# Harmonic and non-Harmonic Based Noisy Reverberant Speech Enhancement in Time
Domain

G. Zucatelli, and R. Coelho R. Coelho is partially supported by the National
Council for Scientific and Technological Development (CNPq) 308155/2019 and
Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ) 203075/2016
research grants. This work is also supported by the Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Grant Code
001. The authors are with the Laboratory of Acoustic Signal Processing,
Military Institute of Engineering (IME), Rio de Janeiro, RJ 22290-270, Brazil
(e-mail: coelho@ime.eb.br).

###### Abstract

This paper introduces the single step time domain method named HnH-NRSE, whihc
is designed for simultaneous speech intelligibility and quality improvement
under noisy-reverberant conditions. In this solution, harmonic and non-
harmonic elements of speech are separated by applying zero-crossing and energy
criteria. An objective evaluation of the its non-stationarity degree is
further used for an adaptive gain to treat masking components. No prior
knowledge of speech statistics or room information is required for this
technique. Additionally, two combined solutions, IRMO and IRMN, are proposed
as composite methods for improvement on noisy-reverberant speech signals. The
proposed and baseline methods are evaluated considering two intelligibility
and three quality measures, applied for the objective prediction. The results
show that the proposed scheme leads to a higher intelligibility and quality
improvement when compared to competing methods in most scenarios.
Additionally, a perceptual intelligibility listening test is performed, which
corroborates with these results. Furthermore, the proposed HnH-NRSE solution
attains SRMR quality measure with similar results when compared to the
composed IRMO and IRMN techniques.

###### Index Terms:

Speech Enhancement , Noisy-Reverberant, Non-Stationarity, Speech
Intelligibility, Speech Quality

##  I Introduction

Speech communication in urban environments is an essential challenge in daily
life. In these scenarios, the speech signal is generally submitted to the
presence of acoustic noises and reverberation. This situation may lead to
masking effects on speech signals and can also alter the human auditory
perception [1][2][3][4]. Consequently, noisy-reverberant interferences reduce
the quality and the intelligibility of speech signals by inducing changes on
relevant harmonic structures of speech [5][6].  Studies have demonstrated that
intelligibility and quality are related [7], which motivates combined
solutions that simultaneously improve these characteristics under adverse
conditions.

Methods have been designed to exclusively improve speech intelligibility under
noisy and reverberant conditions [8][9][10][11][12]. Spectral solutions
usually adopt energy equalization to deal with these acoustic distortions. The
goal is to change the speech spectral energy distribution in other to have a
more intelligible signal propagated inside a room. This is the case of the ACO
(Adaptive Compressive Onset-Enhancement) [13], which alters the energy present
in different frequency bands considering prior knowledge on the acoustic noise
and the Room Impulse Response (RIR). A similar approach based on the PDMSE
(Perceptual Distortion Measure-based Speech Enhancement) [9] and FIR (Fast
Inverse Filtering) techniques [14] was proposed in [15] with interesting
objective and subjective intelligibility results. In [16] two solutions are
described to cope with acoustic noise in order to improve speech
intelligibility by spectral equalization strategies. Temporal adaptive methods
focus on preserving speech transient regions that are considered more relevant
for speech intelligibility due to greater phonetic information [17]. The SSS
(Steady-State Suppression) [10] was proposed considering the suppression of
steady-state components that usually indicates frames with greater energy. By
amplitude attenuation, there is a reduction on the masking distortion that
affects transient regions, therefore increasing the intelligibility. Another
temporal approach is the AGC (Adaptive Gain Control) [11], that adjusts the
energy of each frame in order to reduce masking effects on relevant speech
regions by an optimization criteria. In the ARANSD (Adaptive Reverberation
Absorption using Non-Stationarity Detection) method [12] the non-stationarity
of speech regions is accessed by the INS (Index of Non-Stationarity) [18].
This solution is able to restore the natural non-stationary behavior of speech
signals. Furthermore, it also attained an interesting speech quality gain
[12]. This method defines the RGs (Reverberation Groups) regions in order to
observe the non-stationary behavior of speech signals and absorbs masking
components based on the non-stationarity variation.

In the literature, speech enhancement approaches are designed to mainly
improve the speech quality in adverse conditions [19][20][21][22]. These
solutions identify the distortion components based on temporal and spectral
estimators [23] or using decomposition techniques such as the EMD (Empirical
Mode Decomposition) [24]. The OMLSA (Optimally-Modified Log-Spectral
Amplitude) [19] adopts an acoustic noise estimator to access the spectral
noise power and reconstruct the speech signal based on the minimization of the
log-spectral mean square error. Another speech enhancement, the EMDH
(Empirical Mode Decomposition with Hurst exponent) [20][21] is a temporal
method that employs the Hurst exponent [25] to identify and remove Intrinsic
Mode Functions (IMFs) most corrupted by noise from the EMD decomposition. The
NNESE (Nonstationary Noise Estimation for Speech Enhancement) strategy [22] is
a temporal solution that employ the robust DATE estimator [26] on a short-time
basis, being able to account for the non-stationarity of acoustic signals.

In order to simultaneously improve speech intelligibility and quality, the
TSDL (Two-Stage with Deep Learning) [27] approach is proposed based on two DNN
(Deep Neural Network) stages. Each stage is trained in to separately and
consecutively mitigate noise and reverberation for concurrent improvement. The
combined approach presented in [28] is a composition of the acoustic mask IBM
(Ideal Binary Mask) [28] and the OMLSA speech enhancement technique [19] to
provide intelligibility and quality improvement in noisy environments.

In this paper, the method HnH-NRSE (Harmonic and non-Harmonic based Noisy-
Reverberant Speech Enhancement) is proposed as a single approach to attain
simultaneous intelligibility and quality improvement of noisy-reverberant
speech signals. The solution takes into account the non-stationarity
variations of speech for the adaptive gain approach and consider an harmonic
and non-harmonic decomposition, since the harmonic components are naturally
related to the intelligibility [6][29]. Additionally, two combined solutions
IRMO (Ideal Reverberant Mask with OMLSA) and IRMN (Ideal Reverberant Mask with
NNESE) are proposed for the noisy-reverberant scenario. These methods adopt
the Ideal Reverberant Mask (IRM) [30] in composition with speech enhancement
techniques OMLSA and NNESE, and is also designed for both intelligibility and
quality improvement.

Extensive experiments are conducted to objectively evaluate the proposed
solutions for speech intelligibility and speech quality gain. The noisy-
reverberant scenario is composed of two real reverberant rooms from the AIR
[31] and LASP_RIR111Available at www.lasp.ime.eb.br databases and two
background non-stationary acoustic noises extracted from RSG-10 [32] and
DEMAND [33] databases. The intelligibility assessment in performed considering
the objective measures STOI [34] and ASIIST [35]. The SRMR [36], PESQ [37] and
f2-model PEAQ [38] [39] measures are adopted for quality objective evaluation.
A subjective intelligibility listening test is also performed and results show
that the proposed HnH-NRSE method outperforms competing techniques in terms of
speech intelligibility. Moreover, the HnH-NRSE is able to achieve similar SRMR
values when compared to the proposed combined approaches.

The main contributions of this work are:

  * •

introduction of the HnH-NRSE single method for simultaneous intelligibility
and quality improvement of noisy-reverberant speech signals in time domain.

  * •

design of composite solutions IRMN for noisy-reverberant speech signals
improvement.

  * •

definition of composite solution IRMO for speech improvement under noisy-
reverberant environments.

The remainder of this paper is organized as follows. In Section II, the noisy-
reverberant effect is defined and analyzed considering the speech harmonic
structure and non-stationarity. The single stage HnH-NRSE solution and
proposed combined methods IRMO and IRMN are introduced in Section III. The
objective measures used to evaluate the techniques on speech quality and
intelligibility are briefly described in Section IV. The experiments are
exposed in Section V. Then, the results for improvements on speech
intelligibility and quality are presented and discussed for all methods.
Finally, Section VI concludes this work.

##  II Noisy-Reverberant Acoustic Effects

The reverberation effect is usually defined as a linear filtering process such
that, given a RIR $h(t)$, the reverberated signal can be obtained by
convolution. In real environments, acoustic noises are also a common
distortion, which means that the resultant noisy-reverberant speech signal
$r(t)$ can be obtained by

$r(t)=s(t)*h(t)+w(t),$ (1)

where $s(t)$ is the clean speech signal and $w(t)$ is the background noise.

(a)

(b)

(c)

Figure 1: Spectrogram and INS of clean (a), reverberant (b) and noisy-
reverberant (c) speech signals for LASP2 room and Babble noise at $10$ dB.

Fig. 1 depicts the spectrogram and INS [18] of the clean, reverberant and
noisy-reverberant speech signal considering the LASP2 room11footnotemark: 1
and the Babble noise at $10$ dB. It can be seen that the distinction between
silence and voice active regions present on the direct signal is masked by the
noisy and reverberant effects. Moreover, the harmonic structure of the speech
signal is altered, changing observable formants, specially for lower
frequencies. This illustrate how the noisy-reverberant effect alter the
harmonic structure of the speech signals, change human auditory perception and
reduce acoustic intelligibility and quality [1][2][6]. Moreover, the
corresponding INS values show how the natural non-stationary behavior of
speech signals is altered in these conditions.

Figure 2: Block diagram for the proposed HnH-NRSE method. The harmonic
structure of the speech signal is adopted as a criteria for signal
decomposition and adaptive absorption of noisy-reverberant masking components.

This objective measure compares the target signal with stationarity references
called surrogates for different time scales $T_{h}/T$, where $T_{h}$ is the
short-time spectral analysis length and $T$ is the total signal duration.
Surrogate signals have the same absolute spectrogram values of a target signal
and phase identically distributed in $[-\pi,\pi]$ [18]. For each length
$T_{h}$, the INS is defined using the ratio between the variance of spectral
distances from local to global multitaper spectrograms for the target signal
($\Theta_{1}$) and corresponding surrogates ($\Theta_{0}(k)$) as

$\text{INS}\coloneqq\sqrt{\frac{\Theta_{1}}{\langle\Theta_{0}(k)\rangle}_{k}}.$
(2)

A threshold $\gamma$ is defined to keep the stationarity assumption
considering a $95\%$ confidence degree as

$INS\quad\left\\{\begin{matrix}\leq\gamma,\quad\text{signal is stationary}\\\
>\gamma,\quad\text{signal is non-stationary}.\end{matrix}\right.$ (3)

The harmonic components of the speech signal are altered by both masking
effects that reduce the natural non-stationary behavior of speech signals from
a maximum value of $600$ down to $300$ and $100$ for the reverberant and
noisy-reverberant scenarios, respectively.

##  III Proposed Methods

In this Section, the HnH-NRSE method is presented, which aims to
simultaneously improve intelligibility and quality of noisy-reverberant speech
signals. Moreover, the two composed solutions IRMN and IRMO establish the
intrusive reference as a composition of the Ideal Reverberant Mask (IRM) [30]
and speech enhancement methods NNESE [22] and OMLSA [19].

###  III-A Proposed HnH-NRSE: Harmonic and non-Harmonic based Noisy
Reverberant Speech Enhancement

The HnH-NRSE aims to apply the adaptive absorption based on variations of
speech non-stationarity [12] taking into account the harmonic elements of
speech signal. A block diagram is illustrated in Fig. 2.  The solution can be
described in three main steps: harmonic and non-harmonic decomposition,
adaptive masking absorption and signal reconstruction.

####  III-A1 Harmonic and non-Harmonic Separation

Given a noisy-reverberant signal $r(t)$, consider its $l$-th frame
representation $r(l,t)$ and the maximum energy $EN_{\text{max}}$ observed in
all frames. The harmonic and non-harmonic separation is performed based on the
energy and zero-crossing thresholds [40], $EN_{th}$ and $ZC_{th}$, as

$r_{\text{Harm}}(l,t)=\begin{cases}r(l,t),\quad ZC(l)<ZC_{th}\text{ and }\\\
\quad\quad\quad\quad EN(l)-EN_{\text{max}}>EN_{th}\\\ [0\dots
0],\quad\quad\text{otherwise,}\end{cases}$ (4)

and

$r_{\text{non-Harm}}(l,t)=\begin{cases}[0\dots 0],\quad ZC(l)<ZC_{th}\text{
and }\\\ \quad\quad\quad\quad EN(l)-EN_{\text{max}}>EN_{th}\\\
r(l,t),\quad\quad\text{otherwise,}\end{cases}$ (5)

where $EN(l)$ and $ZC(l)$ are the energy and zero-crossing values for a given
frame, respectively. For low SNR scenarios, when
$EN_{\text{max}}-EN_{\text{min}}$ is greater than $EN_{th}$, the energy
threshold is updated to $EN_{th}=-0,55*(EN_{\text{max}}-EN_{\text{min}})$.
This is important to achieve a better discrimination between harmonic and non-
harmonic frames.

####  III-A2 Adaptive Masking Absorption

For the adaptive absorption of masking components, it is adopted a short-time
absorption procedure based on variations of the non-stationarity of
Reverberation Groups (RG) [12] assessed by the INS [18].

The non-stationarity variation for consecutive RGs is defined by the
normalized distance
$\delta_{INS}(m)=||{\mathbf{v_{INS}}}(m)-{\mathbf{v_{INS}}}(m-1)||/(||{\mathbf{v_{INS}}}(m)||+||{\mathbf{v_{INS}}}(m-1)||)$,
where $m$ stands for the $m$-th RG of a signal and $\mathbf{v_{INS}}$ its INS
vector. A similar approach is also performed on a frame-by-frame bases as
$d(l)\in[0,1]$ to define the frame absorption $A(m,l)$ which is dependent on a
non-stationarity threshold $\theta_{INS}$ as

$\small
A(m,l)=\begin{cases}F(l).\frac{L(m)-S}{1+\exp(-k.(d(l)-d_{0}))}+S,\quad\delta_{INS}\leq\theta_{INS};\\\
\frac{L^{\prime}}{1+\exp(-k^{\prime}.(d(l)-d^{\prime}_{0})},\quad\hskip
36.98866pt\delta_{INS}>\theta_{INS},\end{cases}$ (6)

where $d_{0}$ and $d^{\prime}_{0}$ are the inflection points with growth rates
of $k$ and $k^{\prime}$, $S$ is a minimum shift in order to avoid total
absorption. $L^{\prime}$ and $L(m)=p\delta_{INS}+(1-p)L(m-1)$ define the
maximum absorption values, where $p$ assigns the importance of the present RG
and $F(l)$ is a factor defined in [12] such that $A(m,l)\approx L(m)$ only for
$d(l)\approx 1$.

Considering the harmonic structure of speech signals, the harmonic absorption
approach $A_{\text{HnH}}(m,l)$ is defined as

$\small
A_{\text{HnH}}(m,l)=\begin{cases}(1+f_{\text{HnH}}.(\delta_{INS})^{c_{\text{HnH}}})A(m,l),\quad\text{harmonic}\\\
(1-f^{\prime}_{\text{HnH}}.(\delta_{INS})^{c^{\prime}_{\text{HnH}}})A(m,l),\quad\text{non-
harmonic},\end{cases}$ (7)

where $f_{\text{HnH}}$, $f^{\prime}_{\text{HnH}}$, $c_{\text{HnH}}$ and
$c^{\prime}_{\text{HnH}}$ are free parameters adopted to incorporate the
harmonic and non-harmonic information of each frame based on the non-
stationary variation $\delta_{INS}$. This is performed in order to enhance
harmonic frames that are relevant for speech intelligibility and reduce noisy-
reverberant masking components. Therefore, the processed frame
$r^{\prime}(l,n)$ can be obtained by
$r^{\prime}(l)=Abs_{\text{HnH}}(m,l)*r_{\text{Harm}}(l,n)+Abs_{\text{HnH}}(m,l)*r_{\text{não-
Harm}}(l,n)$.

####  III-A3 Speech Signal Reconstruction

The last step of the HnH-NRSE method is the reconstruction of the speech
signal as a processed version $r^{\prime}(t)$. To this end, resulting frames
$r^{\prime}(l,n)$ are used on the time domain overlap-add signal composition
technique.

For the proposed HnH-NRSE, the harmonic factors and exponent coefficients are
defined as $f_{\text{HnH}}=1.1$, $f^{\prime}_{\text{HnH}}=0.7$,
$c_{\text{HnH}}=0.2$ and $c^{\prime}_{\text{HnH}}=0.1$.

###  III-B Proposed Composite Methods

In this subsection, the composite methods IRMN and IRMO are introduced as a
combination of the acoustic mask IRM and the speech enhancement methods NNESE
and OMLSA.

$1)$ IRMN: The IRM [30] is defined for reverberant environments considering
the SRR (Speech-to-Reverberant Ratio). The main idea is to select TF regions
on frame $l$ and band $j$ that have more energy related to the direct path
signal and remove overlap-masking components due to reverberation inside a
room. Therefore, the IRM can be defined as

$\text{IRM}(l,j)=\begin{cases}1,\quad\text{SRR}(l,j)>\Theta_{Th}\\\
0,\quad\text{otherwise},\end{cases}$ (8)

where $\Theta_{Th}$ is a threshold in dB that defines the selection of TF
regions.

The composite scheme proposed in this paper is based on the temporal speech
enhancement method NNESE [22]. In the NNESE procedure the noise standard
deviation is estimated using the DATE estimator (d-Dimensional Trimmed
Estimator) [26] on a non-overlap frame-by-frame approach. To this end, a
detection threshold $\xi(\rho_{l})$ is defined as

$\xi(\rho_{l})=\frac{1}{2}\rho_{l}+\frac{1}{\rho_{l}}\log\left(1+\sqrt{1-\exp({-\rho_{l}^{2}})}\right)$
(9)

where $\rho_{l}$ is the minimum SNR threshold for the $l$-th frame. For a
Gaussian noise, $\rho_{l}=4$ and $\xi(\rho)=3.4742$.

The DATE estimator organize the $T$ samples of the $l$-th frame $y_{l}(t)$ by
amplitude as $Y_{1}\leq Y_{2}\leq...\leq Y_{T}$. The Bienayme-Chebyshev-Markov
inequality is then applied to calculate $t_{min}$ such that the samples lower
than $Y_{t_{min}}$ are considered noise-only amplitudes.

$t_{min}=\frac{T}{2}-\frac{1}{\sqrt{4T(1-Q)}}T$ (10)

where $Q$ is the confidence degree, which is assumed to be $95$% for a
Gaussian noise.

Then, it is verified the existence of a $b_{l}\in\\{t_{min},\dots,T\\}$ that
satisfies the relation

$||Y_{t-1}||\leq\frac{c\sum_{i=1}^{b_{l}}||Y_{i}||}{t}\leq||Y_{t+1}||,$ (11)

where $c$ is adjustment factor that is dependent on the detection threshold:

$c=\frac{\xi(\rho)\Gamma(0.5)}{\sqrt{2}\Gamma(1.0)}.$ (12)

Otherwise, it is considered to be $b_{l}=t_{min}$. The noise standard
deviation for that frame is then calculated as

$\hat{\sigma}_{l}=\frac{c\sum_{i=1}^{b_{l}}||Y_{i}||}{b_{l}}.$ (13)

The amplitude is modified directly from the corrupted signal considering
attenuation factors $\alpha$ and $\beta$ as

$\tilde{y}_{l}(t)=\begin{cases}y_{l}(t)-\alpha\hat{\sigma}_{l}&,y_{l}(t)>Y(b_{l})\\\
\beta y_{l}(t)&,\text{otherwise}.\end{cases}$ (14)

The IRM is directly considered for the estimation of $\hat{\sigma_{l}}$. After
organizing the samples of a frame $l$, the standard deviation estimate is
selected by the fraction threshold $\Phi_{l}$ of regions dominated by noisy-
reverberant distortions in the local acoustic mask. For a total number of
frequency bands $J$ adopted in the IRM, $\Phi_{l}$ is defined as

$\Phi_{l}=1-\frac{\sum_{j=1}^{J}\text{IRM}(l,j)}{J},\quad\Phi_{l}\in[0,1].$
(15)

This way, the value $b_{l}$ in equation (13) is replaced by $b_{l}=\lfloor
T.\Phi_{l}\rfloor$, where $\lfloor\bullet\rfloor$ is the integer operator.

After the amplitude removal of the distortion components, the acoustic mask is
applied to the enhanced signal. As the NNESE speech enhancement is temporal
designed, the posterior application of IRM can deal with spectro-temporal
regions that were not completely considered in a intrusive fashion, leading to
an overall better speech intelligibility and quality improvement result.

$2)$ IRMO: The second comparative method is a variation of [41] for
environments with the reverberation effect. The proposed adapted strategy
adopts the acoustic mask IRM and OMLSA speech enhancement approaches and is
denominated as IRMO.

An important step of OMLSA technique is the association of the SSPs (Speech
Presence Probabilities) in a given TF region. This is performed in order to
calibrate the gain for the enhancement procedure. In this work, this step is
replaced by a IRM indication of a TF speech region as follows.

Consider $X(l,j)$ and $Y(l,j)$ the respective STFT coefficients of the direct
and corrupted speech signals obtained via overlapping Hamming analysis
windows. Let the presence and absence of speech be considered under hypothesis
$H_{1}(l,j)$ and $H_{0}(l,j)$, respectively. Taking into account the
assumption of statistically independent Gaussian random variables for the STFT
coefficients, the desired gain $G(l,j)$ is defined such that

$G(l,j)|Y(l,j)|=\exp\\{E[\log|X(l,j)||Y(l,j)|]\\},$ (16)

where $E[\cdot]$ is the expectation operator. The OMLSA adopts the IMCRA [42]
to estimate the noise power spectrum. After the estimation, this method
reconstructs the enhanced speech signal by minimizing the mean-square error of
the log-spectral amplitude. Under the constraint of $G_{H_{1}}(l,j)$ assuming
greater values than a threshold $G_{min}$ when speech is absent it is shown
[19] that

$G(l,j)={G_{H_{1}}(l,j)}^{p(l,j)}G_{min}^{q(l,j)},$ (17)

where $G_{H_{1}}(l,j)$ is the gain under the hypothesis that speech is present
in the TF bin, $p(l,j)$ is the SPP and $q(l,j)=1-p(l,j)$ is the probability of
absence of speech.

For the IRMO composite scheme, the values of $q(l,j)$ and $G_{min}$ are both
determined by IRM$(l,j)$, as in

$q(l,j)=\begin{cases}Q^{1},\quad\text{IRM}(l,j)=1\\\
Q^{0},\quad\text{IRM}(l,j)=0,\end{cases}$ (18)

and

$G_{min}(l,j)=\begin{cases}G^{1},\quad\text{IRM}(l,j)=1\\\
G^{0},\quad\text{IRM}(l,j)=0,\end{cases}$ (19)

where $Q^{1}$, $Q^{0}$, $G^{1}$ and $G^{0}$ are defined in [41]. The
composition of these two methods merges the spectral cues present in the mask
that are important to intelligibility with the quality gain imposed by the
speech enhancement method. This way, the approach can obtain simultaneous
intelligibility and quality improvement under noisy-reverberant conditions.
Additionally, the reverberant mask threshold $\theta_{Th}$ is set to $-6$ dB
for composite solutions IRMO and IRMN.

##  IV Objective Evaluation of Speech Intelligibility and Quality

This Section, briefly introduces of the objective measures applied in this
work for speech intelligibility and speech quality prediction.

###  IV-A Short-Time Intelligibility Objective Measure

The short-time objective intelligibility measure (STOI) [34] is a correlation-
based method to compare the spectrum of the clean and the enhanced speech
signals in the frequency domain. The correlation between temporal envelopes of
the clean and noisy speech signals is defined as the intermediate
intelligibility measure STOI(j,l) of each frequency band $j$ and each time
frame $l$. The STOI is then computed as

$\text{STOI}=\frac{1}{15}\sum_{l=1}^{L}\sum_{j=1}^{15}\text{STOI}_{(j,l)}$
(20)

where $L$ is the total number of speech frames.

###  IV-B Short-Time Approximated Speech Intelligibility Index

For approximated short-time versions of the SII [43][35], time-frequency SNR
$\xi(j,l)$ values are computed at each critical frequency band $j$ and time
frame $l$, and than normalized to the $[0,1]$ range. The ASIIST measure
assumes a smooth normalization for the time-frequency SNR, given by

$\small d(j,l)=\frac{\xi(j,l)}{\xi(j,l)+1}.$ (21)

The ASIIST is then computed as a weighted average of all values given in (21):

$\small\text{ASII\textsubscript{ST}}=\frac{1}{L}\sum_{l=1}^{L}\sum_{j=1}^{J}\gamma_{j}d(j,l),$
(22)

where $J$ is the total number of critical bands, and $\gamma_{j}$ are the
critical-band-importance weights. A relevant aspect of these short-time based
measures is the capacity to account for the natural non-stationarity behavior
of acoustic noises.

###  IV-C Perceptual Evaluation of Speech Quality

The PESQ is an objective measure assess quality of narrow-banded speech and
handset telephony [37]. It is also largely adopted to estimate the quality of
speech signals corrupted with noise due to its high correlation to hearing
perceptual evaluation [44].

The PESQ score is calculated following four major steps of pre-processing,
time alignment, loudness perception model transformation and disturbance
computation. Due to the non-stationary behavior of acoustic noises, in this
work the symmetrical ($d_{s,l}$) and asymmetrical ($d_{a,l}$) distances are
computed per frame between the analyzed and reference speech signals

$PESQ_{l}=4.5-0.1d_{s,l}-0.0309d_{a,l},$ (23)

such that the final PESQ score is average over all frames taking into account
the variability of acoustic distortions.

###  IV-D Speech-to-Reverberation Modulation Energy Ratio

The non-intrusive SRMR quality metric [36] estimates the human perceived
reverberation effect on speech signals. This measure uses long-term temporal
dynamics information to identify the presence of reverberation considering a
double modulation per frame approach. An average per-modulation band energy
$\bar{\varepsilon_{j}}$ is obtained from Hilbert temporal envelopes. The SRMR
is then computed as

$\text{SRMR}=\frac{\sum_{j=1}^{4}\bar{\varepsilon_{j}}}{\sum_{j=5}^{J^{*}}\bar{\varepsilon_{j}}},$
(24)

where $J^{*}$ is the upper summation bound that is dependent on the speech
signal under evaluation.

###  IV-E Perceptual Evaluation of Audio Quality

The PEAQ [38] is a measurement of perceptual evaluation of coded audio signals
based on several perceptual features called Model Output Variables (MOVs).
These features are typically computed estimated masking thresholds for the
error signal or comparing the internal ear representation between processed
and reference signals. The measure is finally obtained using the output of a
neural network. In this work, the PEAQ f2-model variation is adopted with
newly parameters and two MOVs: the Average Distorted Blocks (ADB) and the
Average Modulation Difference #1 (AvgModDiff1) as in [39].

##  V Experiments and Discussion

The speech intelligibility and quality results obtained with proposed
solutions and baseline techniques ARANSD [12] and TSDL [27] are presented in
this Section. Each testing scenario considered $100$ speech signals from the
IEEE speech database [45]. A total of $600$ speech signals is selected for the
TSDL training step, with $100$ of these signals being used as validation set.
Acoustic noises Babble and Cafeteria were extracted from Freesound222Available
at www.freesound.org. database identified as “Group Talking” and “Cafe
Ambiance”, respectively. Speech utterances have sampling rate of $16$ kHz and
average time duration of $3$ seconds.

Several noisy-reverberant conditions are used to evaluate the proposed
techniques in terms of speech intelligibility and quality. In order to compose
the noisy-reverberant environment, two real reverberation rooms (LASP2 and
Stairway) and two non-stationary acoustic noises (Babble and Cafeteria) are
considered in the experiments. The LASP2 room is part of the
LASP_RIR333Available at lasp.ime.eb.br. database with a reverberation time of
$T_{60}=0.79$ s. Room Stairway is selected from the AIR database [31] and
presents reverberation times of $T_{60}=1.1$ s. The Babble and Cafeteria
additive background noises are selected, respectively, from the RSG-10 [32]
and DEMAND [33] databases. These noises are classified as non-stationary, with
maximum INS of $40$ and $15$.

Table I: STOI results for LASP2 room and acoustic noises Babble and Cafeteria.

Noise | SNR | UNP | ARANSD | TSDL | HnH-NRSE | IRMO | IRMN
---|---|---|---|---|---|---|---
Babble INS $=40$ | 2.3 | 64.0 | 64.2 | 64.6 | 67.1 | 85.1 | 87.0
0.2 | 60.0 | 60.1 | 61.1 | 64.0 | 83.6 | 85.2
-1.9 | 56.0 | 56.8 | 58.4 | 61.0 | 81.5 | 83.1
-4.3 | 52.0 | 52.2 | 54.6 | 57.9 | 79.0 | 80.6
-7.3 | 48.0 | 48.6 | 50.1 | 54.8 | 75.6 | 77.3
-10.6 | 44.0 | 44.7 | 44.8 | 51.7 | 70.3 | 71.6
Average | 54.0 | 54.4 | 55.6 | 59.4 | 79.2 | 80.8
Cafeteria INS $=15$ | 1.3 | 64.0 | 64.1 | 65.8 | 66.4 | 85.3 | 87.4
-0.8 | 60.0 | 60.2 | 63.7 | 63.4 | 83.7 | 85.8
-2.9 | 56.0 | 56.2 | 61.3 | 60.3 | 81.6 | 83.8
-5.4 | 52.0 | 52.7 | 58.7 | 57.2 | 79.1 | 81.4
-8.3 | 48.0 | 48.3 | 53.7 | 54.1 | 75.6 | 77.8
-12.2 | 44.0 | 41.1 | 46.8 | 51.0 | 70.4 | 73.6
Average | 54.0 | 54.3 | 58.3 | 58.7 | 79.3 | 81.6
Overall Average | 54.0 | 54.4 | 57.0 | 59.1 | 79.2 | 81.4

Table II: STOI results for Stairway room and acoustic noises Babble and
Cafeteria.

Noise | SNR | UNP | ARANSD | TSDL | HnH-NRSE | IRMO | IRMN
---|---|---|---|---|---|---|---
Babble INS $=40$ | 4.2 | 64.0 | 64.3 | 63.5 | 68.6 | 86.9 | 89.0
2.1 | 60.0 | 60.4 | 61.2 | 65.4 | 85.6 | 87.4
0.1 | 56.0 | 56.0 | 58.6 | 62.2 | 83.9 | 85.5
-2.2 | 52.0 | 52.7 | 55.4 | 59.0 | 81.7 | 83.1
-4.8 | 48.0 | 48.3 | 51.3 | 55.7 | 78.9 | 80.0
-8.6 | 44.0 | 44.8 | 45.9 | 52.3 | 74.1 | 75.4
Average | 54.0 | 54.4 | 56.0 | 60.5 | 81.8 | 83.4
Cafeteria INS $=15$ | 2.5 | 64.0 | 64.2 | 66.2 | 68.3 | 87.1 | 89.0
0.6 | 60.0 | 60.2 | 64.2 | 65.2 | 85.6 | 87.4
-1.5 | 56.0 | 56.9 | 61.5 | 62.1 | 83.8 | 85.5
-3.8 | 52.0 | 52.6 | 57.9 | 58.9 | 81.6 | 83.1
-6.5 | 48.0 | 48.4 | 53.2 | 55.6 | 78.4 | 79.8
-12.8 | 44.0 | 44.1 | 47.2 | 52.3 | 73.2 | 75.6
Avg. | 54.0 | 54.4 | 58.4 | 60.4 | 81.6 | 83.4
Overall Average | 54.0 | 54.4 | 57.2 | 60.5 | 81.7 | 83.4

(a)

(b)

Figure 3: Boxplot of ASIIST objective values for (a) LASP2 room and (b)
Stairway room, considering acoustic noises Babble and Cafeteria

Speech signals are corrupted considering SNRs values ranging from $-12.8$ dB
up to $4.2$ dB, where the SNRs are measured between the reverberant speech
signal and the background noise. For each reverberant signal, SNRs are
selected in order to obtain STOI [34] objective intelligibility scores of
{$0.44$, $0.48$, $0.52$, $0.56$, $0.60$ and $0.64$}. The limit values of
$0.44$ and $0.64$ are defined here as thresholds of poor and good
intelligibility for the unprocessed (UNP) noisy-reverberant speech. Results
are organized considering the increasing value of $T_{60}$, i.e., for each
measure rooms LASP2 ($T_{60}=0.79$ s) and Stairway ($T_{60}=1.1$ s) are
presented consecutively with acoustic noises Babble and Cafeteria within each
case.

A listening test [46] with $10$ native Brazilian volunteers ($5$ male and $5$
female) was conducted adopting a closed scenario of phonetic balanced
words22footnotemark: 2. Their ages ranged from $20$ to $24$ years with an
average of $22$. A simulated room with $7.0$ x $5.2$ x $3.0$ m3 and
$T_{60}=1.0$ s was generated by the image source method (ISM) [47]. The SSN
acoustic noise was adopted with SNRs of $-3$ dB, $0$ dB and $3$ dB. $10$ words
were considered for each of test conditions, i.e., $3$ SNR levels for $5$
methods plus the unprocessed case. Participants were introduced to the task in
a training session with $4$ words. The material was diotically presented using
a pair of Sennheiser HD200 headphones. Listeners heard each word once in an
arbitrary presentation order and selected one among five words in order to
compute the Word Recognition Rate (WRR).

###  V-A Objective Speech Intelligibility Evaluation

The proposed and competing methods are first evaluated in terms of STOI
intelligibility scores. Tables I and II show the STOI values for rooms LASP2
and Stairway, respectively. Note that, the HnH-NRSE attains the highest STOI
average results for most SNR conditions and noises when considering the ARANSD
and TSDL methods. This is particularly true for the Stairway room with higher
$T_{60}$, where the proposed solution surpasses the competing methods in all
cases. The HnH-NRSE acquires an average STOI value of $59.1$ and $60.5$ for
rooms LASP2 and Stairway. Equivalent intelligibility scores of $57.0$ and
$57.2$ are observed for the TSDL whereas $54.5$ and $54.4$ are achieved by the
ARANSD. The lower intelligibility acquired by the ARANSD for the STOI measure
can be explained by its processing approach that absorbs frames with dominant
masking components without considerations on the spectral envelope correlation
which are relevant for the STOI intelligibility measures. The highest overall
average intelligibility results are achieved by composed solutions IRMO and
IRMN. For theses reference approaches, the proposed IRMN is able to acquire
the highest intelligibility scores for all scenarios. The IRMN attained, an
average STOI of $81.4$ and $83.4$ for rooms LASP2 and Stairway, respectively.
This indicates intelligibility gains of $27.4$ and $29.4$ for the IRMN
solution against $25.2$ and $27.1$ for the IRMO technique.

Figure 3 depicts the box-plot of ASIIST intelligibility scores for rooms LASP2
and Stairway considering both acoustic noises Babble and Cafeteria. Note that
for this objective measure, the HnH-NRSE surpasses methods ARANSD and TSDL in
all conditions, achieving the highest average ASIIST results. In light of
these solutions, the variance of ASIIST values are similar for all STOI
scenarios. For the LASP2 room, the proposed HnH-NRSE overall average is $42.8$
in comparison with $37.8$ for the TSDL and $29.2$ for the ARANSD. Equivalent
scores for Stairway room are $39.9$, $35.4$ and $26.0$, respectively. This
implies an overall average intelligibility gain of $15.9$ for HnH-NRSE, $11.1$
for TSDL and $6.8$ for ARANSD, which corroborates the capacity of the proposed
method to increase speech intelligibility under noisy-reverberant conditions.
Furthermore, this indicates that the proposed approach is suited for real
acoustic effects with different non-stationary behavior. For the composed
techniques, the IRMN attained overall averages of $74.9$ and $74.4$ for rooms
LASP2 and Stairway in comparison to $74.8$ and $74.4$ for the IRMO,
respectively. The similarity of these results can be partially explained by
the direct adoption of local time-frequency SNR for the ASIIST computation and
the usage of the IRM acoustic mask for both composite measures, as the IRM
imposes on the enhanced speech signal the spectral cues present in the mask
that are important to intelligibility.

###  V-B Perceptual Intelligibility Evaluation

Figure 4: Intelligibility results and standard deviation for perceptual
evaluation.

A perceptual evaluation is also performed in this work. The average
intelligibility scores and standard deviations values for each method are
presented in Fig. 4. The HnH-NRSE improves the intelligibility under all
conditions over competing techniques. The proposed method achieves
intelligibility improvements of $22$ p.p., $28$ p.p. and $15$ p.p. against
$14$ p.p., $14$ p.p. and $6$ p.p. for the TSDL and $12$ p.p., $6$ p.p. and
$10$ p.p. for ARANSD under SNR values of $-3$ dB, $0$ dB and $3$ dB,
respectively. Furthermore, composed solutions attained overall average
intelligibility improvements of $47$ for IRMO and $53$ for IRMN. This result
corroborates with previous objective findings on the proposed strategies
capacity to provide intelligibility improvement.

Table III: PESQ results for LASP2 room and acoustic noises Babble and
Cafeteria.

Noise | SNR | UNP | ARANSD | TSDL | HnH-NRSE | IRMO | IRMN
---|---|---|---|---|---|---|---
Babble INS $=40$ | 2.3 | 1.93 | 1.79 | 1.95 | 2.01 | 3.02 | 3.02
0.2 | 1.85 | 1.65 | 1.87 | 1.90 | 2.90 | 2.87
-1.9 | 1.72 | 1.69 | 1.82 | 1.78 | 2.74 | 2.73
-4.3 | 1.64 | 1.61 | 1.70 | 1.68 | 2.58 | 2.58
-7.3 | 1.50 | 1.60 | 1.63 | 1.57 | 2.38 | 2.41
-10.6 | 1.44 | 1.58 | 1.41 | 1.49 | 2.16 | 2.18
Average | 1.68 | 1.66 | 1.72 | 1.74 | 2.63 | 2.63
Cafeteria INS $=15$ | 1.3 | 1.95 | 1.97 | 1.98 | 2.10 | 3.12 | 3.11
-0.8 | 1.90 | 1.98 | 1.92 | 1.99 | 3.00 | 2.98
-2.9 | 1.84 | 1.86 | 1.87 | 1.88 | 2.86 | 2.84
-5.4 | 1.76 | 1.77 | 1.82 | 1.77 | 2.69 | 2.69
-8.3 | 1.56 | 1.74 | 1.78 | 1.67 | 2.51 | 2.52
-12.2 | 1.54 | 1.63 | 1.62 | 1.58 | 2.29 | 2.31
Average | 1.76 | 1.82 | 1.83 | 1.83 | 2.74 | 2.74
Overall Average | 1.72 | 1.74 | 1.77 | 1.78 | 2.68 | 2.68

Table IV: PESQ results for Stairway room and acoustic noises Babble and
Cafeteria.

Noise | SNR | UNP | ARANSD | TSDL | HnH-NRSE | IRMO | IRMN
---|---|---|---|---|---|---|---
Babble INS $=40$ | 4.2 | 2.00 | 1.97 | 2.03 | 2.26 | 3.12 | 3.12
2.1 | 1.93 | 1.94 | 1.99 | 2.13 | 3.03 | 3.02
0.1 | 1.90 | 1.88 | 1.95 | 2.01 | 2.92 | 2.90
-2.2 | 1.81 | 1.85 | 1.93 | 1.89 | 2.79 | 2.78
-4.8 | 1.71 | 1.82 | 1.84 | 1.78 | 2.63 | 2.65
-8.6 | 1.66 | 1.78 | 1.74 | 1.68 | 2.48 | 2.48
Average | 1.83 | 1.87 | 1.91 | 1.96 | 2.83 | 2.83
Cafeteria INS $=15$ | 2.5 | 2.10 | 2.12 | 2.26 | 2.39 | 3.11 | 3.11
0.6 | 2.03 | 2.10 | 2.22 | 2.27 | 3.03 | 3.02
-1.5 | 2.00 | 2.01 | 2.16 | 2.16 | 2.93 | 2.91
-3.8 | 1.94 | 1.93 | 2.11 | 2.04 | 2.80 | 2.79
-6.5 | 1.91 | 1.90 | 2.06 | 2.01 | 2.63 | 2.66
-12.8 | 1.78 | 1.88 | 2.01 | 1.96 | 2.44 | 2.55
Average | 1.96 | 1.99 | 2.14 | 2.14 | 2.82 | 2.84
Overall Average | 1.89 | 1.93 | 2.02 | 2.05 | 2.82 | 2.84

###  V-C Speech Quality Measures

The PESQ quality results are presented in Tables III and IV. In this case, the
UNP speech signal attained an overall average quality of $1.72$ and $1.89$ for
rooms LASP2 and Stairway. The highest scores are obtained by HnH-NRSE
corresponding values of $1.78$ and $2.05$, followed by TSDL with $1.77$ and
$2.02$ and ARANSD with $1.74$ and $1.93$. This indicates that the proposed
method outperforms competing non-composite strategies for the majority of
noisy-reverberant scenarios. This can be particularly noted for higher SNR
values, where the HnH-NRSE accomplishes the highest PESQ gains. For the
composite techniques, the IRMN and IRMO attained the same overall average
$2.68$ for room LASP2. Considering the most reverberant room Stairway, the
IRMN achieves a better PESQ score of $2.83$ against $2.82$ for the IRMO.

(a) (b)

Figure 5: Boxplot of SRMR objective values for (a) LASP2 room and (b) Stairway
room, considering acoustic noises Babble and Cafeteria

In this work, proposed and competing methods are also evaluated in terms of
SRMR quality scores. Figure 5 depicts the SRMR box-plot values for the noisy-
reverberant scenarios. Note that, the HnH-NRSE attains the highest SRMR
results for most STOI conditions when considering the ARANSD and TSDL methods.
This can be noted specially for higher STOI results. The HnH-NRSE attains an
average quality of $1.80$ for LASP2 room against $1.76$ and $1.05$ for TSDL
and ARANSD. The same behavior can be noted for the Stairway room, with
equivalent quality average scores of $1.73$, $1.68$ and $1.06$. Note that the
proposed HnH-NRSE solution attains SRMR quality scores that are close to
composite approaches IRMO and IRMN. This is particularly true for higher STOI
scores. For the condition of STOI $=0.64$, the HnH-NRSE attains SRMR of $1.92$
which is the same as the composite method IRMO, which indicates that the
proposed approach is able to improve quality of speech signals under adverse
noisy-reverberant conditions. Regarding composed methods, the IRMN attained
the highest SRMR scores in all noisy-reverberant scenarios, with the average
scores of $2.04$ and $1.93$ for rooms LASP2 and Stairway, respectively. The
competing IRMO acquired $1.97$ and $1.81$ average SRMR values for the same
acoustic environments.

(a) (b)

Figure 6: PEAQ quality improvement for LASP2 room with (a) Babble and (b)
Cafeteria acoustic noises.

For the PEAQ objective measure, the quality improvement attained by each
method is depicted in Figures 6 and 7 for rooms LASP2 and Stairway,
respectively. The HnH-NRSE outperforms competing methods in most scenarios,
leading to an overall average gain of $0.21$ for LASP2 and $0.26$ for
Stairway. Considering the most non-stationary Babble noise and the greater
$T_{60}$ of $1.1$ s for the Stairway room, the proposed method acquired the
highest $\Delta$PEAQ gain in all SNR conditions, with an average $\Delta$PEAQ
of $0.24$. This result corroborates the capacity of the proposed method to
deal with real acoustic effects of different non-stationary natures, being
able to provide both quality and intelligibility under diverse noisy-
reverberant environments. As means of comparison, the TSDL and ARANSD
strategies achieved average $\Delta$PEAQ values of $0.18$ and $0.13$ for the
same condition. Note that quality gains attained by HnH-NRSE are in most cases
the closest to the composite methods IRMO and IRMN scores. The IRMN obtains
the highest quality improvements for most environments regarding composite
methods. This reinforces that the combined usage of IRM and NNESE on noisy-
reverberant scenarios simultaneously provide the best quality and
intelligibility gains.

(a) (b)

Figure 7: PEAQ quality improvement for Stairway room with (a) Babble and (b)
Cafeteria acoustic noises.

##  VI Conclusion

In this work, the HnH-NRSE method was proposed as a single time domain
solution for simultaneous speech intelligibility and quality improvement under
noisy-reverberant conditions. To this end, the technique performs an harmonic
decomposition and an objective assessment of the non-stationarity variations
of speech signal in order to define an adaptive gain to deal with masking
components. Two other composite solutions IRMO and IRMN were presented as
combined methods for simultaneous improvement on noisy and reverberant
scenarios, combining the IRM mask with OMLSA and NNESE speech enhancement
solutions. The HnH-NRSE attained an overall average intelligibility
improvement of $5.8$ p.p. for the STOI measure and $15.9$ p.p. for the ASIIST
measure. The highest STOI improvement was achieved by the composed solution
IRMN with $28.4$ p.p. intelligibility gain. Objective quality results
demonstrated that the HnH-NRSE is also able to increase speech quality, with
the highest PESQ increment of $0.26$ for the Stairway room with Babble noise
at SNR of $4.2$ dB. The proposed HnH-NRSE also obtained average SRMR quality
scores of $1.92$ for the Stairway room, which is similar to the reference
composed approache IRMO, whereas the IRMN attained $2.11$ in this scenario. A
perceptual intelligibility listening test was further considered and
corroborated the objective results performed in this work.

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# Topological insulating phase arising in transition metal dichalcogenide
alloy

F. Crasto de Lima<EMAIL_ADDRESS>Ilum School of Science, CNPEM,
Campinas, SP, Brazil  B. Focassio Ilum School of Science, CNPEM, Campinas, SP,
Brazil  Center for Natural and Human Sciences, Federal University of ABC,
Santo Andre, SP, Brazil  R. H. Miwa<EMAIL_ADDRESS>Instituto de Física,
Universidade Federal de Uberlândia, Uberlândia, MG, Brazil  A. Fazzio
<EMAIL_ADDRESS>Ilum School of Science, CNPEM, Campinas, SP,
Brazil  Center for Natural and Human Sciences, Federal University of ABC,
Santo Andre, SP, Brazil

###### Abstract

Transition metal dichalcogenides have been the subject of numerous studies
addressing technological applications and fundamental issues. Single-layer
PtSe2 is a semiconductor with a trivial bandgap, in contrast, its counterpart
with $25\%$ of Se atoms substituted by Hg, Pt2HgSe3 (jacutingaite, a naturally
occurring mineral), is a 2D topological insulator with a large bandgap. Based
on ab-initio calculations, we investigate the energetic stability, and the
topological transition in Pt(HgxSe1-x)2 as a function of alloy concentration,
and the distribution of Hg atoms embedded in the PtSe2 host. Our findings
reveal the dependence of the topological phase with respect to the alloy
concentration and robustness with respect distribution of Hg. Through a
combination of our ab-initio results and a defect wave function percolation
model, we estimate the random alloy concentration threshold for the
topological transition to be only $9\%$. Our results expand the possible
search for non-trivial topological phases in random alloy systems.

##  I Introduction

Finding new candidates for topological materials has become a critical point
in the applications and technology advances [1], given the need for (i) large
topological gaps – dictating the electronic picture robustness; (ii)
structural stability under atmospheric conditions; and (iii) accessible
synthesis routes for scaling the production. The search for (new) materials
fulfilling all those characteristics is an active area. Recently, a naturally
occurring Pt-based mineral, Pt2HgSe3, was discovered, and later was
characterized as hosting a large bulk bandgap 2D topological insulating phase
[(i)] [2; 3; 4]. This natural occurrence suggests that Pt2HgSe3 (also known as
jacutingaite) is structurally stable under atmospheric conditions [(ii)]. Its
structure is made up of a transition metal dichalcogenide (TMD) PtSe2 backbone
with 25 % of the chalcogen atoms substituted by Hg [4]. This implies that
similar routes yielding PtSe2, or post-processing involving the incorporation
of Hg onto the PtSe2 matrix, can be used to obtain a large-scale production
[(iii)] of such a 2D topological material [5; 6; 7]. Furthermore, the
experimental knowledge on the synthesis, modification, and engineering of TMD-
based devices makes this class of materials an obvious candidate for the
transition from fundamental research to applications. For example, the
synthesis of a fully metalized source/drain device based on 2D van der Waals
heterostructures of PtSe2 [8].

Defects in 2D TMDs provide a controllable degree of freedom for tuning their
electronic and structural properties [9]. The incorporation of Hg atoms in the
PtSe2 matrix [$\rightarrow$ Pt(HgxSe1-x)2] can result in localized defects and
disorder. Disorder effects may play a deleterious role in the topological
phase in Bi2Se3 [10]. On the other hand, it can regulate the transport
mechanism [11] or induce topological phases in 2D TMDs [12]. Indeed, the
knowledge of the alloying and disordered effects at the atomic scale is a
critical issue for predicting and synthesizing topological materials based on
2D TMDs.

In this paper, we have studied the quantum spin-Hall (QSH) phase in PtSe2
monolayer (ML) upon the inclusion of substitutional Hg${}_{\text{Se}}$
impurities, resulting in ordered and random Pt(HgxSe1-x)2 alloys. We start our
investigation through the calculation of the Hg${}_{\text{Se}}$ substitutional
energy in the PtSe2 host. We have discussed the energetic preference for the
ordered $\rightarrow$ random transition as a function of the temperature and
alloy concentration. Topological characterization of the ordered phase,
performed by the calculation of the $\mathcal{Z}_{2}$ invariant, reveals that
the Pt(HgxSe1-x)2 alloys are topological as predicted by the Kane-Mele model,
for $x$ between 0.25 and 0.75. Whereas, at the edge concentrations, we found
the $\mathcal{Z}_{2}$-metallic phase [13] for $x$ = 1.0 (PtHg2), and a trivial
insulator for $x$ = 0, PtSe2 host. The topological character of the random
alloys, examined through the calculation of the spin-Bott index, reveals that
the topological phase of the Pt(HgxSe1-x)2 alloys has been preserved even upon
a random distribution of the Hg${}_{\text{Se}}$ impurities. In the sequence,
the topological non-trivial $\leftrightarrow$ trivial transition has been
examined by the calculation of the percolation threshold of the orbital
overlap of the Hg${}_{\text{Se}}$ wave functions, randomly, embedded in the
PtSe2 host.

##  II Methods

The calculations were performed on the Viena ab-initio simulation package
(VASP) [14], within a plane-wave base with a cutoff energy of $400$ eV. The
spin-orbit coupling was taken into account in all calculations, where the
exchange and correlations were treated with the Perdew-Burk-Ernzehof
functional [15]. The electron-ion interactions were described within the
projected augmented wave (PAW) [16] method, with all atoms allowed to relax
until each atom resultant force was lower than $10^{-2}$ eV/Å. For the
pristine unity cell, the total energy was calculated on the 2D Brillouin zone
with a regular K-point grid of $5\times 5$ and $7\times 7$ for the atom’s
relaxation and self-consistent charge density, respectively. The same K-point
density was taken on the larger supercell calculations. The topological
invariant ($\mathcal{Z}_{2}$ [17; 18] and spin-Bott index [19]) and semi-
infinite ribbon calculations were performed after extracting a Wannier
function tight-biding model from the ab-initio calculations through the
Wannier90 code [20].

For the random alloy, we have generated the alloy structure through the
Special-quasirandom Structure (SQS) [21] for cells with 48 and 27 atoms
achieving 1st-neighbor pair correlation (PC) exactly random, and 2nd-neighbor
and 3rd-neighbor PC exactly random for most structures. Such structures
represent systems without translational symmetry, their reciprocal space is
undefined, and therefore a different approach for computing the
$\mathcal{Z}_{2}$ invariant is needed. Recently an approach involving the
evaluation of the spin Bott index was successful to characterize non-periodic
systems [19; 10], which we computed for the random alloy structures.

##  III Results

###  III.1 Hg-Hg-interaction embedded in PtSe2 matrix

The formation of jacutingaite, Pt2HgSe3, can be viewed as a (partial)
substitution of Se with Hg atoms in the 1T phase of the Pt2Se matrix [2; 3;
4]. In this case, the substitutional energy ($E_{s}$) of $n$ Se atoms by Hg in
the PtSe2 is

$E_{s}=(E_{\rm system}+n\times\mu_{\rm Se}-E_{\rm matrix}-n\times\mu_{\rm
Hg})/n,$ (1)

with $E_{\rm system}$ the total energy of the final (Hg${}_{\text{Se}}$ doped)
system, $E_{\rm matrix}$ the total energy of the 1T Pt2Se matrix; $\mu_{\rm
Se}$ and $\mu_{\rm Hg}$ are the chemical potentials of Se and Hg atoms,
respectively.

Let us start with a single substitutional impurity of Hg in the Se site,
Hg${}_{\text{Se}}$. The calculations were performed by using a Pt2Se
tetragonal supercell with dimensions of $18.7$ Å and $32.4$ Å, schematically
shown in Fig. 1(a), in order to avoid spurious interactions between the
periodic images. Our results of $E_{s}$, presented in Fig. 1(b), indicates
that the formation of a single Hg${}_{\text{Se}}$ [indicated by a blue-circle
in Fig. 1(a)] is an endothermic process, $E_{s}>0$, with higher occurrence
rate at the Se-poor (Pt-rich) condition. Although the positive values of
$E_{s}$, it is worth noting that the combination of the energetic preference
of formation of Se vacancies (V${}_{\text{Se}}$) in PtSe2 at the Se-poor
condition [22; 23], and the experimental realization of doping of TMDs
(including PtSe2) by filling the chalcogen vacancies created by electron-
irradiation [5; 7], makes the formation of Hg${}_{\text{Se}}$ in PtSe2 a quite
likely process.

Figure 1:  (a) Substitutional sites for Hg on PtSe2 matrix. (b) Se
substitutional energy for a single Hg (Eq. 1) as a function of Se chemical
potential. (c) Substitutional energy for two Hg atoms separated by a distance
$d_{s}$, the number labels the site of figure (a) with red/black curves of the
pair of Hg lying on the same/opposite side of the PtSe2 layer.

In the sequence, the higher mobility of V${}_{\text{Se}}$, induced by an
electron beam, promotes the formation of energetically stable paired
divacancies and vacancy-line structures in single layer PtSe2 [24; 25]. Since,
as discussed above, these intrinsic defects can be filled up by foreign atoms,
it is worth investigating the interaction between the Hg${}_{\text{Se}}$
impurities embedded in PtSe2. Here, we have calculated the formation energy of
Hg${}^{0}_{\text{Se}}$-Hg${}^{i}_{\text{Se}}$ pair configurations, with $i$ =
1-10 as depicted in Fig. 1(a). Black (Red) circles indicate
Hg${}^{i}_{\text{Se}}$ atoms lying on the same (opposite) side of
Hg${}^{0}_{\text{Se}}$ (blue circle). Our results of $E_{s}$ as a function of
the distance between the impurities, Fig. 1(c), reveal that the
Hg${}^{0}_{\text{Se}}$-Hg${}^{3}_{\text{Se}}$ configuration will be the most
stable (likely) one, which is characterized by in-line Hg-Pt-Hg bonds sharing
the same Pt atoms. Interestingly, such a Hg-Pt-Hg bonding configuration is
exactly that found in the Pt2HgSe3 crystals, which suggests that
Hg${}^{0}_{\text{Se}}$-Pt-Hg${}^{3}_{\text{Se}}$ pairs act as seeds for the
formation of topological insulator jacutingaite crystals. On the other hand,
in contrast with the Pt2HgSe3 crystals, we may have (i) other concentrations,
as well as (ii) periodic or random distribution of Hg${}_{\text{Se}}$
impurities embedded in PtSe2. In the sequence, we will discuss the importance
of (i) and (ii) to the electronic properties of Hg-doped PtSe2 monolayers.

###  III.2 Ordered and random alloys

Figure 2:  Ordered configuration (a) atomic structure, (b) formation energy,
and (c) band structure for Pt(HgxSe1-x)2. The band shown in black (red) are
for calculation without (with) SOC. Figure 3:  Edge states for ordered
structure. All of the explored systems, except the x=0 and x=1, are
topologically non-trivial. Figure 4:  Free energy as a function of temperature
with configurational entropy accounted for the ordered (dotted lines) and SQS
(continuous lines) alloys.

We start by examining ordered Pt(HgxSe1-x)2 alloys, with $x$ $=$ 0, 0.25,
0.50, 0.75, and 1.0 created by incorporating Hg${}_{\text{Se}}^{0}$-Pt-
Hg${}_{\text{Se}}^{3}$ units [Fig. 1] in the PtSe2 host, Fig. 2(a). Pristine
PtSe2 ($x$ = 0) is energetically stable, characterized by negative values of
formation energy, $E^{\rm for}$, $-0.48$ eV/atom, Fig. 2(b), followed by the
jacutingaite phase ($x$ = 0.25), which also present $E^{\rm for}$ $<$ 0\.
These results are compatible with the natural formation of both the PtSe2 and
Pt2HgSe3 systems. Whereas for $x$ = 0.5 we find that PtHgSe lies on the verge
of positive values of $E^{\rm for}$.

Figure 5:  Disordered SQS structures (upper panels) and respective topological
gaps (lower panels).

The electronic band structures of the ordered Pt(HgxSe1-x)2 alloys, presented
in Fig. 2(c), show that the pristine PtSe2 host ($x$ = 0) is a semiconductor
with a band gap of $1.2$ eV, while at the other end ($x$ = 1) we find that
PtHg2 presents a metallic character. It is important to note that the SOC
controls the semiconducting character in these Pt(HgxSe1-x)2 alloys, which
were computed for $x$ = 0.25, 0.50, and 0.75. As seen by the solid black lines
in Fig. 2(c), such alloys become semi-metallic by turning off the SOC,
resulting in the linear crossing of energy bands at the Fermi level. Such a
bandgap ($E_{g}$) opening, mediated by the SOC, is in consonance with the QSH
effect as predicted by the Kane-Mele model. In particular, for jacutingaite,
$x$ = 0.25, our DFT-SOC results of $E_{g}$ (=0.14 eV) are in good agreement
with the one obtained by Marrazzo et al. [3]. Meanwhile, for the other
concentrations, we found $E_{g}$ of 0.11 and and 0.20 eV for $x$ = 0.50, and
0.75, respectively. Interestingly, the SOC-induced bandgaps do not exhibit a
nearly linear behavior with respect to the alloy concentration, in contrast to
the results obtained for the formation energy, Fig. 2(b) and (c).

In order to provide further confirmation of the non-trivial topology of
Pt(HgxSe1-x)2, we have calculated the $\mathcal{Z}_{2}$ invariant by
extrapolating the system to a Wannier base representation and analyzing its
spatial evolution through the Brillouin zone (BZ) [18]. Our results reveal
that the Pt(HgxSe1-x)2 ordered alloys present a non-trivial topological phase
[except the pristine $x=0$ case]. Here, although the full Hg-covered system is
metallic, we considered that the negative dispersive band at the M-point is
fully occupied, while the positive dispersive band at K-point is fully empty,
with an arising $\mathcal{Z}_{2}$-metallic topological phase [13]. Finally, as
a signature of the SOC-induced topological gaps in Pt(HgxSe1-x)2, in Fig. 3 we
present the projected edge states which are characterized by the emergence of
(chiral) spin-polarized Dirac cones characteristic of the QSH phase.

According to Refs. [10; 26; 27; 12; 28], disordered systems can also exhibit
non-trivial topological phases. Here, focusing on the Pt(HgxSe1-x)2 alloys, we
might have a random distribution of the Hg${}_{\text{Se}}$ impurities
preserving the host’s (1T) backbone structure. Such randomness can alter the
electronic/topological properties/identities of Pt(HgxSe1-x)2 alloys as well
as the energetic stability for each alloy concentration.

In order to provide a realistic description of random alloys, the distribution
of the Hg${}_{\text{Se}}$ impurities in Pt(HgxSe1-x)2 were generated by using
the SQS approach [21]. We have generated SQS structures ranging from $x$ =
0.25 to $x$ = 0.75. By taking into account the configurational entropy,
$S=k_{B}\ln\left(\Omega\right)$, with $\Omega$ the number of equivalent
configurations, we have determined the free energy ($F=U-TS$) as a function of
the temperature ($T$) and alloy concentration, $x$. Our results of free
energy, summarized in Fig. 4, reveal that (i) there is an energetic preference
for the pristine PtSe2 ($x=0$) for temperatures with $k_{B}T<15$ meV; while
(ii) the random counterpart of jacutingaite becomes more stable than its
ordered phase for $k_{B}T>6$ meV; where (iii) such an energetic preference for
$x$ = 0.25 has been maintained up to temperatures with $k_{B}T<32$ meV; and
(iv) above that, we find the PtHgSe random alloy will present the lowest free
energy.

As discussed above, we found that Pt(HgxSe1-x)2 ordered alloys, with $x$ =
0.25, 0.50, and 0.75, present the QSH phase characterized by the SOC-induced
non-trivial bandgaps at the Fermi level, Fig. 2(c). In the sequence, we will
examine the topological identities of Pt(HgxSe1-x)2 random alloys within the
same range of alloy concentrations. Given the non-periodicity of the random
alloys, we have computed a real space invariant (Spin Bott-index) which is
equivalent to the spin-Chern number [19]. In Fig. 5 we highlight (in gray) the
topological energy gaps of the random alloy systems. Our results reveal that,
although the random distribution introduces trivial bandgaps close to the
Fermi energy, the topological gaps are always present within an energy
interval of $\pm$ 0.5 eV around the Fermi level. Here, although the non-
uniform distribution of the Hg${}_{\text{Se}}$ impurities gives rise to
sensible differences in the energy localization and the width of the
topological gaps, our findings indicate that the Pt(HgxSe1-x)2 system will
always exhibit non-trivial bandgaps for alloy concentrations between 0.25 and
0.75, regardless of how (ordered or randomly) the Hg${}_{\text{Se}}$
impurities are distributed in the PtSe3 host. This result is in line with that
obtained by Lee et al. [29]; the authors verified the stability of the
$\mathcal{Z}_{2}$-topological phase upon the inclusion of aleatory vacancies,
through disordered potentials, in the Kane-Mele model on the honeycomb
lattice. Back to our Pt(HgxSe1-x)2 system, since PtSe2 ($x$ = 0) is a trivial
insulator, while Pt2HgSe3 ($x$ = 0.25, jacutingaite) is a topological
insulator, it is expected a trivial $\leftrightarrow$ non-trivial topological
transition for alloy concentrations between 0 and 0.25.

###  III.3 Topological limit

Lowering the Hg${}_{\text{Se}}$ concentration, $x\rightarrow 0$, will lead the
Pt(HgxSe1-x)2 system to a topologically trivial phase, PtSe2. In this
subsection, we will look at a trivial $\leftrightarrow$ non-trivial
topological transition in light of the spatial localization of the
Hg${}_{\text{Se}}$ impurities (electron percolation limit).

The topological character can be predicted by the spin Chern number,
$c^{(s)}=c_{+}-c_{-}$, with $c_{j}$ the Chern number on the subspace of
$j=+,-$ spin. This is dependent on the integration over a closed surface of
the Berry connection,

$c_{j}=\sum_{n}^{\rm occup}\oint_{C}\vec{A}_{n}^{(j)}\cdot d\vec{k},$ (2)

with
$\vec{A}_{n}^{(\sigma)}=i\braket{n,\vec{k},\sigma}{\nabla_{k}}{n,\vec{k},\sigma}$.
Here, $i\nabla_{k}$ is the definition of the position operator on the momentum
space. If the system is composed only of fully localized states (atomic
limit), that is, eigenfunctions of the position operator,

$c_{j}=\sum_{n}^{\rm occup}x_{n}\oint_{C}dk=0.$ (3)

Therefore, a non-zero Chern number requires the electronic states to not be
completely localized. This imposes a necessary (however not sufficient)
condition for the emergence of the QSH phase. That is, in Pt(HgxSe1-x)2
alloys, the Hg${}_{\text{Se}}$ impurity wave functions (IWFs) should interact
with one another above a given percolation limit, i.e. a threshold random
alloy concentration which leads to the trivial $\leftrightarrow$ non-trivial
topological transition.

Here we can estimate such a threshold concentration in Pt(HgxSe1-x)2 random
alloys based upon the percolation threshold of the IWFs. In this case, the
topological transition will be dictated by the electronic percolation through
impurity sites embedded in the PtSe2 host. That is if the overlap of
neighboring IWFs enables the electronic percolation through a large (infinite)
clustered set of Hg${}_{\text{Se}}$ atoms, the energy dispersion between the
impurity states will lead to the non-trivial topological phase. On the other
hand, if the neighboring IWFs do not overlap (atomic limit), the absence or
nearly zero energy dispersion will result in a trivial phase. In this case,
based on the random distribution of Hg${}_{\text{Se}}$ in a hexagonal lattice
(on the Se sites), the percolation threshold can be written as [30],

$\phi_{c}=1-e^{-n_{C}}=0.69,$ (4)

where $n_{C}$ is the critical ratio between the area ($A$) occupied by the
IWFs and that of the unity cell (UC), $n_{C}=A_{\text{I}}/A_{\text{UC}}$.
Here, the interaction between Hg${}_{\text{Se}}$ impurities was estimated
based on the substitutional energy, $E_{s}$, and electronic interaction
between the impurity atoms. As shown in Fig. 1(c), the
Hg${}^{0}_{\text{Se}}$-Hg${}^{8}_{\text{Se}}$,
Hg${}^{0}_{\text{Se}}$-Hg${}^{9}_{\text{Se}}$, and
Hg${}^{0}_{\text{Se}}$-Hg${}^{10}_{\text{Se}}$ configurations have nearly the
same $E_{s}$ values, i.e. substitutional energy differences ($\Delta E_{s}$)
of about $0.05$ eV/Hg${}_{\text{Se}}$-atom, which corresponds to
Hg${}_{\text{Se}}$-Hg${}_{\text{Se}}$ distances, $d_{s}$, between $10$ and
$11$ Å. For $d_{s}$ of $11$ Å, we found an electronic (HOMO-LUMO) interaction
of $\sim 5$ meV between the Hg${}_{\text{Se}}$ impurities,
$\Delta\varepsilon=5$ meV, which is smaller than the SOC induced non-trivial
bandgaps. Thus, we can assume that $11$ Å is the lower limit for the orbital
localization of the IWFs, i.e. for $d_{s}>11$ Å we will have $\Delta
E_{s}<0.05$ eV/Hg${}_{\text{Se}}$-atom and $\Delta\varepsilon<5$ meV. In
Pt(HgxSe1-x)2, an averaged distance of $11$ Å between
Hg${}_{\text{Se}}$-Hg${}_{\text{Se}}$ impurities results in a trivial
$\leftrightarrow$ topological threshold concentration $x_{t}=0.1$.

Here, our simulations going down to $x=0.25$ for the random alloy confirm the
persistence of the topological phases within the threshold concentration. It
is worth pointing out that simulating realistic disordered systems, within the
SQS approach for concentrations below $x_{t}=0.09$, increases the
computational costs considerably. That is, to achieve concentrations near the
pristine case, the supercell size needs to be increased to achieve a good
correlation towards real disordered systems. However, note that the
Jacutingaite phase (ordered $x=0.25$) is the Kane-Mele model’s realization. It
is shown that randomly vacancies on the honeycomb Kane-Mele model (replacing
Hg by Se) a possible topological phase is preserved down to $x\sim 0.2$ [29].

##  IV Conclusion

Based on ab-initio calculations, we have studied the energetic stability, and
the trivial $\rightarrow$ non-trivial topological phase transition in single
layer PtSe2 mediated by substitutional Hg${}_{\text{Se}}$ atoms, Pt(HgxSe1-x)2
alloys. We found an energetic preference for a random distribution of
Hg${}_{\text{Se}}$ with $x=25\%$ (jacutingaite’s stoichiometry), with respect
to the PtSe2 host, for temperatures with $k_{B}T>15$ meV. Meanwhile, random
alloys with $x=50\%$, PtHgSe, become more stable for $k_{B}T>32$ meV. The
robustness of the QSH phase against the random distribution of
Hg${}_{\text{Se}}$ substitutional atoms has been verified for alloy
concentrations between $25\%$ and $75\%$. In addition, based on a combination
of ab-initio results and a percolation model, we estimate a threshold
concentration, of about 10%, for the topological non-trivial $\leftrightarrow$
trivial transition in Pt(HgxSe1-x)2 random alloys.

###### Acknowledgements.

The authors acknowledge financial support from the Brazilian agencies FAPESP
(grants 19/20857-0, 19/04527-0, and 17/02317-2), FAPEMIG, INCT-Nanocarbono,
and Laboratório Nacional de Computação Científica for computer time (project
ScafMat2).

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