Binaural dereverberation based on delayed widely linear predictionin the time domain
Xin Leng(1) , Jingdong Chen(1) , and Jacob Benesty(2)(1)

CIAIC, Northwestern Polytechnical University, China, llengxin@126.com, jingdongchen@ieee.org(2)
INRS-EMT, University of Quebec, Canada, benesty@emt.inrs.ca

Abstract
Sound spatial information benefits human listeners in reverberant environments. This paper deals with the
problem of binaural dereverberation, which reduces reverberation and meanwhile preserves the sound spatial
information at the binaural outputs. A widely linear (WL) filtering framework is adopted where the multiple
real microphone signals are merged into complex signals. The desired binaural outputs are also converted
into complex signals with one channel being its real part, and the other channel being its imaginary part.
By doing so, we transform the problem of binaural dereverberation to one of monaural dereverberation. In
such a framework, the complex late reverberation is modeled using the multichannel delayed WL prediction
by fully taking advantage of the noncircularity of the complex signals. A maximum likelihood method isthen developed to estimate the optimal prediction filter with the speech signal of interest being modeled by
a complex normal distribution. The relationship between the proposed method and the weighted prediction
error (WPE) method is also discussed. Finally, simulation results are provided to justify the effectiveness ofthe developed method.

Keywords: Binaural dereverberation, delayed linear prediction, widely linear estimation, spatial information preservation.

1

INTRODUCTION

In various speech-related communication and human-machine interface systems, dereverberation is a criticalproblem since room reverberation may significantly degrade the speech quality and speech intelligibility as
well as the recognition performance of automatic speech recognition (ASR) systems. As a result, tremendous
research efforts have been devoted to this problem over the past few decades [1–8]. Most such efforts
so far are focused on producing a single-channel dereverberated speech output. However, in hearing aid,
virtual/augmented reality, and many other applications, in addition to reducing the reverberation effect, a
key challenge is also to preserve the spatial information of the sound source such that after dereverberation,
the human listeners can better localize the sound source with a binaural output. This problem is calledbinaural dereverberation [9–13]. The spatial information also benefits other signal processing techniques suchas source separation, beamforming, and speech enhancement, to make them more effective in reverberant
environments, where dereverberation is often used as a preprocessor [14–16].
It is known that early reflections (approximately within the first 40 ms of arrival) may help add warmth tosound due to the colorization effect, while the late reverberation smears the speech temporal and spectral
information, thereby deteriorating the speech quality and intelligibility [13, 14]. To preserve the spatial information in binaural dereverberation, a straightforward way is to apply a same weighting factor at the two
outputs while suppressing the reverberation with the spectral subtraction-based approach [9, 11] or based ona sigmoidal coherence-to-gain mapping [13] in the power spectral domain. In [9], a second stage coherencebased Wiener filter was also suggested to further attenuate the residual reverberation by taking into accountthe shadowing effects of the human head. However, this method requires good estimation of the spatial coherence and the dereverberation process corrupts the linear relationship between the source and microphonesignals.
In this paper, a widely linear (WL) filtering approach is adopted to achieve binaural dereverberation using a
microphone array in the time domain. The WL filtering framework has shown its potential to preserve thesound spatial information in binaural noise reduction [17, 18] and stereophonic acoustic echo cancellation[19–21]. In such a framework, the multiple real microphone signals and binaural outputs are merged into
complex signals. The complex late reverberation is then modeled by a multichannel delayed WL prediction.
The estimation of the WL prediction coefficients is achieved through a maximum likelihood method in which4924

the speech signal of interest is modeled by a complex normal distribution [22, 23]. Note that the optimal
prediction filter can also be estimated in the real domain with a proper formulation where the binaural output
speech signals are modeled with the multivariable normal distribution, which is also discussed in this paper.
We also analyze the relationship between the proposed method and the popularly used weighted prediction
error (WPE) method proposed in [24], where speech dereverberation is performed using the delayed linear
prediction.

2

SIGNAL MODEL

Consider a linear microphone array consisting of 2M sensors, which is used to capture a signal of interest
in some reverberant and noisy environment. The received signal at the mth (m = 1, 2, . . . , 2M ) microphone
can then be expressed asxm (t) = gm (t) ∗ s(t) + vm (t)

(1)

Lg −1

=

X

gm,k s(t − k) + vm (t),k=0where ∗ denotes linear convolution, s(t) is the unknown source signal, gm (t) is the room impulse response
from the source to the mth microphone, and vm (t) is the additive noise at the mth microphone. We assume
that all the signals xm (t) and vm (t) are real, broadband, and zero mean, and gm (t), m = 1, 2, . . . , 2M shareno common zeros.
Let us first neglect the noise term vm (t) for simplicity as in [24, 25]. The reverberant speech signal xm (t)can then be written as the sum of two components, i.e.,xm (t) = em (t) + rm (t),

(2)

where
em (t) ,D−1
Xgm,k s(t − k),

(3)

k=0
Lg −1

rm (t) ,X

gm,k s(t − k),

(4)

k=D

are, respectively, the direct sound plus early reflections and late reverberation, and D is the time durationof the early reflections. In dereverberation, em (t) and rm (t) are often assumed to be uncorrelated with eachother [14].
Following (4), we can rewrite xm (t) as [24]Txm (t) = em (t) + gl,ms(t − D)T
= em (t) + gl,m
G+ Gs(t − D)T= em (t) + amx(t − D),

where the superscript T is the transpose operator, + denotes the Moore-Penrose pseudo-inverse,Tgl,m , gm,D gm,D+1 · · · gm,Lg −1 0 · · · 0 ,Ts(t) , s(t) s(t − 1) · · · s(t − Ls + 1) ,TTx(t) , x1T (t) x2T (t) · · · x2M(t)= Gs(t),T
xm (t) , xm (t) xm (t − 1) · · · xm (t − L + 1) ,T +am = (G ) gl,m ,

(5)

(6)
(7)
(8)
(9)
(10)

gl,m and s(t) are, respectively, the late impulse response and source signal, both of length Ls , xm (t), oflength L, is the vector form of xm (t), x(t) and am are, respectively, the observation signal vector andprediction filter, both of length 2M L,T
G , GT1 GT2 · · · GT2M ,
(11)

4925

 T
gm
 0

Gm ,  .
 ..

0
T
gm
..
.


0
0 

..  ,
. 

···
···
..
.

0
0
..
.

T
0
0 · · · 0 gmT
gm , gm,0 gm,1 · · · gm,Lg −1 ,

(12)

(13)

gm , of length Lg , is the vector form of gm (t), and G and Gm are matrices of size 2M L × Ls and L × Ls
respectively, Ls = L + Lg − 1. Here, we assume Ls 6 2M L. Assume that gm (t), m = 1, 2, . . . , 2M shareno common zeros, then rank(G) = Ls , and G+ G = ILs , where ILs is the identity matrix of size Ls × Ls .
From (5), one can see that the late reverberation rm (t) can be modeled by a delayed linear prediction withthe prediction filter am .

3

WIDELY LINEAR MODEL

To perform binaural dereverberation, it is necessary to produce at least two output signals from the observations at the microphone array. This can be done by using the WL filtering framework, where the multiple
real microphone signals are merged into complex signals [17, 18], i.e.,√yi (t) , xi (t) + xM +i (t) = hi (t) ∗ s(t),(14)

where  = −1 is the imaginary unit, and hi (t) , gi (t) + gM +i (t) is the complex room impulse responseof the ith (i = 1, 2, . . . , M ) complex channel. It follows immediately according to (5) thatTyi (t) = di (t) + hl,is(t − D)T
= di (t) + hl,i
H+ Hs(t − D)eH y= di (t) + hi e (t − D),(15)

where the superscript H stands for the conjugate-transpose operator, di (t) , ei (t) + eM +i (t) is the complex
direct sound plus early reflections, hl,i , gl,i + gl,M +i is the complex late impulse response,
e i = (HH )+ h∗ = 1 TM L (GT )+ h∗ ,
h
l,i
l,i
2


y(t)
e(t) , ∗y= Hs(t) = TM L x(t),
y (t)

H
H
y(t) , y1H (t) y2H (t) · · · yM
(t) ,

(16)
(17)
(18)

the superscript ∗ denotes complex conjugation, yi (t) is defined analogously to xi (t),H = TM L G,

(19)

and
TM L ,


IM L
IM L

IM L
−IM L
(20)

is the real-to-complex transformation matrix, H+ H = ILs .
Inspecting (15) and (17), one can see that both y(t − D) and y∗ (t − D) are used to model the late reverberation. Without loss of generality, we choose d1 (t) = e1 (t) + eM +1 (t) as the desired signal to recover. An
estimate of d1 (t) can then be obtained aseH ye(t − D),
db1 (t) = y1 (t) − h(21)e is the complex prediction filter to be estimated. From (21), one can see that the objective of binaural
where he that can best estimate d1 (t) from y1 (t) and ye(t − D).
dereverberation is to find an optimal filter h

4

DELAYED WIDELY LINEAR PREDICTION

The dereverberation filter proposed in [24] assumes that em (t) is a Gaussian process with a time-varying
variance. Since we deal with complex signals, the complex normal distribution should be exploited to model
d1 (t) [22, 23]. Therefore, the probability density function (pdf) of d1 (t) is




1
1 H
−1
p d1 (t) =
exp
−
d
(t)Γ
(t)d
(t)
,
(22)
1
2 1
π[det Γ(t)]1/2

4926

whereT
d1 (t) = d1 (t) d∗1 (t) ,1γd1 (t)Γ(t) = σd21 (t) ∗,γd1 (t)
1

(23)
(24)



σd21 (t) , E |d1 (t)|2 is the variance of d1 (t), E[·] denotes mathematical expectation, andE d21 (t)γd1 (t) = 
E |d1 (t)|2(25)

is the second-order circularity quotient [17, 18] of d1 (t). The absolute value of γd1 (t) satisfies 0 6 |γd1 (t)| 6
1. If γd1 (t) = 0, d1 (t) is second-order circular since its pdf can be well modeled by its variance. Otherwise,
it is non-circular.
Let us check the circularity of d1 (t). According to (25), we have






E e21 (t) − E e2M +1 (t) + 2E e1 (t)eM +1 (t)


.
(26)
γd1 (t) ,E |d1 (t)|2According to the signal model given in (1)–(3) and the assumption that em (t), m = 1, 2, . . . , 2M , are fromthe same source s(t), we should have E e1 (t)eM +1 (t) 6= 0. Therefore, with our signal model, we can statethat γd1 (t) 6= 0, and d1 (t) is noncircular [17, 18].
4.1 Delayed Widely Linear Prediction Methode Γ(t) can be estimated by maximizing the log likelihood
With the pdf of d1 (t), the parameters θ = h,function [24]:TXlog p d1 (t) ,(27)
L(θ) =
t=1

where T is the time duration of the signal of interest. The solution can be derived iteratively as follows[24].
1) Initialize
b =
Γ(t)

Lf
X

d1 (t + τ )dH
1 (t + τ )(28)

τ =−Lfwith a frame length of 2Lf + 1 and d1 (t) = y1 (t).
e and Γ(t)b2) Repeat huntil convergencee(a) Update h

"

# 
e
R
h
=
H
∗
e
−C
h

−C
RT

+ 


r−c
,
r∗ − c∗

(29)

where
R=

T
X
e(t − D)eyyH (t − D)
t=1

C=

c=

,

T
X
γ
bd∗ (t)ey(t − D)eyT (t − D)
1

t=1

r=

σ
bd21 (t)(1 − |γd1 (t)|2 )σbd21 (t)(1 − |γd1 (t)|2 )TXe(t − D)y1∗ (t)y,σb2 (t)(1 − |γd1 (t)|2 )t=1 d1TX
γbd∗ (t)ey(t − D)y1 (t)1σb2 (t)(1 − |γd1 (t)|2 )t=1 d1

.

(30)

,

(31)

(32)

(33)

(b) Update (28) usingeH ye(t − D).
db1 (t) = y1 (t) − h

(34)

4927

4.2 Delayed Linear Prediction Method
e by using the WL filtering framework in the complex
The previous subsection derived the prediction filter hdomain. In this subsection, we derive an equivalent method in the real domain. Let us denote x(t) =[x1 (t) xM +1 (t)]T and e(t) = [e1 (t) eM +1 (t)]T . Using (5), we have
 T ax(t) = e(t) + T 1
x(t − D),
(35)
aM +1


1 
(36)
d1 (t) =e(t).
1 −Therefore, an estimate of e(t) can then be obtained asbe(t) = x(t) − AT x(t − D),(37)bM +1 ] is the estimated prediction filter.
where A = [ba1 aSubstituting (36) into (22), we get


1
1 T
−1
exp − e (t)Λ (t)e(t) ,p [e(t)] =22π[det Λ(t)]1/2(38)

where Λ(t) = E[e(t)eT (t)] is the correlation matrix of e(t). The estimation of A and Λ(t) can be achievedin a similar way as in [15, 16] by maximizing the log likelihood function (27).
4.3 Special CaseLet us consider the particular case where γd1 (t) = 0. In such a scenario, we have
"
#


1
|d1 (t)|2
p d1 (t) =exp − 2.
πσd21 (t)σd1 (t)(39)

Using (39), (27) can be rewritten as
L(θ) = −

T
X

"

t=1

#
2
eH y
e(t − D)y1 (t) − h+ log σd21 (t) + const.,
σd21 (t)(40)

and (29) becomese = R+ r.
h(41)

Combining (17) and (41) and substituting the result into (21), we getT
b1T x(t − D) − bdb1 (t) = y1 (t) − aaM+1 x(t − D)= eb1 (t) + beM +1 (t),
where
bm =
a

(42)

" T
#+ T
X x(t − D)xT (t − D)X x(t − D)xm (t)t=1

σbe2m (t)t=1

σbe2m (t),(43)

with σe2m (t) , E[e2m (t)] being the variance of em (t). From (42), one can see that this output fully corresponds to the delayed linear prediction as in [24]. In other words, the method developed in [24] is aparticular case of the method developed in this paper.

5

SIMULATIONS

Simulations are conducted using the impulse responses measured in the Bell Labs Varechoic Chamber [26,27]. Two microphones at (3.237, 0.500, 1.400) (in meters) and (3.437, 0.500, 1.400) are used, and the clean
speech source is placed at the positions (1.337:1.000:5.337, 1.938, 1.600) to simulate a moving source, i.e.,the source stays at one position for 2 s and then moves to the next position. This moving source is used
to evaluate the performance of spatial information preservation of the developed dereverberation method. We
consider two sets of reverberation conditions: a light reverberation condition with the reverberation time
T60 being approximately 0.24 s and a moderate reverberation condition with T60 being approximately 0.58 s.

4928

Lag Time

8

1.0

4

0.5

0

0

-4

-0.5

(a)

Lag Time

-8
8

1.0

4

0.5

0

0

-4

-0.5

(b)

Lag Time

-8
8

-1.0
1.0
0.5

4
0

0

-4

-0.5

(c)

-8
8
Lag Time

-1.0

-1.0
1.0
0.5

4
0

0

-4

-0.5

(d)

-8
0

1

2

3

4

5
6
Time (s)

-1.0

7

8

9

10

Figure 1. Contours of the computed short-time CCFs in two room reverberant conditions: (a) between thetwo reverberant signals with T60 = 0.24 s, (b) between the signals after dereverberation of the signals in (a),(c) between the two reverberant signals with T60 = 0.58 s, and (d) between the signals after dereverberationof the signals in (c).
The clean speech signals used are taken from the TIMIT database [28]. We consider dereverberation of
narrowband speech with 4 kHz bandwidth and 8 kHz sampling rate. So, all the signals are resampled fromits original sampling frequency to 8 kHz. The reverberant signals are generated by convolving the source
signal with the corresponding measured room impulse responses.
The parameters for all simulations are set as follows: D = 240 and Lf = 160. L is set to 1600 for thelight reverberation condition and 3200 for the moderate reverberation condition. The number of iterations
is limited to 1 as in [24] in all cases. Note that the pre-whitening technique was also exploited before
estimating the filter coefficients as in [14].
The cross-correlation function (CCF) is computed to visualize the spatial sound information before and after
dereverberation. The CCF is computed using a short time average every 100 ms without any overlap asin [17], whose maximal value stands for the current source direction. Figure 1 plots the contours of thecomputed CCF between the two reverberant inputs as well as that between the two dereverberant outputs inthe two studied reverberantion conditions. Comparing Fig. 1(a) and (c), one can clearly see that reverberation
deteriorates the source spatial information. But the spatial information is well recovered by the developed
method as seen in Fig. 1(b) and (d).
To further evaluate the dereverberation performance of the developed method, we compute the average cepstral distance (CD) over short-time frames according to [24]. Specifically, the CD (in dB) between two
signals at each short-time frame is defined as
v
u
12
X
10 u
t
2
CD =
(β̂0 − β0 ) + 2(β̂k − βk )2 ,
(44)
ln 10
k=1

where β̂k and βk are the real cepstral coefficients [29] of the speech signal under evaluation and desired
signal (direct sound plus early reverberation), respectively. We use 20 sentences from different speakers (10male and 10 female), and compute the average CDs for the left and right output channels separately witha frame length of 32 ms, and 75% overlap. The length of speech are all 2 s, and the room reverberation
condition is T60 = 0.58 s. With our experimental setup, the average CDs of the left- and right-channel
reverberant speech are 2.69 and 2.72, respectively, while that of the dereverberant speech are 2.43 and 2.45,
respectively.

4929

4

1.0 4

3

3

2

0.5 2

1

1
(a)

0
4

3

2

0.5 2

1

1
(c)
0

0.4

0.8
1.2
Time (s)

0.5

(b)

0 0
1.0 4

3

0

1.0

0
1.6 2.0

0
1.0

0.5

(d)

0
0

0.4

0.8
1.2
Time (s)0
1.6 2.0

Figure 2. Spectrograms of the reverberant signals [(a) left channel, (b) right channel] and the dereverberated
signals [(c) left channel, (d) right channel]. The room reverberation time is T60 = 0.58 s.
To give an example, Figure 2 plots the spectrograms of the left- and right-channel reverberant and dereverberated speech signals. One can see from this figure that the developed method has effectively reduced
the reverberation component from the reverberant speech. Meanwhile, the speech spectrum and the spatial
information are well preserved.

6

CONCLUSIONSThis paper studied the binaural dereverberation problem using a microphone array in the time domain. By
adopting the WL framework, we merge the multiple real microphone inputs and binaural outputs into complex signals. The late reverberation is then modeled using the delayed WL prediction, which takes into
account the noncircular property of the complex speech signals. The optimal prediction filter coefficients are
estimated by maximizing the log likelihood function where the complex normal distribution is used to model
the speech signal of interest. The relation of the developed method with the WPE method developed in theliterature was also discussed. Simulation results showed that this dereverberation filter can effectively reduce
the reverberation effect and meanwhile preserves the spatial information of the sound source.

ACKNOWLEDGEMENTS
This work was supported in part by the National Science Foundation of China (NSFC) and the Israel Science Foundation (ISF) joint research program (grant No. 61761146001), the NSFC key program (grant No.
61831019), the NSFC Distinguished Young Scientists Fund (grant No. 61425005).

REFERENCES[1] M. Miyoshi and Y. Kaneda, “Inverse filtering of room acoustics,” IEEE Trans. Acoust., Speech, Signal
Process., vol. 36, no. 2, pp. 145–152, 1988.
[2] P. A. Naylor and N. D. Gaubitch, Speech dereverberation.
Business Media, 2010.

Berlin, Germany: Springer Science &[3] T. Yoshioka, A. Sehr, M. Delcroix, K. Kinoshita, R. Maas, T. Nakatani, and W. Kellermann, “Making
machines understand us in reverberant rooms: Robustness against reverberation for automatic speech
recognition,” IEEE Signal Process. Mag., vol. 29, no. 6, pp. 114–126, Nov. 2012.
[4] J.-M. Yang and H.-G. Kang, “Online speech dereverberation algorithm based on adaptive multichannel
linear prediction,” IEEE/ACM Trans. Audio, Speech, Lang. Process., vol. 22, no. 3, pp. 608–619, Mar.
2014.
[5] K. Han, Y. Wang, D. Wang, W. S. Woods, I. Merks, and T. Zhang, “Learning spectral mapping forspeech dereverberation and denoising,” IEEE/ACM Trans. Audio, Speech, Lang. Process., vol. 23, no. 6,
pp. 982–992, Jun. 2015.

4930

[6] K. Kinoshita, M. Delcroix, S. Gannot, E. A. P. Habets, R. Haeb-Umbach, W. Kellermann, V. Leutnant,
R. Maas, T. Nakatani, B. Raj, A. Sehr, and T. Yoshioka, “A summary of the REVERB challenge: stateof-the-art and remaining challenges in reverberant speech processing research,” EURASIP J. Adv. Signal
Process., vol. 2016, no. 1, Jan 2016.
[7] D. S. Williamson and D. Wang, “Time-frequency masking in the complex domain for speech dereverberation and denoising,” IEEE/ACM Trans. Audio, Speech, Lang. Process., vol. 25, no. 7, pp. 1492–1501,
Jul. 2017.
[8] S. Watanabe, M. Delcroix, F. Metze, and J. R. Hershey, New Era for Robust Speech Recognition. Berlin,
Germany: Springer-Verlag, 2017.
[9] M. Jeub, M. Schafer, T. Esch, and P. Vary, “Model-based dereverberation preserving binaural cues,”
IEEE Trans. Audio, Speech, Lang. Process., vol. 18, no. 7, pp. 1732–1745, Sep. 2010.
[10] M. Jeub and P. Vary, “Binaural dereverberation based on a dual-channel Wiener filter with optimized
noise field coherence,” in Proc. IEEE ICASSP, 2010, pp. 4710–4713.
[11] A. Tsilfidis, E. Georganti, and J. Mourjopoulos, “Binaural extension and performance of single-channel
spectral subtraction dereverberation algorithms,” in Proc. IEEE ICASSP, 2011, pp. 1737–1740.
[12] A. Tsilfidis, A. Westermann, J. M. Buchholz, E. Georganti, and J. Mourjopoulos, “Binaural dereverberation,” in The technology of binaural listening. Berlin, Germany: Springer, 2013, pp. 359–396.
[13] A. Westermann, J. M. Buchholz, and T. Dau, “Binaural dereverberation based on interaural coherence
histograms,” J. Acoust. Soc. Amer., vol. 133, no. 5, pp. 2767–2777, May 2013.
[14] K. Kinoshita, M. Delcroix, T. Nakatani, and M. Miyoshi, “Suppression of late reverberation effect on
speech signal using long-term multiple-step linear prediction,” IEEE Trans. Audio, Speech, Lang. Process.,
vol. 17, no. 4, pp. 534–545, May 2009.
[15] T. Yoshioka, T. Nakatani, M. Miyoshi, and H. G. Okuno, “Blind separation and dereverberation of
speech mixtures by joint optimization,” IEEE Trans. Audio, Speech, Lang. Process., vol. 19, no. 1, pp.
69–84, Jan. 2011.
[16] T. Yoshioka and T. Nakatani, “Generalization of multi-channel linear prediction methods for blind
MIMO impulse response shortening,” IEEE Trans. Audio, Speech, Lang. Process., vol. 20, no. 10, pp.
2707–2720, Dec. 2012.
[17] J. Benesty, J. Chen, and Y. Huang, “Binaural noise reduction in the time domain with a stereo setup,”
IEEE Trans. Audio, Speech, Lang. Process., vol. 19, no. 8, pp. 2260–2272, Nov. 2011.
[18] J. Chen and J. Benesty, “On the time-domain widely linear LCMV filter for noise reduction with astereo system,” IEEE Trans. Audio, Speech, Lang. Process., vol. 21, no. 7, pp. 1343–1354, Jul. 2013.
[19] J. Benesty, C. Paleologu, T. Gänsler, and S. Ciochină, A perspective on stereophonic acoustic echo cancellation. Springer Science & Business Media, 2011, vol. 4.
[20] C. Stanciu, J. Benesty, C. Paleologu, T. Gänsler, and S. Ciochină, “A widely linear model for stereophonic acoustic echo cancellation,” Signal Process., vol. 93, no. 2, pp. 511–516, 2013.
[21] C. Paleologu, J. Benesty, and S. Ciochină, “Widely linear general Kalman filter for stereophonic acoustic
echo cancellation,” Signal Process., vol. 94, pp. 570–575, 2014.
[22] A. van den Bos, “The multivariate complex normal distribution-a generalization,” IEEE Trans. Inform.
Theory, vol. 41, no. 2, pp. 537–539, Mar. 1995.
[23] B. Picinbono, “Second-order complex random vectors and normal distributions,” IEEE Trans. Signal
Process., vol. 44, no. 10, pp. 2637–2640, Oct. 1996.
[24] T. Nakatani, T. Yoshioka, K. Kinoshita, M. Miyoshi, and B.-H. Juang, “Speech dereverberation basedon variance-normalized delayed linear prediction,” IEEE Trans. Audio, Speech, Lang. Process., vol. 18,
no. 7, pp. 1717–1731, Sep. 2010.
[25] A. Jukic and S. Doclo, “Speech dereverberation using weighted prediction error with Laplacian modelof the desired signal,” in Proc. IEEE ICASSP, 2014, pp. 5172–5176.
[26] W. C. Ward, G. Elko, R. Kubli, and W. McDougald, “The new varechoic chamber at AT&T Bell Labs,”
in Proc. Wallace Clement Sabine Centenn. Symp., 1994.
[27] A. Härmä, “Acoustic measurement data from the varechoic chamber,” Agere Systems, Tech. Memo., Nov.
2001.
[28] J. Garofolo, L. Lamel, W. Fisher, J. Fiscus, D. Pallett, N. Dahlgren, and V. Zue, TIMIT acoustic-phonetic
continuous speech corpus. Linguistic Data Consort., 1993.
[29] R. Kubichek, “Mel-cepstral distance measure for objective speech quality assessment,” in Proc. IEEE
Pacific Rim Conf. Commun. Comput. Signal Process., vol. 1, 1993, pp. 125–128.

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