Using partial a priori knowledge of relative transfer functions to design an
MVDR beamformer for a binaural hearing assistive device with external
microphones
Randall ALI(1) , Toon vanWATERSCHOOT(2) , Marc MOONEN(3)(1) KU Leuven, Dept. of Electrical Engineering (ESAT-STADIUS), Leuven, Belgium, randall.ali@esat.kuleuven.be(2) KU Leuven, Dept. of Electrical Engineering (ESAT-ETC/STADIUS), Leuven, Belgium, tvanwate@esat.kuleuven.be(3) KU Leuven, Dept. of Electrical Engineering (ESAT-STADIUS), Leuven, Belgium, marc.moonen@esat.kuleuven.be

Abstract
This paper considers a binaural hearing assistive device (HAD) equipped with a separate local microphone array(LMA) for the left and right ear, as well as external microphones (XMs) that may be located within the vicinity ofthis HAD. For such a system, a binaural minimum variance distortionless response (BMVDR) beamformer may
be used for noise reduction, and for the preservation of the relevant binaural speech cues, provided that a reliable
estimate of the left and right ear relative transfer function (RTF) vectors pertaining to all the microphones canbe obtained. In this paper, an alternative approach is considered, which makes use of available partial a priori
knowledge of these RTF vectors, i.e., known separate left and right ear RTF vectors for the respective LMAs
on the binaural HAD. The procedure for this approach will be discussed, which requires the estimation ofan appropriate scaling between the left and right ear RTF vectors, and the missing part of these RTF vectors
pertaining to the XMs. An experiment involving a dummy head, two behind-the-ear dummy hearing aids, and
XMs is also performed in order to evaluate the benefit of the proposed approach.
Keywords: Binaural MVDR, External Microphones, Hearing Assistive Device

1

INTRODUCTION

In noisy environments, speech intelligibility is inevitably degraded for individuals that suffer with a hearing
impairment and hence hearing assistive devices (HADs) such as hearing aids (HAs) or cochlear implants (CIs)must perform speech enhancement tasks. In addition to the fundamental task of noise reduction, preservationof the binaural cues, i.e., the interaural time differences (ITDs) and interaural level differences (ILDs) is also
important to maintain the spatial perception of the auditory scene.
For a binaural HAD equipped with a separate local microphone array (LMA) for the left and right ear, and acommunication link between them, the binaural minimum variance distortionless response beamformer (BMVDR)(1) is known to exhibit substantial noise reduction and to preserve the ITD and ILD of a target speaker1 . In
recent work (2, 3), such a binaural HAD has also been supplemented with an external microphone (XM) (e.g.
a wearable microphone or the microphone on a mobile device) and it was demonstrated that the XM could
contribute to additional noise reduction and preserve the relevant binaural cues. For the successful operation of
the BMVDR in this case, an estimate of the entire vector of transfer functions from the target signal at a left
ear reference microphone to all the other microphones, i.e., the left ear relative transfer function (RTF) vector,
and a corresponding right ear RTF vector is required. However, obtaining such estimates becomes increasingly
challenging in adverse acoustic conditions.
Therefore in this paper, generalising the system to include more than one XM, an alternative approach is considered, which makes use of available partial a priori knowledge of these RTF vectors, i.e., known separate left
and right ear RTF vectors for the respective LMAs on the binaural HAD (4). In such a case, it is only the
1 Although, the BMVDR beamformer preserves the binaural cues for the target speaker, it distorts the binaural cues for the noise. However, in(1), several remedies have been proposed, and hence this work will focus only on the preservation of the binaural cues for the target speaker.

6815

estimation of an appropriate scaling between the left and right ear RTF vectors, and the missing part of these
RTF vectors pertaining to the XMs that need to be estimated.
The paper is organised as follows. In Section 2, the data model and notation are described. In Section 3, thestate of the art procedure for estimating the entire RTF vector is reviewed. In Section 4, the proposed procedurethat makes use of the partial a priori knowledge of the RTF vector is discussed. In Section 5, the proposed
procedure is evaluated using recorded audio data, and conclusions are drawn in Section 6.

2

DATA MODEL

The scenario as depicted in Figure 1 is considered, in which a user of a binaural HAD is listening to one
target speaker of interest in a noisy, reverberant environment. The binaural HAD consists of an LMA with Ma
microphones for the left ear and an LMA with Ma microphones for the right ear. Additionally, there are Me
XMs randomly placed within the room (Me = 2 in Fig.1). In the short-time Fourier transform (STFT) domain,
TARGET
SPEAKER, s(k,l)

XM, ye,1(k,l)

XM, ye,2(k,l)

ya,L(k,l)ya,R(k,l)

LEFT LMA

RIGHT LMA

Figure 1. Scenario with a user of a binaural HAD having access to XMs, listening to the target speaker.
the microphone signals at one frequency, k, and one time frame, l, can be stacked into a vector and represented
as follows:
 




ya,L (k, l)aa,L (k, l)na,L (k, l)y(k, l) = a(k, l)s(k, l) + n(k, l) =⇒ ya,R (k, l) = aa,R (k, l) s(k, l) + na,R (k, l)
(1)
|
{z
}
y
(k,
l)
a
(k,
l)
n
(k,
l)
e
e
e
x(k,l)
where2 aa,L = [aa1,L , aa2,L , . . . aaMa ,L ]T , aa,R = [aa1,R , aa2,R , . . . aaMa ,R ]T , and ae = [ae1 , ae2 , . . . aeMe ]T are theacoustic transfer functions (ATFs) from the target speaker to the microphones on the left LMA, the right LMA,
and the XMs respectively. Furthermore, s is the target speaker, and na,L , na,R , and ne are the noise contributions
similarly defined as aa,L , aa,R , and ae respectively. Without loss of generality, the first microphone in each of
the LMAs is also chosen as the reference microphone:
ya1,L = eTL y = sa1,L + na1,L

ya1,R = eTR y = sa1,R + na1,R(2)

where sa1,L = aa1,L s, sa1,R = aa1,R s, which are the speech components that need to be estimated, and eL andeR are all-zero vectors except for a one in the left and right LMA reference microphone position respectively.
In order to perform the estimation, it is firstly convenient to re-define eq. (1) in terms of a relative transfer
function (RTF) vector, as opposed to the ATF vector, a. The RTF vector is simply the ATF vector normalised
to a reference microphone. Therefore in the binaural context, a separate RTF vector can be defined for the left
ear and another for the right ear. Hence, eq. (1) can be expressed as follows:y = hL sa1,L + n
2 The dependence on (k, l) is dropped for notational convenience.

6816

y = hR sa1,R + n(3)

where hL and hR are the RTF vectors defined as:

 

aa,L
ha,L
1 
aa,R  = ϕ ha,R 
hL =
aa1,L
ae
he,L
a


 1
ha,L
aa,L
ϕ
1 


aa,R  =  ha,R 
hR =
aa1,R
1
ae
ϕ he,L


(4)

a

a,R
a,L
and ha,R = aa1,R
are the individual RTF vectors corresponding to each of the left and rightwhere ha,L = aa1,L

aLMAs and the complex scaling, ϕ = aa1,R. It should be noted that the part of the RTF vector pertaining to thea1,Lae
XMs in hR is a scaled version of that in hL , where he,L = aa1,L. In fact, it can be seen that hL = ϕhR , whichmeans that the RTF vectors are parallel.
The speech-plus-noise spatial correlation matrix, Ryy , the noise-only correlation matrix, Rnn , and the speechonly correlation matrix, Rxx , all ∈ C(2Ma +Me )×(2Ma +Me ) , are given respectively as:
Ryy = E{yyH }; Rnn = E{nnH }; Rxx = E{xxH }(5)

where E{.} is the expectation operator, H is the Hermitian transpose, and Rxx is a rank-1 correlation matrix:2
H
Rxx = E{xxH } = σs2a1,L hL hH
L = σsa1,R hR hR(6)where σs2a1,L = E{|sa1,L |2 } and σs2a1,R = E{|sa1,R |2 } are the speech powers in the reference left and right microphone respectively. It is also assumed that the speech components are uncorrelated with the noise components,and hence Ryy = Rxx + Rnn . A perfect communication link is additionally assumed among the left and right
LMAs in the binaural HAD, and the XMs, with no bandwidth constraints and synchronous sampling.
The estimate of the speech component in the reference microphone of the left and right LMAs, i.e. the estimateof sa1,L and sa1,R is then obtained through the linear filtering of the microphone signals, with the complex-valued
filters, wL and wR respectively:(7)ŝa1,L = wHŝa1,R = wH
Ly
Ry
The BMVDR beamformer filters, wL and wR , are then given by:
wL =

R−1
nn hL
H
hL R−1
nn hL

wR =

R−1
nn hR
H
hR R−1nn hR(8)Consequently, in order to compute these filters, estimates are required for Rnn , and the RTFs, hL , and hR .
Typically, Rnn can be estimated during periods of noise only with recursive averaging (3). Hence this paper
focuses on the estimation of hL and hR .

3

ESTIMATING THE ENTIRE RTF VECTOR

Given R̂yy and R̂nn , which are estimates of Ryy and Rnn respectively, a generalised eigenvalue decomposition(GEVD) (5) or what is equivalently known as covariance whitening (3) can be used to estimate hL and hR . A
spatial pre-whitening operation can be firstly defined from R̂nn using the Cholesky decomposition:
1/2

H/2

R̂nn = R̂nn R̂nn

(9)

1/2

where R̂nn is a lower triangular matrix. Spatial pre-whitening is then performed by pre-multiplying the signal−1/2
vector of interest by R̂nn . For an autocorrelation matrix, spatial pre-whitening is performed by pre-multiplying
−1/2
−H/2
it by R̂nn and post-multiplying it by R̂nn . Using the definition of Rxx from eq. (6) and that Ryy = Rxx + Rnn ,the following optimisation problem can be considered to estimate hL (and hR by an appropriate scaling):
min

σs2

a1,L

, hL

−1/2

−H/2

2
||R̂nn ((R̂yy − R̂nn ) − σs2a1,L hL hHL )R̂nn ||F

(10)

6817

where ||.||F is the frobenius norm. The solution to eq. (10) then follows from an eigenvalue decomposition
−1/2
−H/2
(EVD) of R̂nn R̂yy R̂nn or equivalently, GEVD of the matrix pencil {R̂yy , R̂nn }:
ΣU−1
R̂−1
nn R̂yy = UΣ(11)where Σ is a diagonal matrix of the generalised eigenvalues arranged in descending order, and U is an invertible matrix containing the corresponding generalised eigenvectors. The GEVD is also equivalent to a joint
diagonalisation of R̂yy and R̂nn :Σy QHR̂yy = QΣΣ n QHR̂nn = QΣ(12)where Σ y and Σ n are diagonal matrices, and Q = U−H is an invertible matrix. A rank-1 approximation toΣy − Σ n )QH yields an estimate for Rxx , R̂xx = Qe1 eT1 (ΣΣy − Σ n )e1 eT1 QH , where e1 ∈ C2Ma +Me(R̂yy − R̂nn ) = Q(Σis an all-zero vector except for a one as the first element (and it is noted that e1 = eL ). It can be shown (5)that this corresponds to the rank-1 approximation sought from eq. (10) so that the estimates to hL and hR thenfollow as:
Qe1
Qe1
ĥR = TĥL = T
(13)
eL Qe1eR Qe1
Finally, a substitution of R̂nn from eq. (12) and ĥL and ĥR from eq. (13) into eq. (8) results in the corresponding BMVDR filters:(14)ŵL = Ue1 eT1 QH eLŵR = Ue1 eT1 QH eR

4

USING PARTIAL A PRIORI KNOWLEDGE OF THE RTF VECTOR

As opposed to estimating the entire RTF vectors, hL , and hR , an alternative procedure may be followed if there
is a priori knowledge of the RTF vectors for the separate left and right LMA, i.e., if a suitable approximation
to ha,L and ha,R is available. For instance, such an approximation may be the measured RTF vectors for the
separate left and right LMA in an anechoic room or RTF vectors from an existing binaural noise reduction
ea,L and hea,R respectively,
system that uses only the LMAs. Denoting this approximation to ha,L and ha,R as hand recalling the definitions from eq. (4), an alternative optimisation problem to eq. (10) can be considered:
ea,L hhi −HeH ϕ ∗ heH hH )R̂nn/2 ||2F
ea,R  hmin||R̂nn ((R̂yy − R̂nn ) − σs2a1,L ϕ ha,La,R e,L
σs2 , ϕ, he,L
a1,L
he,L


−1/2

(15)

where now it is only the scaling, ϕ, and the RTF vector for the XMs, he,L , which need to be found as opposedto the entire hL as in eq. (10). As will be discussed in the following, the solution can be realised in the block
scheme of Figure 2, which consists of compressing the left and right LMA signals, an orthogonalisation operation, and finally a GEVD on a lower dimensional (C(Me +2)×(Me +2) ) matrix pencil. In order to solve eq. (15),the following blocking matrices, Ca ∈ C2Ma ×(2Ma −2) , Ca,L ∈ CMa ×(Ma −1) , Ca,R ∈ CMa ×(Ma −1) , fixed beamformers,
Fa ∈ C2Ma ×2 , fa,L ∈ CMa , fa,R ∈ CMa , and transformation matrix, T ∈ C(2Ma +Me )×(2Ma +Me ) are firstly defined:




Ca,L
0
fa,L
0


Fa =
Ca =
Ca F a
0
0
Ca,R
0
fa,R
T=
(16)
0
0 IMe
H e
H e
H e
e
h
=
0;
C
h
=
0
f
h
=
1;
f
h
=
1
CH
a,R
a,L
a,R
a,L
a,L
a,R
a,L
a,R
where IMe ∈ CMe ×Me is an identity matrix. The first two blocks in Fig. 2 apply the transformation, TH , to y to
2Ma −2 , two compressed signals, fH y
Hyield a set of blocking matrix signals, CHa ya ∈ Ca,L a,L and fa,R ya,R resulting
from the left and right fixed beamformers respectively, and the unaltered set of XM signals, ye . An alternative
spatial pre-whitening operation can then be defined by applying the transformation to R̂nn :TH R̂nn TH = LLH

6818

(17)

ya,L
ya,R

H
fa,L
ya,L

Fa

+

yL

-

yR

H
fa,R
ya,R

+
-

ŵL

ga,L
ga,R

Ca

+

ge,1

-

ye,1
ge,Me

ye,Me

Speech estimate
for LEFT ear

ye,1
+

ye,Me

-

ŵR

Speech estimate
for RIGHT ear

Figure 2. Block scheme for a BMVDR that uses partial a priori knowledge of the RTF vectors.
where L is a lower triangular matrix. It can then be shown (6) that eq. (15) can be equivalently re-written as:ea,L hhieH ϕ ∗ heH hH )T L−H ||2F
ea,R  hmin||L−1 TH ((R̂yy − R̂nn ) − σs2a1,L ϕ h
(18)
a,L
a,R
e,L
σs2 , ϕ, he,L
a1,L
he,L
from which the solution follows eventually from a GEVD of a matrix pencil consisting of the lower dimensional C(Me +2)×(Me +2) correlation matrices (6) (7) corresponding to the Me + 2 compressed signals, y =[yL yR ye,1 . . . ye,Me ]T ∈ CMe +2 . These compressed signals can be computed by performing an orthogonalisation
involving the previously transformed signals, or equivalently, with a GSC beamformer, A:
T
y = AH y = FH y − GH CH
aE y



Fa
0
and G ∈ C(2Ma −2)×(Me +2) , which can be computed as follows:with A = F−ECa G, F ∈ C(2Ma +Me )×(Me +2) defined as F =

(19)


0
, the selection matrix, E = [I2Ma | 0(2Ma ×Me ) ]T ,IMeT
−1 H TG = (CHa E R̂nn ECa ) Ca E R̂nn F = [ga,L ga,R ge,1 . . . ge,Me ](20)

is comprised of the CMe +2 orthogonalisation filters in each column. As depicted in Fig. 2, ga,L and ga,RH
will orthogonalise the noise components of the signals fH
a,L ya,L and fa,R ya,R onto the noise components of theblocking matrix signals, and [ge,1 . . . ge,Me ] will orthogonalise the noise components of each of the respective
XMs onto the noise components of the blocking matrix signals.
A new signal model that resembles eq. (3) can in fact then be realised by substituting eq. (3) into eq. (19):y = hL sa1,L + n

y = hR sa1,R + n

(21)

h
iT

T
where hL = AH hL = 1 ϕ hTe,L , hR = AH hR = ϕ1 1 ϕ1 hTe,L , n = AH n, and the associated correlation matrices,
Ryy = E{yyH } and Rnn = E{nnH }. These compressed signals, y, can now be used to design the BMVDR as
opposed to y. Estimates of sa1,L and sa1,R can be obtained by directly filtering y with the BMVDR filters,wL ∈ CMe +2 and wR ∈ CMe +2 respectively (as in eq. (7) and depicted in Fig. 2):
wL =

R−1
h
nn L
hH
R−1 h
L nn L

wR =

R−1
h
nn R
hH
R−1 h
R nn R

(22)

6819

Hence, similar to eq. (8), estimates are now required for Rnn , hL , and hR in order to compute these filters.
With the estimates of the correlation matrices, R̂yy = AH R̂yy A and R̂nn = AH R̂nn A, the GEVD procedure from
Section 3 then follows from the matrix pencil {R̂yy , R̂nn } to estimate hL and hR :−1R̂yy = Q Σ y QH ;R̂nn R̂yy = U Σ U−1 ;R̂nn = Q Σ n QH

(23)

where Σ is a diagonal matrix of the generalised eigenvalues arranged in descending order, and U contains thecorresponding generalised eigenvectors, Σ y , and Σ n are also diagonal matrices, and Q = U−H is an invertible
matrix. The estimates for hL and hR then follow as:
Qe

ĥL =

1
eTL Q e1

Qe

ĥR =

1
eTR Q e1(24)

where the selection vectors, e1 = eL = [1, 0, 0(1×Me ) ]T and eR = [0, 1, 0(1×Me ) ]T . Finally, the substitution of R̂nn
from eq. (23) and ĥL and ĥR from eq. (24) into eq. (22) results in the corresponding BMVDR filters:ŵL = U e1 eT1 QH eL

5ŵR = U e1 eT1 QH eR(25)

RESULTS AND DISCUSSION

In order to evaluate these algorithms, audio recordings were made in an L-shaped room of height 3.8 m, whose
longer dimensions were approximately 6 m × 5.5 m, with an estimated broadband reverberation time of 1.5 s.
Similar to Fig. 1, a Neumann KU-100 dummy head was placed in a central location of the room and equippedwith two behind-the-ear (BTE) hearing aids (HAs), each consisting of an LMA with two microphones spaced
approximately 1.3 cm apart. Denoting 0° as the azimuth direction directly in front of the dummy head andpositive angles as clockwise, two XMs (AKG-CK-97-O) were positioned such that one was at 60° and 1 m awayfrom the dummy head, and the other was at 20 cm below the dummy head. A Genelec 8030C loudspeaker, 1 mfrom the dummy head, was used to generate the speech signal from a target male speaker (8). Four of the same
loudspeakers, facing away from the dummy head toward opposite corners, were used to generate uncorrelated
excerpts of babble noise to approximate a diffuse noise field. The speech and noises were recorded separately,
but mixed together at an input signal-to-noise ratio (SNR) of 0 dB at the reference (frontal) microphone on theright LMA.
Four algorithms were then evaluated: (i) the BMVDR from Section 3 (BMVDR−ĥ), (ii) the BMVDR from Sece (iii) the BMVDR from Section 3 but only using the left and right LMA (BMVDR−ĥa ),tion 4 (BMVDR−h),ea ). For the processand (iv) the BMVDR as from Section 4 but only using the left and right LMA (BMVDR−hing of the algorithms, the Weighted Overlap and Add (WOLA) method (9), with a Discrete Fourier Transform(DFT) size of 128 samples, 50 % overlap, and a sampling frequency of 16 kHz was used. Using the speechpresence probability (SPP) (10), periods for which the speech was active were extracted if the SPP > 0.6, andnot active if the SPP < 0.4. R̂yy and R̂nn were then estimated accordingly via recursive averaging with anea,L and hea,R were computed from the DFT of 128 samples
averaging time of 1 s. Finally, the a priori RTFs, h(including only the direct path component) of the measured impulse responses (IRs) from the target speaker
position to each of the left and right LMAs. The metrics used to evaluate the following experiments were thechange in speech intelligibility-weighted SNR improvement (11) (∆SI-SNR) from the input SI-SNR at LM1,and change in binaural cue errors, ∆ITD and ∆ILD, using the auditory model from (12). As the room wasquite reverberant, the reference binaural signal for computing ∆ITD and ∆ILD was the speech signal of theea,L and hea,R . All metrics weretarget speaker convolved with the time domain direct path IRs used to define hcomputed in 2 s time frames with a 50 % overlap.

6820

0

2

4

6

8

10

12

4
2
0

0

2

4

6

8

Time (s)

10

12

14

e
BMVDR−h

0.2

0

2

4

6

8

10

12

0.2
0

0

2

4

6

8

Time (s)

10

12

14

(c)

6
4
2
0

14
(e)

0.4

BMVDR−ĥ8∆ILD (dB)

(b)

0.4

0

14
(d)

6

BMVDR−ĥa

0

2

4

6

8

10

12

8
∆ILD (dB)2∆ITD (ms)40∆SI-SNR (dB)(a)

6∆ITD (ms)∆SI-SNR (dB)ea
BMVDR−h

14
(f)

6
4
2
0

0

2

4

6

8

10

12

14

Time (s)Figure 3. Performance of the various BMVDR algorithms for a target speaker located at 45°. The upper plots((a),(b),(c)) are the results for accurately estimated correlation matrices, while the lower plots ((d),(e),(f)) are
the results for less accurately estimated correlation matrices.
Figure 3 displays the results for an input signal of 13 s when the target speaker was at 45°. The upper plots((a),(b),(c)) display the results when the SPP was computed on the speech-only signal as received by the reference microphone of the right LMA, and represents a case when R̂yy and R̂nn have been accurately estimated.
In the lower plots ((d),(e),(f)), the SPP was computed using the actual noisy signal as received by the same
microphone and represents a case when R̂yy and R̂nn have not been as accurately estimated.
e and BMVDR−ĥ (i.e., algorithms thatFirstly, in terms of ∆SI-SNR, it can be observed that the BMVDR−hea and BMVDR−ĥause the LMAs along with the XMs) always demonstrate an improvement over BMVDR−h(i.e., algorithms that use the LMAs only). Furthermore, the ∆ITD and ∆ILD remain relatively constant between
ea and BMVDR−h,e as well as between BMVDR−ĥ and BMVDR−ĥ, suggesting that the inclusion
BMVDR−hof the XMs does not contribute to any additional distortion in the binaural cues.
The impact of accurate estimation of R̂yy and R̂nn is also evident. In the upper plots ((a),(b),(c)), where R̂yye and BMVDR−ĥ perform quite similarly. In
and R̂nn have been accurately estimated, both the BMVDR−h
the lower plots ((d),(e),(f)), where R̂yy and R̂nn have been estimated less accurately, the absolute values of all
metrics now indicate a generally lower performance for all algorithms. However, in such a case, it is seen
e offers an improved ∆SI-SNR and maintenance of the binaural cues in comparison withthat the BMVDR−h
the BMVDR−ĥ. Therefore, in cases of inaccurate estimation of R̂yy and R̂nn , the approach proposed in thispaper is seen to be beneficial. The resulting audio files from this experiment may be listened to for subjective
evaluation (13).

6

CONCLUSIONS

An approach to designing a BMVDR beamformer for a binaural HAD consisting of separate LMAs for the left
and right ear, as well as XMs has been developed. As opposed to the conventional approach of estimating entire
RTF vectors, partial a priori knowledge of these RTF vectors, i.e., known separate left and right ear RTF vectors
for the LMAs on the binaural HAD was incorporated so that only the computation of an appropriate scaling andthe missing part of these RTF vectors pertaining to the XMs was required. An experiment with a dummy head,two behind-the-ear dummy hearing aids, and XMs indicated that the proposed approach is especially beneficial
in cases where there may be inaccurate estimation of the relevant correlation matrices needed for the BMVDR
beamformer.

6821

ACKNOWLEDGEMENTS
This research work was carried out at the ESAT Laboratory of KU Leuven, in the frame of KU Leuven C216-00449 ’Distributed Digital Signal Processing for Ad-hoc Wireless Local Area Audio Networking’, and KU
Leuven Internal Funds VES/19/004. The research leading to these results has received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation program / ERC
Consolidator Grant: SONORA (no. 773268). This paper reflects only the authors’ views and the Union is notliable for any use that may be made of the contained information. The scientific responsibility is assumed by
its authors.

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