that is an invertible matrix whose
determinant is $1$.

######  Corollary 5.

Suppose that $A$ is a $2\times 2$ matrix with $\det A=1$. The HRT conjecture
is true for any $0\neq f\in L^{2}(\mathbb{R})$ and any set of $N+1$ point of
the form $\left\\{\left(x_{k},y_{k}\right):1\leq k\leq
N\right\\}\cup\\{(\alpha,\beta)\\}$ where
$\left\\{\left(x_{k},y_{k}\right):1\leq k\leq N\right\\}\subset
A\mathbb{Z}^{2}$, and $(\alpha,\beta)\in\mathbb{R}^{2}$.

###### Proof.

When $(\alpha,\beta)\in A\mathbb{Z}^{2},$ we are back in the lattice case,
which was already settled by Linell. Therefore, we may assume that
$(\alpha,\beta)\in\mathbb{R}^{2}\setminus A\mathbb{Z}^{2}$, and suppose that
there, HRT was false for a function $f$ and one such set of $N+1$ points. We
may then use the fact that $A$ is a symplectic matrix to find a function $g$
(image of $f$ under a metaplectic transformation) such that $g$ and the set of
points $\left\\{\left(x^{\prime}_{k},y^{\prime}_{k}\right):1\leq k\leq
N\right\\}\cup\\{(\alpha^{\prime},\beta^{\prime})\\}$ where
$\left\\{\left(x_{k}^{\prime},y_{k}^{\prime}\right):1\leq k\leq
N\right\\}\subset\mathbb{Z}^{2}$. But this will contradict Theorem 3. ∎

###  2.1. Discussion of the proof of Theorem 3

As we will see, the concept of convergence and divergence of infinite products
will also be of central importance in our proofs. As such, we would like to
review here some of the important notions needed in the sequel, and we refer
to [1] for details. Let $\left(a_{j}\right)_{j\geq 1}$ be a sequence of
nonzero real or complex numbers. If the sequence of partial products

$\left({\displaystyle\prod\limits_{j=1}^{n}}a_{j}\right)_{n\in\mathbb{N}}$

converges to a nonzero limit then we say that the infinite product
${\displaystyle\prod\limits_{j=1}^{\infty}}a_{j}$ is convergent. Suppose a
finite number of factors $a_{j}$ are equal to zero, and the infinite product
obtained by removing these factors converges. In that case, we say that the
infinite product converges to zero. Otherwise, we define the infinite product
as divergent to zero. We note that if
${\displaystyle\prod\limits_{j=1}^{\infty}}a_{j}$ is convergent, then
$\lim_{j\to\infty}a_{j}=1$, see [1].

To prove Theorem 3, we note that any finite set of time-frequency operators
parametrized by a collection of finite points on an integer lattice in
$\mathbb{R}^{2}$, forms a collection of pairwise commuting operators. As such,
the Spectral Theorem [13] guarantees the existence of a unitary operator which
diagonalizes the operator

$J={\displaystyle\sum\limits_{k=1}^{N}}c_{k}M_{y_{k}}T_{x_{k}}.$

In our case, this unitary operator is the Zak transform
$Z:L^{2}(\mathbb{R})\rightarrow L^{2}([0,1]^{2})$ defined formally, by

(2) $Zf\left(t,\omega\right)=\sum_{k\in\mathbb{Z}}f\left(t+k\right)e^{-2\pi
k\omega i}.$

We summarize below the main properties of the Zak transform and refer to [12]
for a complete introduction to this transform.

######  Lemma 1.

The Zak transform $Z$ satisfies the following properties.

  1. (1)

$Z$ is a unitary operator $L^{2}\left(\mathbb{R}\right)$ onto
$L^{2}\left(\left[0,1\right)^{2}\right).$

  2. (2)

For $\varphi\in L^{2}(\mathbb{R})$ and $(t,\omega)\in[0,1)^{2}$, we have:

    1. (a)

$\left[ZT\varphi\right]\left(t,\omega\right)=e^{-2\pi\omega
i}\cdot\left[Z\varphi\right]\left(t,\omega\right).$

    2. (b)

$\left[ZM\varphi\right]\left(t,\omega\right)=e^{-2\pi
it}\cdot\left[Z\varphi\right]\left(t,\omega\right).$

    3. (c)

For each $\alpha,\beta>0$,
$\left[ZM_{\alpha}T_{\beta}\varphi\right]\left(t,\omega\right)=e^{-2\pi
i\alpha t}\cdot\left[Z\varphi\right]\left(t-\beta,\omega+\alpha\right).$

    4. (d)

For any integer $j,$
$\left[ZT\varphi\right]\left(t+j,\omega\right)=e^{2\pi\omega
ji}\left[ZT\varphi\right]\left(t,\omega\right).$

    5. (e)

For any integer $j,$
$\left[ZT\varphi\right]\left(t,\omega+j\right)=\left[ZT\varphi\right]\left(t,\omega\right).$

Applying Zak transform to the linear dependence equation
$Jf=M_{\alpha}T_{\beta}f,$ and letting $F=Zf,$ for almost every
$(t,w)\in[0,1]^{2}$, we have

(3) ${\displaystyle\sum\limits_{k=1}^{N}}c_{k}e^{-2\pi iy_{k}t}e^{-2\pi
ix_{k}\omega}\cdot F\left(t,\omega\right)=e^{-2\pi i\alpha t}\cdot
F\left(t-\beta,\omega+\alpha\right),$

or equivalently,

(4) $p\left(t,\omega\right)\cdot F\left(t,\omega\right)=e^{-2\pi i\alpha
t}\cdot F\left(t-\beta,\omega+\alpha\right),$

where $p:\mathbb{R}^{2}\rightarrow\mathbb{C}$ is the non-zero trigonometric
polynomial given by

(5) $p\left(t,\omega\right)={\displaystyle\sum\limits_{k=1}^{N}}c_{k}e^{-2\pi
iy_{k}t}e^{-2\pi
ix_{k}\omega}={\displaystyle\sum\limits_{k=1}^{N}}c_{k}e^{-2\pi
i\langle(t,\omega),(y_{k},x_{k})\rangle}.$

It follows that for almost every $(t,w)\in[0,1]^{2},$

$\left|p\left(t,\omega\right)\right|\cdot\left|F\left(t,\omega\right)\right|=\left|F\left(t-\beta,\omega+\alpha\right)\right|\iff|F(z+\gamma)|=|p(z)|\cdot|F(z)|,$

where, for simplicity in the notations, we set
$\gamma=\left(-\beta,\alpha\right)$ and $z=\left(t,\omega\right).$ Iterating,
the last equation, we see that for almost every $z\in\left[0,1\right]^{2}$ and
for any positive integer $n\geq 1,$

(6)
$\left|F\left(z+n\gamma\right)\right|=\left|{\displaystyle\prod\limits_{j=0}^{n-1}}p\left(z+j\gamma\right)\right|\cdot\left|F\left(z\right)\right|.$

We let $S_{F}$ be the essential support of $F$ in $[0,1]^{2}$ and let
$K_{F}=\left[0,1\right]^{2}\backslash S_{F}$ be its complement. By Lusin’s
theorem, there exists a set of positive measure $\Lambda\subset S_{F},$ such
that the restriction $|F|_{|\Lambda}$ of $|F|$ onto $\Lambda$ is continuous.
Finally, owning to the analyticity of $|p(z)|^{2},$ we may assume that
$\Lambda$ contains no zeroes of $p(z)$.

The final tool needed in our proof is the measure-preserving map
$\sigma:\left[0,1\right)^{2}\rightarrow\left[0,1\right)^{2}$ such that for any
$z\in\left[0,1\right)^{2},$ $\sigma\left(z\right)$ is the unique element in
$\left[0,1\right)^{2}$ which is a solution to the equation
$z+\gamma\mathbb{Z}^{2}=\sigma\left(z\right)+\mathbb{Z}^{2}.$ In other words,
from here on, we will regard the range of $\sigma$ as a connected transversal
for the $\mathbb{Z}^{2}$-cosets in $\mathbb{R}^{2}.$

Observe that, (6) suggests that, at least for continuous functions, the
behavior of $|F|$ at $z+n\gamma$ is fully determined by its value at a single
point such as $z.$ Thus, if for instance, $F$ is assumed to be continuous,
since $p$ is predetermined; for any unknown function $f$ for which the HRT
conjecture is true, the restriction of $|F|$ to the set
$\\{z+n\gamma:n\in\mathbb{N}\\}$ is completely determined by its value at a
single element such as $z.$ Even if, $F$ is not assumed to be continuous,
since the Zak transform converges almost everywhere and since (6) imposes a
rather stringent condition on $F,$ and, in principle,
$Jf=M_{\alpha}T_{\beta}f$ should be incompatible with the square-integrability
of nonzero $f.$ Our main concern in this section is to fully substantiate
these claims.

###  2.2. Proof of Theorem 3

###### Proof of Theorem 3.

The proof is twofold. First, we assume that $\left\\{1,\alpha,\beta\right\\}$
is linearly independent over the rationals. Secondly, we deal with the
situation where this set is dependent over the rationals.

Case 1: Let us assume that $\left\\{1,\alpha,\beta\right\\}$ is linearly
independent over the rationals. It follows that for any $z\in[0,1]^{2},$ the
set $\\{\sigma(z+n\gamma):n\in\mathbb{N}\\}$ is dense in $[0,1]^{2},$ and that
$\sigma$ defines an ergodic map on the unit square (viewed as a fundamental
domain of an integer lattice). Furthermore, according to Poincaré Recurrence
Theorem, for almost every $\lambda\in\Lambda,$
$\sigma^{n}\left(\lambda\right)\in\Lambda$ for infinitely many $n.$

Next, assume that that both $S_{F}$ (the support of $F$) and its complement
$K_{F}$ in $S,$ have positive Lebesgue measure as subsets of the unit square.
It follows then, that there exists $n\in\mathbb{N},$ such that
$\sigma^{n}\left(S_{F}\right)\cap K_{F}$ has positive Lebesgue measure in
$\left[0,1\right]^{2},$ which is a contradiction (see Equation (6)). Since
$F\neq 0$, we conclude that $|S_{F}|=1$ (or that $|K_{F}|=0$), where $|E|$
denotes the Lebesgue measure of $E\subset\mathbb{R}^{2}$.

We now see that $n\geq 1$ and any $z\in S_{F}$, (6) is equivalent to

(7)
$\left|\frac{F\left(\sigma^{n}\left(z\right)\right)}{F\left(z\right)}\right|=\left|{\displaystyle\prod_{j=0}^{n-1}}p\left(\sigma^{j}\left(z\right)\right)\right|.$

Let $z_{0}\in S_{F}$ and fix $\lambda_{0}\in\Lambda.$ Choose a subsequence
$\\{\sigma^{n_{k}}(\lambda_{0})\\}_{k\geq 1}\subset\Lambda$ such that
$\lim_{k\to\infty}\sigma^{n_{k}}(\lambda_{0})=z_{0}.$ Appealing to (7), we
derive the following:

$\left|\dfrac{F\left(z_{0}\right)}{F\left(\lambda_{0}\right)}\right|=\lim_{k\rightarrow\infty}\left|\frac{F\left(\sigma^{n_{k}}\left(\lambda_{0}\right)\right)}{F\left(\lambda_{0}\right)}\right|=\lim_{k\rightarrow\infty}\left|{\displaystyle\prod_{j=0}^{n_{k}-1}}p\left(\sigma^{j}\left(\lambda_{0}\right)\right)\right|.$

Moreover, applying the natural log to each side of the previous equation, the
following is immediate:

$\lim_{k\rightarrow\infty}\sum_{j=0}^{n_{k}-1}\ln\left|p\left(\sigma^{j}\left(\lambda_{0}\right)\right)\right|=\ln\left|\frac{F\left(z_{0}\right)}{F\left(\lambda_{0}\right)}\right|<\infty.$

Set
$s_{k}=\sum_{j=0}^{n_{k}-1}\ln\left|p\left(\sigma^{j}\left(\lambda_{0}\right)\right)\right|$
and note that $\lim_{k\to\infty}s_{k+1}-s_{k}=0$, which implies that

$\sum_{k=1}^{\infty}(s_{k+1}-s_{k})=\lim_{K\to\infty}\sum_{k=1}^{K}(s_{k+1}-s_{k})=\ln\left|\frac{F\left(z_{0}\right)}{F\left(\lambda_{0}\right)}\right|-s_{1}<\infty.$

Now the sequence $\\{s_{k+1}-s_{k}\\}_{k\geq 1}$ can be written as
$\\{\ln\left|p\left(\sigma^{m_{k}}\left(\lambda_{0}\right)\right)\right|\\}_{k\geq
1}$ where the increasing sequence $\\{m_{k}\\}$ is the enumeration of
$\\{n_{k}+\ell:\ell=0,1,2,\ldots,n_{k+1}-1-n_{k},\,k=1,2,\ldots\\}.$ We
observe that the original sequence
$\\{\ln\left|p\left(\sigma^{n_{k}}\left(\lambda_{0}\right)\right)\right|\\}_{k\geq
1}$ is a subsequence of the new sequence
$\\{\ln\left|p\left(\sigma^{m_{k}}\left(\lambda_{0}\right)\right)\right|\\}_{k\geq
1}$. We can then write that

$\sum_{k=1}^{\infty}\ln\left|p\left(\sigma^{m_{k}}\left(\lambda_{0}\right)\right)\right|<\infty.$

This implies that

$\lim_{k\to\infty}\ln\left|p\left(\sigma^{m_{k}}\left(\lambda_{0}\right)\right)\right|=0\implies\lim_{k\to\infty}\ln\left|p\left(\sigma^{n_{k}}\left(\lambda_{0}\right)\right)\right|=0$

and as such,
$\lim_{k\to\infty}\left|p\left(\sigma^{n_{k}}\left(\lambda_{0}\right)\right)\right|=\left|p\left(z_{0}\right)\right|=1.$

We conclude that $|p(z)|=1$ on $\left[0,1\right]^{2}$. Note however, that
$|p|$ is not a constant function, since $|p|^{2}$ is real-analytic (a
trigonometric polynomial, more precisely.) This produces the desired
contradiction.

Case 2: Assume now that the set $\\{1,\alpha,\beta\\}$ is rationally
dependent. Since the case where the parameters $\alpha,\beta$ are rationals
(see [19, Proposition 6.3] and [18]) has already been settled, to complete the
proof, it remains to address the subcase where either $\alpha$ or $\beta$ is
irrational. Without loss of generality, assume that
$0\neq\beta\not\in\mathbb{Q}$ and write $\alpha=r+s\beta$ for some rational
numbers $r,s.$ For each $\lambda\in[0,1)^{2}$, let

$L(\lambda)=\left\\{\lambda+t\gamma\mod 1:t\in\mathbb{R}\right\\}$

be the toral line passing through $\lambda$ and $\gamma.$ Then $L(\lambda)$
can be identified with a finite union of parallel line segments inside the
two-dimensional unit square. Note that these toral lines partition the unit
square $[0,1)^{2}$.

Figure 2. An illustration of several distinct toral lines in $[0,1)^{2}$

In this case, $\sigma$ fails to act ergodically on the (entire) unit square.
Next, we aim to show that the trigonometric polynomial $p$ in (5) is invariant
under the action of $\sigma$. Indeed, for $t\in\mathbb{R},$ let
$z_{t}\in[0,1)^{2}$ be the unique element such that $\lambda+t\gamma-
z_{t}=q\in\mathbb{Z}^{2}$. It follows that $\sigma(\lambda+t\gamma)=z_{t},$
and that

$\displaystyle p(\sigma(\lambda+t\gamma))$
$\displaystyle=\sum_{k=1}^{N}c_{k}e^{-2\pi
i\langle\sigma(\lambda+t\gamma),(y_{k},x_{k})\rangle}=\sum_{k=1}^{N}c_{k}e^{-2\pi
i\langle z_{t},(y_{k},x_{k})\rangle}$
$\displaystyle=\sum_{k=1}^{N}c_{k}e^{-2\pi i\langle
z_{t}+q,(y_{k},x_{k})\rangle}=p(\lambda+t\gamma).$

By the continuity of $p,$ we conclude that $p(\sigma(z))=p(z)$ for each $z\in
L(\lambda)$. In addition, since the action of $\sigma$ is ergodic (with
respect to a normalized arclength measure) on the toral lines $L(\lambda),$ it
follows that $p$ must be constant on each of these lines.

We proceed as in the first case and choose a set of positive measure
$\Lambda\subset S_{F}$ such that the restriction $|F|_{|\Lambda}$ of $|F|$
onto $\Lambda$ is continuous. Let $\lambda\in\Lambda$. Using the fact that
$\sigma$ is measure-preserving, once again, we may apply Poincaré Recurrence
Theorem to assert the existence of a subsequence
$\left(\sigma^{n_{k}}\left(\lambda\right)\right)_{k\in\mathbb{N}}\subset\Lambda$
such that
$\lim_{k\rightarrow\infty}\sigma^{n_{k}}\left(\lambda\right)=\lambda_{0}\in\Lambda.$
Using the same arguments as in Case 1, we conclude that

$\left|\dfrac{F\left(\lambda_{0}\right)}{F\left(\lambda\right)}\right|=\lim_{k\rightarrow\infty}\left|\frac{F\left(\sigma^{n_{k}}\left(\lambda\right)\right)}{F\left(\lambda\right)}\right|=\lim_{k\rightarrow\infty}\left|{\displaystyle\prod_{j=0}^{n_{k}-1}}p\left(\sigma^{j}\left(\lambda\right)\right)\right|,$

which implies that
$\lim_{k\rightarrow\infty}\left|p\left(\sigma^{n_{k}}\left(\lambda\right)\right)\right|=\left|p\left(\lambda_{0}\right)\right|=1.$
We now use the fact that
$\left(\sigma^{n_{k}}\left(\lambda\right)\right)_{k\in\mathbb{N}}\subset
L(\lambda)$ to conclude that $\lambda_{0}\in L(\lambda)$. It follows that
$|p(\lambda)|=\left|p\left(\lambda_{0}\right)\right|=1.$

Since the collection of toral lines $L(\lambda)$ is a partition of the unit
square (up to a set of Lebesgue measure zero) and since $\Lambda$ is assumed
to have positive Lebesgue measure, then $|p|^{2}=1$ on a set of positive
Lebesgue measure. Hence $|p|^{2}$ (being real analytic), and $|p|$ must be
constant everywhere on the unit square. This would imply that all but one of
the coefficients $c_{k}s$ is nonzero. This produces a desired contradiction. ∎

##  3\. The HRT for $(1,n)$-configurations

In this section, we prove the HRT conjecture for all $(1,n)-$configurations.
In particular, the result settles the $(1,3)-$configurations and should be
compared to the results in [4, 5, 20, 21].

######  Theorem 6.

Suppose that $\\{x_{k}\\}_{k=1}^{N}\subset\mathbb{R}$ is a set of $N$ distinct
points. Assume that there exists a function $f\in L^{2}(\mathbb{R})$ such that

(8) ${\displaystyle\sum\limits_{k=1}^{N}}c_{k}T_{x_{k}}f=Mf$

for some nonzero complex coefficients $c_{1},\cdots,c_{N}$. Then $f$ must be
the zero function.

###  3.1. Discussion of the proof of Theorem 6

Since the case where all translation parameters $x_{k}$ are rational is
already established [19, 18, Proposition 6.3], without loss of generality, we
may assume that

$\dim_{\mathbb{Q}}\left(\mathbb{Q+Q}x_{1}+\cdots+\mathbb{Q}x_{N}\right)>1.$

Although the rationality condition above implies that at least, one of the
translation parameters must be irrational, it is perhaps worth pointing out
that it is not necessarily the case that
$\left\\{1,x_{1},\cdots,x_{N}\right\\}$ is linearly independent over the
rationals. Applying Fourier transform to each side of Equation (8) and letting
$F$ be the Fourier transform of $f,$ it follows that

(9) $\left(\sum_{k=1}^{N}c_{k}e^{-2\pi ix_{k}\xi}\right)\cdot
F\left(\xi\right)=F\left(\xi+1\right)\text{ a.e. }$

Next, we consider the trigonometric polynomial

$p\left(\xi\right)=\sum_{k=1}^{N}c_{k}e^{-2\pi ix_{k}\xi}.$

The following will be quite convenient for the proof of our main result.

######  Lemma 2.

$p$ is a matrix coefficient of an $N$-dimensional unitary representation
$\mathbb{R}.$

###### Proof.

Indeed, it is not hard to verify that $p\left(\xi\right)=\left\langle e^{\xi
D}c,v\right\rangle$ where

$D=\left[\begin{array}[c]{ccc}-2\pi ix_{1}&&\\\ &\ddots&\\\ &&-2\pi
ix_{N}\end{array}\right]\text{ }e^{\xi D}=\left[\begin{array}[c]{ccc}e^{-2\pi
i\xi x_{1}}&&\\\ &\ddots&\\\ &&e^{-2\pi i\xi x_{N}}\end{array}\right]:=A.$

Additionally

$v=\left[\begin{array}[c]{c}1\\\ \vdots\\\
1\end{array}\right],c=\left[\begin{array}[c]{c}c_{1}\\\ \vdots\\\
c_{N}\end{array}\right].$

∎

Each quantity $p\left(\xi\right)=\left\langle e^{\xi D}c,v\right\rangle$ is
obtained by letting $\mathbb{R}$ acts on $\mathbb{C}^{N}$ via the action
$\left(\xi,c\right)\mapsto e^{\xi D}c$ and this produces a family of scalars
$\left(\left\langle e^{\xi D}c,v\right\rangle\right)_{\xi\in\mathbb{R}}$
obtained by projecting $e^{\xi D}c$ continuously on the function $v$ described
above. As $\xi$ runs through the set of real numbers, $e^{\xi D}c$ traces out
a curve wrapping itself densely around a torus in $\mathbb{C}^{N}$ passing
through, $c.$

Observe, that generally, the countable group generated by matrix $A=e^{D}$ is
not closed in the subspace topology of the General Linear Group which $A$
belongs to. In what follows, let $G$ be the topological closure of the
countable group

$\Gamma=\\{A^{j}=e^{jD}:j\in\mathbb{Z}\\}$

generated by $A.$ Additionally, let $H$ be the topological closure (as a
subset of the General Linear Group) of the $1$-parameter group $\left\\{e^{\xi
D}:\xi\in\mathbb{R}\right\\}.$ We refer to [11] for more background.

######  Lemma 3.

$G$ is a compact abelian group which is either homeomorphic to a torus or to a
finite union of translates of tori of the same positive dimension.

###### Proof.

By assumption, since
$\dim_{\mathbb{Q}}\left(\mathbb{Q+Q}x_{1}+\cdots+\mathbb{Q}x_{N}\right)>1,$
the set $\left\\{x_{1},\cdots,x_{N}\right\\}$ must contain at least one
irrational number. Without loss of generality, we may also assume that
$x_{1}\not\in\mathbb{Q}$, and additionally,
$\left\\{1,x_{1},\cdots,x_{m}\right\\}$ is a basis for the rational space
$\mathbb{Q}x_{1}+\cdots+\mathbb{Q}x_{m}+\mathbb{Q}.$ In other words, we may
rewrite the remaining parameters $x_{m+1},\cdots,x_{N}$ as follows:

$\displaystyle x_{m+1}$
$\displaystyle=u_{m+1}+\sum_{l=1}^{m}d_{m+1,l}x_{l}\text{ for
}\left(u_{m+1},d_{m+1,1,\cdots}d_{m+1,m}\right)\in\mathbb{Q}^{m+1}$
$\displaystyle\vdots$ $\displaystyle x_{N}$
$\displaystyle=u_{N}+\sum_{l=1}^{m}d_{N,l}x_{l}\text{ for
}\left(u_{N},d_{N,1,\cdots}d_{N,m}\right)\in\mathbb{Q}^{m+1}.$

Furthermore, we also assume that every member of the set
$\left\\{1,x_{1},\cdots,x_{N}\right\\}$ has a rationalized denominator. Next,
let $L$ be the least common multiple of all of these denominators. For every
integer $j,$ let $j=qL+r$, that is $r=j\operatorname{mod}L$. It follows that
the diagonal entries of $A^{j}$ where $A=e^{D}$ are

$\left(e^{-2\pi ix_{1}\left(qL+r\right)},\cdots,e^{-2\pi
ix_{m}\left(qL+r\right)},e^{-2\pi
i\left(u_{m+1}+\sum_{l=1}^{m}d_{m+1,l}x_{l}\right)\left(qL+r\right)},\cdots,e^{-2\pi
i\left(u_{N}+\sum_{l=1}^{m}d_{N,l}x_{l}\right)\left(qL+r\right)}\right).$

Letting $z_{1}=e^{-2\pi ix_{1}},\cdots,z_{m}=e^{-2\pi ix_{m}}$, we obtain
$A^{j}=A^{qL+r}=A^{r}D_{\left(u,q,L\right)}$ where $D_{\left(u,q,L\right)}$ is
a diagonal matrix with diagonal entries

$z_{1}^{qL},\cdots,z_{m}^{qL},e^{-2\pi
i\left(u_{m+1}\right)\left(qL\right)}\left({\displaystyle\prod\limits_{l=1}^{m}}z_{l}^{d_{m+1,l}qL}\right),\cdots,e^{-2\pi
i\left(u_{N}\right)\left(qL\right)}\left({\displaystyle\prod\limits_{l=1}^{m}}z_{l}^{d_{N,l}qL}\right).$

In summary, the closure of the countable group generated by $A$ is a
topological closed subgroup of the General Linear Group which is also of the
form

$G={\displaystyle\bigcup\limits_{r=0}^{L-1}}A^{r}K,$

where $K$ is a connected component of the identity of the group. ∎

######  Remark 7.

In addition to Lemma 3, we also need the following facts.

  1. (1)

Since $G$ is a locally compact group, up to normalization, $G$ admits a unique
measure which is invariant under left translations by elements of $G.$
Moreover, this invariant measure on $G$ is the sum of measures
$\frac{1}{L}\sum_{k=1}^{L}\mu_{k}$ where each $\mu_{k}$ is the push-forward of
the invariant measure $\mu_{0}$ on the connected component $K$ to $A^{k}K$ via
the translation map $g\mapsto A^{k}g.$

  2. (2)

Generally $H$ and $G$ do not coincide and since $G$ is a closed subgroup of
$H,$ there exists a subset $Z$ of $H$ such that $\left\\{zG:z\in Z\right\\}$
is a transversal for $H/G$ [9]. Additionally, the transversal $Z$ admits up to
normalization, a unique $H$-invariant measure. We endow each set $zG$ with the
following measure. For a suitable subset $zE$ of $zG,$
$\mu_{zG}\left(zE\right)=\mu_{G}\left(E\right)$ where $\mu_{G}$ is a fixed
invariant measure on $G.$ Additionally, we shall assume that $\mu_{G}$ is
normalized to satisfy $\mu_{G}\left(G\right)=1.$

  3. (3)

Note that the group generated by $e^{D}$ is generally, not a dense subset of
$H.$ Therefore, rotation by $e^{D}$ is generally not an ergodic map on $H.$
However, $e^{D}$ does act ergodically on every coset of $G$ in $H.$ Precisely,
for each fixed $z$ in a transversal $Z,$ of $H/G,$ $G$ acts on $zG$ as
follows. Given $g\in G$ and $zh\in zG,$ $gzh=zgh=z\left(gh\right)\in zG.$
Since the countable subgroup $\left\\{e^{jD}:j\in\mathbb{Z}\right\\}$ is dense
in $G,$ for every fixed $z\in Z$ where $Z\subseteq H$ is a measurable
transversal for $H/G,$ the map $\sigma:zG\rightarrow zG$ defined by
$\sigma\left(zh\right)=e^{D}zh=ze^{D}h$ is an ergodic map with respect to the
measure $\mu_{zG}.$

We are now ready to prove Theorem 6.

###  3.2. Proof of Theorem 6

###### Proof.

Let $S$ be the support of $F$ and assume that $S$ does not contain any zero of
the polynomial $p$ (this is possible since the set of these zeroes is at most
countable.) For each $\xi\in S$, we can iterate (9) so that for any positive
integer $n,$ the following holds true:

(10)
$|F\left(\xi+n\right)|=\left({\displaystyle\prod\limits_{j=0}^{n-1}}\left|p\left(\xi+j\right)\right|\right)|F\left(\xi\right)|\text{
a.e.}$

and additionally

(11)
$|F\left(\xi-n\right)|=\left({\displaystyle\prod\limits_{j=1}^{n}}\left|p\left(\xi-j\right)\right|\right)^{-1}|F\left(\xi\right)|\text{
a.e.}$

Since $F$ is square-integrable,

$\infty>\int_{S}\left|F\left(\xi\right)\right|^{2}d\xi=\int_{S\cap\left[0,1\right]}{\displaystyle\sum\limits_{k=-\infty}^{\infty}}\left|F\left(\xi+k\right)\right|^{2}d\xi.$

This implies that for a.e. $x\in S\cap[0,1],$ we have

$\displaystyle{\displaystyle\sum\limits_{k=-\infty}^{\infty}}\left|F\left(\xi+k\right)\right|^{2}=\left|F\left(\xi\right)\right|^{2}+{\displaystyle\sum\limits_{k=1}^{\infty}}\left(\left|F\left(\xi+k\right)\right|^{2}+\left|F\left(\xi-k\right)\right|^{2}\right)<\infty.$

Appealing to (10) and (11), we derive the following

(12)
${\displaystyle\sum\limits_{k=-\infty}^{\infty}}\left|F\left(\xi+k\right)\right|^{2}=\left|F\left(\xi\right)\right|^{2}\cdot\left(1+{\displaystyle\sum\limits_{k=1}^{\infty}}\left({\displaystyle\prod\limits_{j=0}^{k-1}}\left|p\left(\xi+j\right)\right|^{2}+{\displaystyle\prod\limits_{j=1}^{k}}\left|p\left(\xi-j\right)\right|^{-2}\right)\right)<\infty.$

By Egorov’s theorem, there exists a set $\Lambda\subset S\cap\left[0,1\right]$
of positive Lebesgue measure such that

$\lim_{k\to\infty}{\displaystyle\prod\limits_{j=0}^{k-1}}\left|p\left(\xi+j\right)\right|^{2}={\displaystyle\prod\limits_{j=0}^{\infty}}\left|p\left(\xi+j\right)\right|^{2}=\lim_{k\to\infty}{\displaystyle\prod\limits_{j=1}^{k}}\left|p\left(\xi-j\right)\right|^{2}={\displaystyle\prod\limits_{j=1}^{\infty}}\left|p\left(\xi-j\right)\right|^{-2}=0$

uniformly on $\Lambda.$

Moving forward, formally define functions $P,Q$ on set the $e^{\Lambda D}$
such that

$P\left(e^{\xi
D}\right)={\displaystyle\prod\limits_{j=0}^{\infty}}\left|p\left(\xi+j\right)\right|\text{
and }Q\left(e^{\xi
D}\right)={\displaystyle\prod\limits_{j=1}^{\infty}}\left|p\left(\xi-j\right)\right|^{-1}.$

Since $X=\overline{e^{\Lambda D}}$ has positive measure in $H$, we may extend
the definition of $P$ and $Q$ to $X$ such that $Q=P=0$ on $X$. Let $Z$ be a
measurable transversal for $H/G$ in $G.$ Since $X$ has positive
$\mu_{H}$-measure, there exists a unique $H$-invariant measure $\mu_{Z}$ [11,
Theorem 2.51] on $Z$ such that

(13)
$\mu_{H}\left(X\right)=\int_{H}1_{X}\left(h\right)d\mu_{H}\left(h\right)=\int_{Z}\int_{G}1_{X}\left(zg\right)d\mu_{G}\left(g\right)d\mu_{Z}\left(z\right)>0.$

Equation (13) gives a disintegration of the Haar measure on $H$ into a family
of measures, parameterized by the transversal $Z$ and defined over the cosets
of $G.$ Additionally, by virtue of (13), there exists at least one coset $zG$
of $G$ such that $zG\cap X$ has positive measure in $zG.$ Fixing such a coset,
the matrix $h\in zG\cap X,$ viewed as an element of $H$ may be factored
uniquely such that $h=zg$, $z\in Z$ and additionally,

$\displaystyle
P\left(h\right)=P\left(zg\right)={\displaystyle\prod\limits_{j=0}^{\infty}}\left|\left\langle
zge^{jD}c,v\right\rangle\right|=Q\left(h\right)=Q\left(zg\right)={\displaystyle\prod\limits_{j=1}^{\infty}}\left|\left\langle
zge^{-jD}c,v\right\rangle\right|^{-1}=0$

for all $g$ in a set $\Omega$ of positive measure in $G$.

Fix $\epsilon\in\left(0,1\right).$ Choose a natural number $L,$ sufficiently
large to ensure that for all $n>L,$ and $zg\in z\Omega,$

$\begin{cases}{\displaystyle\prod\limits_{j=0}^{n-1}}\left|\left\langle
zge^{jD}c,v\right\rangle\right|<\epsilon\\\
{\displaystyle\prod\limits_{j=1}^{n}}\left|\left\langle
zge^{-jD}c,v\right\rangle\right|^{-1}<\epsilon.\end{cases}$

Since for $z\in Z,$ the map $\sigma:zG\rightarrow zG$ defined by
$\sigma\left(z_{1}g_{1}\right)=e^{D}z_{1}g_{1}$ is ergodic, and by the Ergodic
theorem, there exists $n>L$ such that
$\mu_{zG}\left(\sigma^{n}\left(z\Omega\right)\cap z\Omega\right)>0.$ For
$g\in\sigma^{n}\left(z\Omega\right)\cap z\Omega,$ we may write $g=z\omega$ and

${\displaystyle\prod\limits_{j=1}^{n}}\left|\left\langle z\omega
e^{-jD}c,v\right\rangle\right|^{-1}={\displaystyle\prod\limits_{j=1}^{n}}\left|\left\langle
ge^{-jD}c,v\right\rangle\right|^{-1}<\epsilon$

for $n>L.$ Similarly, since $\sigma^{-n}\left(g\right)\in z\Omega,$ the
following inequality is satisfied:

${\displaystyle\prod\limits_{j=0}^{n-1}}\left|\left\langle\sigma^{-n}\left(g\right)e^{jD}c,v\right\rangle\right|={\displaystyle\prod\limits_{j=0}^{n-1}}\left|\left\langle
ge^{\left(j-n\right)D}c,v\right\rangle\right|<\epsilon.$

The change of variables $\ell=j-n$ yields

${\displaystyle\prod\limits_{\ell=-n}^{-1}}\left|\left\langle ge^{\ell
D}c,v\right\rangle\right|={\displaystyle\prod\limits_{\ell=1}^{n}}\left|\left\langle
ge^{-\ell D}c,v\right\rangle\right|<\epsilon.$

In summary, we have found $n>L$ such that

$\begin{cases}{\displaystyle\prod\limits_{j=1}^{n}}\left|\left\langle
ge^{-jD}c,v\right\rangle\right|^{-1}<\epsilon\\\
{\displaystyle\prod\limits_{j=1}^{n}}\left|\left\langle
ge^{-jD}c,v\right\rangle\right|<\epsilon,\end{cases}$

which is a contradiction, since $0<\epsilon<1.$ ∎

######  Remark 8.

  1. (a)

We say that two configurations $\Lambda_{1},\Lambda_{2}$ of $n$ distinct
points in $\mathbb{R}^{2}$ are equivalent if and only if there exists a
symplectic matrix $A\in SL(2,\mathbb{R})$ such that
$\Lambda_{2}=A\Lambda_{1}$. This relation defines an equivalence relation on
the set of all $n$-point configurations. As a result, we can partition the
collection of all sets of $n$ distinct points in $\mathbb{R}^{2}$ under this
equivalence relation. Therefore, when we choose an $n$-point configuration, we
may assume without any loss of generality that it is a representative for a
corresponding equivalence class. Furthermore, when $n\geq 3$ we may assume
that the chosen (representative) $n$-point configuration includes the set
$\\{(0,0),(0,1),(a,0)\\}$ where $a\neq 0$, [10, 12, 16].

  2. (b)

We now consider the case where $n=4$. By Theorem 6, the assumption
$g\in\mathcal{S}(\mathbb{R})$ can be removed from part (2) of Proposition 1.
Consequently, to settle the conjecture for sets of $4$ points, one needs to
consider configurations that are equivalent to
$\Lambda_{g}=\\{(0,0),(0,1),(a,0),(\alpha,\beta)\\}$ that are neither
$(1,3)-$configuration, $(2,2)-$configuration, nor
$(1,\mathbb{Z}^{2})-$configurations. Let $0\neq g\in L^{2}(\mathbb{R})$ and
suppose that $\Lambda_{g}$ is the collection of all $4$-point configurations
for which the HRT conjecture fails for this function $g$. The ultimate goal is
to prove that such collection is empty. Thus far, we can only say that it does
not contain any $(1,3)-$configuration, $(2,2)-$configuration, nor a
$(1,\mathbb{Z}^{2})-$ configuration. A question that has not received much
attention is to understand the structures of $\Lambda_{g}$. For example, is it
true that if $g_{1}\neq g_{2}\in L^{2}(\mathbb{R})$, then $\Lambda_{g_{1}}$
and $\Lambda_{g_{2}}$ are disjoint? In particular, suppose that
$\mathcal{G}(g,\Lambda_{g})$ is linearly dependent, and let $\Lambda^{\prime}$
be equivalent to $\Lambda_{g}$, then is it the case that
$\mathcal{G}(g,\Lambda^{\prime})$ is also linearly dependent?

## Acknowledgment

K.A.O. was partially supported by a grant from the National Science Foundation
grant DMS 1814253.

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# Speech-dependent Modeling of Own Voice Transfer Characteristics for In-ear
Microphones in Hearables

###### Abstract

Many hearables contain an in-ear microphone, which may be used to capture the
own voice of its user in noisy environments. Since the in-ear microphone
mostly records body-conducted speech due to ear canal occlusion, it suffers
from band-limitation effects while only capturing a limited amount of external
noise. To enhance the quality of the in-ear microphone signal using algorithms
aiming at joint bandwidth extension, equalization, and noise reduction, it is
desirable to have an accurate model of the own voice transfer characteristics
between the entrance of the ear canal and the in-ear microphone. Such a model
can be used, e.g., to simulate a large amount of in-ear recordings to train
supervised learning-based algorithms. Since previous research on ear canal
occlusion suggests that own voice transfer characteristics depend on speech
content, in this contribution we propose a speech-dependent system
identification model based on phoneme recognition. We assess the accuracy of
simulating own voice speech by speech-dependent and speech-independent
modeling and investigate how well modeling approaches are able to generalize
to different talkers. Simulation results show that using the proposed speech-
dependent model is preferable for simulating in-ear recordings compared to
using a speech-independent model.

Keywords: hearables, own voice, system identification, acoustic modeling,
relative transfer function

##  1 Introduction

Hearables, i.e. smart earbuds containing a loudspeaker and one or more
microphones, are often used in everyday noisy environments. Although hearables
are frequently used to enhance the voice of a person the hearable user is
communicating with in a noisy environment, the scenario we are considering in
this paper is to enhance the own voice of the user while talking in a noisy
environment (e.g., to be transmitted via a wireless link to a mobile phone or
another hearable). In-ear microphones may offer benefits for own voice pickup
since external noise is attenuated due to ear canal occlusion. However, own
voice recorded inside the occluded ear suffers from amplification below 1 kHz
and heavy attenuation above 2 kHz, leading to a limited bandwidth [1]. The
occlusion effect is affected by the ratio between the airborne and body-
conducted component of own voice, which depends on the phonemes uttered by the
user [2, 3]. Body conduction from different places of excitation and mouth
movements during articulation likely influence this transmission behavior,
which we refer to as transfer characteristics in this work. Additionally,
body-produced noise (e.g., breathing sounds, heartbeats) may be recorded by an
in-ear microphone [4]. To enhance the quality of the in-ear microphone signal,
several approaches have been proposed aiming at bandwidth extension,
equalization and/or noise reduction, either based on signal processing [1] or
supervised learning [5]. For supervised learning-based approaches large
amounts of training data are typically required, which may be hard to obtain
for realistic in-ear recordings.

Similar requirements have been addressed in supervised learning-based speech
enhancement approaches using bone-conduction sensors. In [6], it has been
proposed to use a device-specific corpus of recordings to train a deep neural
network (DNN) combining bone- and air-conducted speech recordings. In [7], a
semi-supervised training scheme has been utilized to jointly train a DNN
simulating bone-conduction and a multi-modal enhancement DNN together. In [8],
it has been proposed to convert airborne to bone-conducted speech using a DNN
that accounts for individual differences between talkers using a speaker
identification system. In previous work, we have proposed to estimate the
transfer characteristics between the entrance of the ear canal and the in-ear
microphone using a time-invariant linear model to simulate short segments of
in-ear speech for data augmentation in DNN training [5].

In this paper, we propose to model own voice transfer characteristics using a
phoneme-dependent system identification approach, where for each phoneme a
different linear filter is estimated. The proposed approach can be utilized to
simulate speech at an in-ear microphone from regular speech recordings.

##  2 Signal Model

Fig. 1 depicts the considered scenario, where a talker is wearing a hearable
device equipped with an in-ear microphone and an outer microphone, denoted by
subscript $\mathrm{i}$ and $\mathrm{o}$, respectively. In the short time
Fourier transform (STFT) domain the recorded own voice signal at the outer
microphone of talker $a$ is denoted by $Y_{\mathrm{o}}^{a}(k,l)$ with $k$ the
frequency bin index and $l$ the time frame index. The signal recorded at the
in-ear microphone $Y_{\mathrm{i}}^{a}$ is assumed to contain an in-ear own
voice speech component $S_{\mathrm{i}}^{a}$ and a body-noise component
$V_{\mathrm{i}}^{a}$, i.e.

$Y_{\mathrm{i}}^{a}(k,l)=S_{\mathrm{i}}^{a}(k,l)+V_{\mathrm{i}}^{a}(k,l)$ (1)

where $S_{\mathrm{i}}^{a}$ and $V_{\mathrm{i}}^{a}$ are assumed to be
uncorrelated. The own voice speech at the in-ear microphone
$S_{\mathrm{i}}^{a}(k,l)$ is related to the own voice speech at the outer
microphone $Y_{\mathrm{o}}^{a}(k,l)$ by transfer characteristics
$T^{a}\\{\cdot\\}$, i.e.

$S_{\mathrm{i}}^{a}(k,l)=T^{a}\left\\{Y_{\mathrm{o}}^{a}(k,l)\right\\}.$ (2)

We assume these transfer characteristics to be linear, time-varying and
individual. The goal in this paper is to obtain an accurate transfer
characteristics model, which is robust against talker mismatch.

$Y_{\mathrm{o}}^{a}(k,l)$$Y_{\mathrm{i}}^{a}(k,l)$$S_{\mathrm{o}}^{a}(k,l)$$S_{\mathrm{i}}^{a}(k,l)$$V_{\mathrm{i}}^{a}(k,l)$$T^{a}$

Figure 1: The signal model of own voice transfer characteristics considered in
this paper.

##  3 Modeling Own Voice Transfer Characteristics

In this section, we present two system identification-based approaches to
model the transfer characteristics (see Fig. 2).

Figure 2: Overview of the identification and simulation steps for modeling own
voice transfer characteristics.

During identification, recordings of talker $a$ are used to obtain the model
$\hat{T}^{a}$. During simulation, this model is used to obtain an estimate
$\hat{S}_{\mathrm{i}}^{b}(k,l)$ of the own voice of talker $b$. If $a=b$, the
model is applied to the same talker as in identification. If $a\neq b$, the
model is applied to a different talker, i.e. talker mismatch is present. In
Section 3.1, we present a time-invariant speech-independent model; in Section
3.2 we propose a time-varying speech-dependent model.

###  3.1 Speech-independent model

Assuming a linear time-invariant filter, the transfer characteristics $T^{a}$
of the speech-independent model are modeled as a relative transfer function
(RTF) between the outer microphone and the in-ear microphone. Since the outer
microphone signal does not contain any additive noise, the RTF
$\hat{H}^{a}(k)$ can be estimated using the well-known least squares approach
[9], i.e.

$\hat{H}^{a}(k)=\frac{\sum_{l}Y_{\mathrm{o}}^{a,*}(k,l)\cdot
Y_{\mathrm{i}}^{a}(k,l)}{\sum_{l}|Y_{\mathrm{o}}^{a}(k,l)|^{2}}$ (3)

where $\cdot^{*}$ denotes complex conjugation. For simulation, own voice
speech of talker $b$ recorded at the outer microphone is filtered in the STFT
domain as

$\hat{S}_{\mathrm{i}}^{b}(k,l)=\hat{H}^{a}(k)\cdot Y_{\mathrm{o}}^{b}(k,l).$
(4)

###  3.2 Speech-dependent model

Since own voice transfer characteristics likely depend on speech content, we
propose a time-varying speech-dependent model for the transfer characteristics
$T^{a}$. Using a phoneme recognition system, we first obtain a frame-wise
phoneme annotation $p(l)\in 1,\dots,P$ of a recorded speech signal, with $P$
possible phoneme classes. For each unique phoneme $p^{\prime}$, an RTF is then
estimated over all detected occurrences of this phoneme within the
identification utterances of talker $a$, i.e.

$\hat{H}_{p^{\prime}}(k)=\frac{\sum_{p(l)=p^{\prime}}Y_{\mathrm{o}}^{a,*}(k,l)\cdot
Y_{\mathrm{i}}^{a}(k,l)}{\sum_{p(l)=p^{\prime}}|Y_{\mathrm{o}}^{a}(k,l)|^{2}}.$
(5)

In total, the speech-dependent model hence consists of a database of $P$ RTFs.

For simulation, the phoneme sequence $p^{b}(l)$ is first determined on the own
voice speech of talker $b$ recorded at the outer microphone. For each frame,
the corresponding RTF $\hat{H}_{p(l)}^{a}(k)$ is then used to filter this
signal in the STFT domain, i.e.

$\hat{S}_{\mathrm{i}}^{b}(k,l)=\hat{H}_{p^{b}(l)}^{a}(k)\cdot
Y_{\mathrm{o}}^{b}(k,l).$ (6)

##  4 Technical Evaluation

In this section, the previously described transfer characteristics models are
evaluated in terms of their accuracy in predicting own voice signals at an in-
ear microphone.

###  4.1 Evaluation data and setup

A dataset of own voice speech from 14 native German talkers with approximately
30 minutes of speech in total is utilized in the evaluation. The hearable
device used for recording is the closed-vent variant of the Hearpiece [10].
Approximately 23 utterances per talker were recorded, resulting in a total of
329 utterances. For the speech-dependent model, an in-house proprietary
phoneme recognition system with $P=62$ phoneme classes was utilized, which was
trained on German speech. Both the speech-independent and the speech-dependent
model were estimated on all utterances per talker. For simulation, in-ear
speech was predicted for each utterance of 5 s length. No voice activity
detection was employed so that utterances may contain small pauses between
words. The estimation accuracy is computed in two conditions: estimating
speech of the same talker ($a=b$) for each utterance of each talker, and
estimating speech with talker mismatch ($a\neq b$) each utterance of each
talker is simulated using a randomly assigned different talker model.

The simulations were carried out at a sampling frequency of $5$ kHz to account
for the reduced bandwidth of the in-ear speech. An STFT framework with a frame
length of $K=256$ (corresponding to 51.2 ms), a frame overlap of 50 % and a
square-root Hann window for analysis and synthesis was used. Since body-
conducted speech travels faster than airborne speech, a prediction delay of 11
samples was applied to the in-ear signals prior to identification and
simulation to enable causal modeling. As evaluation metric, we utilize the
log-spectral distance (LSD) [11] between the real and simulated in-ear
recordings, where a lower value indicates a more accurate estimate.

###  4.2 Results and discussion

The results for the same talker-condition are shown in Fig. 3. It can be
observed that the in-ear speech signals can be predicted much better using the
time-varying speech-dependent model than using the time-invariant speech-
independent model. Since in-ear recordings are assumed to also contain body
noise uncorrelated to the outer microphone signals, remaining modeling errors
are expected due to models not accounting for body-noise.

Figure 3: LSD results for the same talker-condition.

Figure 4: LSD results for the talker mismatch-condition.

The results for the talker mismatch-condition are shown in Fig. 4. It can be
observed for both models that the performance is worse than in the same talker
condition, where especially for the speaker-independent model there is a
larger spread of LSD scores in the talker mismatch condition than in the same-
talker condition. Nevertheless, also for the talker mismatch condition the
speech-dependent model clearly outperforms the speech-independent model.

##  5 Conclusion

In this paper, two approaches to model own voice transfer characteristics in
hearables have been investigated. Results indicate that speech-dependent
modeling is beneficial compared to speech-independent modeling.

##  6 Acknowledgments

The Oldenburg Branch for Hearing, Speech and Audio Technology HSA is funded in
the program »Vorab« by the Lower Saxony Ministry of Science and Culture (MWK)
and the Volkswagen Foundation for its further development. Part of this work
was funded by the German Ministry of Science and Education BMBF FK 16SV8811.
This work was partly funded by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) - Project ID 352015383 (SFB 1330 C1).

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