Binaural Direction-of-Arrival Estimation in Reverberant Environments Using
the Direct-path Dominance test
Hanan BEIT-ON(1) , Boaz RAFAELY(2)(1) Ben-Gurion University of the Negev, Israel, hananb@post.bgu.ac.il(2) Ben-Gurion University of the Negev, Israel, br@bgu.ac.il

Abstract
Abstract— Speaker localization in reverberant environments is currently an important challenge in audio signal
processing, and several methods have recently been proposed to address it. One such family of methods isbased on the direct-path dominance (DPD) test that was developed specifically for spherical arrays and recently
extended to arbitrary arrays. In this paper, an extension of the DPD test for a binaural array is examined. Thisextension uses a focusing technique to remove the frequency dependence of the steering matrix within frequency
bands. This facilitates frequency smoothing, which is necessary for decorrelating coherent reflections. Unlike
other focusing approaches, the proposed focusing approach does not rely on the initial estimation of source
directions, facilitating its application in reverberant environments. However, in the binaural case, there is no
guarantee of focusing performance. This paper presents an analysis of this focusing process for the binaural caseand shows that good focusing performance can be obtained in the binaural case. A simulation study examines
the effect of the focusing and smoothing and demonstrates the proposed method’s capabilities in overcoming
adverse reverberation conditions.

1

Introduction

Binaural direction-of-arrival (DOA) estimation of sound from speakers in a room is important for speech enhancement and source separation in applications such as hearing aids and robot audition. DOA estimation
methods for sources, in general, include maximum likelihood estimation [16], beamforming techniques [16]and subspace methods, such as MUSIC [16]. Because speakers in a room generate coherent signals due toroom reflection, DOA estimation methods that are designed for coherent signals are more suitable in this case.
One such method is the coherent signal subspace method (CSSM) [17], which employs focusing and frequency
smoothing to decorrelate the sources, enabling the DOA estimation of coherent signals using subspace methods.
Coherent processing of the CSSM has recently been applied in a DOA estimation method for speech signals,
called the direct-path dominance (DPD) test [10]. This method tackles the issue of reverberation by using only
the direct-path bins in the short-time Fourier transform (STFT) domain for the estimation task. The need offocusing matrices has been circumvented by employing spherical arrays, for which the frequency dependence ofthe steering matrix is removed by the plane-wave decomposition operation [11]. Although the DPD test method
has shown robustness to diverse reverberation and noise conditions [12], its application is limited to sphericalarrays.
An extension of the DPD test for arbitrary arrays was recently proposed [3]. This extension incorporates afocusing process that does not require an initial DOA estimation facilitating its application in reverberant environments. Unlike the bin selection method proposed in [15], which is also available for arbitrary arrays, theDPD test employs coherent processing. The coherent processing increases the accuracy of the spatial spectrum [6], and this, in turn, leads to an accurate estimation when using a subspace localization method such
as MUSIC. Unlike the binaural localization method proposed in [9], the DPD test does not require a training
phase. Furthermore, in contrast to the speaker localization method proposed in [8], the DPD test does not
require speech-free signal segments for estimating the noise statistics and can operate on short time segments.

3305

Although the proposed extension of the DPD test is useful, it has limitations. In particular, poor focusing
performance may result in DOA estimation errors. For binaural arrays, the focusing matrices are obtained
as the least squares solution to an over-determined system, thus incorporating a fitting error. Moreover, thedevelopment of the focusing matrices assumes knowledge of the spherical harmonics coefficients matrix. For
binaural arrays, this matrix is obtained via the discrete spherical Fourier transform (DSFT) of the head relatedtransfer function (HRTF) and its accuracy depends on the sampling of the HRTF. In this paper, the performanceof the focusing for binaural arrays is investigated, and guidelines for sampling the HRTF are suggested. A
simulation study examines the effect of focusing and smoothing and demonstrates binaural DOA estimation
with the DPD test under adverse reverberation conditions.

2

System model

Consider an acoustic scene of a single speaker in a reverberant environment. Also, consider a sound fieldcomposed of L far field sources, where a source can represent a direct-sound or, for example, a reflection
due to room boundaries. Employing the multiplicative transfer function (MTF) approximation [2], according towhich if the length of the time windows are sufficiently large with respect to the length of the HRTF (in time),
the binaural signal can be expressed in the STFT domain asp (τ, ν) = H (ν, ψ) s (τ, ν) + n (τ, ν) ,(1)
T

where τ and ν are the time frame and the frequency indices, respectively. p (τ, ν) = [pl (τ, ν) , pr (τ, ν)] , withpl (τ, ν) and pr (τ, ν) denoting the signal at the left and the right binaural microphones, respectively. s (τ, ν) =[s1 (τ, ν) , ..., sL (τ, ν)]T is the source signal vector, where sl (τ, ν) is the complex amplitude of the l’th sourceat the origin. H (ν, ψ) = [h (ν, ψ1 ) , ..., h (ν, ψL )]T is a 2 × L HRTF matrix, where h (ν, ψl ) is the HRTF that
corresponds to the l’th source with DOA ψl = (θl , φl ), where θl and φl are the elevation and the azimuth
angles, respectively. n (τ, ν) = [nl (τ, ν) , nr (τ, ν)]T is the additive sensor noise, which is assumed to be spatiallywhite and uncorrelated with s(τ, ν).

3

The DPD test for binaural arrays

The first stage of the DPD test method is to construct the smoothed spectrum at each time frequency (TF) bin.
To construct the smoothed spectrum in bin (τ0 , ν0 ), focusing is applied to a rectangular window of Jτ timeframes and Jν frequencies around (τ0 , ν0 ), followed by an averaging of the spectrum over the window. The
averaging over time frames approximates the expectation operation, and the averaging over frequencies implements frequency smoothing. Focusing is applied to remove the frequency dependence of the HRTF matrices
within the averaging window to preserve the spatial information in the smoothed HRTF matrix. Focusing is
performed by aligning the HRTF matrices within the averaging window to the HRTF matrix from the central
frequency using a focusing matrices T (ν, ν0 ) that satisfies
T (ν, ν0 ) H (ν, ψ) = H (ν0 , ψ) .

(2)

The focusing is applied by multiplying the binaural signal p (τ, ν) with the corresponding focusing matrix
T (ν, ν0 ), such that the transformed binaural signal p̃ (τ, ν) can be expressed asp̃ (τ, ν) = T (ν, ν0 ) p (τ, ν) = H (ν0 , ψ) · s (τ, ν) + ñ (τ, ν) ,(3)

where ñ (τ, ν) = T (ν, ν0 ) n (τ, ν). The smoothed spectrum is then constructed by averaging the spectrum within
the averaging window. Assuming s (τ, ν) and n (τ, ν) are uncorrelated, the smoothed spectrum can be expressedasS p̃ (τ0 , ν0 ) = H (ν0 , ψ) Ss (τ0 , ν0 ) H H (ν0 , ψ) + Sñ (τ0 , ν0 ) ,

3306

(4)

where Ss (τ0 , ν0 ) and Sñ (τ0 , ν0 ) are the smoothed covariance matrices of the source and noise signals, respectively. The frequency smoothing is employed to decorrelate coherent sources, such that after smoothing thematrix Ss (τ0 , ν0 ) is assumed to be of full-rank with a low condition number. This facilitates the application ofsubspace localization methods such as MUSIC.
The DPD test selects TF bins with one dominant source and assumes that these bins contain a significant
contribution from the direct-sound and insignificant contributions from room reflections. Thus, the DOA of theactual source can be correctly estimated using these bins. The group of bins selected by the DPD test isσ1 (τ0 , ν0 )ADPD = (τ0 , ν0 ) :> THDPD ,σ2 (τ0 , ν0 )where σ1 (τ0 , ν0 ) and σ2 (τ0 , ν0 ) are the singular values of S p̃ (τ0 , ν0 ) and THDPD is a chosen threshold. Several
approaches for estimating the speaker DOA from the selected bins have been proposed, including MUSIC withcoherent and incoherent integration of the signal sub-spaces from the different bins [10], and bin-wise DOA
estimation followed by a statistical analysis to fuse the estimates [14].
The formulation of the focusing matrices proposed in [3] assumes an order-limited HRTF, i.e. the HRTF canbe fully represented using spherical harmonics up to order N. As an order-limited function, the decompositionof the HRTF matrix using spherical harmonics is given by
H (ν, ψ) = Hnm (ν)Y (ψ) ,

(5)

where
Y00 (ψ1 )
Y00 (ψ2 ) Y −1 (ψ1 ) Y −1 (ψ2 )1
 1
Y (ψ) = 
..
..

.
.


YNN (ψ1 )

YNN (ψ2 )
· · · Y00 (ψL )· · · Y1−1 (ψL ) 


..
..

.
.
···

YNN (ψL )is an (N + 1)2 × L matrix of the spherical harmonics functions Ynm (ψ) of order n and degree m [11] and
Hnm (ν) =hl0,0 (ν) hl1,−1 (ν) · · · hlN,N (ν)
hr0,0 (ν) hr1,−1 (ν) · · · hrN,N (ν)

is a 2 × (N + 1)2 matrix of the spherical harmonics coefficients hln,m (ν), hrn,m (ν) of order n and degree m of the
HRTF of the left (l) and right (r) binaural microphones, respectively. Substituting (5) into (2) it follows that a
focusing matrix that aligns H (ν, ψ) to H (ν0 , ψ) for any set of sources should satisfy
T (ν, ν0 ) Hnm (ν) = Hnm (ν0 ) .

(6)

The general solution to this system is given by
T (ν, ν0 ) = Hnm (ν0 ) Hnm † (ν) ,(7)

where (·)† denotes the pseudo-inverse operation.

4

Focusing formulation for binaural arrays

The development of the focusing matrices presented in the previous section assumes an order-limited HRTF
and knowledge of the spherical harmonics matrix Hnm (ν). Conditions for which these assumptions hold areformulated in this section.

3307

The resemblance of the human head to a rigid sphere suggests that, like a rigid sphere, the HRTF is nearlyorder-limited with order N = kr, where k = 2πc f is the wave number with c denoting the speed of sound, andr is the sphere radius. Thus, the assumption according to which the HRTF can be represented with sphericalharmonics coefficients up to order N is approximately maintained for kr < N.
While for some arrays, such as spherical arrays with a rigid-sphere, the spherical harmonics coefficients matrix follows an analytical expression, in the binaural case, this matrix is computed using the discrete spherical
Fourier transform (DSFT) of the HRTF matrix. The DSFT is given by [11]Hnm (ν) = H (ν, ψ)Y † (ψ) ,(8)

T

where ψ = [ψ1 , ..., ψL ] is a set of sampling points of the HRTF over a sphere of directions. Such samples
can be obtained with a numerical simulation or using measurements. Accurate computation of Hnm (ν), requires
more samples than coefficients, i.e. L ≥ (N + 1)2 , and a matrix Y (ψ) of full-rank [11]. Table 1 details the necessary number of required samples L for several sampling schemes in order that these conditions are satisfied.
Table 1. Number of samples for accurate computation of Hnm (ν) for several sampling schemes
Sampling scheme
Number of samples

equal angle
4 (N + 1)2

Gaussian
2 (N + 1)2

uniform(N + 1)2

Even if the assumptions discussed above approximately hold, there is no guarantee for ideal focusing in thebinaural case. This is because the system in (6) is over-determined for all N (excluding N = 0 for which the
HRTF contains no directional information); hence, the solution in (7) is obtained in the least squares sense, thus
incorporating a fitting error. The next section examines the focusing performance for a binaural array.

5

Focusing performance analysis

To evaluate the performance of focusing for simple binaural arrays, an array of two microphones mounted ona rigid sphere was employed. The array steering function from a set of L = 338 points in a Gaussian sampling
scheme was obtained according to [11] up to order N = 12, with a sampling frequency of Fs = 16 kHz and witha frequency resolution of 512 points. The mean (over directions) of the normalized focusing error is definedas [7]
ε f (ν0 ) =1 L kT (ν, ν0 ) h (ν, ψl ) − h (ν0 , ψl )k2
,
∑
L l=1
kh (ν0 , ψl )k2(9)where (·) denotes averaging over the frequency within the focusing window.
The focusing matrices were computed according to (7) with several spherical harmonics orders N for a focusing
windows of 15 frequencies. Figure 1 shows ε f (ν0 ) as a function of the central frequency ν0 . To examine theeffect of the order-limited assumption, the values of kr are depicted at the top of figure 1. The dashed line
denotes the values of ε f (ν0 ) when the focusing matrices were chosen to be an identity matrix, i.e. no focusing
was employed.
Figure 1 shows that, although the focusing computation incorporates the solution of an over-determined system,
good performance can still be achieved, and a significant improvement is evident with respect to the case of
no focusing. In addition, Figure 1 shows that the performance of focusing matrices of order N deteriorates
for kr > N. This is because in this range a truncated representation of the HRTF is used in (7), resulting in
an increased focusing error. Nevertheless, good focusing performance is obtained at frequencies in the range
kr < N. This implies that in the case for which the DPD test does not exhibit high sensitivity to focusing errors,

3308

5

1

2

3

4

7

6

5

8

9

10

11

12

0
-5
-10
-15
2
4

-20

6

8
10
12

-25
-30
0

1

2

3

4

5

6

7

8

Figure 1. Focusing error for several spherical harmonics orders N = 2, 4, 6, 8, 10, 12 and for focusing window of
15 frequencies. The dashed line depicts the error when no focusing is employedsparse sampling of the HRTF may be sufficient for good performance. The next section examines the effect offocusing performance on the DOA estimation.

6

Simulation study

In this section the application of the DPD test for the binaural case is examined through simulation. A simple
acoustic condition of single a speaker and a single room reflection is initially assumed in Section 6.1 in order toexamine the effect of smoothing and focusing. Then, a more complex acoustic condition of a highly reverberant
room is assumed in Section 6.2 in order to examine the effect of smoothing and focusing in a more realistic
condition and to demonstrate the DOA estimation with the DPD test under adverse reverberation conditions.
Azimuth estimation only was considered, which is more natural for the binaural array.
The Numman KU-100 [4] was used as a binaural array. The speaker, simulated as a point source, was positioned 1.5 m away from the array at the array frontal horizontal plane. A four-seconds segment of a speech
signal from the TIMIT database [5] was used as speech material. The spherical harmonics coefficients up toorder N = 12 of the plane waves’ density at the array position were synthesized using the image method [1] ata sampling frequency of 16 kHz. The binaural signal was simulated according to [13].
The DPD test was applied to the simulated signal with the following parameters: STFT with Hann windowof 512 samples length (32 ms) and 50% overlap between adjacent frames, Jτ = 3 time frames and Jν = 15
frequencies were used for averaging, and the focusing matrices were computed with spherical harmonics orderof N = 12. The threshold was selected separately for each frequency such that 5% of the TF bins from eachfrequency would pass the test. TF bins from frequencies below 1 kHz and beyond 6.5 kHz were automaticallyrejected by the test.
6.1 A single reflection
A simple acoustic condition of a single speaker and a single room reflection is studied to examine the effectof smoothing and focusing on DOA estimation with the DPD test. The speaker and the reflection DOAs were
ψ1 = (90◦ , 70◦ ) and ψ2 = (90◦ , 20◦ ), respectively. The amplitude ratio at the array between the reflection and the
direct-path (speech signal) was 0.6, and the reflection arrives at the array with a delay of 5 ms, which represents
the delay of a typical early room reflection. A noise free condition was assumed.
To examine the effect of smoothing and focusing, the DPD test was applied with and without focusing andsmoothing. Under all conditions, the spectrum of MUSIC was computed for the frontal azimuth direction ateach of the selected TF bins. An incoherent integration of the spectrum is then applied by averaging the3309

spectra from the different bins. Figure 2 shows the mean MUSIC spectrum under the different conditions. The
true azimuth is marked by a dashed black line.
0

0

0

-20

-20

-20

-40

-40

-40

-60

-60

-60

-80

-80
-50

0

50

-80
-50

(a)

0

50

-50

(b)

0

50

(c)

Figure 2. Mean MUSIC spectrum from the selected TF bins in each condition: (a) without smoothing, (b) withsmoothing without focusing, (c) with smoothing and focusing. The dashed line represent the direction of the
speaker.
Figure 2a shows that without smoothing the peak in the MUSIC spectrum is away from the speaker azimuth.
This is because, with no smoothing the spatial spectrum in all TF bins is of unit rank; consequently, a significant
number of selected TF bins contain DOA information that is distorted by the reflection. Figures 2b, 2c showthat the MUSIC spectra for both of the conditions that apply smoothing have a peak very close to the speakerazimuth. Nevertheless, the peak in the condition with focusing is sharper and more prominent. Incorporating
focusing better preserves the spatial information in the smoothed HRTF matrix and, consequently, improves the
DOA estimation from the selected TF bins.
6.2 Reverberant room
The binaural signal due to a speaker in a reverberant room is simulated. A room of dimensions 8 × 5 × 3 m
and with a reverberation time of T60 = 1 sec was considered. The array position was selected to be (x, y, z) =(2, 1.5, 1.7) m. A sensor noise of 40 dB SNR was simulated. Figure 3 shows the mean MUSIC spectra under
the different conditions.
0

0

0

-20

-20

-20

-40

-40

-60

-60

-40
-60
-80

-80
-50

0

(a)

50

-80
-50

0

(b)

50

-50

0

50

(c)

Figure 3. Mean MUSIC spectrum from the selected TF bins in each condition: (a) without smoothing, (b) withsmoothing without focusing, (c) with smoothing and focusing. The dashed line represent the direction of the
speaker.
The results suggest that without smoothing, TF bins that contain a significant contribution from reflections are
selected. Figures 2b, 2c shows that the spectrum for both conditions that apply smoothing are quite similar,

3310

excluding the area around the speaker azimuth. In this area the condition with focusing has a peak very close
to the speaker azimuth, and the condition without focusing exhibits a couple of peaks with deviation from the
speaker azimuth. This result suggests that focusing have a positive effect the accuracy of the DOA estimationfrom the selected TF bins.
Figure 4 shows the spectrogram of the signal obtained in the left microphone and the TF bins that were selectedby the DPD test in black.
8

0

-5

-10

4

-15

Magnitude [dB]

6

-20

2

-25

0
0.5

1

1.5

2

2.5

3

3.5

Figure 4. Spectrogram of reverberant speech with selected bins in black
Figure 4 shows that the majority of the selected TF bins are located at the onset of the speech signal. Thisis because the direct-sound arrives earlier at the array than reflections that have a longer traveling time. This
result supports the validity of the used measure in selecting TF bins in which the direct-path is dominant.
Figure 5a shows the histogram of the azimuth estimates from the selected TF bins and figure 5b shows thehistogram of the azimuth estimates from all TF bins. The histograms were normalized such that the area below
the graph will equal 1.

(a)

(b)

Figure 5. Histogram of azimuth estimates from (a) all bins, (b) selected bins
Comparing the histograms, it is clear that using the TF bins selection with the DPD test improves the accuracy3311

of the DOA estimation.

7

Conclusions

In this paper, the DPD test method for speaker localization is examined for the binaural case. Central to the
DPD test is a focusing process. The assumptions at the basis of this focusing process are examined for thebinaural case, and conditions for which these assumptions hold are formulated. It has been shown that good
focusing performance can be obtained in the binaural case. An experiment with simulated data verifies thenecessity of focusing and smoothing and demonstrates the capabilities of the proposed method of overcoming
adverse reverberation conditions with the DPD test.

REFERENCES[1] J. B. Allen and D. A. Berkley. Image method for efficiently simulating small-room acoustics. The Journal of the AcousticalSociety of America, 65(4):943–950, 1979.
[2] Y. Avargel and I. Cohen. On multiplicative transfer function approximation in the short-time fourier transform domain. IEEE
Signal Processing Letters, 14(5):337–340, 2007.
[3] H. Beit-On and B. Rafaely. Speaker localization using the direct-path dominance test for arbitrary arrays. In 2018 IEEE
International Conference on the Science of Electrical Engineering in Israel (ICSEE), pages 1–4. IEEE, 2018.
[4] B. Bernschütz. A spherical far field hrir/hrtf compilation of the neumann ku 100. In Proceedings of the 40th Italian (AIA)
annual conference on acoustics and the 39th German annual conference on acoustics (DAGA) conference on acoustics,
page 29. AIA/DAGA, 2013.
[5] J. S. Garofolo, L. F. Lamel, W. M. Fisher, J. G. Fiscus, and D. S. Pallett. Darpa timit acoustic-phonetic continous speech
corpus cd-rom. nist speech disc 1-1.1. NASA STI/Recon technical report n, 93, 1993.
[6] H. Hung and M. Kaveh. Focussing matrices for coherent signal-subspace processing. IEEE Transactions on Acoustics,
Speech, and Signal Processing, 36(8):1272–1281, 1988.
[7] T.-S. Lee. Efficient wideband source localization using beamforming invariance technique. IEEE Transactions on Signal
Processing, 42(6):1376–1387, 1994.
[8] X. Li, L. Girin, R. Horaud, and S. Gannot. Estimation of the direct-path relative transfer function for supervised sound-source
localization. IEEE/ACM Transactions on Audio, Speech, and Language Processing, 24(11):2171–2186, 2016.
[9] N. Ma, T. May, and G. J. Brown. Exploiting deep neural networks and head movements for robust binaural localization ofmultiple sources in reverberant environments. IEEE/ACM Transactions on Audio, Speech and Language Processing (TASLP),
25(12):2444–2453, 2017.
[10] O. Nadiri and B. Rafaely. Localization of multiple speakers under high reverberation using a spherical microphone array andthe direct-path dominance test. IEEE/ACM Transactions on Audio, Speech, and Language Processing, 22(10):1494–1505,
2014.
[11] B. Rafaely. Fundamentals of spherical array processing, volume 8. Springer, 2015.
[12] B. Rafaely and K. Alhaiany. Speaker localization using direct path dominance test based on sound field directivity. Signal
Processing, 143:42–47, 2018.
[13] B. Rafaely and A. Avni. Interaural cross correlation in a sound field represented by spherical harmonics. The Journal of the
Acoustical Society of America, 127(2):823–828, 2010.
[14] B. Rafaely, C. Schymura, and D. Kolossa. Speaker localization in a reverberant environment using spherical statisticalmodeling. The Journal of the Acoustical Society of America, 141(5):3523–3523, 2017.
[15] V. Tourbabin, D. L. Alon, and R. Mehra. Space domain-based selection of direct-sound bins in the context of improved
robustness to reverberation in direction of arrival estimation. 2018.
[16] H. L. Van Trees. Optimum array processing: Part IV of detection, estimation, and modulation theory. John Wiley & Sons,
2004.
[17] H. Wang and M. Kaveh. Coherent signal-subspace processing for the detection and estimation of angles of arrival of multiple
wide-band sources. IEEE Transactions on Acoustics, Speech, and Signal Processing, 33(4):823–831, 1985.

3312